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Full text of "Report of the British Association for the Advancement of Science"

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REPORT 




SIXTEENTH MEETING 



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BRITISH ASSOCIATION 



ADVANCEMENT OF SCIENCE; 



HELD AT SOUTHAMPTON IN SEPTEMBER 1S46. 



LONDON: 

JOHN MURRAY, ALBEMARLE STREET. 

1847. 



PRINTED BY RICHARD AND JOHN E. TAYLOR, 
RED LION COURT, FLEET STREET. 




CONTENTS. 



Objects and Rules of the Association v 

Places of Meeting and Officers from commencement viii 

Table of Council from commencement x 

Treasurer's Account xii 

Officers and Council xiv 

Officers of Sectional Committees xv 

Corresponding Members xvi 

Report of Council to the General Committee xvi 

Proceedings of the General Committee at Southampton xix 

Recommendations for Additional Reports and Researches in Science xix 

Synopsis of Money Grants xxi 

Arrangement of the General Evening Meetings xxvi 

Address of the President xxvii 

REPORTS OF RESEARCHES IN SCIENCE. 

Report on Recent Researches in Hydrodynamics. By G. G. Stokes, 

M.A., Fellow of Pembroke College, Cambridge 1 

Sixth Report of a Committee, consisting of H. E. Strickland, Esq., 
Prof. Daubeny, Prof. Henslow and Prof. Lindley, appointed to 

continue their Experiments on the Vitality of Seeds 20 

On the Colouring Matters of Madder. By Dr. Schunck 24 

On the Physiological Action of Medicines. By J. Blake, M.B., F.R.C.S., 

&c. &c 27 

Report on the Actinograph. By Mr. Robert Hunt 31 

Notices on the Influence of Light on the Growth of Plants. By Mr. 

Robert Hunt 33 



IV CONTENTS. 

Page 
On the Recent Progress of Analysis (Theory of the Comparison of 

Transcendentals). By R. L. Ellis, M.A 34 

On Comparative Analytical Researches on Sea Water. By Prof. 

FORCHHAMMER 90 

On the Calculation of the Gaussian Constants for 1829. By A. Erman 92 

On the Progress, present Amount, and probable future Condition of the 
Iron Manufacture in Great Britain. By G. R. Porter, F.R.S 99 

Third Report on Atmospheric Waves. By William Radcliff Birt 119 

Report on the Archetype and Homologies of the Vertebrate Skeleton. 
By Prof. Owen, F.R.S 169 

On Anemometry. By John Phillips, F.R.S., F.G.S 340 

Report on the Crystalline Slags. By John Percy, M.D 351 



4 



OBJECTS AND RULES 



THE ASSOCIATION. 



OBJECTS. 

The Association contemplates no interference with the ground occupied by 
other Institutions. Its objects are, — To give a stronger impulse and a more 
systematic direction to scientific inquiry, — to promote the intercourse of those 
who cultivate Science in different parts of the British Empire, with one an- 
other, and with foreign philosophers, — to obtain a more general attention to 
the objects of Science, and a removal of any disadvantages of a public kind 
which impede its progress. 

RULES. 

ADMISSION OF MEMBERS AND ASSOCIATES. 

All Persons who have attended the first Meeting shall be entitled to be- 
come Members of the Association, upon subscribing an obligation to con- 
form to its Rules. 

The Fellows and Members of Chartered Literary and Philosophical So- 
cieties publishing Transactions, in the British Empire, shall be entitled, in 
like manner, to become Members of the Association. 

The Officers and Members of the Councils, or Managing Committees, of 
Philosophical Institutions, shall be entitled, in like manner, to become Mem- 
bers of the Association. 

All Members of a Philosophical Institution recommended by its Council 
or Managing Committee, shall be entitled, in like manner, to become Mem- 
bers of the Association. 

Persons not belonging to such Institutions shall be elected by the General 
Committee or Council, to become Life Members of the Association, Annual 
Subscribers, or Associates for the year, subject to the approval of a General 
Meeting. 

COMPOSITIONS, SUBSCRIPTIONS, AND PRIVILEGES. 

Life Members shall pay, on admission, the sum of Ten Pounds. They 
shall receive gratuitously the Reports of the Association which may be pub- 
lished after the date of such payment. They are eligible to all the offices 
of the Association. 

Annual Subscribers shall pay, on admission, the sum of Two Pounds, 
and in each following year the sum of One Pound. They shall receive 
gratuitously the Reports of the Association for the year of their admission 
and for the years in which they continue to pay without intermission their 
Annual Subscription. By omitting to pay this Subscription in any particu- 
lar year. Members of this class (Annual Subscribers) lose for that and all 
future years the privilege of receiving the volumes of the Association gratis : 

h 



VZ RULES OF THE ASSOCIATION. 

but they may resume their Membership and other privileges at -x^.y sub- 
sequent Meetin<f of the Association, paying on each such occasion the sum of 
One Pound. 'I'hey are eligible to all the Offices of the Association. 

Associates for the year shall pay on admission the sum of One Pound. 
They shall not receive gratuitously the Reports of the Association, nor be 
eligible to serve on Committees, or to hold any office. 

The Association consists of the following classes : — 

1. Life Members admitted from 1831 to 1845 inclusive, who have paid 
on admission Five Pounds as a composition. 

2. Life Members who in 1846, or in subsequent years, have paid on ad- 
mission Ten Pounds as a composition. 

'3. Annual Members admitted from IS."! to 1839 inclusive, subject to the 
payment of One Pound annually, [may resume their Membership after inter- 
mission of Annual Payment.] 

4: Annual iMembers admitted or to be admitted in any year since 18S9, 
subject to the payment of Two Pounds for the first year, and One Pound in 
each following year, [may resume their membership after intermission of 
Annual Payment.] 

5. Associates for the year, subject to the payment of One Pound. 

6. Corresponding Members nominated by the Council. 

And the Members and Associates will be entitled to receive the annual 
volume of Reports, gratis, or to purchase it at reduced (or Members') price, 
according to the following specification, viz. : — 

1. Gratis. — Old Life Members who have paid Five Pounds as a compo- 

sition for Annual Payments, and Two Pounds as a Book Subscrip- 
tion. 

New Life Members who shall have paid Ten Pounds as a composition. 

Annual Members who have not intermitted their Annual Subscription. 

2. /it reduced or Members' Prices. — Old Life Members who have paid 

Five Pounds as a composition for Annual Payments, but no Book 
Subscription. 
Annual Members, who, having paid on admission Two Pounds, have 

intermitted their Annual Subscription in any subsequent year. 
Associates for the year. [Privilege confined to the volume for that 
year only.] 
Subscriptions shall be received by the Treasurer or Secretaries. 

MEETINGS. 

The Association shall meet annually, for one week, or longer. The place 
of each Meeting shall be appointed by the General Committee at the pre- 
vious Meeting ; and the Arrangements for it shall be entrusted to the Offi- 
cers of the Association. 

GENERAL COMMITTEE. 

The General Committee shall sit during the week of the Meeting, or 
longer, to transact the business of the Association. It shall consist of the 
following persons : — 

1. Presidents and Officers for the present and preceding years, with au- 
thors of Reports in the Transactions of the Association. 

2. Members who have communicated any Paper to a Philosophical Society, 
which has been printed in its Transactions, and which relates to such subjects 
as are taken into consideration at the Sectional Meetings of the Association. 

3. Office-bearers for the time being, or Delegates, altogether not exceed- 
ing three in number, from any Philosophical Society publishing Transactions. 



RULES OP THE ASSOCIATION. VU 

4. Office-bearers for the time being, or Delegates, not exceeding three, 
from Philosophical Institutions established in the place of Meeting, or in any 
place where the Association has formerly met. 

5. Foreigners and other individuals whose assistance is desired, and who 
are specially nominated in writing for the meeting of the year by the Presi- 
dent and General Secretaries. 

6. The Presidents, Vice-Presidents, and Secretaries of the Sections are ex 
officio members of the General Committee for the time being. 

SECTIONAL COMMITTEES. 

The General Committee shall appoint, at each Meeting, Committees, con- 
sisting severally of the Members most conversant with the several branches 
of Science, to advise together for the advancement thereof. 

The Committees shall report what subjects of investigation they would 
particularly recommend to be prosecuted during the ensuing year, and 
brought under consideration at the next Meeting. 

The Committees shall recommend Reports on the state and progress of 
particular Sciences, to be drawn up from time to time by competent persons, 
for the information of the Annual Meetings. 

COMMITTEE OF RECOMMEHDATIONS. 

The General Committee shall appoint at each Meeting a Committee, which 
shall receive and consider the Recommendations of the Sectional Committees, 
and report to the General Committee the measures which they would advise 
to be adopted for the advancement of Science. 

All Recommendations of Grants of Money,^ Requests for Special Re- 
searches, and Reports on Scientific Subjects, shall be submitted to the Com- 
mittee of Recommendations, and n'ot taken into consideration by the General 
Committee, unless previously recommended by the Committee of Recommen- 
dations. 

LOCAL COMMITTEES. 

Local Committees shall be formed by the Officers of the Association to 
assist in making arrangements for the Meetings. 

Local Committees shall have the power of adding to their numbers those 
Members of the Association whose assistance they may desire. 



A President, two or more Vice-Presidents, one or more Secretaries, and a 
Treasurer, shall be annually appointed by the General Committee. 

COUNCIL. 

In the intervals of the Meetings, the affairs of the Association shall be 
managed by a Council appointed by the General Committee. The Council 
may also assemble for the despatch of business during the week of the 
Meeting. 

PAPERS AND COMMUNICATIONS. 

The Author of any paper or communication shall be at liberty to reserve 
his right of property therein. 

ACCOUNTS. 

The Accounts of the Association shall be audited annually, by Auditors 
appointed by the Meeting. 

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03 



MEMBERS OP COUNCIL. 



II. Table showing the Names of Members of the British Association who 
have served on the Council in former years. 

Acland, Sir Thomas D., Bart., M.P., F.R.S. 
Adamson, J., F.L.S. 

Adare. Viscount, M.P., F.R.S. 

Airy, G.B.,D.C.L.,F.R.S., Astronomer Royal. 

Ainslie, Rev. Gilbert, D.D., Master of Pem- 
broke Hall, Cambridge. 

Ansted, Professor D. T., M.A., F.R.S. 

Arnott, Neil, M.D., F.R.S. 

Ashburton, Lord, D.C.L. 

Babbage, Charles, F.R.S. 

Babington, C. C, F.L.S. 

Bailv, Francis, F.R.S. 

Barker, George, F.R.S. 

Bengough, George. 

Bentham, George, F.L.S. 

Bigge, Charles. 

BlaUston, Peyton, M.D., F.R.S. 

Brewster, Sir David, K.H., LL.D., F.R.S. 

Breadalbane, The Marquis of, F.R.S. 

Brisbane, Lieut.-GeneralSirThomasM., Bart., 
K.C.B., G.C.H., D.C.L., F.R.S. 

Brown, Robert, D.C.L., F.R.S. 

Brunei, Sir M. I., F.R.S. 

Buckland, Very Rev. WilUam, D.D., Dean of 
AVestminster, F.R.S. 

Burlington, The Earl of, M.A., F.R.S., Chan- 
cellor of the University of London. 

Carson, Rev. Joseph. 

Cathcart, The Earl, K.C.B., F.R.S.E. 

Chalmers, Rev. T., D.D., Professor of Di- 
vinity, Edinburgh. 

Christie, Professor S. H., M.A., Sec.R.S. 

Clare, Peter, F.R.A.S. 

Clark, Rev, Professor, M.D., F.R.S. (Cam- 
bridge). 

Clark, Henry, M.D. 

Clark, G. T. 

Clift, WiUiam, F.R.S. 

Colquhoun, J. C, M.P. 

Couybeare, Very Rev. W. D., Dean of Llandaff, 
M.A., F.R.S. 

Corrie, John, F.R.S. 

Currie, William Wallace. 

Dalton, John, D.C.L., F.R.S. 

Daniell, Professor J. F., F.R.S. 

Dartmouth, The Earl of, D.C.L., F.R.S. 

Daubenv, Professor Charles G.B., M.D., 
F.R.S. 

De la Beche, Sir Henry T., F.R.S., Director- 
General of the Geological Survey of the 
United Kingdom. 

Driukwater, J. E. 

Durham, The Bishop of, F.R.S. 

Egerton, Sir Philip de M. Grey, Bart., F.R.S. 

Eliot, Lord, M.P. 

Ellesmere, The Earl of, F.G.S. 

Faraday, Professor, D.C.L., F.R.S. 

Fitzwiliiam, The Earl, D.C.L., F.R.S. 

Fleming, H., M.D. 

Forbes, Charles. 

Forbes, Professor Edward, F.R.S. 

Forbes, Professor J. D., F.R.S. 



Fox, Robert Were, F.R.S. 

Gilbert, Davies, D.C.L., F.R.S. 

Graham, Rev. John, D.D., Master of Christ's 

College, Cambridge. 
Graham, Professor Thomas, M.A., F.R.S. 
Gray, John E., F.R.S. 
Gray, Jonathan. 
Gray, WUliam, jun., F.G.S. 
Green, Professor Joseph Henry, F.R.S. 
Greenough, G. B., F.R.S. 
Hallam, Henry, M.A., F.R.S. 
Hamilton, W. J., M.P., Sec.G.S. 
Hamilton, Sir William R., Astronomer Royal 

of Ireland, M.R.LA. 
Harcourt, Rev. William Vernon, M.A., F.R.S. 
Hardwicke, The Earl of. 
Harford, J. S., D.C.L., F.R.S. 
Harris, W. Snow, F.R.S. 
Hatfeild, WUliam, F.G.S. 
Henslow, Rev. Professor, M.A., F.L.S. 
Henry, W. C, M.D., F.R.S. 
Herbert, Hon. and Very Rev. William, Dean 

of Manchester, LL.D., F.L.S. 
Herschel, Sir John F. W., Bart.,D.C.L.,F.R.S. 
Heywood, Sir Benjamin, Bart., F.R.S. 
Hey wood, James, F.R.S. 
Hodgkin, Thomas, M.D. 
Hodgkinson, Eaton, F.R.S. 
Hodgson, Joseph, F.R.S. 
Hooker, Sir WiUiam J., LL.D., F.R.S. 
Hope, Rev. F. W., M.A., F.R.S. 
Hopkins, WUliam, M.A., F.R.S. 
Horner, Leonard, F.R.S., F.G.S. 
Hovenden, V. F., M.A. 
Hutton, Robert, F.G.S. 
Hutton, WUliam, F.G.S. 
Jameson, Professor R., F.R.S. 
Jenyns, Rev. Leonard, F.L.S. 
Jerrard, H. B. 

Johnston, Professor J. F. W., M.A., F.R.S. 
Keleher, WiUiam. 
Lardner, Rev. Dr. 
Lee, R., M.D., F.R.S. 

Lansdowne, The Marquis of, D.C.L., F.R.S. 
Lefevre, Right Hon. Charles Shaw, Speaker 

of the House of Commons. 
Lemon, Sir Charles, Bart., M.P., F.R.S. 
Liddell, Andrew. 
Liudley, Professor, Ph.D., F.R.S. 
Listowel, The Earl of. 
Lloyd, Rev. Bartholomew, D.D., Provost of 

Trinity CoUege, Dublin. 
Lloyd, Rev. Professor, D.D., F.R.S. 
Lubbock, Sir John W., Bart., M.A., F.R.S. 
Lubv, Rev. Thomas. 
Lyeil, diaries, jun., M.A., F.R.S. 
MacCuUagh, Professor, D.C.L., M.R.LA. 
Macfarlane, The Very Rev. Principal. 
MacLeay, WUliam Sharp, F.L.S. 
Mac^^eiil, Professor Sir John, F.R.S. 
MeyncU, Thomas, Jun., F.L.S. 
Miller, Professor W. H., M.A., F.R.S. 



MEMBERS OF COUNCIL. 



Moilliet, J. L. 

Moody, T. C, Esq. 

Moody, T. F. 

Morley, The Earl of. 

Morpeth, Viscount, F.G.S. 

Moseley, Rev. Henry, M.A., F.R.S. 

Mount Edgecumbe, The Earl of. 

Murchison, Sir Roderick I., G.C.S., F.R.S. 

Neill, Patrick, M.D., F.R.S.E. 

Nicol, Rev. J. P., LL.D. 

Northampton, The Marquis of, President of 

the Royal Society. 
Northumberland, The Duke of, K.G., M.A., 

F.R.S. 
Norwich, The Bishop of, President of the 

Linnaean Society, F.R.S. 
Ormerod, G. W., F.G.S. 
Orpen, Thomas Herbert, M.D. 
Owen, Professor Richard, M.D., F.R.S. 
Oxford, The Bishop of, F.R.S., F.G.S. 
Osier, FoUett, F.R.S. 
Palmerston, Viscount, G.C.B., M.P. 
Peacock, Very Rev. George, D.D., Dean of 

Ely, V.P.R.S. 
Pendarves, E., F.R.S. 
PhiUips, Professor John, F.R.S. 
Powell, Rev. Professor, M.A., F.R.S. 
Prichard, J. C, M.D., F.R.S. 
Ramsay, Professor W., M.A. 
Rennie, George, V.P.&Treas.R.S. 
Rennie, Sir John, F.R.S., President of the 

Institute of Civil Engineers. 
Richardson, Sir John, M.D., F.R.S. 
Ritchie, Rev. Professor, LL.D., F.R.S. 
Robinson, Rev. J., D.D. 
Robinson, Rev. T. R., D.D. 
Robison, Sir John, Sec.R.S.Edin. 
Roche, James. 

Roget, Peter Mark, M.D., Sec.R.S. 
Ross, Capt. Sir James C, R.N., F.R.S. 
Rosse, The Earl of, F.R.S. 
Royle, Professor John F., M.D., F.R.S. 



Russell, James. 

Sabine, Lieut.-Colonel Edward, R.A., For. 

Sec.R.S. 
Sanders, William, F.G.S. 
Sandon, Lord. 

Scoresby, Rev. "W., D.D., F.R.S. 
Sedgwick, Rev. Professor, M.A., F.R.S. 
Selby, Prideaux John, F.R.S.E. 
Smith, Lt.-Colonel C. Hamilton, F.R.S. 
Staunton, Sir George T., Bart., M.P., D.C.L., 

F.R.S. 
Stevelly, Professor John, LL.D. 
Strang, John. 
Strickland, H. E., F.G.S. 
Sykes, Lieut.-Colonel W. H., F.R.S. 
Talbot, W. H. Fox, M.A., F.R.S. 
Tayler, Rev. J. J. 
Taylor, John, F.R.S. 
Taylor, Richard, jun., F.G.S. 
Thompson, WUliam, F.L.S. 
Traill, J. S., M.D. 
Turner, Edward, M.D., F.R.S. 
Turner, Samuel, F.R.S., F.G.S. 
Turner, Rev. W. 
Vigors, N. A., D.C.L., F.L.S. 
Walker, James, F.R.S. 
Walker, J. N., F.G.S. 
Warburton, Henry, M.A., M.P., F.R.S. 
Washington, Captain, R.N. 
West, WiUiam, F.R.S. 
Wheatstone, Professor, F.R.S. 
Whewell, Rev. William, D.D., Master of 

Trinity College, Cambridge. 
Williams, Professor Charles J.B.,M.D.,F.R.S. 
WilUs, Rev. Professor, M.A., F.R.S. 
Winchester, The Marquis of. 
Woollcombe, Henry, F.S.A. 
Wortley, The Hon. John Stuart, B.A., M.P., 

F R S 
Yarrell, WUliam, F.L.S. 
Yarborough, The Earl of, D.C.L. 
Yates, James, M.A., F.R.S. 



BRITISH ASSOCIATION FOR THE 



THE GENERAL TREASURER'S ACCOUNT from 19th of June 



RECEIPTS. 

£ s. d. 
To Life Compositions received at the Cambridge Meeting and since 

To Annual Subscriptions -..Ditto Ditto Ditto 

To Associates' TickeU Ditto Ditto Ditto 

To Ladies' Tickets Ditto Ditto Ditto 

To Book Compositions Ditto Ditto Ditto 

To Dividends on Stock Ditto Ditto Ditto 

To Sale of £1000 in the 3 per cent. Consols 

To Cash from Cambridge Local Fund Committee 

To Ditto portion of Grant returned 

To received from the Sale of Publications : — 

of 1st volume 2 2 5 

of 2nd volume 2 12 

of 3rd volume 5 3 

oi'4th volume 3 5 8 

of 5th volume 2 IS 5 

of Cth volume 4 4 

of 7th volume 3 15 

of 8th volume 5 15 2 

of 9th volume 7 9 

of 10th volume 9' 11 8 

of 11th volume 7 3 6 

of 12th volume 14 ]3 7 

of 13th volume 93 17 jo 

of 14th volume 10 4 

of Lithograph Signatures 10 5 



£ 


s. 


d. 


185 








161 








407 








172 








64 








225 


14 


6 


956 


5 





73 


5 


7 


6 









Balance carried on . 



173 12 
125 3 1 



£2549 2 



To Balance due to the General Treasurer 237 16 10 

To Ditto due from Local Treasurers 7 19 6 

To Ditto in the Bankers' hands 104 14 3 112 13 9 



£123 3 1 



The General Treasurer in Account 

To Balance of Grant brought on from last Account 634 2 



£634 2 



T. H. SHADWELL CLERKE, " 
D. T. ANSTED, 



■ Audilors. 



ADVANCEMENT OF SCIENCE. 



1845 (at Cambridge) to the 10th of September 1846 (at Southampton). 



PAYMENTS. 

£ s. 
By Balance in advance on the General Account brought on... 
By Sundry Disbursements by Treasurer and Local Treasurers, 
including the Expenses of the Meeting at Cambridge, 

Advertising, Sundry Printing, &c 

By Printing, &c. of the 14th Report (13th vol.) 

By Salaries to Assistant General Secretary, Accountant, &c. 

IS months to Midsummer 1846 

By Paid to the order of Committees on Account of Grants 
for Scientific purposes, viz. for — 

British Association Catalogue of Stars 1844 211 15 

Fossil Fishes of the London Clay 100 

Computation of the Gaussian Constants for 1839 50 

Maintaining the Establishment at Kew Observatory 146 IC 



Experiments on the Strength of Materials 

Researches in Asphyxia 

Examination of Fossil Shells 

Vitality of Seeds 1844 

Ditto ditto 1845 

Marine Zoology of Cornwall 

Ditto Ditto Britain 

Exotic Anoplura 1844 

Expenses attending Anemometers 

Anemometers' Repairs 

Researches on Atmospheric Waves 

Captive Balloons 1844 

Varieties of the Human Race 1844 

Statistics of Sickness and Mortality at York 




16 2 


15 10 
12 3 



£ s. 
360 10 



203 11 

774 2 



525 



685 16 



£2549 2 



By Balance in advance brought down as per contra 



£125 3 1 



tvitk the Government Grant. 

By Amount Paid on Account of the Printing of Lalande and 

Lacaille's Catalogues 

Balance in Treasurer's hands. 



553 5 
81 1 7 



£634 2 



XIV OFFICERS AND COUNCIL. 

OFFICERS AND COUNCIL, 1846-47. 



Trustees (permanent).— Sir Roderick Impey Murcliison, G.C.S'.S., F.R.S. 
John Taylor, Esq., F.R.S. The Very Reverend George Peacock, D.D., 
Dean of Ely, F.R.S. 

President.— Sir Roderick Impey Murchison, G.C.S*.S., F.R.S. 

Vice-Presidents. — The Marquis of Winchester. The Earl of Yarborough, 
D.C.L. Viscount Palmerston, G.C.B., M.P. Lord Ashburton, D.C.L. The 
Bishop of Oxford, F.R.S., F.G.S. The Right Hon. the Speaker, Charles 
Shaw Lefevre, M.P., F.G.S. Sir George T. Staunton, Bart., M.P., D.C.L., 
F.R.S. Professor Owen, M.D., F.R.S. Rev. Professor Powell, F.R.S. 

President Elect.— Sir Robert Harry Inglis, Bart., D.C.L., F.R.S., M.P. 

for the University of Oxford. 

Vice-Presidents Elect.— The Earl of Rosse, F.R.S. The Lord Bishop of 
Oxford, F.R.S. The Vice-Chancellor of the University of Oxford. 
Thomas G. Bucknall Estcourt, Esq., D.C.L., M.P. for the University of Ox- 
ford. The Very Rev. The Dean of Westminster, D.D., F.R.S., Professor of 
Geology and Mineralogy, Oxford. Charles G. JB. Daubeny, M.D., F.R.S., 
Professorof Chemistry and Botany, Oxford. The Rev. Baden Powell, M.A., 
F.R.S., Savilian Professor of Geometry, Oxford. 

General Secretary.— Lie\xt.-Co\, Sabine, For. Sec. R.S., Woolwich. 

Assistant General Secretary. — John Phillips, Esq., F.R.S., York. 

General Treasurer.— Johr\ Taylor, Esq., F.R.S., 2 Duke Street, Adelphi, 
London. 

Secretaries for the Oxford Meeting in 1847. — Rev. Robert Walker, M.A., 
F.R.S., Reader in Experimental Philosophy, Oxford. Henry Wentworth 
Acland, Esq., B.M., F.R.S,, Lee's Reader in Anatomy, Oxford. 

Treasured to the Oxford Meeting in 1847. — Rev. Edward Hill, M.A., 
F.G.S., Christ Church, Oxford. 

Council. — Professor Ansted. Sir H. T. De la Beche. Major Shadwell 
Gierke. Professor E. Forbes. Dr. Fitton. Professor T. Graham. W. R. 
Grove, Esq. W. J. Hamilton, Esq. Sir John F. W. Herschel, Bart. 
James Heywood, Esq. William Hopkins, Esq. Leonard Horner, Esq. 
Robert Hutton, Esq. Capt. Ibbotson. Dr. Latham. Sir Charles Lemon, 
Bart. The Marquis of Northampton. G. R. Porter, Esq. Sir John Ri- 
chardson, M.D. Rev. Dr. Robinson. Dr. Roget. Captain Sir James 
Ross, R.N. Prof. J. Forbes Royle, M.D. H. E. Strickland, Esq. Lieut.- 
Col. Sykes. T. Tooke, Esq. William Thompson, Esq. Professor Wheat- 
stone. C. J. B. Williams, M.D. Professor Willis. 

Local Treasurers — W. Gray, jun., Esq., York. Rev. E. Hill, Oxford. 
C. C. Babington, Esq., Cambridge. J. H. Orpen, LL.D., Dublin. Charles 
Forbes, Esq., Edinburgh. Professor Ramsay, Glasgow. William Sanders, 
Esq., Bristol. Samuel Turner, Esq., Liverpool. G. W. Ormerod, Esq., 
Manchester. James Russell, Esq., Birmingham. William Hutton, Esq., 

Newcastle-on-Tyne. , Plymouth. James Roche, 

Esq., Cork. J. Sadleir Moody, Esq., Southampton. 

Auditors. — Professor Ansted. Professor Willis. Major Shadwell Clarke. 



OFFICERS OF SECTIONAL. COMMITTEES. XV 

OFFICERS OF SECTIONAL COMMITTEES AT THE 
SOUTHAMPTON MEETING. 

SECTION A. MATHEMATICAL AND PHYSICAL SCIENCE. 

President.—Sir John F. W. Herschel, Bart., F.R.S., &c. 

Vice-Presidents.— Sir D. Brewster, F.R.S. L. & E. Professor Wheat- 
stone, F.R.S. Col. Colby, R.E., F.R.S. & M.R.I. A. The Master of Tri- 
nity College, Cambridge. 

Secretaries. — Dr. Stevelly. G. G. Stokes, Esq, John Drew, Esq. 

SECTION B. CHEMICAL SCIENCE, INCLUDING ITS APPLICATION TO 

AGRICULTURE AND THE ARTS. 

President. — Michael Faraday, D.C.L., F.R.S. 

Vice-Presidents. — Professor W. R. Grove, F.R.S. Dr. Andrews, F.R.S. 
Professor Johnston, F.R.S. Dr. Daubeny, F.R.S. 

Secretaries. —Dr. Miller, F.R.S. Robert Hunt, Esq. Wm. Randall, Esq. 

SECTION C. GEOLOGY AND PHYSICAL GEOGRAPHY. 

President Leonard Horner, F.R.S., Pres. of Geological Society. 

Vice-Presidents. — The Very Rev. Dr. Buckland, Dean of Westminster. 
Sir Henry De la Beche, F.R.S., Director-General of the Geological Survey of 
the United Kingdom. W'illiam Henry Fitton, M.D., F.R.S. William Hop. 
kins, F.R.S. {For Geographij) G. B. Greenough, F.R.S. 

Secretaries. — Robert A. Austen, F.G.S. Professor Oldham, M.R.I.A., 
F.G.S. J. H. Norton, M.D. {For Geographij) Charles T. Beke, Ph.D. 

SECTION D. ZOOLOGY AND BOTANY. 

President. — Sir John Richardson, M.D., F.R.S. 

Vice-Presidents. — Charles Darwin, M.A., F.R.S. Dr. Robert Brown, 
F.R.S., V.P.L.S. ProfessorE. Forbes, F.R.S. H.E. Strickland, M. A., F.G.S. 

Secretaries.— Br. Lankester, F.R.S., F.L.S. T. V. Wollaston, B.A., 
F.C.P.S. H. W'ooldridge, Esq. 

SECTION E, — PHYSIOLOGY. 

President. — Professor Owen, F.R.S. 

Vice-Presidents. — Sir James Clark, F.R.S. Dr. Roget. Dr. J. Forbes. 
Dr. Fowler. 

Secretaries. — Dr. Sargent. Dr. Laycock. C. P. Keele, Esq. 

SECTION F. — STATISTICS. ^ 

President G. R. Porter, F.R.S. 

Vice-Presidents. — Sir Charles Lemon, Bart., F.R.S. Col. Sykes, F.R.S, 
James Heywood, F.L.S. Edward Nightingale, Esq. 

Secretaries, — W. Cooke Taylor, LL.D. Joseph Fletcher, Esq. F. G. P. 
Neison, Esq. Rev. T. L. Shapcott. 

SECTION G. MECHANICS. 

SL President. — Rev, Professor Willis, F.R.S. 

t Vice-Presidents. — Rev, Dr, Robinson, F,R.S. George Rennie, F.R.S. 

^^Scott Russell, F.R.S, W, Snow Harris, F,R.S. 

^^LSecretaries. — Charles Manby, Sec. Inst. C.E. William Belts, jun. 

^^B SUBSECTION OF ETHNOLOGY. 

^^pPresident. — Dr. Prichard, 

~ Vice-Presidents. — Admiral Sir Charles Malcolm, P,Eth.Soc, Dr. R. G. 
I Latham. Dr. Hodgkin. 
Secretary. — Dr. King. 



Xvi REPORT — 1846. 

CORRESPONDING MEMBERS. 
Professor Agassiz, Neufchatel. M. Arago, Paris. Dr. A. D, Baclie, Phi- 
ladelphia. Professor Berzelius, Stockholm. Professor H. von Boguslavvski, 
Breslau. Monsieur Boutignyd'Evreux, Paris. Professor Braschmann, Mos- 
cow. M. De la Rive, Geneva. Professor Dove, Berlin. Professor Dumas, 
Paris. Professor Ehrenberg, Berlin. Dr. Eisenlohr, Carlsruhe. Professor 
Encke, Berlin. Dr. A. Erman, Berlin. Professor Forchhammer, Copen- 
hagen. Professor Henry, Princeton, United States. Professor Kreil, Prague. 
M. Kupffer, St. Petersburg. Dr. Langberg, Christiania. Baron de Selys 
Longchamps, Liege. M. Frisiani, Milan. Baron Alexander von Humboldt, 
Berlin. M. Jacobi, St. Petersburg. Professor Jacobi, Konigsberg, Dr. La- 
mont, Munich. Baron von Liebig, Giessen. Professor Link, Berlin. Profes- 
sor Matteucci, Pisa. Professor MiddendorfF, St. Petersburg, Dr. OErsted, 
Copenhagen. Chevalier Plana, Turin. M. Qaetelet, Brussels. Professor 
C. Ritter, Berlin. Professor H. Rose, Berlin. Professor Schumacher, 
Altona. Baron Senftenbeig, Bohemia. Dr. Svanberg, Stockholm. Baron 
Sartorius von Waltershausen, Gotha. Professor Wartmann, Lausanne. 



Report of the Proceedings of the Council in 1845-46, presented to the 
General Committee at Southampton, Wednesday, Sept. 9, 184(5. 

Report of the Council to the General Committee. 

1. The Council have the very satisfactory duty to perform, of reporting to 
the General Committee that the resolutions of the Magnetical and Meteoro- 
logical Conference, adopted by the General Committee at Cambridge, on 
the 25th of June 1845, were submitted to the Right Hon. Sir Robert Peel, 
Bart., by the President Sir John Herschel, Bart., accompanied by a commu- 
nication from the Marquis of Northampton, President of the Royal Society, 
conveying the concurrence of that body in the recommendations contained 
therein ; that they received a very favourable consideration from Her Ma- 
jesty's Government, and that the recommendations connected with the British 
observatories, both at home and in the Colonies, are in progress of being 
carried out. The resolutions relating to the East Indian observatories and 
surveys have met with an equally favourable reception from the Hon. Court 
of Directors of the East India Company, and the recommendations which they 
contained have been approved and sanctioned. In accordance with the re- 
solutions passed at Cambridge, therefore, the magnetic observatory at Green- 
wich is permanently continued upon the most extensive and efficient scale. 
The magnetical and meteorological observations are constituted a permanent 
branch of the duties of the astronomical observatories at the Cape of Good 
Hope, Bombay and Madras, and arrangements are in progress for inaking 
them also a permanent branch of the observations to be made at the Obser- 
vatory at Paramatta. The detachment of the Royal Artillery, by whom the 
duties at the Cape of Good Hope have been hitherto performed, has been 
relieved by a permanent increase in jhe civil strength of the Astronomical Ob- 
servatory at that station, and in like manner the officers of the Royal Navy, 
who now form the establishment of tlie observatory at Van Diemen Island, 
will be relieved as soon as the civil establishment at Paramatta is completed. 
The Ordnance Observatories at Toronto and St. Helena are continued until 
the close of 1848. 

With reference to the recommendations relating to magnetic surveys, a 



REPORT OF THE COUNCIL. XVll 

magnetic survey of the Indian Seas by Lieut, Elliot, of the Madras Engineers, 
lias received tlie sanction of the Hon. Court of Directors of the East India 
Company, and is now in progress. Also in the present summer, Lieut. Moore, 
of the Royal Navy, proceeded under the direction of the Lords of the Ad- 
miralty to Hudson's Bay, in one of the vessels belonging to the Hudson's Bay 
Company, for the purpose of connecting the observations of the Canadian 
Survey with those which the Expedition under Sir John Franklin is making 
in the seas to the north of the American Continent. 

In accordance with the recommendation concerning the co-operation of 
foreign magnetical and meteorological observatories, communications were 
made, on the application of the President, by the Earl of Aberdeen, Her Ma- 
jesty's principal Secretary of State for Foreign Affairs, to the governments of 
Russia, Austria, Prussia, Belgium, Sweden and Spain, from all of whom very 
favourable replies have been received. 

2. The resolution passed by the General Committee, to the effect " that 
it is highly desirable to encourage, by specific pecuniary reward, the im- 
provement of self-recording magnetical and meteorological apparatus, and 
that the Presidents of the Royal Society and of the British Association be 
requested to solicit the favourable consideration of Her Majesty's Govern- 
ment to this subject," has been brought under the notice of Government, 
and arrangements have been made to carry the recommendation into effect. 
Whilst on this subject the Council has also much pleasure in noticing that 
the President and Council of the Royal Society have granted £50 from the 
Wollaston Donation Fund to assist in the construction of apparatus devised 
by Mr. Ronalds for the self-registry of magnetical and meteorological instru- 
ments ; which apparatus is in progress of completion at the Observatory of 
the British Association at Kew. The Council are persuaded that the Gene- 
ral Committee will view with satisfaction this co-operaticn of the Royal So- 
ciety and British Association for objects common to both, and for which the 
Observatory at Kew furnishes a very convenient locality. 

3. The General Committee at Cambridge having passed a resolution, 
" That it be rcferi'ed to the Council to take into consideration, before the 
next Meeting of the Association, the expediency of discontinuing the Kew 
Observatory," — the Council appointed a Committee, consisting of the Presi- 
dent (Sir John Herschel), the Dean of Ely, the Astronomer Royal, Professors 
Graham and Wheatstone, and Lieut. -Colonel Sabine, to collect information 
on the scientific purposes which the Kew Observatory has served, and on its 
general usefulness to science and to the Association ; from whom they re- 
ceived the following report : — 

" Kew Observatory, May 7. 1846. Present,—Sir J. F. W. Herschel, Bart., 
the Astronomer Royal, Professors Graham and Wheatstone, and Lieut.- 
Colonel Sabine. • 

" After an attentive examination of the present state of the establishment, 
and of other matters connected therewith, the following resolutions were 
unanimously adopted, viz. — 

" That it be recommended to the General Committee that the establish-, 
ment at Kew, the occupancy of which lias been granted by Her Majesty 
to the British Association, be maintained in its present state of effi- 
ciency: — 

" 1. Because it affords, at a very inconsiderable expense, a local 
habitation to the Association; and a convenient depository for 
its books, manuscripts and apparatus. 
" 2. Because it has afforded to Members of the Association the means 



xviii REPORT — 1846. 

of prosecuting many pliysical inquiries which otherwise would 
not have been entered upon. 

" 3. Because the establishment has already become a point of interest 
to scientific foreigners, several of whom have visited it. 

" 4. Because the grant of the occupancy of tlie building by Her Ma- 
jesty, at the earnest request of the British Association, is an in- 
stance of Her Majesty's interest in, and approval of, the objects 
of the Association. 

" 5. Because, if the Association at the present time relinquish the 
establishment, it will probably never again be available for the 
purposes of science. 

*' 6. Because it appears, both from the publications of the British As- 
sociation and from the records in progress at the establishment, 
that a great amount of electrical and meteorological observation 
has been and continues to be made, and that a systematic inquiry 
into the intricate subject of atmospheric electricity has been car- 
ried out by Mr. Ronalds, which has been productive of very ma- 
terial improvements in that subject, and has in effect furnished 
the model of the processes conducted at the Royal Observatory ; 
and because these inquiries are still in progress under local cir- 
cumstances extremely favourable. 

" 7. Because other inquiries into the working of self-registering ap- 
paratus, both meteorological and magnetical, are in actual pro- 
gress at the establishment, and that there is a distinct prospect 
of the facilities it affords being speedily much more largely pro- 
fited by. 

" 8. Because the access to the Observatory from London to Members 
of the Association will shortly be greatly improved by railroads, 
and because the local facilities and conveniences of the esta- 
blishment have been very greatly enhanced by alterations in its 
relations to the Commissioners of Woods and Forests. 

"J. F. W. Herschel, Chairman." 

In presenting this Report to the General Committee, the Council requests 
that it may be understood to convey also the opinion of the Council. 

4. The Council has received a letter from the honorary Secretary of the 
Literary and Philosophical Institution at Cheltenham, expressing, on the part 
of the Members of that Institution, deep regret that " circumstances have 
arisen which render uncertain their being able to give the British Association 
that welcome and generous reception which it would be their desire to do, 
and which they last year felt that they would have done had the Association 
been so circumstanced as to have accepted the invitation for the summer of 
1846." 

5. The Council has been informed by a letter from W. H. Grove, Esq., 
F.R.S., that a deputation has been appointed by the Mayor and Corporation 
of Swansea, the principal inhabitants, magistracy and country gentlemen of 
the neighbourhood, and by the Members of the Royal Institution of South 
Wales, to attend the Meeting at Southampton, for the purpose of inviting 
the British Association to hold their annual Meeting at Swansea at as early 
a period as may suit their convenience. 

Southampton, September 9, 1846. 



RESEARCHES IN SCIENCE. XIX 

Recommendations adopted by the Gekeral Committee at the 
Southampton Meeting in September 1846. 

Involving Applications to Government and Public Institutions. 

That Her Majesty's Government be requested to direct the publication of 
the Meteorological Observations made by the Officers of the Irish Trigono- 
metrical Survey at Mountjoy and the Pigeon House since the year 1834. 

That application be made to Her Majesty's Government, to direct that 
during the progress of the Ordnance Trigonometrical Surveys in the North 
of Scotland, the so-called parallel roads of Glen Roy and the adjoining coun- 
try be accurately surveyed, with the view of determining whether they are 
truly parallel and horizontal, the intervening distances, and their elevations 
above the present Sea-level. 

Recommendations for Reports and Researches not involving Grants of Money, 

That Mr. Hopkins be requested to furnish, at the next Meeting of the As- 
sociation, a Report on the Theory of such movements and displacements of 
the Earth's Crust, as may be connected with Earthquakes. 

That Mr. Mallet be requested to furnish, at the same time, a Report of 
the Static and Dynamic Facts which have been observed to be the results of 
Earthquakes, or connected with them. 

That Mr. Ellis be requested to continue his Report on the recent Progress 
of Analysis. 

That Professor Edward Forbes be requested to prepare a Report on the 
State of our knowledge of the Acalephae. 

That Mr. J. Scott Russell be I'equested to prepare a Report on the pre- 
sent condition of the Science of Naval Construction, including Steam Navi- 
gation. 

That Mr. Robert Mallet be requested to continue his inquiry into the 
Corrosion of Iron Rails in and out of use. 

That Mr. Robert Hunt be requested to continue his investigations with 
the Actinograph, and that Mr. Ronalds be associated with him (the instru- 
ment to be placed at Kew). 

That Mr. Robert Hunt be requested to continue his investigations on the 
Influence of Light on the Growth of Plants. 

That a Committee, consisting of The Master of Trinity College, Cam- 
bridge, and Capt. Sir James C. Ross, R.N., be requested to draw up a Plan 
for a Naval Expedition for the purpose of completing our knowledge of the 
progress of the Tides. 

That the Master of Trinity College, Cambridge be requested to draw up 
brief Instructions for Tide-observations by Voyagers and Surveyors, with a 
view to a speedy determination of the course of the Tide-wave. 

That Professor Forchhammer's paper on Sea-Currents be printed entire 
among the Reports of the Association. 

That Professor Owen's paper on the Homologies of the Cranial Vertebrae 
be printed entire among the Reports of the Association. 



XX REPORT — 1846. 

Recommendations of Special Researches in Science, involving Grants of 

Money. 

KEW OBSERVATORY. 

That the sum of £150 be placed at the disposal of the Council for the 
purpose of maintaining the establishment in Kew Observatory. 

MATHEMATICAL AND PHYSICAL SCIENCE. 

That £50 be placed at the disposal of Professor Erman for continuing and 
completing the computation of the Gaussian Constants for 1839, and that he 
be requested to superintend the same. 

That Mr. Birt be requested to endeavour to procure a repetition of the 
Barometric Observations made during the month of November in former 
years ; and that the sum of £10 be placed at his disposal for the purpose. 

That a provisional grant of £70 be placed at the disposal of the Committee 
for the publication of the Catalogues of Lalande and Lacaille, to enable them 
to complete the publication and distribution of those Catalogues. 

That a grant of £10 be made for a new Anemometer of the Rev. Dr. Ro- 
binson's improved construction, for the use of the Observatory at Kew ; and 
that Dr. Robinson be requested to superintend its construction. 

CHEMICAL SCIENCE. 

That Dr. Percy and Professor Miller be requested to continue the exa- 
mination of Crystalline Slags, and the quantity of impurities which perfect 
Crystals may contain ; and that £20 be placed at their disposal for the 
purpose. 

That Dr. Schunck be requested to continue his investigations on Colour- 
ing Matters, and that the sum of £10 be placed at his disposal for the 
purpose. 

ZOOLOGY AND BOTANY. 

That Mr. H. E. Strickland, Dr. Daubeny, Professor Lindley and Professor 
Henslow, be requested to continue their experiments on the Vitality of Seeds, 
and that the sum of £10 be placed at their disposal for the purpose. 

ThatCapt. Portlock, R.E., be requested to continue his investigations into 
the Marine Zoology of Corfu by means of the dredge, and that the sum of 
£10 be placed at his disposal for the purpose. 

That a Committee, consisting of Sir Charles Lemon and Mr. Couch, be 
requested to aid Mr. Peach in his Researches into the Marine Zoology of 
Cornwall, and that the sum of £10 be placed at their disposal for the 
purpose. 

That a Committee, consisting of Prof. E. Forbes, Mr. Goodsir, Mr. Pat- 
terson, Mr. Thompson, Mr. Ball, Mr. J. Smith, Mr. Couch, Dr. AUman, 
Mr. M'Andrew, Mr. Alder, and the Rev. F. W. Hope, be requested to con- 
tinue their investigations into the Marine Zoology of Britain by means of the 
dredge, and that the sum of £10 be placed at their disposal for tlie purpose. 

That Sir Philip Egerton, Professor Owen, and Professor E. Forbes, be a 
Committee for aiding Mr. Price in his Researches into the Habits of Marine 
Animals, and that the sum of £10 be placed at their disposal for the purpose. 

That Mr. Newport be requested to draw up a Report on tlie Scorpionidae 
and Tracheary Arachnidae, and that the sum of £10 be placed at the dis- 
posal of Mr. Spence and Mr. WoUaston for the purpose. 

That Professor Owen and Mr. R. Taylor be a Committee to superintend 
the publication of Tabular Forms in reference to the Report of periodical phae- 



SYNOPSIS. XXI 

nomena of Animals and Vegetables, and that £10 be placed at their disposal 
for the purpose. 

MEDICAL SCIENCE. 

That a Committee, consisting of Dr. J. Blake and Professor Sharpey, be 
requested to continue their Researches into the Action of Medicines, and 
that the sum of £20 be placed at their disposal for the purpose. 

That the second and third parts of Dr. Carpenter's Report on the Micro- 
scopic structure of Shells, be illustrated by lithographic plates, not exceed 
ing twenty. 



Synopsis of Grants of Money appropriated to Scientific Objects by the 

General Committee at the Southampton Meeting, September 1845, 
with the Name of the Member, who alone or as the First of a Com- 
mittee, is entitled to draw for the Money. 

Kew Observatory. £ s. d. 

For maintaining the establishment in Kew Observatory under 

the direction of the Council 150 

Mathematical and Physical Science. 

Erman, a. — For Computation of the Gaussian Constants for 

1839 50 

BiRT, W. — Researches on Atmospheric Waves 10 

Robinson, Rev, Dr. — Construction of a New Anemometer. ... 10 

(Committee). — Completion of Catalogue of Stars 70 

Chemical Science, 

Percy, Dr On Crystalline Slags, &c 20 

ScHONCK, Dr — On Colouring Matters 10 

Zoology and Botany, 

Strickland, H. E. — Vitality of Seeds 10 

PoRTLOCK, Capt. — Marine Zoology of Corfu 10 

Lemon, Sir Charles, Bart. — Marine Zoology of Cornwall 10 

Forbes, Prof. E. — Marine Zoology of Britain 10 

Egerton, Sir Philip, Bart. — Habits of Marine Animals ...... 10 

Spence, William. — On Scorpionidae and Arachnidae 10 

Owen, Prof — Tabular Forms for Registering Periodical Phse- 

nomena 10 

Physiology. 

Blake, Dr. — Physiological Action of Medicines 20 

Total of Grants £410 

Dr. Carpenter's Report on the Microscopic Structure of Shells, &:c., to 
be illustrated by Plates, not exceeding Twenty in number. 



1846. 



XXll 



REPORT — 184G 



General Statement of Sums which have been paid on Account of Grants for 
Scientific Purposes. 



1834. 



Tide Discussions 



£ 
20 







1835. 

Tide Discussions .... 62 

BritisliFossil Ichthyolog y 105 

£16 7 

1836. 

Tide Discussions 163 

BritishFossillchtliyology 105 
Thermometric Observa- 
tions, &c 50 

Experiments on long- 
continued Heat .... 17 1 

Rain Gauges 9 13 

Refraction Experiments 15 

Lunar Nutation 60 

Thermometers 15 6 

£434 14 

1837. 

Tide Discussions 284 1 

Chemical Constants .. 24 13 6 

Lunar Nutation 70 

Observations on Waves. 100 12 

Tides at Bristol 150 

Meteorology and Subter- 
ranean Temperature . 89 5 
VitrificationExperiments 150 
Heart Experiments. .. . 8 4 6 
Barometric Observations SO 

Barometers 11 18 6 

£918 14 6 

1838. 

Tide Discussions 29 

British Fossil Fishes ..100 

Meteorological Observa- 
tions and Anemometer 

(construction) 100 

Cast Iron (strength of) . 60 

Animal and Vegetable 
Substances (preserva- 
tion of) 19 1 10 

Carried forward £308 1 10 



£ s. d. 

Brought forward 308 110 

Railway Constants .... 41 12 10 

Bristol Tides 50 

Growth of Plants .... 7500 

Mud in Rivers 3 6 6 

Education Committee . . 50 
Heart Experiments. ... 530 
Land and Sea Level . . 207 8 7 
Subterranean Tempera- 
ture 8 6 

Steam-vessels 100 

Meteorological Commit- 
tee 31 9 5 

Thermometers 16 4 

£956 12 2 



1839. 
Fossil Ichthyology • • • • 1 ' 
Meteorological Observa- 
tions at Plymouth . . 63 
Mechanism of Waves . . 144 

Bristol Tides 35 

Meteorology and Subter- 
ranean Temperature . 21 
VitrificationExperiments 9 
Cast Iron Experiments . 100 
Railway Constants .... 28 
Land and Sea Level . . 274 
Steam-Vessels' Engines. 100 
Stars in Histoire Celeste 331 

Stars in Lacaille 11 

Stars in R.A.S. Catalogue 
Animal Secretions .... 10 
Steam-engines in Corn- 
wall 50 

Atmospheric Air 16 

Cast and Wrought Iron. 40 
Heat on Organic Bodies 5 
Gases on Solar Spec- 
trum 22 

Hourly Meteorological 
Observations, Inver- 
ness and Kingussie . . 49 

Fossil Reptiles 118 

Mining Statistics 50 

£1595^ 







10 





o 





18 


6 


11 





4 


7 








7 


2 


1 


4 








18 


6 








16 





10 











1 






















7 8 
2 9 




11 



GENERAL STATEMENT. 



XXUl 



£ 
1840. 

Bristol Tides 100 

Subterranean Tempera- 
ture < 13 

Heart Experiments. ... IS 
Lungs Experiments . . 8 
Tide Discussions. . . , . . 50 
Land and Sea Level . . 6 
Stars (Histoire Celeste) 2412 

Stars (Lacaille) 4 

Stars (Catalogue) .... 2G4 

Atmospheric Air 15 

Water on Iron 10 

Heat on Organic Bodies 7 
MeteorologicalObserva- 

tions 52 

Foreign Scientific Me- 
moirs 112 

Working Population .. 100 

School Statistics 50 

Forms of Vessels .... ] 84 
Chemical and Electrical 

Phsenomena 40 

Meteorological Observa- 
tions at Plymouth . . 80 
Magnetical Observations 185 
£1546 



s. d. 







13 


6 


19 





13 











11 


1 


10 





15 











15 
















17 





7 








13 .0 



16 4 



1841. 

Observations on Waves. 30 
Meteorologyand Subter- 
ranean Temperature . 8 8 

Actinometers 10 

Earthquake Shocks . . 17 7 

Acrid Poisons 6 

Veins and Absorbents. . 3 

Mud in Rivers. .• 5 

Marine Zoology 15 12 8 

Skeleton Maps 20 

Mountain Barometers. . G 18 6 

Stars (Histoire Celeste). 185 

Stars (Lacaille) 79 5 

Stars (Nomenclature of) 17 19 6 

Stars (Catalogue of) . . 40 

Water on Iron 50 

Meteorological Observa- 
tions at Inverness . . 20 
Meteorological Observa- 
tions (reduction of) .. 25 

Carried forward £539 10 8 



£ 8. d. 

Brought forward 539 10 8 

Fossil Reptiles •. 50 

Foreign Memoirs .... 62 
Railway Sections .... 38 1 6 
Forms of Vessels .... 193 12 
Meteorological Observa- 
tions at Plymouth . . 55 
Magnetical Observations 61 18 8 
Fishes of the Old Red 

Sandstone 100 

Tides at Leilh 50 

Anemometer at Edin- 
burgh 69 1 10 

Tabulating Observations 9 6 3 

Races of Men 5 

Radiate Animals 2 

£1235 10 11 



1842. 
Dynamometric Instru- 
ments 113 

Anoplura Britanniae . . 52 

Tides at Bristol 59 

Gases on Light 30 

Chronometers 26 

Marine Zoology 1 

British Fossil Mammalia 100 

Statistics of Education . . 20 
Marine Steam-vessels' 

Engines 28 

Stars (Histoire Celeste) 59 
Stars (British Associa- 
tion Catalogue of) .. 110 
Railway Sections. .... . 161 

British Belemnites .... 50 

Fossil Reptiles (publica- 
tion of Report) .... 210 

Forms of Vessels. .... . 180 

Galvanic Experiments on 

Rocks 5 

Meteorological Experi- 
ments at Plymouth. . 68 
Constant Indicator and 
Dynamometric Instru- 
ments 90 

Force of Wind 10 

LightonGrowthof Seeds 8 

Vital Statistics 50 

Vegetative Power of 
Seeds 

Carried forward £1442 



11 


2 


12 





8 





14 


7 


17 


6 


5 



































10 
























8 6 















8 1 11 

8 8 



GKNERAIi STATEMENT* 



£ s. d. 



Brought forward 1442 8 
Questions on Human 

Race 7 9 







£1449 17 8 















^A 


8 









184.^3. 

Revision of the Nomen- 
clature of Stars 2 

Reduction of Stars, Bri- 
tish Association Cata- 
logue 25 

Anomalous Tides, Frith 
of Forth 120 

Hourly Meteorological 
Observations at Kin- 
gussie and Inverness 77 

Meteorological Observa- 
tions at Plymouth . . 55 

Whewell's Meteorolo- 
gical Anemometer at 
Plymouth 10 

Meteorological Observa- 
tions, Osier's Anemo- 
meter at Plymouth . . 20 

Reduction of Meteorolo- 
gical Observations . . 30 

Meteorological Instru- 
ments and Gratuities 39 G 

Construction of Anemo- 
meter at Inverness .. 56 12 2 

Magnetic Co-operation . 10 8 10 

Meteorological Recorder 

for Kevv Observatory 50 

Action of Gases on Light 18 IG 1 

Establishment at Kew 
Observatory, Wages, 
Repairs,Furniture,and 
Sundries ..133 4 7 

Experiments by Captive 

Balloons 81 8 

Oxidation of the Rails 

of Railways.. 20 

Publication of Report on 

Fossil Reptiles 40 

Coloured Drawings of 

Railway Sections. .. . 147 18 3 

Registration of Earth- 
quake Shocks 30 

Report on Zoological 

Nomenclatiu-e 10 

Carried forward £977 6 7 



£ 
Brought forward 977 

Uncovering Lower Red 
Sandstone near Man- 
chester 4 

Vegetative Power of 
Seeds 5 

MarineTestacea (Habits 
of) 10 

Marine Zoology 10 

Marine Zoology 2 

Preparation of Report 
on British Fossil Mam- 
malia 100 

Physiological operations 
of Medicinal Agents 20 

Vital Statistics 36 

Additional Experiments 
ontheFormsofVessels 70 

Additional Experiments 
ontheFormsofVessels 100 

Reduction of Observa- 
tions on the Forms of 
Vessels 100 

Morin's Instrument and 
Constant Indicator . . 69 

Experiments on the 
Strength of Materials 60 

£1565 



s. d. 

6 7 



4 6 

3 8 





14 11 













5 


8 




















4 


10 









10 2 



1844. 

Meteorological Observa- 
tions at Kingussie and 
Inverness 12 

CompletingObservations 

at Plymouth 35 

Magnetic and Meteoro- 
logical Co-operation. . 25 8 4 

Publication of tlie Bri- 
tish Association Cata- 
logue of Stars 35 

Observations on Tides 
on the East Coast of 
Scotland 100 

Revision of the Nomen- 
clature of Stars. . 1842 2 9 6 

Maintaining the Esta- 
blislmient in Kew Ob- 
servatory 117 17 3 

Instruments for Kew Ob- 
servatory 56 7 3 

Carried forward £384 2 4 



GENERAL STATEMENT. 



o 


4 














17 


6 








11 


10 



£ 
Brought forward 384 
Influence of light on 
Plants 10 

Subterraneous Tempera- 
ture in Ireland 5 

Coloured Drawings of 
Railway Sections .... 15 

Investigation of Fossil 
Fishes of the Lower 
Tertiary Strata .... 100 

Registering the Shocks 
of Earthquakes, 1842 23 

Researches into the 
Structure of Fossil 
Shells 20 

Radiata and Mollusca of 
the JEgean and Red 
Seas 1842 100 

Geographical distribu- 
tions of Marine Zo- 
ology 1842 10 

Marine Zoology of De- 
von and Cornwall .. 10 

Marine Zoology of Corfu 10 

Experiments on the Vi- 
tality of Seeds 9 

Experiments on the Vi- 
tality of Seeds. . 1842 8 

Researches on Exotic 
Anoplura 15 

Experiments on the 
Strength of Materials 100 

Completing Experiments 
on the Forms of Ships 100 

Inquiries into Asphyxia 10 

Investigations on the in- 
ternal Constitution of 
Metals 50 

Constant Indicator and 
Morin's Instrument, 
1842 10 



£981 







5 

7 3 











_3 6 

12 8 



1845. 

Publication of the British 
Association Catalogue 
of Stars 351 14 6 

Meteorological Observa- 
tions at Inverness ., 30 18 11 

Magnetic and Meteoro- 
logical Co-operation 16 16 8 

Carried forward £399 10 1 



Brought forward 399 

Meteorological Instru- 
ments at Edinburgh 18 

Reduction of Anemome- 
trical Observations at 
Plymouth 25 

Electrical Experiments 
at Kew Observatory 43 

Maintaining the Esta- 
blishment in Kew Ob- 
servatory 149 

For Kreil's Barometro- 
graph 25 

Gases from Iron Fur- 
naces 50 

Experiments on the Ac- 
tinograph 1 ^ 

Microscopic Structure of 
Shells 20 

Exotic Anoplura . .1843 10 

Vitality of Seeds.. 1843 2 

Vitality of Seeds.. 1844 7 

Marine Zoology of Corn- 
wall 10 

Physiological Action of 
Medicines 20 

Statistics of Sickness and 
Mortalit:y in York . . 20 

Registration of Earth- 
quake Shocks . .1843 15 



s. d. 

10 1 

11 9 









17 


8 


15 































7 






















1^ 



£831 9 9 



1846. 

British Association Ca- 
talogue of Stars, 1844 211 15 

Fossil Fishes of the Lon- 
don Clay 100 

Computation of the Gaus- 
sian Constants for 1839 50 

Maintaining the Esta- 
blishment at Kew Ob- 
servatory 146 16 

Experiments on the 
Strength of Materials 

Researches in Asphyxia 

Examination of Fossil 
Shells 

Vitality of Seeds,. 1844 

Vitality of Seeds. . 1845 

Marine Zoology of Corn- 
wall 



60 
6 



10 












16 2 


15 10 



7 12 



10 



Carried forward £605 15 10 



XXVI 



REPORT — 1846. 



Brought foivvard 
Marine Zoology of Bri- 
tain 

Exotic Anoplnra. . 1844 
Expenses attending Ane- 
mometers 

Anemometers' Repairs • 

Carried forward 



£ 


s. 


d. 


G05 


\5 


10 


10 








25 








11 


7 


6 


2 


3 


6 



G54 6 10 



£ s. d. 
Brought forward 654 6 10 
Researches on Atmo- 
spheric Waves 3 3 3 

Captive Balloons.. 1844 8 19 8 
Varieties of the Human 

Race 1844 7 6 3 

Statistics of Sickness and 

Mortality at York .. 12 

£6«.5 16 



Extracts from Resolutions of the General Coynmiltee. 

Committees and individuals, to whom grants of money for scientific pur- 
poses have been entrusted, are required to present to each following meeting 
of the Association a Report of the progress which has been made ; with a 
statement of tlie sums which have been expended, and the balance which 
remains disposable on each grant. 

Grants of pecuniary aid for scientific purposes from the funds of the Asso- 
ciation expire at the ensuing meeting, unless it shall appear by a Report that 
the Recommendations have been acted on, or a continuation of them be ordered 
by the General Committee. 

In each Committee, the Member first named is the person entitled to call 
on the Treasurer, John Taylor, Esq., 2 Duke Street, Adelphi, London, for 
such portion of the sum granted as may from time to time be required. 

In "rants of money to Committees, the Association does not contemplate 
the payment of personal expenses to the Members, 

In all cases where additional grants of money are made for the continua 
tion of Researches at the cost oi the Association, the sum named shall be 
deemed to include, as a part of the amount, the specified balance which may 
remain unpaid on the former grant for the same object. 



On Thursday evening, September 10th, at 8 p.m., in the Victoria Rooms, 
Southampton, the late President, Sir John F. W. Herschel, Bart., F.R.S., 
resigned his office to Sir Roderick Impey Murchison, G.C.S'.S., F.R.S.,who 
took the Chair at the General Meeting, and delivered an Address, for which 
see p. xxvii. 

On Friday evening, September 11th, in the same room, Professor Owen, 
F.R.S., delivered a Discourse on the Fossil Mammalia of the British Islands. 

On Monday evening, in the same room, Charles Lyell, Esq., F.R.S., de- 
livered a Discourse on the Valley and Delta of the Mississippi, and other 
points in the Geology of the United States, from observations made in the 
years 1845-46. 

On Tuesday evening, in the same room, W. R. Grove, Esq. explained 
the properties of the Explosive Substance recently discovered by Dr. Schon- 
bein ; and communicated some recent researches of his own, on the Decom- 
position of Water into its constituent Gases by Heat. 

On Wednesday evening, at 8 p.m., in the same room, the Concluding 
General Meeting of the Association was held, v.hen the Proceedings of the 
General Committee, and the grants of money for scientific purposes, were 
explained to the Members. 

The Meeting was adjourned to Oxford, on the 24th of June, 1847. 



ADDRESS 



SIR RODERICK IMPEY MURCHISON, G.C.St.S., F.R.S., V.P.G.S. 

&c. &c. 



Gentlemen, — After fifteen years of migration to various important cities 
and towns in the United Kingdom, you are for the first time assembled in the 
South-Eastern districts of England, at the solicitation of the authorities and 
inhabitants of Southampton. Easily accessible on all sides to the cultivators 
of science, this beautiful and flourishing sea-port is situated in a tract so 
adorned by nature, and so full of objects for scientific contemplation, that, 
supported as we are by new friends in England, and by old friends from 
distant parts of Europe, we shall indeed be wanting to ourselves, if our 
proceedings on this occasion should not support the high character which 
the British Association has hitherto maintained. 

For my own part, though deeply conscious of my inferiority to my eminent 
predecessor in the higher branches of science, I still venture to hope, that 
the devotion I have manifested to this Association from its origin to the pre- 
sent day, may be viewed by you as a guarantee for the zealous execution of 
my duties. Permit me then. Gentlemen, to offer you my warmest acknow- 
ledgements for having placed me in this enviable position ; and to assure 
you, that I value the approbation which it implies as the highest honour 
which could have been bestowed on me — an honour the more esteemed from 
its being conferred in a county endeared to me by family connexions, and in 
which I rejoice to have made my first essay as a geologist. 

The origin, progress and objects of this our " Parliament of Science " have 
been so thoroughly explained on former occasions by your successive Pre- 
sidents, particularly in reference to that portion of our body which cultivates 
the mathematical, chemical and mechanical sciences, that after briefly allu- 
ding to some of the chief results of bygone years, with a view of impressing 
upon our new members the general advances we have made, I shall in tins 
address dwell more particularly on the recent progress and present state of 
natural history, the department of knowledge with which my own pursuits 
have been most comiected, whilst I shall also incidentally advert to some of 
the proceedings which are likely to occupy our attention during this Meeting. 

No sooner, Gentlemen, had this Association fully estal)lished its character 
as a legitimate representative of the science of the United Kingdom, and by 
its published Reports, the researches which it instituted, and the other sub- 
stantial services which it rendered to science, had secured public respect, 
than it proceeded towards the fulfilment of the last of the great objects which 
a Brewster and a Harcourt contemplated at its foundation, by inviting the , 
attention of the Government to important national points of scientific inter- 
est. At the fourth Meeting held in Edinburgh, the Association memorialized 
the Government to increase the forces of the Ordnance Geographical Survey 
of Britain, and to extend speedily to Scotland the benefits which had been 
already applied by that admirable establishment to the South of England, 
Wales and Ireland. From that time to the present it has not scrupled to call 



xxviii REPORT — 1846. 

the notice of the Ministers of the day to everj"^ great scientific measure which 
seemed, after due consideration, likely to promote the interests or raise the 
character of the British nation. Guided in the choice of these applications 
by a committee selected from among its members, it has sedulously avoided 
the presentation of any request which did not rest on a rational basis, and 
our rulers, far from resisting such appeals, have uniformly and cordially 
acquiesced in them. Thus it was when, after pajing large sums from our 
own funds for the reduction of masses of astronomical observations, we 
represented to the Government the necessity of enabling the Astronomer 
Royal to perform the same work on the observations of his predecessors 
which had accumulated in the archives of Greenwich, our appeal was an- 
swered by arrangements for completing so important a public object at the 
public expense. Thus it was, when contemplating the vast accession to pure 
science as well as to useful maritime knowledge, to be gained by the ex- 
ploration of the South Polar regions, that we gave the first impulse to the 
project of the great Antarctic expedition, which, supported by the influence 
of the Royal Society and its noble President, obtained the full assent of the 
Government, and led to results which, through the merits of Sir James 
Ross and his companions, have shed a bright lustre on our country, by 
copious additions to geography and natural history, and by affording nu- 
merous data for the development of the laws that regulate the magnetism of 
the earth. 

The mention of terrestrial magnetism brings with it a crowd of recol- 
lections honourable to the British Association, from the perspicuous manner 
in which every portion of fresh knowledge on this important subject has 
been stoi'ed up in our volumes, with a view to generalization, by Colonel 
Sabine and others ; whilst a wide field for its diffusion and combination has 
been secured by the congress held at our last meeting, at which some of the 
most distinguished foreign and British magneticians were assembled under 
the presidency of Sir John Herschel. 

It is indeed most satisfactory for us to know, that not only did all the 
recommendations of the Association on this subject which were presented 
to our Government meet Avith a most favourable reception, but that in 
consequence of the representations made by Her Majesty's Secretary of. 
State for Foreign Affairs to the public authorities of other countries which 
had previously taken part in the system of cooperative observation, the 
Governments of Russia, Austria, Prussia and Belgium have notified their 
intention of continuing their respective magnetical and meteorological ob- 
servations for another term of three years. 

In passing by other instances in which public liberality has been directed 
to channels of knowledge which required opening out, I must not omit to 
notice the grant obtained from our gracious Sovereign, of the Royal Obser- 
vatory at Kew, which, previously dismantled of its astronomical instruments, 
has, under the suggestions of Professor Wheatstone, been converted by us 
into a station for observations purely physical, and especially for those de- 
tails of atmospheric phcenomena which are so minute and numerous, and 
require sucii unremitting attention, that they imperiously call for separate 
establishments. In realizing this principle, we can now refer British and 
foreign philosophers to our own observatory at Kew, where I have the au- 
thority of most adequate judges for saying they will find that a great amount 
of electrical and meteorological observation has been made, and a system- 
atic inquiry into the intricate subject of atmospheric electricity carried 
out by Mr. Ronalds, to which no higher praise can be given, than that it 
has, in fact, furnished the model of the processes conducted at the Royal 



ADDRESS. XXIX 

Observatory of Greenwich. This establishment is besides so useful through 
the facilities which it offers for researches into the working of self-register- 
ing instruments whicii are there constructed, that I earnestly hope it may 
be sustained as heretofore by annual grants from our funds, particularly 
as it is accomplishing considerable results at very small cost. 

Our volume for the last year contains several communications on physical 
subjects from eminent foreign cultivators of science, whom we have the 
pleasure of reckoning amongst our corresponding members, and whose com- 
munications, according to the usage of the Association, have been printed 
entire amongst the reports. In a discussion of the peculiarities by which the 
great comet of IS^S was distinguished. Dr. Von Bogusiawski of Breslau has 
taken the occasion to announce the probability, resting on calculations which 
will be published in Schumacher's ' Astronomische Nachrichten,' of the iden- 
tity of this comet with several of a similar remarkable character recorded in his- 
tory, commencing with the one described by Aristotle, which appeared in the 
year 371 before our a^i-a : should his calculations be considered to establish 
this fact. Dr. Von Bogusiawski proposes that the comet should hereafter be 
distinguished by the name of " Aristotle's Comet." This communication 
contains also some highly ingenious and important considerations relating to 
the physical causes of the phsenomena of the tails of comets. 

Dr. Paul Erman of Berlin, father of the adventurous geographical explorer 
and magnetician who was one of the active members of the magnetic con- 
gress at Cambridge, has communicated through his son some interesting ex- 
periments on the electro-dynamic effects of the friction of conducting sub- 
stances, and has pointed out the differences between these and normal 
thermo-electric effects. Baron von Senftenberg (who is an admirable ex- 
ample of how much may be done by a liberal zeal for science combined 
with an independent fortune) has published an account of the success with 
which self-registering meteorological instruments have been established at his 
observatory at Senftenberg, as well as at the national observatory at Prague. 

Of our own members, Mr. Birt has contributed a second report on Atmo- 
spheric Waves, in continuation of the investigation which originated in the 
discussion by Sir John Herschel, of the meteorological observations which, 
at his suggestion, were made in various parts of the globe, at the periods of 
the equinoxes and solstices, commencing with the year 1834. 

In a communication to the Meeting of the Association at York, Colonel 
Sabine traced with great clearness (from the hourly observations at Toronto) 
the effect of the single diurnal and single annual progressions of tempera- 
ture, in producing on the mixed vapourous and gaseous elements of the atmo- 
sphere, the well-known progressions of daily and yearly barometrical pressure. 
To the conclusions which he then presented, and which apply, perhaps gene- 
rally, to situations not greatly elevated in the interior of large tracts of land, 
the same author has added, in the last volume, a valuable explanation of 
the more complicated phaenomena which happen at points where land and 
sea breezes, flowing with regularity, modify periodically and locally the con- 
stitution and pressure of the atmosphere. Taking for his data the two-hourly 
observations executed at the observatory of Bombay by Dr. Buist, Colonel 
Sabine has succeeded in demonstrating for this locality a double daily pro- 
gressiou of gaseous pressure, in accordance with tlie flow and re-flow of the 
air from surfaces of land and water which are unequally affected by heat. 
And thus the diurnal variation of the daily pressure at a point within the 
tropics, and on the margin of the sea, is explained by the same reasoning 
which was suggested by facts observed in the interior of the vast continent 
of North America. 



XXX REPORT — 1846. 

Among the many useful national objects which have been promoted by 
the physical researches of the British Association, there is one which calls 
for marked notice at this time, in the proposal of Mr. Robert Stephenson to 
carry an iron tube or suspended tunnel over the Menai Straits to sustain 
the great railway to Holyhead. This bold proposal could never have been 
realized, if that great engineer had not been acquainted with the progress 
recently made in the knowledge of the strength of materials, and specially of 
iron ; such knowledge being chiefly due to investigations in which the Asso- 
ciation has taken and is still taking a conspicuous share, by the devotion of 
its friends and the employment of its influence — investigations which have, 
as you know, been prosecuted with great zeal and success by its valued mem- 
bers Mr. Hodgkinson and Mr. Fairbairn. I may further state, that in the 
recent improvements in railways the aid of scientific investigations having 
been called for by the civil engineer, to assist him in determining with accu- 
racy the power to be provided for attaining the high velocities of fifty and 
sixty miles an hour; it was found and admitted by the most eminent en- 
gineers, that the very best data for this purpose, and indeed the only experi- 
ments of any practical value, were those which had been provided for some 
years ago by a Committee of the British Association, as published in our 
Transactions. Let such results as these be our answer when we are asked, 
what have been the useful objects attained by the British Association! 

However imperfect my knowledge of experimental philosophy may be, I 
must now notice that the last volume of our Reports contains two contribu- 
tions to it, in which subjects of the deepest theoretical and practical interest 
have been elucidated, at the request of the Association, by the labours of 
our foreign coadjutors. 

That some substance of a peculiar kind everywhere exists, or is formed in 
the atmosphere by electrical agency, both natural and artificial, had long been 
suspected, especially from the persistency of the odour developed by such 
agency, and its transference by contact to other matter. Professor Schon- 
bein, to whom I shall hereafter advert as the author of an important practical 
discovery, is, however, the first philosopher who undertook to investigate the 
nature of that substance ; and though the investigation is not yet complete, 
he has been enabled to report no inconsiderable progress in this difficult and 
refined subject of research. 

A request from the Association to Professor Bunsen of Marburg, and our 
countryman Dr. Lyon Playfair, coupled with a contribution of small amount 
towards the expenses involved in the undertaking, has produced a report on 
the conditions and products of iron-furnaces which is of considerable value in 
a commercial view to one of the most important of our manufactures, and 
possesses at the same time a very high interest to chemical science in some 
of the views which it developes. On the one hand it exhibits an entirely 
new theory of the reduction, by cyanogen gas as the chief agent, of iron 
from the ore; on the other it shows, that in addition to a vast saving of fuel, 
about 2 cwt. of sal-ammoniac may daily be collected at the single establish- 
ment of Alfreton, where the experiments were made ; thus leading us to 
infer that in the iron-furnaces of Britain there may be obtained from vapour 
which now passes away, an enormous quantity of this valuable substance, 
which would materially lessen the dependence of our agriculturists on foreign 
guano. It is, indeed, most gratifying to observe, that in pursuing this inquiry 
into the gaseous contents of a blazing furnace of great height, our associates 
traced out, foot by foot, the most recondite chemical processes, and described 
the fiery products with the same accuracy as if their researches had been made 
on the table of a laboratory. Weighed however only in the scales of absolute 



ADDRESS. XXXI 

and immediate utility, the remarkable results of these skilful and elaborate 
experiments give them a character of national importance, and justly entitle 
the authors and the body which has aided them to the public thanks. 

After this glance at the subjects of purely physical science treated of in 
the last volume of our Transactions, let us now consider the domains of 
natural history ; and as one of the cultivators of a science which has derived 
its main support and most of its new and enlarged views from naturalists, 
let rae express the obligation which geologists are under to this Association, 
for having aided so effectively in bringing forth the zoological researches 
of Owen, Agassiz, and Edward Forbes. These three distinguished men 
have themselves announced, that in default of our countenance and assist- 
ance, they would not have undertaken, and never could have completed, 
some of their most important inquiries. Agassiz, for example, had not 
otherwise the means of comparing the ichthyolites of the British Isles with 
those of the continent of Europe. Without this impulse, Owen would 
not have applied his profound knowledge of comparative anatomy to British 
fossil saurians ; and Edward Forbes might never have been the explorer of 
the depths of the JEgean, nor have revealed many hitherto unknown laws 
of submarine life, if his wishes and suggestions had not met with the warm 
support of our body, and been supported by its strongest recommendations 
to the Naval authorities. 

Such allusions to naturalists, whose works have afforded the firmest sup- 
ports to geology, might lead me to dilate at length on the recent progress of 
this science ; but as the subject has been copiously treated at successive 
anniversaries of the Geological Society of London, and has had its recent 
advances so clearly enunciated by the actual President of that body who now 
presides over our Geological Section *, I shall restrain my " esprit de corps " 
whilst I briefly advert to some of the prominent advances which geologists 
have made. When our associate Conybeare reported to us at our second 
meeting, on the actual state and ulterior prospects of what he well termed the 
" archaeology of the globe," he dwelt with justice on the numerous reseai'ches 
in different countries which had clearly established the history of a descent 
as it were into the bowels of the earth — which led us, in a word, downwards 
through those newer deposits that connect high antiquity with our own 
period, into those strata which support our great British coal-fields. Beyond 
this however the perspective was dark and doubtful — 

" Res alta terra et caligiue mersas." 

Now, however, we have dispersed this gloom, and by researches first 
carried out to a distinct classification in the British Isles, and thence ex- 
tended to Russia and Arnerica, geologists have shown that the records of 
succession, as indicated by the entombment of fossil animals, are as une- 
quivocally developed in these very ancient or palaeozoic strata as in any of 
the overlying or more recently formed deposits. After toiling many years in 
this department of the science, in conjunction with Sedgwick, Lonsdale, De 
Verneuil, Keyserling, and others of ray fellow-labourers, I have arrived at 
the conclusion, that we have reached the very genesis of animal life upon 
the globe, and that no separate and clearly definable " vestigia retrorsum " 
will be found beneath that protozoic or Lower Silurian group, in the great 
inferior mass of which scarcely a vertebrated animal has yet been detected, 
amid the profusion of the lower orders of marine animals entombed in it. 
But however this may be, it is certain that in the last few years all Central 

* Mr. Leonard Homer. 



XXxii REPORT — 1846. 

and Eastern Europe, including Turkej'*, and even parts of Siberia have been 
brought into accordance M'ith typical strata. France has been accurately 
classified and illustrated by the splendid map of Elie de Beaumont and Du- 
frenoy ; and whilst, by the labours of Deshayes and others, its tertiary fossils 
have been copiously described, the organic remains of its secondary strata are 
now undergoing a complete analysis in the beautiful work of M. Alcide d'Or- 
bigny. Belgium, whose mineral structure and geological outlines have been 
delineated by D'Omalius d'Halloy and Dumont, has produced very perfect 
monographs of its palaeozoic and tertiary fossils ; the first in the work of 
M. de Koningk, the second in the recently published monograph of M. Nyst. 
Germany, led on by Von Buch, has shown that she can now as materially 
strengthen ti»8 zoological and botanical groundworks of the science, as in the 
days of Werner she was eminent in laying tliose mineralogical foundations 
M'hich have been brought so near to perfection by the labours of several living 
men. So numerous in fact have been the recent German contributions, that 
I cannot permit myself to specify the names of individuals in a country w'hich 
boasts so many excellent geologists. As distinctly connected, however, with 
the objects of this Meeting, I must be permitted to state that the ingenious 
botanist Gdppert, whose works, in combination with those of Adolphe Brong- 
niart in France, have shed so much light on fossil plants, has just sent to me, 
for communication to our Geological Section, the results of his latest inqui- 
ries into the formation of the coal of Silesia — results which will be the more 
interesting to Dr. Buckland and the geologists of England, who have most 
attended to this subject, because they are founded on data equally new and 
original. Italy has also, to a great extent, been presented to us in its 
general geological facies, through the labours of Sismonda, Marmora, Pareto, 
Pasiui, Catullo and others ; whilst our kinsmen of the far West, with the 
true enterprise of the Saxon race, have so laid open the structure of their 
wide-spreading States, that our countryman Lyell has informed us, that the 
instructive map which accompanies his work upon North America is simply 
the grouping together of data prepared by native State geologists, which he 
has paralleled with our well-known British types. 

If then the astronomer has, to a vast extent, expounded the mechanism of 
the heavens ; if lately, through his large, new telescope, our associate the 
Earl of Rosse has assigned a fixity and order to bodies which were pre- 
viously viewed as mere nebulse floating in space, and has also inferred that the 
surface-cavities in our nearest neighbour of the planetary system are analo- 
gous to the volcanic apertures and depressions of the earth ; the geologist, 
contributing data of another order to the storehouse of natural knowledge, 
has determined, by tangible proofs, the very manner in which our planet has 
been successively enveloped in divers cerements, each teeming with peculiar 
forms of distinct life, and has marked the revolutions which have interfered 
witli these successive creations, from the earliest dawn of living things to the 
limits of the historic sera. In short, the fundamental steps gained in geology 
since the early days of the British Association, are so remarkable and so 
numerous, that the time has now come for a second report upon the progress 
of this science, which may I trust be prepared for an approaching Meeting. 

Intimately connected with these broad views of the progress of geology is 
the appearance of the first volume of a national work by Sir Henry De la 
Beche and his associates in the Geological Survey of Great Britain. Fol- 
lowing, as it does, upon the issue of numerous detailed coloured maps and 

* See the geological map of Tui'key by Bouc, and that of Russia and the Ural Mountains 
by my coadjutors and myself. 



ADDRESS. XXXm 

sections, which for beauty of execution and exactness of detail are unrivalled, 
I would specially direct your attention to this new volume, as affording the 
clearest evidence that geology is now strictly brought within the pale of the 
fixed sciences. In it are found graphic descriptions of the strata in the South- 
West of England and South Wales, whose breadth and length are accurately 
measured, whose mineral changes are chemically analysed, and whose im- 
bedded remains are compared and determined by competent palaeontologists. 
The very statistics of the science are thus laid open, theory is made rigorously 
to depend on facts, and the processes and produce of foreign mines are com- 
pared with those of Britain. 

When we know how intimately the Director- General of this Survey and 
his associates have been connected with the meetings of the British Associa- 
tion, and how they have freely discussed with us many parts of their re- 
searches — when we recollect that the geologist of Yorkshire, our invaluable 
Assistant General Secretary, around whom all our arrangements since our 
origin have turned, and to whom so much of our success is due, occupies 
his fitting place among these worthies — that Edward Forbes, who passed as 
it were from this Association to the iEgean, is the palaeontologist of this 
Survey ; and again when we reflect, that if this Association had not repaired 
to Glasgow, and there discovered the merits of the delineation of the Isle of 
Arran by Mr. Ramsay, that young geologist would never have become a 
valuable contributor to the volume under consideration — it is obvious from 
these statements alone, that the annual visits of our body to different parts 
of the Empire, by bringing together kindred spirits, and by testing the natural 
capacity of individuals, do most effectually advance science and benefit the 
British community. 

Whilst considering these labours of the Government geologists, I shall now 
specially speak of those of Professor E. Forbes, because he here makes him- 
self doubly welcome, by bringing to us as it were upon the spot the living 
specimens of submarine creatures, which through the praiseworthy en- 
thusiasm of Mr. M'^ Andrew, one of our members, who fitted out a yacht 
for natural-history researches, have been dredged up this summer by these 
naturalists from the southern coast, between the Land's End and South- 
ampton. As a favourite yachting port like Southampton may, it is hoped, 
afford imitators, I point out witli pleasure the liberal example of Mr, M'^An- 
drew, who although not professing to describe the specimens he collects, has 
now, as on former occasions, placed them in the hands of the members best 
qualified to do them justice, and is thus a substantial promoter of science. 

The memoir, then, of Edward Forbes in the Government Geological Sur- 
vey to which I now allude, is, in truth, an extension of his views re- 
specting the causes of the present distribution of plants and animals in the 
British Isles, made known crt the last meeting of tlie British Association. As 
this author has not only shown the application of these ideas to the re- 
searches of the British Geological Survey, but also to the distribution of 
animals and plants over the whole earth, it is evident that these views, in 
great part original, will introduce a new class of inquiries into natural history, 
which will link it on more closely than ever to geology and geography. In 
short, this paper may be viewed as the first attempt to explain the causes 
of the zoological and botanical features of any region anciently in connexion. 
Among the new points which it contains, I will now only mention, that it 
very ingeniously (and I trust satisfactorily) explains the origin of the pecu- 
liar features of the botany of Britain — the theory of tlie origin of Alpine 
Floras distributed far apart — the peculiarity of the zoology of Ireland as 
compared with that of England — the presence of the same species of marine 



XXxiv REPORT — 1846. 

animals on the coasts of America and Europe — the specialities of the marine 
zoology of the British seas called for by this Association — the past and pre- 
sent distribution of the great Mediterranean Flora ; — and lastly, it applies 
the knowledge we possess of the distribution of plants to the elucidation of 
the history of the superficial detritus, termed by geologists the " Northern 
Drift." 

Amid the numerous subjects for reflection which the perusal of this me- 
moir occasions, I must now restrict myself to two brief comments. First, 
to express my belief that even Humboldt himself, who has written so much 
and so admirably on Alpine floras, will admit that our associate's explanation 
of the origin of identity removes a great stumbling-block from the path of 
botanical geographers. Secondly, having myself for some years endeavoured 
to show, that the Alpine glacialists had erroneously applied their views, as 
founded on terrestrial phaenomena, to large regions of Northern Europe, 
which must have been under the sea during the distribution of erratic blocks, 
gravel and boulders, I cannot but consider it a strong confirmation of that 
opinion, when I find so sound a naturalist as Edward Forbes sustaining the 
same view by perfectly independent inferences concerning the migration of 
plants to isolated centres, and by a studious examination and comparison of 
all the sea shells associated with these transported materials. And if I mis- 
take not, my friend Mr. Lyell will find in both the above points, strong evi- 
dences in support of his ingenious climatal theories. 

Recent as the blocks and boulders to which I have alluded may seem 
to be, they were however accumulated under a glacial sea, whose bottom 
was first raised to produce that connexion between the continent and 
Britain, by which the land animals migrated from their parent East to our 
western climes ; a connexion that was afterwards broken through by the se- 
paration of our islands, and by the isolation in each of them of those terres- 
trial races which had been propagated to it. This latter inference was also, 
indeed, thoroughly sustained by the researches of Professor Owen, commu- 
nicated to this Association; first, in the generalization by which his report 
on the Extinct Mammals of Australia is terminated, and still more in de- 
tailed reference to our islands in his recently published work ' On the Ex- 
tinct Fossil British Mammalia,' — a work which he has stated in his dedication 
originated at the call of the British Association. Professor Owen, who fills a 
Vice-President's chair, adds, indeed, greatly to the strength of our present 
Meeting, by also acting as the President of one of our Sections, which having 
in its origin been exclusively occupied in the study of Medicine, is now more 
peculiarly devoted to the cultivation of Physiology. Under such a leader I 
have a right to anticipate, that this remodelled Section will exhibit evidences 
of fresh vigour, and will clearly define the vast progress that has been made 
in general and comparative anatomy since the daj's of Hunter and of Cuvier, 
for so large a part of which we are indebted to our eminent associate. I 
may, indeed, confidently announce, from what I know of the communica- 
tions about to be made to us by Professor Owen on anatomical homologies, 
that our Members will be highly gratified in seeing our next volume en- 
hanced with subjects from his pen, which hitherto have almost exclusively 
occupied the attention of continental anatomists. 

Assembled in a county which has the good fortune to have been illus- 
trated by the attractive history of the naturalist of Selborne, I am confident 
that our Fourth Section, to whose labours I would next advert, will yield a 
rich harvest, the more so as it is headed by that great zoologist who has en- 
riched the adjacent Museum of the Naval Hospital at Haslar with so many 
animals from various parts of the world, and has so arranged them as to render 



ADDRESS. XXXV 

them objects well-worthy of your notice. The report of Sir John Richardson 
in the last volume, on the Fishes of China, Japan and New Zealand, when 
coupled with his account in former volumes of the Fauna of North America, 
may be regarded as having completely remodelled our knowledge of the 
geographical distribution of fishes ; first by affording the data, and next by 
explaining the causes through which a community of ichthyological characters 
is in some regions widely spread, and in others restricted to limited areas. 
We now know, that just as the lofty mountain is the barrier which separates 
different animals and plants, as well as peculiar varieties of man, so the deepest 
seas are limits which peremptorily check the wide diffusion of certain genera 
and species of fishes ; whilst the interspersion of numerous islands, and still 
more the continuance of lands throughout an ocean, ensures the distribution 
of similar forms over many degrees of latitude and longitude. 

The general study, indeed, both of zoology and botany has been sin- 
gularly advanced by the labours of the Section of Natural History. I cannot 
have acted for many years as your General Secretary without observing, 
that by the spirit in which this Section has of late years been conducted, 
British naturalists have annually become more philosophical, and have given 
to their inquiries a more physiological character, and have more and more 
studied the higher questions of structure, laws and distribution. This 
cheering result has mainly arisen from the personal intimacy brought about 
among various individuals, who, living at great distances from each other, 
were previously never congregated ; and from the mutual encouragement 
imparted by their interchange of views and their comparisons of specimens. 
Many active British naturalists have in fact risen up since these Meetings 
commenced, and many (in addition to the examples already mentioned) have 
pursued their science directly under the encouragement we have given them. 
The combination of the enthusiastic and philosophic spirit thus engendered 
among the naturalists, has given popularity to their department of science, 
and this Section, assuming an importance to which during our earliest Meet- 
ings it could show comparatively slender claims, has vigorously revived the 
study of natural history, and among other proofs of it, has given rise to that 
useful publishing body the Ray Society, which holds its anniversary du- 
ring our sittings. Any analysis of the numerous original and valuable re- 
ports and memoirs on botanical and zoological subjects which have occupied 
our volumes is forbidden by the limits of this address, but I cannot omit to 
advert to the extensive success of Mr. H. Strickland's report on Zoological 
Nomenclature, which has been adopted and circulated by the naturalists of 
France, Germany, Sweden and America, and also by those of Italy headed 
by the Prince of Canino. In each of these countries the code drawn up by 
the Association has been warmly welcomed, and through it we may look 
forward to the signal advantage being gained, of the adoption of an uniform 
zoological nomenclature all o^er the globe. 

Whilst investigations into the geographical distribution of animals and plants 
have occupied a large share of the attention of our Browns and our Darwins, 
it is pleasing to see that some members, chiefly connected with physical 
researches, are now bringing these data of natural history to bear upon 
climatology and physical geography. A committee of our naturalists, to 
whom the subject was referred, has published in our last volume a good 
series of instructions for the observation of the periodical phsenomena of 
animals and plants, prepared by our foreign associate M. Quetelet, the 
Astronomer Royal of Belgium. Naturalists have long been collecting ob- 
servations on the effects produced by the annual return of the seasons, but 
their various natural-history calendars being local, required comparison and 



XXXvi REPORT — 1846. 

concentration, as originally suggested by Linnajus, This has now for the 
first time been executed by the Belgian Astronomer, who following out a 
plan suggested by himself at our Plymouth Meeting, has brought together 
the contributions and suggestions of the naturalists of his own country. 
When M. Quetelet remarks, " that the phases of the smallest insect are 
bound up with the phases of the plant that nourishes it ; that plant itself 
beins in its gradual development the product, in some sort, of all anterior 
modffications of the soil and atmosphere," he compels the admission, that 
the study which should embrace all periodical phaenomena, both diurnal and 
annual, would of itself form a science as extended as instructive. 

Referring you to M. Quetelet's report for an explanation of the dependence 
of the vegetable and animal kingdoms on the meteorology and physics of 
the flobe, and hoping that the simultaneous observations he inculcates will 
be followed up in Britain, I am happy to announce, that the outline of a 
memoir on physical geography was some months ago put into my hands 
by Mr. Cooley, which in a great degree coinciding with the system of 
M. Quetelet, has ultimately a different object. M. Quetelet chiefly aims 
at investigating the dependence of organized bodies on inorganized matter, 
by observing the periodical phfenomena of the former. Mr. Cooley seeks to 
obtain an acquaintance with the same phsenomena for the sake of learning 
and registering comparative climate as an element of scientific agriculture. 
Speaking to you in a county which is so mainly dependent on the produce 
of the soil, I cannot have a more favourable opportunity for inculcating the 
value of the suggestions of this British geographer. The complete esta- 
blishment of all the data of physical geography throughout the British 
Islands ; i. e. the registration of the mean and extremes of the temperature of 
the air and of the earth ; the amount of conduction, radiation, moisture and 
magnetism; the succession of various phases of vegetation, &c. (with their 
several local corrections for elevation and aspect), must certainly advance the 
cause of science, and promote the material interests of our country. 

A minute knowledge of all the circumstances of climate cannot but be of 
importance to those whose industry only succeeds through the co-operation of 
nature, and it may therefore be inferred, how a report like that with which 
I trust Mr. Cooley will favour us, if completed by the addition of tables, 
must prove to be a useful public document. Imbibing the ardour of that 
author, I might almost hope that such researches in physical geography may 
enable us to define, in the language of the poet, 

" Et quid quaeque ferat regio, et quid qiiseque recuset." 

At all events, they will tend to raise physical geography in Britain towards 
the level it has attained in Prussia under the aegis of Humboldt and Ritter, 
and through the beautiful maps of Berghaus. 

Though our countryman, Mr. Keith Johnston, is reproducing, in attrac- 
tive forms, the comparative maps of the last-mentioned Prussian author, much 
indeed still remains to be done in Britain, to encourage the study of physical 
geography and to place it on a basis worthy of tiiis great exploring and colo- 
nizing nation ; and as one of the elementary aids to the training of the youth- 
ful mind to acquire some perception of the science, I commend the spirited 
project of M. Guerin of Paris, to establish in London a georama of vast size, 
the objects and details of which he intends to explain during this week to the 
geographers present. 

Reverting to ceconomical views and the improvement of lands, I would 
remind our agricultural members, that as their great practical Society was 
founded on the model of the British Association, we hope they will always 



ADDRESS. XXXVn 

oome to our Sections for the solution of any questions relating to their pur- 
suits to which can be given a purely scientific answer. If they ask for the 
explanation of the dependence of vegetation upon subsoil or soil, our geo- 
logists and botanists are ready to reply to them. Is it a query on the com- 
parison of the relative value of instruments destined to ceconomize labour, 
the mechanicians now present are capable of answering it. And if, above all, 
they wish us to solve their doubts respecting the qualities of soils and the re- 
sults of their mixtures, or the effects of various manures upon them, our che- 
mists are at hand. One department of our Institution is, in fact, styled the 
Section of Chemistry and Mineralogy, with their applications to Agriculture 
and the Arts, and is officered in part by the very men, Johnston, Daubeny and 
Piayfair, to whom the agriculturists have, in nearly all cases, appealed. The 
first-mentioned of these was one of our earliest friends and founders ; the 
second had the merit of standing by the British Association at its first meet- 
ing, and there inviting us to repair to that great University where he is so 
much respected, and where he is now steadily determining, by elaborate 
experiments, the dependence of many species of plants on soil, air and 
stimulus ; whilst the third has already been alluded to as one of our best 
actual contributors. 

If in reviewing our previous labours I have endeavoured to gain your at- 
tention by some incidental allusions to our present proceedings, I have yet to 
assure you, that the memoirs communicated to our Secretaries are sufficiently 
numerous to occupy our Sections during the ensuing week with all the vigour 
which has marked our opening day. Among the topics to which our as- 
sembling at Southampton gives peculiar interest, I may still say that if 
geologists should find much to interest them in the Isle of Wight, the same 
island contains a field for a very curious joint discussion between them and 
the mathematicians, with which I became acquainted in a previous visit to 
this place. It is a discovery by Colonel Colby, the Director of the Trigo- 
nometrical Survey, of the existence of a notable attraction of the plumb-line 
to the south, at the trigonometrical station called Dunnose, on Shanklin 
Down. The details of this singular phsenomenon, which has been verified 
by observations with the best zenith sectors, will be laid before the Sections. 
In the meantime, we may well wonder, that on the summit of a chalk hill of 
low altitude which is bounded on the south by the sea (near whose level the 
deviation is scarcely perceptible), there should exist an attraction of more 
than half the intensity of that which was registered by Maskelyne, when he 
suspended a plummet at the side of the lofty Scottish mountain of Schehal- 
lion ! If those of our geologists, who like Mr. Hopkins of Cambridge have 
entered boldly into the field of geological dynamics, cannot explain this re- 
markable fact, by connecting it with the ridge of dislocated strata that runs 
through the island as a back-bone from west to east, may we venture to 
refer it to dense plutonic masses, of rock ranging beneath the surface, parallel 
to the line of displacement of the deposits ? 

Another local subject — one indeed of positive practical interest — that 
stands before us for discussion, is, whether, by persevering in deepening the 
large shaft which they have sunk so deep into the chalk near this tqwn, the 
inhabitants of Southampton may expect to be eventually repaid, like those 
of Paris, by a full supply of subterranean water, which shall rise to the 
surface of the low plateau on which the work has been undertaken ? On 
no occasion, I must observe, could this town be furnished with a greater 
number of willing counsellors, whose opinions will, it is hoped, be ade- 
quately valued by the local authorities. The question whether this work 
ought to be proceeded with or not, will however, I apprehend, be most 
184.6. d 



xxxviii REPORT — 1846. 

effectively answered by those geologists who are best acquainted with the 
sections in the interior of this county, and with the levels at which the upper 
greensand and subcretaceous strata there crop out and receive the waters, 
which thence flow southwards beneath the whole body of chalk, of the hills 
in the south of Hampshire. 

Again, as we are now assembled in the neighbourhood of our great 
naval arsenal — as some of its functionaries, including the Admiral on the 
station, have honoured us with their support, and as, further, I am now 
speaking in a town whose magnificent new docks may compete with any 
for bold and successful engineering, I must say a few words on our naval 
architecture, the more so as we have here a Mechanical Section, presided 
over by the eminent mechanician Professor Willis, assisted by the great 
dynamical mathematician Dr. Robinson, and that sound engineer George 
Rennie. Duly impressed with the vast national importance of this subject, 
and at the same time of its necessary dependence on mathematical principles, 
the British Association endeavoured in its earliest days to rouse attention to 
the state of ship-building in England, and to the history of its progress in 
France and other countries, through a memoir by the late Mr. G. Harvey. 
It was then contended, that notwithstanding the extreme perfection to which 
the internal mechanism of vessels had been brought, their external forms 
or lines, on which their sailing so much depends, were deficient as to ad- 
justment by mathematical theory. Our associate Mr. Scott Russell has, 
as you know, ably developed this view. Experimenting upon the resistance 
of water, and ascertaining with precision the forms of vessels which would 
pass through it with the least resistance, and consequently with the greatest 
velocity, he has contributed a most valuable series of memoirs, accompanied 
by a great number of diagrams, to illustrate his opinions and to show the 
dependence of naval architecture on certain mathematical lines. Employed, 
in the meantime, by merchants on their own account, to plan the construction 
of sailing ships and steamers, Mr. Scott Russell has been so successful in 
combining theory with practice, that we must feel satisfied in having at 
different meetings helped him onwards by several money grants ; our only 
regret being, that our means should not have permitted us to publish (he 
whole number of diagrams of the lines prepared by this ingenious author. 

But however desirous to promote theoretical knowledge on this point, the 
men of science are far from wishing not to pay every deference to the skilful 
artificers of our wooden bulwarks, on account of their experience and practi- 
cal acquaintance with subjects they have so long and so successfully handled. 
We are, indeed, fully aware, that the naval architects of the Government, 
who construct vessels carrying a great weight of metal and requiring 
much solidity and capacious stowage, have to solve many problems with 
which the owners of trading vessels or packets have little concern. All that 
we can wish for is, that our naval arsenals should contain schools or public 
boards of ship-building, in which there might be collected all the " constants 
of the art," in reference to capacity, displacement, stowage, velocity, pitching 
and rolling, masting, the effect of sails and the resistance of fluids. Having 
ourselves expended contributions to an extent which testify, at all events, 
our zeal in this matter, we are, I think, entitled to express a hope, that the 
data derived from practice by our eminent navigators may be effectively 
combined with the indications of sound theory prepared by approved culti- 
vators of mathematical and mechanical science. 

I cannot thus touch upon such useful subjects without saying, that our Sta- 
tistical Section has been so well conducted by its former presidents, that its 
subjects, liable at all times to be diverted into moral considerations and thence 



ADDRESS. XXXIX 

into politics, have been invariably restricted to the branch of the science 
which deals in facts and numbers ; and as no one individual has contributed 
more to the storehouse of such valuable knowledge than Mr. George Porter 
(as evidenced even by his report in our last volume), so may we believe that 
in this town, with which he is intimately connected, he will contribute to 
raise still higher the claims of the Section over which he is so well qualified 
to preside. 

If in this discourse I have referred somewhat more largely to those 
branches of science which pertain to the general division of natural history, 
in which alone I can venture to judge of the progress made by others, let me 
however say, that no member of this body can appreciate more highlj^ than 
I do, the claims of the mathematical and experimental parts of philosophy, 
in which my friend Professor Baden Powell of Oxford, who supports me as 
a Vice-President of this meeting, has taken so distinguished a part. No one 
has witnessed with greater satisfaction the attendance at our former meetings 
of men, from all parts of Europe, the most eminent in these high pursuits. 
No one can more glory in having been an officer of this Association when 
it was honoured with the presence of its illustrious correspondent Bessel, 
than whom the world has never produced a more profound astronomer. 
If among his numerous splendid discoveries he furnished astronomers with 
■what they had so long and so ardently desired — a fixed and ascertained point 
in the immensity of space, beyond the limits of our own sidereal system, it is 
to Bessel, as 1 am assured by a contemporary worthy of him, that Englishmen 
owe a debt of gratitude for his elaborate discussion of the observations of their 
immortal Bradley, which, in his hands, became the base of modern astronomy. 

Passing from this recollection, so proud yet so mournful to us all as 
friends and admirers of the deceased Prussian astronomer, can anyone see with 
more delight than myself the brilliant concurrence at our present Meeting of 
naturalists, geologists, physiologists, ethnologists and statists, with mathemati- 
cians, astronomers, mechanicians, and experimental philosophers in physics and 
in chemistry? Surely then I may be allowed to signalize a particular ground of 
gratification among so many, in the presence at this Meeting of two individuals 
in our Experimental Sections, to one of whom, our eminent foreign associate 
Oersted, we owe the first great link between electric and magnetic phsenomena, 
by showing the magnetic properties of the galvanic current ; whilst the other, 
our own Faraday, among other new and great truths which have raised the 
character of English science throughout the world, obtained the converse 
proof by evoking electricity out of magnets. And if it be not given to the 
geologist whom you have honoured with this chair, to explain how such arcana 
have been revealed, still, as a worshiper in the outer portico of the temple of 
physical science, he may be permitted to picture to himself the delight which 
the Danish philosopher must have felt, when on returning to our shores, after 
an absence of a quarter of a century, he found that the grand train of dis- 
covery of which he is the progenitor, had just received its crowning accession 
in England from his former disciple, who, after a long and brilliant series 
of investigations peculiarly his own, has shown that magnetic or dia-magnetic 
forces are distributed throughout all nature. 

And thus shall we continue to be a true British Association, with cosmo- 
polite connexions, so long as we have among us eminent men to attract such 
foreign contemporaries to our shores. If then at the last assembly we ex- 
perienced the good effects which flowed from a concentration of mathe- 
maticians and magneticians, drawn together from different European king- 
doms — if then also the man* of solid learning, who then represented the 

* Mr. Everett. 

rf2 



xl REPORT — 1846. 

United States of America, and who is now worthily presiding over the Cam- 
bridge University of his native soil, spoke to us with chastened eloquence 
of the benefits our Institution was conferring on mankind ; let us rejoice 
that this Meeting is honoured by the presence of foreign philosophers as 
distinguished as those of any former year. 

Let us rejoice that we have now among us men of science from Den- 
mark, Sweden, Russia, Prussia, Switzerland, Italy and France. The King 
of Denmark, himself personally distinguished for his acquaintance with 
several branches of natural history, and a warm patron of science, has 
honoured us by sending hither, not only the great discoverer Oersted, who 
evincing fresh vigour in his mature age brings with him new communications 
on physical science, but also my valued friend, the able geologist and chemist 
Forchhammer, who has produced the first geological map of Denmark, and 
who has presented to us a lucid memoir on the influence exercised by marine 
plants on the formation of ancient crystalline rocks, on the present sea and 
on agriculture. 

As these two eminent men and their associates of the North received me 
as the General Secretary of the British Association with their wonted cor- 
diality at the last Scandinavian Scientific Assembly, I trust we may convince 
them that the sentiment is reciprocal, and that Englishmen are akin to them 
in the virtues of friendship and hospitality which so distinguish the dwellers 
within the circle of Odin. 

Still adverting to Scandinavia, we see here a deputy from the country of Lin- 
n£Eus in the person of Professor Svanberg, a successful young experimenter in 
physics, who represents his great master Berzelius — that profound chemist and 
leader of the science of the North of Europe, who established on a firm basis 
the laws of atomic weights and definite proportions, and who has personally 
assured me, that if our Meeting had not been fixed in the month of September, 
when the agriculturists of Sweden assemble at Stockholm, he would as- 
suredly have repaired to us. And if the same cause has prevented Nilsson 
from coming hither, and has abstracted Retzius from us (who was till within 
these few days in England), I cannot mention these distinguished men, who 
earnestly desired to be present, without expressing the hope, that the memoirs 
they communicate to us may give such additional support to our British ethno- 
logists, as will enable this new branch of science, which investigates the origin 
of races and languages, to take the prominent place in our assemblies to 
which it is justly entitled. 

The Royal Academy of Berlin, whose deputies on former occasions have 
been an Ehrenberg, a Buch, and an Erman, has honoured us by sending 
hither M. Heinrich Rose, whose work on chemical analysis is a text-book 
even for the most learned chemists in every country ; and whilst his researches 
on the constitution of minerals, like those of his eminent brother Gustave on 
their form, have obtained for him so high a reputation, he now brings to us 
the description of a new metal which he has discovered in the Tantalite of 
Bavaria. 

Switzerland has again given to us that great master in palaeontology, Agassiz, 
who has put arms into the hands of British geologists with which they have 
conquered vast regions, and who now on his road to new fields in America, 
brings to us his report on the fossil fishes of the basin of London, which will, he 
assures me, exceed in size all that he has ever written on ichthyolites. 

From the same country we have our warm friend Professor Schonbein, who, 
in addition to his report on Ozone, to which I have already referred, has now 
brought to us a discovery which promises to be of vast practical importance. 
The " gun-cotton" of Schonbein, the powers of which he will exhibit to his 



ADDRESS. 



xli 



colleagues, is an explosive substance, which is stated to exercise a stronger 
projectile force than gunpowder, to possess the great advantages over it of 
producing little or no smoiie or noise, and of scarcely soiling fire-arms ; whilst 
no amount of wet injures this new substance, which is as serviceable after 
being dried as in its first condition. The mere mention of these properties, 
to which our associate lays claim for his new material, is sufficient to sug- 
gest its extraordinary value in warlike affairs, as also in every sort of sub- 
terranean blasting, and may well lead me to say, that this discovery, which 
may almost rival the invention of the substance which it is destined to sup- 
plant, will signally mark this meeting at Southampton. But, as if British 
chemistry were not to be outdone, here also there will be promulgated, for 
the first time, the very rema;rkable discovery of our countryman Mr. Grove, 
of the decomposition of water by heat. 

Professor Matteucci of Modena, who joined us at the York meeting, and 
then explained his various new and delicate investigations in electro-phy- 
siology, again favours us with a visit, as the representative of the Italian 
Philosophical Society of Modena and of the University of Pisa. This 
ingenious philosopher, who has measured the effect of galvanic currents in 
exciting through the nerves mechanical force in the muscles, doubtless brings 
with him such interesting contribution as will add great additional interest 
to the proceedings of the Physiological Section. 

Among these sources of gratification, no one has afforded me sincerer 
pleasure than to welcome hither the undaunted Siberian explorer. Professor 
von Middendorff. Deeply impressed as I am with the estimation in 
which science is held by the illustrious ruler of the empire of Russia, I 
cannot but hope that the presence of this traveller, so signalized by his 
enterprising exploits, may meet with a friend in every Englishman who is 
acquainted with the arduous nature of his travels. To traverse Siberia 
from south to north and from west to east ; to reach by land the extreme 
northern headland of Taimyr ; to teach us, for the first time, that even 
to the latitude of 72° north, trees with stems extend themselves in that 
meridian ; that crops of rye, more abundant than in his native Livonia, grow 
beyond Yakutsk, on the surface of that frozen subsoil, the intensity and 
measure of cold in which he has determined by thermometric experiments; 
to explain, through their language and physical form, the origin of tribes now 
far removed from their parent stock ; to explore the far eastern regions of 
the Sea of Ohkotsk and of the Shantar Isles ; to define the remotest north- 
eastern boundary between China and Russia; and finally to enrich St. Peters- 
burgh with the natural productions, both fossil and recent, of all these wild 
and untrodden lands, are the exploits for which the Royal Geographical So- 
ciety of London has, at its last meeting, conferred its Gold Victoria Medal on 
this most successful explorer. Professor von Middendorff now visits us to con- 
verse with our naturalists most able to assist him, and to inspect our museums, 
in which, by comparison, he can best determine the value of specific cha- 
racters before he completes the description of his copious accumulations; and 
I trust that during his stay in England he will be treated with as much true 
hospitality as I have myself received at the hands of his kind countrymen. 

It is impossible for me to make this allusion to the Russian empire, without 
assuring you that our allies in science on the Neva, who have previously sent 
to us a Jacobi and a Kupffer, are warmly desirous of continuing their good 
connexion with us. It was indeed a source of great pleasure to me to have 
recently had personal intercourse in this very town with that eminent scientific 
navigator Admiral Liitke, in whose squadron His Imperial Highness the 
Grand Duke Constantine was acquiring a knowledge of his maritime duties. 



Xlii REPORT — 1846. 

Besides the narrative of his former voyages, Liitke has since published an 
account of the periodical tides in the Great Northern Ocean and in the 
Glacial Sea, which I have reason to think is little known in this country. 
Having since established a hypsalographe in the White Sea, and being also 
occupied from time to time in observations in Behring's Straits, the Russians 
will soon be able to provide us with other important additions to our 
knowledge of this subject. Separated so widely as Admiral Liitke and Dr. 
Whewell are from each other, it is pleasing to see, that the very recommenda- 
tion which the last-mentioned distinguished philosopher of the tides has re- 
cently suggested to me, as a subject to be encouraged by this Association, has 
been zealously advocated by the former. Let us hope then that this Meeting 
will not pass away without powerfully recommending to our own Government, 
as well as to that of His Imperial Majesty, the carrying out of systematic and 
simultaneous investigations of the tides in the Great Ocean, particularly in the 
Northern Pacific, — a subject (as Admiral Liitke well observes) which is not 
less worthy of special expeditions and of the attention of great scientific bodies, 
than the present inquiries into terrestrial magnetism ; and one which, I may 
add, this Association will doubtless warmly espouse, since it has such strong 
grounds for being satisfied with the results which it has already contributed to 
obtain through its own grants, and by the researches of several of its associates. 

Lastly, in alluding to our foreign attendants, let us hope that our nearest 
neighbours may respond to our call, and may prove by their affluence to 
Southampton, that in the realms of science there is that " entente cordiale " 
between their great nation and our own, of which, at a former meeting, we 
were assured by the profound Arago himself. No sooner was it made known 
that the Chair of Chemistry at this Meeting was to be filled by Michael 
Faraday, than a compeer worthy of him in the Academy of Sciences of Paris 
was announced in the person of M. Dumas *, by a letter from that philoso- 
pher to myself. To M. Dumas it is well known that we owe, not only the 
discovery of the law of substitution of types, which has so powerfully aided 
the progress of organic chemistry, but also the successful application of his 
science to the arts and useful purposes of life ; his great work on that sub- 
ject, ' La Chiniie appliquee aux Arts,' being as familiar in every manufactory 
in England as it is upon the Continent. 

Nor, if we turn from chemistry to geology, will such of us as work among 
the rocks be backward in welcoming any French geologists who may come 
to examine, in our own natural sections of the Isle of Wight, the peculiar 
development of their Paris basin, the identity of their chalk and our own, 
the fine sections of our greensand and of the Wealden formation of Mantell, 
and to determine with us in situ the strict relations of their Neocomian rocks 
with those peculiar strata which at Atherfield, in the Isle of Wight, have 
been so admirably illustrated by Dr. Fitton and other native geologists, and 
of which such beautiful and accurate diagrams have been prepared by 
Captain Ibbetson. 

Will it not then be admitted, that the gathering together of such foreign 
philosophers, as those above mentioned, with our own men of science, must 
be productive of good results? Putting aside even the acknov/ledged fact, 
that numerous memoirs of value are published in one country which are 
unknown in another, where is the person, acquainted with the present acce- 
lerated march of science, who can doubt that the germs of discovery which 
are floating in the minds of distant contemporaries, must often be brought 
to maturity by the interchange of such thouglits? The collision of these 

* The resolution of M. Dumas to visit the Meeting was arrested by a sudden illness, and 
his apology only reached the President towards the close of the Meeting. 



ADDRESS. xliii 

thoughts may indeed be compared to the agency of the electric telegraph of 
our own Wheatstone, which concentrates knowledge from afar, and at once 
unites the extremities of kingdoms in a common circle of intelligence. 

But although the distinguished foreigners to whom I have adverted, and 
others, including our welcome associate M. Wartmann, the Founder of the 
Vaudois Society, and M. Prevost of Geneva, on whose merits I would 
willingly dilate if time permitted it, are now collected around us ; many, 
among whom I must name M. de Caumont, the President of the French 
Society for the Advancement of Science, have been prevented from ho- 
nouring us with their presence, because the national meetings in their 
several countries also occur in the month of September. To remedy this in- 
convenience, 1 ventured, when addressing you six years ago at the Glasgow 
meeting, to express the hope, that each of the European societies might 
be led to abstain during one year from assembling in its own country, for 
the purpose of repairing by its own deputies to a general congress, to be 
held at Frankfort or other central city under the presidency of the universal 
Humboldt. Had the preparation of the ' Cosmos ' and other avocations of 
that renowned individual permitted him to accept this proposition, which 
the British Association would doubtless have supported, many benefits to 
science must have resulted, and each national body, on re-assembling the 
following year in its native land, would, I am convinced, have more vigorously 
resumed its researches. 

But whether it be considered desirable or not to suspend the national 
scientific meetings during one year, I call on my countrymen and their foreign 
friends now present, to sustain the proposal for centralizing in a future year 
the representatives of the various branches of science of different countries, 
when they may at once learn the progress made in each nation, and when, at 
all events, they can so appoint the periods of their respective assemblies, 
as to prevent those simultaneous meetings in France, Germany, Scandinavia, 
Italy, Switzerland and England, which are so much to be deprecated as in- 
terfering with a mutual intercourse. 

Finally, my fellow-labourers in science, if by our united exertions we have 
done and are doing good public service, let me revert once more to the place 
in which we are assembled, and express on your part the gratification I know 
you experience in being on this occasion as well supported by the noblemen, 
clergymen, and landed proprietors around Southampton, as by its inhabitants 
themselves — an union which thus testifies that the British Association em- 
braces all parties and all classes of men. 

Seeing near me Her Majesty's Secretary of State for Foreign Affairs, 
the Speaker of the House of Commons, and several persons of high station 
and great influence, who willingly indicate by their presence the sense 
they entertain of the value of our conferences and researches, let us wel- 
come these distinguished individuals, as living evidences of that good opinion 
of our countrymen, the possession of which« will cheer us onward in our 
career. And above all, let us cherish the recollection of this Southampton 
Meeting, which will be rendered memorable in our annals by the pre- 
sence of the illustrious Consort of our beloved Sovereign, who participating 
in our pursuits, in some of which His Royal Highness is so well-versed, 
thus demonstrates that our Association is truly national, and enjoys the 
most general and effectual support throughout British society, from the 
humblest cultivators of science to the highest personages in the realm. 



REPORTS 



ON 



THE STATE OF SCIENCE. 



Report on Recent Researches in Hydrodynamics, 
By G. G, Stokes, M.A., Fellow of Pembroke College, Cambridge. 

At the meeting of the British Association held at Cambridge last year, the 
Committee of the Mathematical Section expressed a wish that a Report on 
Hydrodynamics should be prepared, in continuation of the reports which 
Prof. Challis had already presented to the Association on that subject. Prof. 
Challis having declined the task of preparing this report, in consequence of 
the pressure of other engagements, the Committee of the Association did 
me the honour to entrust it to me. In accordance with the wishes of the 
Committee, the object of the present report will be to notice researches in 
this subject subsequent to the date of the reports of Prof. Challis. It will 
sometimes however be convenient, for the sake of giving a connected view 
of certain branches of the subject, to refer briefly to earlier investigations. 

The fundamental hypothesis on which the science of hydrostatics is based 
may be considered to be, that the mutual action of two adjacent portions of 
a fluid at rest is normal to the surface which separates them. The equality 
of pressure in all directions is not an independent hypothesis, but a necessary 
consequence of the former. This may be easily proved by the method given 
in the Exercices of M. Cauchy*, a method which depends on the considera- 
tion of the forces acting on a tetrahedron of the fluid, the dimensions of which 
are in the end supposed to vanish. This proof applies equally to fluids at 
rest and fluids in motion ; and thus the hypothesis above-mentioned may be 
considered as the fundamental hypothesis of the ordinary theory of hydro- 
dynamics, as well as hydrostatics. This hypothesis is fully confirmed by ex- 
periment in the case of the equilibrium of fluids ; but the comparison of theory 
and experiment is by no means so easy in the case of their motion, on account 
of the mathematical difficulty of treating the equations of motion. Still 
enough has been done to show that the ordinary equations will suffice for 
the explanation of a great variety of phsenomena ; while there are others the 
laws of which depend on a tangential force, which Is neglected in the common 
theory, and in consequence of which the pressure is different in different 
directions about the same point. The linear motion of fluids in uniform 
pipes and canals is a simple instance. In the following report I shall first 
consider the common theory of hydrodynamics, and then notice some theo- 
ries which take account of the inequality of pressure in different directions. 
It will be convenient to consider the subject under the following heads : — 

I. General theorems connected with the ordinary equations of fluid motion. 

II. Theory of waves, including tides. 

* Tom. ii. p. 42. 

1846. B 



2 REPORT — 1846. 

III. The discharge of gases through small orifices. 

IV. Theory of sound. 

V. Simultaneous oscillations of fluids and solids. 

VI. Formation of the equations of motion when the pressure is not sup- 
posed equal in all directions. 

I. Although the common equations of hydrodynamics have been so long 
known, their comi)lexity is so great that little has been done with them 
except in the case in which the expression usually denoted by 

udx + vdy+wdz (A.) 

is the exact differential of a function of the independent variables x, 7/, z*. 
It becomes then of the utmost importance to inquire in what cases this sup- 
position may be made. Now Lagrange enunciated two theorems, by virtue 
of which, supposing them true, the supposition may be made in a great 
number of important cases, in fact, in nearly all those cases which it is most 
interesting to investigate. It must be premised that in these theorems the 
accelerating forces X, Y, Zare supposed to be such that Xdx+ Ydy + Zdz is 
an exact differential, supposing the time constant, and the density of the fluid is 
supposed to be either constant, or a function of the pressure. The theorems are — 

First, that (A.) is approximately an exact differential when the motion is 
BO small that squares and products of u, v, w and their differential coefficients 
may be neglected. By calling (A.) approximately an exact differential, it is 
meant that there exists an expression uflx-\-\\dy-\-wflz, which is accurately 
an exact differential, and which is such that m,, v^, lo, differ from u, v, w 
respectively by quantities of the second order only. 

Secondly, that (A.) is accurately an exact differential at all times when it 
is so at one instant, and in particular when tiie motion begins from rest. 

It has been pointed out by Poisson that the first of these theorems is not 
truef . In fact, the initial motion, being arbitrary, need not be such as to 
render (A.) an exact differential. Thus those cases coming under the first 
theorem in which the assertion is true are merged in those which come under 
the second, at least if we except the case of small motions kept up by dis- 
turbing causes, a case in which we have no occasion to consider initial motion 
at all. Tliis case it is true is very important. 

The validity of Lagrange's proof of the second theorem depends on the 
legitimacy of supposing u, v and w capable of expansion according to posi- 
tive, integral powers of the time t, for a sufficiently small value of that varia- 
ble. This proof lies open to objection ; for there are functions of t the 
expansions of whlcli contain fractional powers, and there are others which 
cannot be expanded according to ascending powers of t, integral or fractional, 
even though they may vanish when < = 0. It has been shown by Mr. Power 
that Lagrange's proof is still applicable if w, ?; and w admit of expansion 
according to ascending powers of ^ of any kind;]:. The second objection 
however still remains: nor does the proof which Poisson has substituted for 
Lagrange's in his ' Traite de Mecanique' appear at all more satisfactory. 
Besides, it does not appear from tliese proofs what becomes of the theorem if 
it is only for a certain portion of the fluid that (A.) is at one instant an exact 
differential. 

M. Cauchy has however given a proof of the theorem §, which is totally 
different from either of the former, and perfectly satisfactory. M. Cauchy 

* In nearly all the investigations of Mr. Airy it will be found that (A.) is an exact dififeren- 
tial, although he docs not start with assuming it to be so. 

+ Memoires de rAciidemie des Sciences, torn. x. p. 55'1. 

X Transactions of the Cambridge Philosophical Society, vol. vii. p. 455. 

§ Memoires des Savans Ltraiigers, torn. i. p. 40. 



ON RECENT RESEARCHES IN HYDRODYNAMICS. 3 

first eliminates the pressure by differentiation from the three partial differential 
equations of motion. He then changes the independent variables in the 
three resulting equations from x, y, z, t to a, b, c, t, where a, b, c are the 
initial co-ordinates of the particle whose co-ordinates at the time t are x, y, z. 
The three transformed equations admit each of being once integrated with 
respect to t, and the arbitrary functions of a, b, c introduced by integration 
are determined by the initial motion, which is supposed to be given. The 
theorem in question is deduced without difficulty from the integrals thus 
obtained. It is easily proved that if the velocity is suddenly altered by 
means of impulsive forces applied at the surface of the fluid, the alteration is 
such as to leave (A.) an exact differential if it were such before impact. 
M. Cauchy's proof shows moreover that if (A.) be an exact differential for 
one portion of the fluid, although not for the whole, it will always remain so 
for that portion. It should be observed, that although M. Cauchy has proved 
the theorem for an incompressible fluid only, the same method of proof 
applies to the more general case in which the density is a function of the 
pressure. 

In a paper read last year before the Cambridge Philosophical Society, I 
have given a new proof of the same theorem*. This proof is rather simpler 
than M. Cauchy's, inasmuch as it does not require any integration. 

In a paper published in the Philosophical Magazine f, Prof. Challis has 
raised an objection to the application of the theorem to the case in which 
the motion of the fluid begins from rest. According to the views contained 
in this paper, we are not in general at liberty to suppose (A.) to be an exact 
differential when u, v and %u vanish : this supposition can only be made when 
the limiting value of if-* (udx+vdy+wdz) is an exact differential, where 
a is so taken as that one at least of the terms in this expression does not 
vanish when t vanishes. 

It is maintained by Prof. Challis that the received equations of hydro- 
dynamics are not complete, as regards the analytical principles of the science, 
and he has given a new fundamental equation, in addition to those received, 
which he calls the equation of continuity of the motionX. On this equation 
Prof. Challis rests a result at which he has arrived, and which all must allow 
to be most important, supposing it correct, namely that whenever (A.) is an 
exact differential the motion of the fluid is necessarily rectilinear, one peculiar 
case of circular motion being excepted. As I have the misfortune to differ 
from Professor Challis on the points mentioned in this and the preceding 
paragraph, for reasons which cannot be stated here, it may be well to apprise 
the reader that many of the results which will be mentioned farther on as 
satisfactory lie open to Prof. Challis's objections. 

By virtue of the equation of continuity of a homogeneous incompressible 
fluid, the expression udy — vdx will always be the exact differential of a 
function of x and y. In the Cambridge Philosophical Transactions § there 
will be found some applications of this function, and of an analogous function 
for the case of motion which is symmetrical about an axis, and takes place 
in planes passing through the axis. The former of these functions had been 
previously employed by Mr. Earnshaw. 

II. In the investigations which come under this head, it is to be understood 
that the motion is supposed to be very smallj so that first powers only of 
small quantities are retained, unless the contrary is stated. 

* Transactions of the Cambridge Philosophical Society, vol. viii. p. 307. 
t Vol. xxiv. New Series, p. 94. 

X Transactions of the Cambridge Philosophical Society, vol. viii. p. 31 ; and Philosophical 
Magazine, vol. xxvi. New Series, p. 425. § Vol. vii. p. 439. 

B 2 



4 REPORT — 1846. 

The researches of MM. Poisson and Cauchy were directed to the inves- 
tigation of the waves produced by disturbing causes acting arbitrarily on a 
small portion of ihe fluid, which is then left to itself. The mathematical 
treatment of such cases is extremely difficult ; and after all, motions of this 
kind are not thoi^e which it is most interesting to investigate. Consequently 
it is the simpler cases of wave motion, and those which are more nearly con- 
nected with tlie phsenomena which it is most desirable to explain, that have 
formed the principal subject of more recent investigations. It is true that 
there is one memoir by IVI. Ostrogradsky, read before the French Academy 
in 1826*, to which this character does not apply. In this memoir the author 
has determined the motion of the fluid contained in a cylindrical basin, sup- 
posing the fluid at first at rest, but its surface not horizontal. The interest 
of the memoir however depends almost exclusively on the mathematical 
processes employed ; for the result is very complicated, and has not been 
discussed by the authoi*. There is one circumstance mentioned by M. Plana+ 
which increases the importance of the memoir in a mathematical point of 
view, which is that Poisson met with an apparent impossibility in endea- 
vouring to solve the same problem. I do not know whether Poisson's attempt 
was ever published. 

Theory of Long Waves. — When the length of the waves whose motion is 
considered is very great compared with the depth of the fluid, we may without 
sensible error neglect the difterence between the horizontal motions of dif- 
ferent particles in the same vertical line, or in other words suppose the par- 
ticles once in a vertical line to remain in a vertical line : we may also neglect 
the vertical, compared with the horizontal effective force. These considera- 
tions extremely simplify the problem ; and the theory of long waves is very 
important from its bearing on the theory of the tides. Lagrange's solution 
of the problem in the case of a fluid of uniform depth is well known. It is 
true that Lagrange fell into error in extending his solution to cases to which 
it does not apply ; but there is no question as to the correctness of his result 
when properly restricted, that is when applied to the case of long waves only. 
There are however many questions of interest connected with this theory 
which have not been considered by Lagrange. For instance, what will be 
the velocity of propagation in a uniform canal whose section is not rectan- 
gular? How will the form of the wave be altered if the depth of the fluid, 
or the dimensions of the canal, gradually alter ? 

In a paper read before the Cambridge Philosophical Society in May 1 837 +, 
the late Mr. Green has considered the motion of long w aves in a rectangular 
canal whose depth and breadth alter very slowly, but in other respects quite 
arbitrarily. Mr. Green arrived at the following results : — If /3 be the breadth, 

and y the depth of the canal, then the height of the wave 0C/3~^y~^, the 
horizontal velocity of the particles inagiven phaseof their motion OC (8~ y~*' 
the length of the wave OC y'^, and the velocity of propagation = '*/gy- With 
respect to the height of the wave, Mr. Russell was led by his experiments to 
the same law of its variation as regards the breadth of the canal, and with 
respect to the eff'ect of the depth he observes that the height of the wave 
increases as the depth of the fluid decreases, but that the variation of the 
height of the wave is very slow compared with the variation of the depth of 
the canal. 

In another paper read before the Cambridge Philosophical Society in 

• Mumoiies des Savans Etrangers, torn. iii. p. 23. 

1 Turin Memoirs for 1835, p. 253. 

X T ransaclions of tlie Cambridge Philosophical Society, vol. vi. p. 457. 



ON RECENT RESEARCHES IN HYDRODYNAMICS. O 

February 1839*, Mr. Green has given the theory of the motion of long waves 
in a triangular canal with one side vertical. Mr. Green found the velocity of 
propagation to be the same as that in a rectangular canal of half the depth. 
In a memoir read before the Royal Society of Edinburgh in April 1839t> 
Prof. Kelland has considered the case of a uniform canal whose section is of 
any form. He finds that the velocity of propagation is given by the very 

simple formula a /^--, where A is the area of a section of the canal, and 

b the breadth of the fluid at the surface. This formula agrees with the ex- 
periments of Mr. Russell, and includes as a particular case the formula of 
Mr. Green for a triangular canal. 

Mr. Airy, the Astronomer Royal, in his excellent treatise on Tides and 
Waves, has considered the case of a variable canal with more generality than 
Mr. Green, inasmuch as he has supposed the section to be of any form J. If 
A, b denote the same things as in the last paragraph, only that now they are 
supposed to vary slowly in passing along the canal, the coefficient of horizontal 

displacement OC A~^ b^, and that of the vertical displacement OC A~^ 6~*> 
while the velocity of propagation at any point of tlie canal is that given by 
the formula of the preceding paragraph. Mr. Airy has proved the latter 
formula § in a more simple manner than Prof. Kelland, and has pointed out 
the restrictions under which it is true. Other results of Mr. Airy's will be 
more conveniently considered in connection with the tides. 

Theory of Oscillatory Waves. — When the surface of water is covered with 
an irregular series of waves of different sizes, the longer waves will be con- 
tinually overtaking the shorter, and the motion will be very complicated, and 
will offer no regular laws. In order to obtain such laws we must take a 
simpler case : we may for instance propose to ourselves to investigate the 
motion of a series of waves which are propagated with a constant velocity, 
and without change of form, in a fluid of uniform depth, the motion being in 
two dimensions and periodical. A series of waves of this sort may be taken 
as the type of oscillatory waves in general, or at least of those for which the 
motion is in two dimensions : to whatever extent a series of waves propagated 
in fluid of a uniform depth deviates from this standard form, to the same ex- 
tent they fail in the characters of uniform propagation and invariable form. 

The theory of these waves has long been known. In fact each element of 
the integrals by which MM. Poisson and Cauchy expressed the disturbance 
of the fluid denotes what is called by Mr. Airy a standing oscillation, and a 
progressive oscillation of the kind under consideration will result from the 
superposition of two of these standing oscillations properly combined. Or, 
if we merely replace products of sines and cosints under the integral signs 
by sums and differences, each element of the new integrals will denote a 
progressive oscillation of the standard kind. The theory of these waves how- 
ever well deserves a more detailed investigation. The most important formula 
connected with them is that which gives the relation between the velocity of 
propagation, the length of the waves, and the depth of the fluid. If c be the 
velocity of propagation, X the length of the waves, measured from crest to 

crest, h the depth of the fluid, and m = ^^, then 

A 

mh — mh 

c,=£'-^ (B.) 

* Transactions of the Cambridge Philosophical Society, vol. vii. p. 87. 

t Transactions of the Royal Society of Edinburgh, vol. xiv. pp. 524, 530. 

J Encyclopaedia Metropolitana, article ' Tides and Waves.' Art. 260 of the treatise. 

§ Art. 218, &c. 



6 REPORT 1846. 

If the surface of the fluid be cut by a vertical plane perpendicular to the 
ridges of the waves, the section of the surface will be the curve of sines. 
Each particle of the fluid moves round and round in an ellipse, whose major 
axis is horizontal. The particle is in its highest position when the crest of 
the wave is passing over it, and is then moving in the direction of propaga- 
tion of the wave ; it is in its lowest position when the hollow of the wave is 
passing over it, and is then moving in a direction contrary to the direction 
of propagation. At the bottom of the fluid the ellipse is reduced to a right 
line, along which the particle oscillates. When the length of waves is very 
small compared with the depth of the fluid, the motion at the bottom is in- 
sensible, and all the expressions will be sensibly the same as if the depth were 

Infinite. On this supposition the expression for c reduces itself to a /■§— • 

The ellipses in which the particles move are replaced by circles, and the 
motion in each circle is uniform. The motion decreases with extreme rapid- 
ity as the point considered is further removed from the surface ; in fact, 
the coefficients of the horizontal and vertical velocity contain as a factor the 
exponential £-"'3', where y is the depth of the particle considered below the 
surface. When the depth of the fluid is finite, the laio of the horizontal and 
vertical displacements of the particles is the same as when the depth is infi- 
nite. When the length of the waves is very great compared with the depth 
of the fluid, the horizontal motion of different particles in the same vertical 
line is sensibly the same. The expression for c reduces itself to ^'gju the 
same as would have been obtained directly from the theory of long waves. 
The whole theory is given very fully in the treatise of Mr. K\x^*. The 
nature of the motion of the individual particles, as deduced from a rigorous 
theory, was taken notice of, I believe for the first time, by Mr. Green t> who 
has considered the case in which the depth is infinite. 

The oscillatory waves just considered are those which are propagated uni- 
formly in fluid of which the depth is everywhere the same. When this con- 
dition is not satisfied, as for instance when the waves are propagated in a 
canal whose section is not rectangular, it is desirable to know how the velo- 
city of propagation and the form of the waves are modified by this circum- 
stance. There is one such case in which a solution has been obtained. In 
a paper read before the Royal Society of Edinburgh in January 1841, Prof. 
Kelland has arrived at a solution of the problem in the case of a triangular 
canal whose sides are inclined at an angle of 45° to the vertical, or of a canal 
with one side vertical and one side inclined at an angle of 45°, in which the 
motion will of course be the same as in one half of the complete canal J. The 
velocity of propagation is given by the formula (B.), which applies to a rectan- 
gular canal, or to waves propagated without lateral limitation, provided M'e 

Ik 
take for h half the greatest depth in the triangular canal, and for X, or — , a 

quantity less than the length of the waves in the triangular canal in the ratio 
of 1 to V 2. As to tlie form of the waves, a section of the surface made by 
a vertical plane parallel to the edges of the canal is the curve of sines ; a 
section made by a vertical plane perpendicular to the former is the common 
catenary, with its vertex in the plane of the middle of the canal (supposed 
complete), and its concavity turned upwards or downwards according as the 
section is taken where the fluid is elevated or where it is depressed. Thus 

• Tides and V/aves, art. 160, &c. 

t Transactions of tlie Cambridge Philosophical Society, vol. vii. p. 95. 
X Transactions of the Royal Society of Edinburgh, vol. xv. p. 121. 



ON RECENT RESEARCHES IN HYDRODYNAMICS. 7 

the ridges of the waves do not bend forwards, but are situated in a vertical 
plane, and they rise higher towards the slanting sides of the canal than in 
the middle. I shall write down the value of <p, the integral of (A.), and then 
any one who is familiar with the subject can easily verify the preceding re- 
sults. In the following expression x is measured along the bottom line of 
the canal, y is measured horizontally, and z vertically upwards : — 

^=A(£*2' + £-''2')(e*'=+£-"~)sin v/2'a(x-cO (C) 

I have mentioned these results under the head of oscillatory waves, be- 
cause it is to that class only that the investigation strictly applies. The 
length of the waves is however perfectly arbitrary, and when it bears a large 
ratio to the depth of the fluid, it seems evident that the circumstances of the 
motion of any one wave cannot be materially affected by the waves which 
precede and follow it, especially as regards the form of the middle portion, or 
ridge, of the Avave. Now the solitary waves of Mr. Russell are long com- 
pared with the depth of the fluid ; thus in the case of a rectangular canal he 
states that the length of the wave is about six times the depth. Accordingly 
Mr. Russell finds that the form of the ridge agrees well with the results of 
Prof. Kelland. 

It appears from Mr. Russell's experiments that there is a certain limit to 
the slope of the sides of a triangular canal, beyond which it is impossible to 
propagate a wave in the manner just considered. Prof. Kelland has arrived 
at the same result from theory, but his mathematical calculation does not 
appear to be quite satisfactory. Nevertheless there can be little doubt that 
such a limit does exist, and that if it be passed, the wave will be either con- 
tinually breaking at the sides of the canal, or its ridge will become bow- 
shaped, in consequence of the portion of the wave in the middle of the canal 
being propagated more rapidly than the portions which lie towards the sides. 
When once a wave has become suflSciently curved it may be propagated 
without further change, as Mr. Airy has shown*. Thus the case of motion 
above considered is in nowise opposed to the circumstance that the tide 
wave assumes a curved form when it is propagated in a broad channel in 
which the water is deepest towards the centre. 

It is worthy of remark, that if in equation (C) we transfer the origin to 
either of the upper edges of the canal (supposed complete), and then suppose 
h to become infinite, having previously written A g ~*'' for A, the result 
will express a series of oscillatory waves propagated in deep water along the 
edge of a bank having a slope of 45°, the ridges of the waves being perpen- 
dicular to the edge of the fluid. It is reraarkablte that the disturbance of the 
fluid decreases with extreme rapidity as the perpendicular distance from the 
edge increases, and not merely as the distance from the surface increases. 
Thus the disturbance is sensible only in the immediate neighbourhood of the 
edge, that is at a distance from it, which is a small multiple of A. The for- 
mula may be accommodated to the case of a bank having any inclination by 
merely altering the coeflicients of ?/ and z, without altering the sum of the 
squares of the coeflicients. If i be the inclination of the iDank to the verti- 
cal, it will be easily found that the velocity of propagation is equal to 

(■o— C08Z j . When i vanishes these waves pass into those already men- 
tioned as the standard case of oscillatory waves ; and when i becomes nega- 
tive, or the bank overhangs the fluid, a motion of this sort becomes im- 
possible. 

* Tides and Waves, art. 359. 



8 REPORT — 1846. 

I Iiave had occasion to refer to what Mr. Airy calls a standing oscillation 
or standing tvave. To prevent the possibility of confusion, it may be well 
to observe that Mr. Airy uses the term in a totally different sense from Mr. 
Russell. The standing wave of Mr. Airy is the oscillation which would re- 
sult from the coexistence of two series of progressive waves, which are equal 
in every respect, but are propagated in opposite directions. With respect to 
the standing wave of Mr. Ilussell, it cannot be supposed that the elevations 
observed in mountain streams can well be made the subject of mathematical 
calculation. Nevertheless in so far as the' motion can be calculated, by 
taking a simple case, the theory does not differ from that of waves of other 
classes. For if we only suppose a velocity equal and opposite to ihat of the 
stream impressed both on the fluid and on the stone at the bottom which 
produces the disturbance, we pass to the case of a forced wave produced in 
still water by a solid dragged through it. There is indeed one respect in 
which the theory of these standing waves offers a peculiarity, which is, that 
the velocity of a current is different at different depths. But the theory of 
such motions is one of great complexity and very little interest. 

Theory of Solitary Waves. — It has been already remarked that the length 
of the solitary wave of Mr. Russell is considerable compared with the depth 
of the fluid. Consequently we might expect tiiat the theory of long waves 
would explain the main phaenomena of solitary waves. Accordingly it is 
found by experiment that the velocity of propagation of a solitary wave in a 
rectangular canal is that given by the formula of Lagrange, the height of the 
wave being very small, or that given by Prof. Kelland's formula when the 
canal is not rectangular. Moreover, the laws of the motion of a solitary 
wave, deduced by Mr. Green from the theory of long waves, agree with the 
observations of Mr. Russell. Thus Mr. Green found, supposing the canal 
rectangular, that the particles in a vertical plane perpendicular to the length 
of the canal remain in a vertical plane ; that the particles begin to move 
when the wave reaches them, remain in motion while the wave is passing 
over them, and are finally deposited in new positions ; that they move in 
the direction of propagation of the wave, or in the contrary direction, ac- 
cording as the wave consists of an elevation or a depression*. But when we 
attempt to introduce into our calculations the finite length of the wave, the 
problem becomes one of great difficulty. Attempts have indeed been made 
to solve it by the introduction of discontinuous functions. But whenever 
such functions are introduced, there are certain conditions of continuity to 
be satisfied at the common surface of two portions of fluid to which different 
analytical expressions apply ; and should these conditions be violated, the 
solution will be as much in fault as it would be if the fluid were made to 
penetrate the bottom of the canal. No doubt, the theory is contained, to a 
first approximation, in the fornmlse of MM. Poisson and Cauchy; but as it 
happens the obtaining of these formulas is comparatively easy, their discus- 
sion forms the principal difficulty. When the height of the wave is not very 
small, so that it is necessary to proceed to a second approximation, the theory 
of long waves no longer gives a velocity of propagation agreeing with expe- 
riment. It follows, in fact, from the investigations of Mr. Airy, th at the velo - 
city of propagation of a long wave is, to a second approximation, yg(h+3k), 
where h is the depth of the fluid when it is in equilibrium, and h+k the 
height of the crest of the wave above the bottom of the canal f. 

* Transactions of the Cambridge Philosophical Society, vol. vii. p. 87. 

t Tides and Waves, art. 208. In applying this formula to a solitary wave, it is necessary 
to take for h the depth of the undisturbed portion of the fluid. In the treatise of Mr. Airy 
the formula is obtained for a particular law of disturbance, but the same formula would have 



ON RECENT RESEARCHES IN HYDRODYNAMICS. 9 

The theory of the two great solitary waves of Mr. Russell forms the sub- 
ject of a paper read by Mr. Earnshaw before the Cambridge Philosophical 
Society in December last*. Mr. Russell found by experiment that the hori- 
zontal motion of the fluid particles was sensibly the same throughout the 
whole of a vertical plane perpendicular to the length of the canal. He attri- 
buted J;he observed degradation of the wave, and consequent diminution of 
the velocity of propagation, entirely to the imperfect fluidity of the fluid, and 
its adhesion to the sides and bottom of the canal. Mr. Earnshaw accordingly 
investigates the motion of the fluid on the hypotheses, — first, that the particles 
once in a vertical plane, perpendicular to the length of the canal, remain in 
a vertical plane ; secondly, that the wave is propagated with a constant velo- 
city and without change of form. It is important to observe that these 
hypotheses are used not as a, foundation for calculation, but as a means of 
selecting a particular kind of motion for consideration. The equations of 
fluid motion admit of integration in this case in finite terms, without any 
approximation, and it turns out that the motion is possible, so far as the 2vave 
itself is concerned, and everything is determined in the result except two 
constants, which remain arbitrary. However, in order that the motion in 
question should actually take place, it is necessary that there should be an 
instantaneous generation or destruction of a finite velocity, and likewise an 
abrupt change of pressure, at the junction of the portion of fluid which con- 
stitutes the wave with the portions before and behind which are at rest, both 
which are evidently impossible. It follows of course that one at least of the 
two hypotheses must be in fault. Experiment showing that the first hypo- 
thesis is very nearly true, while the second (from whatever cause) is sensibly 
erroneous, the conclusion is that in all probability the degradation of the 
wave is not to be attributed wholly to friction, but that it is an essential cha- 
racteristic of the motion. Nevertheless the formula for the velocity of pro- 
}>agation of the positive wave, at which Mr. Earnshaw has arrived, agrees very 
well with the experiments of Mr. Russell ; the formula for the negative wave 
also agrees, but not closely. These two formulas can be derived from each 
other only by introducing imaginary quantities. 

It is the opinion of Mr. Russell that the solitary wave is a phaenomenon 
sui generis, in nowise deriving its character from the circumstances of the 
generation of the wave. His experiments seem to render this conclusion 
probable. Should it be correct, the analytical character of the solitary wave 
remains to be discovered. A-complete theory of this wave should give, not 
only its velocity of propagation, but also the law of its degradation, at least 
of that part of the degradation which is independent of friction, which is 
probably by far the greater part. With respect'to the importance of this 
peculiar wave however, it must be remarked that the term solitary loave, as 
so defined, must not be extended to the tide wave, which is nothing more (as 
far as regards the laws of its propagation) than a very long wave, of which 
the form may be arbitrary. It is hardly necessary to remark thdt the me- 
chanical theories of the solitary wave and of the aerial sound wave are 
altogether difierent. 

Theory of River and Ocean Tides. — The treatiseof Mr. Airy already referred 
to is so extensive, and so full of original matter, that it will be impossible 
within the limits of a report like the present to do more than endeavour to 

been arrived at, by the same reasoning, had the law not been restricted. This formula is 
given as expressing the velocity of propagation of the phase of high water, which it is true is 
not quite the same as the velocity of propagation of the crest of the wave ; but the two velo- 
cities are the same to the second order of approximation. 
• Transactions of the Cambridge Philosophical Society, vol. viii. p. 326. 



10 REPORT — 1846. 

give an idea of the nature of the calculations and methods of explanation 
employed, and to mention some of the principal results. 

On account of the great length of the tide wave, the horizontal motion of 
the water will be sensibly the same from top to bottom. This circumstance 
most materially simplifies the calculation. The partial differential equation 
for the motion of long waves, when the motion is very small, is in the simplest 
case the same as that which occurs in the theory of the rectilinear propaga- 
tion of sound ; and in Mr. Airy's investigations the arbitrary functions which 
occur in its integral are determined by the conditions to be satisfied at the 
ends of the canal in which the waves are propagated, in a manner similar to 
that in which the arbitrary functions are determined in the case of a tube in 
which sound is propagated. When the motion is not very small, the partial 
differential equation of wave motion may be integrated by successive ap- 
proximations, the arbitrary functions being determined at each order of ap- 
proximation as before. 

To proceed to some of the results. The simplest conceivable case of a 
tidal river is that in Avhich the river is regarded as a uniform, indefinite canal, 
without any current. The height of the water at the mouth of the canal will 
be expressed, as in the open sea, by a periodic function of the time, of the 
form a sin (w<+a). The result of a first approximation of course is that 
the disturbance at the mouth of the canal will be propagated uniformly up 
it, with the velocity due to half the depth of the water. But on proceeding to 
a second approximation*, Mr. Airy finds that the form of the wave will alter 
as it proceeds up the river. Its front will become shorter and steeper, and 
its rear longer and more gently sloping. When the wave has advanced suf- 
ficiently far up the river, its surface will become horizontal at one point in 
the rear, and further on the wave Mill divide into two. At the mouth of the 
river the greatest velocities of the ebb and flow of the tide are equal, and 
occur at low and high water respectively ; the time during which the water 
is rising is also equal to the time during which it is falling. But at a station 
up the I'iver the velocity of the ebb-stream is greater than that of the flow- 
stream, and the rise of the water occupies less time than its fall. If the sta- 
tion considered is sufficiently distant from the mouth of the river, and the 
tide sufficiently large, the water after it has fallen some Avay will begin to 
rise again : there will in fact be a double rise and fall of the water at each 
tide. This explains the double tides observed in some tidal rivers. The 
velocity with which the phase of high water travels up the river is found to 
be V <7A (1 +36), A being the depth of the water when in equihbrium, and 
h k the greatest elevation of the water at the mouth of the river above its 
mean level. The same formula will apply to the case of low water if we 
change the sign of b. This result is very important, since it shows that the 
interval between the time of the moon's passage over the meridian of the 
river station and the time of high water will be affected by the height of the 
tide. Mr. Airy also investigates the effect of the current in a tidal river. He 
finds that the difference between the times of the water's rising and falling 
is increased by the current. 

When the canal is stopped by a barrier the circumstances are altered. 
When the motion is supposed small, and the disturbing force of the sun and 
moon is neglected, it is found in this case that the tide-wave is a stationary- 
wave t, so that there is high or low water at the same instant at every point 
of the canal ; but if the length of the canal exceeds a certain quantity, it is 
high water in certain parts of the canal at the instant when it is low water 

* Art. 198, &c. t Art. 307. 



ON RECENT KESEABCHES IN HYDRODYNAMICS. 11 

in the remainder, and vice versa. The height of high water is different in 
different parts of the canal : it increases from the mouth of the canal to its 
extremity, provided the canal's length does not exceed a certain quantitj'. If 
four times the length of the canal be any odd multiple of the length of a 
free wave whose period is equal to that of the tide, the denominator of the 
expression for the tidal elevation vanishes. Of course friction would pre- 
vent the elevation from increasing beyond a certain amount, but still the tidal 
oscillation would in such cases be very large. 

When the channel up which the tide is propagated decreases in breadth 
or depth, or in both, the height of the tide increases in ascending the channel. 
This accounts for the great height of the tides observed at the head of the 
Bristol Channel, and in such places. In some of these cases however the 
great height may be partly due to the cause mentioned at the end of the last 
paragraph. 

When the tide-wave is propagated up a broad channel, which becomes 
shallow towards the sides, the motion of the water in the centre will be of 
the same nature as the motion in a free canal, so that the water will be flow- 
ing up the channel with its greatest velocity at the time of high water. 
Towards the coasts however there will be a considerable flow of water to 
and from the shore ; and as far as regards this motion, the shore will have 
nearly the same effect as a barrier in a canal, and the oscillation will be of 
the nature of a stationary wave, so that the water will be at rest when it is 
at its greatest height. If, now, we consider a point at some distance from 
the shore, but still not near the middle of the channel, the velocity of the 
water up and down the channel will be connected with its height in the same 
way as in the case of a progressive wave, while the velocity to and from the 
shore will be connected with the height of the water in the same way as in a 
stationary wave. Combining these considerations, Mr. Airy is enabled to 
explain the apparent rotation of the water in such localities, which arises 
from an actual rotation in the direction of its motion*. 

When the motion of the water is in two dimensions the mathematical cal- 
culation of the tidal oscillations is tolerably simple, at least when the depth 
of the water is uniform. But in the case of nature the motion is in three 
dimensions, for the water is distributed over the surface of the earth in broad 
sheets, the boundaries of which are altogether irregular. On this account a 
. complete theory of the tides appears hopeless, even in the case in which the 
depth is supposed uniform. Laplace's theory, in which the whole earth is 
supposed to be covered with w ater, the depth of which follows a very pecu- 
liar law, gives us no idea of the effect of the limitation of the ocean by conti- 
nents. Mr. Airy consequently investigates the motion of the water on the 
supposition of its being confined to narrow canals of uniform depth, which 
in the calculation are supposed circular. The case in which the canal forms 
a great circle is especially considered. This method enables us in some de- 
gree to estimate the effect of the boundaries of the sea ; and it has the great 
advantage of leading to calculations which can be worked out. There can 
be no doubt, too, that the conclusions arrived at will apply, as to their general 
nature, to the actual case of the earth. 

With a view to this application of the theory, Mr. Airy calculates the 
motion of the water in a canal when it is under the action of a disturbing 
force, which is a periodic function of the time. The disturbing force at a 
point whose abscissa, measured along the canal from a fixed point, is x, is 
supposed to be expressed by a function of the form A sin (n< — mx + a). 
This supposition is sufficiently general for the case of the tides, provided the 
* Art. 360, &c. 



12 REPORT 1846. 

canal on the earth be supposed circular. In all cases the disturbing force 
will give rise to an oscillation in the water having the same period as the force 
itself. Tills oscillation is called by Mr. A'lry'a forced wave. It will be suf- 
ficient here to mention some of the results of this theory as applied to the 
case of the earth. 

In all cases the expression for the tidal elevation contains as a denominator 
the difference of the squares of two velocities, one the velocity of propagation 
of a free wave along the canal, the other the velocity with which a particular 
phase of the disturbing force travels along the canal, or, which is the same, 
the velocity of propagation of the forced wave. Hence the height of the 
tides will not depend simply on the rtiagnitude of the disturbing force, but 
also on its period. Thus the mass of the moon cannot be inferred directly 
from the comparison of spring and neap tides, since the heights of the solar 
and lunar tides are affected by the different motions of the sun and moon in 
right ascension, and consequently in hour-angle. When the canal under 
consideration is equatorial the diurnal tide vanishes. The height of high water 
is the same at all points of the canal, and there is either high or low water at 
the point of the canal nearest to the attracting body, according as the dejith 
of the water is greater or less than that for which a free wave would be pro- 
pagated with the same velocity as the forced wave. In the general case there 
is both a diurnal and a semidiurnal tide, and the height of high water, as well 
as the interval between the transit of the attracting body over the meridian 
of the place considered and the time of high water, is different at different 
points of the canal. When the canal is a great circle passing through the 
poles, the tide-wave is a stationary wave. When the coefficient of the dis- 
turbing force is supposed to vary slowly, in consequence of the change in 
declination, &'C. of the disturbing body, it is found that the greatest tide oc- 
curs on the day on which the disturbing force is the greatest. 

The preceding results have been obtained on the supposition of the absence 
of all friction ; but Mr. Airy also takes friction into consideration. He sup- 
poses it to be represented by a horizontal force, acting uniformly from top to 
liottom of the water, and varying as the first power of the horizontal velocity. 
Of course this supposition is not exact: still there can be no doubt that 
it represents generally the effect of friction. When friction is taken into 
account, the denominator of the expressions for the tidal elevation is essen- 
tially positive, so that the motion can never become infinite. In the case of 
a uniform tidal river stopped by a barrier, the high water is no longer simul- 
taneous at all points, but the phase of high water always travels up the river. 
But of all the results obtained by considering friction, the most important 
appears to be, that when the slow variation of the disturbing force is taken 
into account, the greatest tide, instead of happening on the day when the 
disturbing force is greatest, will happen later by a certain time, /»,. More- 
over, in calculating the tides, we must use, not the relative positions of the sun 
and moon for the instant for which the tide is calculated, but their relative 
positions for a time earlier by the same interval p^ as in the preceding case. 
The expression for p^ depends both on the depth of the canal and on the 
period of the tide, and therefore its value for the diurnal tide cannot be 
inferred from its value for the semidiurnal. It appears also that the phase of 
the tide is accelerated by friction. 

The mechanical theory of the tides of course belongs to hydrodynamics ; 
but I do not conceive that the consideration of the reduction and discussion 
of tidal observations falls within the province of this report. 

Before leaving the investigations of Mr. Airy, I would call attention to a 
method which he sometimes employs very happily in giving a general expla- 



ON RECENT RESEARCHES IN HYDRODYNAMICS. 13 

nation of phgenomena depending on motions which are too complicated to 
admit of accurate calculation. It is evident that any arbitrary motion may 
be assigned to a fluid, (with certain restrictions as to the absence of abrupt- 
ness,) provided we suppose certain forces to act so as to produce them. The 
values of these forces are given by the equations of motion. In some cases 
the forces thus obtained will closely resemble some known forces ; while in 
others it will be possible to form a clear conception of the kind of motion 
which must take place in the absence of such forces. For example, sup- 
posing that there is propagated a series of oscillatory waves of the standard 
kind, except that the height of the waves increases proportionably to their 
distance from a fixed line, remaining constant at the same point as the time 
varies, Mr. Airy finds for the force requisite to maintain such a motion an 
expression which may be assimilated to the force which wind exerts on water. 
This affords a general explanation of the increase in the height of the waves 
in passing from a windward to a lee shore*. Again, by supposing a series 
of waves, as near the standard kind as circumstances will admit, to be pro- 
pagated along a canal whose depth decreases slowly, and examining the force 
requisite to maintain this motion, he finds that a force must be applied to 
hold back the heads of the waves. In the absence, then, of such a force the 
heads of the waves will have a tendency to shoot forwards. This explains 
the tendency of waves to break over a sunken shoal or along a sloping 
beach f. The word tendency is here used, because when a wave comes at all 
near breaking, but little reliance can be placed in any investigation which 
depends upon the supposition of the motion being small. To take one more 
example of the application of this method, by supposing a wave to travel, 
unchanged in form, along a canal, with a velocity different from that of a free 
wave, and examining the force requisite to maintain such a motion, Mr. Airy 
is enabled to give a general explanation of some very curious circumstances 
connected with the motion of canal boats |, which have been observed by 
Mr. Russell. 

III. In the 16th volume of the ' Journal de I'Ecole Polytechnique§, will be 
found a memoir by MM. Barre de Saint- Venant and Wantzel, containing the 
results of some experiments on the discharge of air through small orifices, 
produced by considerable differences of pressure. The formula for the ve- 
locity of efflux derived from the theory of steady motion, and the supposition 
that the mean pressure at the orifice is equal to the pressure at a distance 
from the orifice in the space into which the discharge takes place, leads to 
some strange results of such a nature as to make ue doubt its correctness. If 
we call the space from which the discharge takes place the first space, and 
that into which it takes place the second space, and understan.d by the term 
reduced velocity the velocity of efflux diminished in the ratio of the density 
in the second space to the density in the first, so that the reduced velocity 
measures the rate of discharge, provided the density in the first space remain 
constant, it follows from the common formula that the reduced velocity va- 
nishes when the density in the second space vanishes, so that a gas cannot be 
discharged into a vacuum. Moreover, if the density of the first space is given, 
the reduced velocity is a maximum when the density in the second space is 
rather more than half that in the first. The results remain the same if we 
take account of the contraction of the vein, and they are not materially al- 
tered if we take into account the cooling of the air by its rapid dilatation. 
The experiments above alluded to Avere made by allowing the air to enter an 
exhausted receiver through a small orifice, and observing simultaneously the 

* An. 265, &c. t Art. 238, &c. 

X Art. 405, &c. § Cahier xxvii. p. 85. 



14 REPORT — 1846. 

pressure and temperature of the air in the receiver, and the time elapsed since 
the opening of the orifice. It was found that when the exhaustion was com- 
plete the reduced velocity had a certain value, depending on the orifice em- 
ployed, and that the velocity did not sensibly change till the pressure of the 
air in the receiver became equal to about flhs of the atmospheric pressure. 
The reduced velocity then began to decrease, and finally vanished when the 
pressure of the air in the receiver became equal to the atmospheric pressure. 

These experiments show tliat when tlie difference of pressure in the first 
and second spaces is considerable, we can by no means suppose that the mean 
pressure at the orifice is equal to the pressure at a distance in the second 
space, nor even that there exists a contracted vein, at which we may suppose 
the pressure to be the same as at a distance. The authors have given an 
empirical fornmla, which represents very nearly the reduced velocity, what- 
ever be the pressure of the air in the space into which the discharge takes place. 
The orifices used in these experiments were generally about one millimetre 
in diameter. It was found that widening the mouth of the orifice, so as to 
make it funnel-shaped, produced a much greater proportionate increase of 
velocity when the velocity of efHux was small than when it was large. The 
authors have since repeated their experiments with air coming from a vessel in 
which the pressure was four atmospheres : they have also tried the effect of 
using larger orifices of four or five millimetres diameter. The general results 
were found to be the same as before*. 

IV. In the 6th volume of the Transactions of the Cambridge Philoso- 
phical Society, p. 403, will be found a memoir by Mr. Green on the re- 
flection and refraction of sound, which is well-worthy of attention. This 
problem had been previously considered by Poisson in an elaborate memoirf. 
Poisson treats the subject with extreme generality, and his analysis is con- 
sequently very complicated. Mr. Green, on the contrary, restricts himself 
to the case of plane waves, a case evidently comprising nearly all the phseno- 
mena connected with tliis subject which are of interest in a physical point of 
view, and thus is enabled to obtain his results by a very simple analysis. In- 
deed Mr. Green's memoirs are very remarkable, both for the elegance and 
rif^our of the analysis, and for the ease with which he arrives at most im- 
portant results. This arises in a great measure from his divesting the pro- 
blems he considers of all unnecessary generality : where generality is really 
of importance he does not shrink from it. In the present instance there is 
one important respect in which Mr. Green's investigation is more general 
than Poisson's, which is, that Mr. Green has taken the case of any two fluids, 
whereas Poisson considered the case of two elastic fluids, in which equal con- 
densations produce equal increments of pressure. It is curious, that Poisson, 
forgetting this restriction, applied his formulee to the case of air and water. 
Ofcourse his numerical result is altogether erroneous. Mr. Green easily 
arrives at the ordinary laws of reflection and refraction. He obtains also a 
very simple expression for the intensity of the reflected sound. If A is the 
ratio of the density of the second medium to that of the first, and B the ratio 
of the cotangent of the angle of refraction to the cotangent of the angle of 
incidence, then the intensity of the reflected sound is to the intensity of the 
incident as A — B to A-fB. In this statement the intensity is supposed to 
be measured by the first power of the maximum displacement. When the 
velocity of propagation in the first medium is less than in the second, and the 
angle of incidence exceeds what may be called the critical angle, Mr. Green 
restricts himself to the case of vibrations following the cycloidal law. He 

* Coraptes Rendus, torn. x^ii. p. 1140. 

f Memoires de I'Academie des Sciences, torn. x. p. 317. 



ON RECENT RESEARCHES IN HYDRODYNAMICS. 15 

finds that the sound suffers total internal reflection. The expression for the 
disturbance in the second medium involves an exponential with a negative 
index, and consequently the disturbance becomes quite insensible at a di- 
stance from the surface equal to a small multiple of the length of a wave. 
The phase of vibration of the reflected sound is also accelerated by a quan- 
tity depending on the angle of incidence. It is remarkable, that when the 
fluids considered are ordinary elastic fluids, or rather when they are such 
that equal condensations produce equal increments of pressure, the expres- 
sions for the intensity of the reflected sound, and for the acceleration of 
phase when the angle of incidence exceeds the critical angle, are the same 
as those given by Fresuel for light polarized in a plane perpendicular to the 
plane of incidence. 

V. Not long after the publication of Poisson's memoir on the simultaneous 
motions of a pendulum and of the surrounding air*, a paper by Mr. Green 
was read before the Royal Society of Edinburgh, which is entitled ' Re- 
searches on the Vibration of Pendulums in Fluid Media f.' Mr. Green does 
not appear to have been at that time acquainted with Poisson's memoir. The 
problem which he has considered is one of the same class as that treated by 
Poisson. Mr. Green has supposed the fluid to be incompressible, a suppo- 
sition, however, which will apply without sensible error to air, in considering 
motions of this sort. Poisson regarded the fluid as elastic, but in the end, in 
adapting his forftiula to use, he has neglected as insensible the terms by 
which the effect of an elastic differs from that of an inelastic fluid. The 
problem considered by Mr. Green is, however, in one respect much more 
general than that solved by Poisson, since Mr. Green has supposed the oscil- 
lating body to be an ellipsoid, whereas Poisson considered only a sphere. 
Mr. Green has obtained a complete solution of the problem in the case in 
which the ellipsoid has a motion of translation only, or in which the small 
motion of the fluid due to its motion of rotation is neglected. The result is 
that the resistance of the fluid will be allowed for if we suppose the mass of 
the ellipsoid increased by a mass bearing a certain ratio to that of the fluid 
displaced. In the general case this ratio depends on three transcendental 
quantities, given by definite integrals. If, however, the ellipsoid oscillates in 
the direction of one of its principal axes, the ratio depends on one only of 
these transcendents. When the ellipsoid passes into a spheroid, the tran- 
scendents above-mentioned can be expressed by means of circular or loga- 
rithmic functions. When the spheroid becomes a sphere, Mr. Green's result 
agrees with Poisson's. It is worthy of remark, that Mr. Green's formula will 
enable us to calculate the motion of an ellipse or ojrcle oscillating in a fluid, 
in a direction perpendicular to its plane, since a material ellipse or circle may 
be considered as a limiting form of an ellipsoid. In this case, however, the 
motion would probably have to be extremely small, in order that the formula 
should apply with accuracy. 

In a paper ' On the Motion of a small Sphere acted on by the Vibrations of 
an Elastic ^Medium,' read before the Cambridge Philosophical Society in April 
184;l:j:, Prof. Challis has considered the motion of a ball pendulum, retaining 
in his solution small quantities to the second order. The principles adopted 
by Prof. Challis in the solution of this problem are at variance with those of 
Poisson, and have given rise to a controversy between him and Mr. Airy, 
which will be found in the 17th, ISth and 19th volumes of the Philosophical 

* Memoires de I'Academie des Sciences, torn. xi. p. 521. 

t This paper was read in December 1833, and is printed in the 13th Tolume of the So- 
ciety's Transactions, p. 54, &c. 
} Transactions of the Cambridge Philosophical Society, vol. vii. p. 333. 



16 REPORT— 1846. 

Magazine (New Series). In the paper just referred to, Prof. Challis finds that 
when the fluid is incompressible there is no decrement in the arc of oscilla- 
tion, except what arises frona friction and capillary attraction. In the case 
of air there is a slight theoretical decrement ; but it is so small that Prof. 
Challis considers the observed decrement to be mainly owing to friction. 
This result follows also from Poisson's solution. Prof. Challis also finds that 
a small sphere moving with a uniform velocity experiences no resistance, and 
that when the velocity is partly uniform and partly variable, the resistance 
depends on the variable part only. The problem, however, referred to in 
the title of this paper, is that of calculating the motion of a small sphere 
situated in an elastic fluid, and acted on by no forces except the pi'essure of 
the fluid, in which an indefinite series of plane condensing and rarefying 
waves is supposed to be propagated. This problem is solved by the author on 
principles similar to those which he has adopted in the problem of an oscil- 
lating sphere. The views of Prof. Challis with respect to this problem, which 
he considers a very important one, are briefly stated at the end of a paper 
published in tlie Philosophical Magazine*. 

In a paper ' On some Cases of Fluid Motion,' published in the Trans- 
actions of the Cambridge Philosophical Society f, I have considered some 
modifications of the problem of the ball pendulum, adopting in the main the 
principles of Poisson, of the correctness of which I feel fully satisfied, but 
supposing the fluid incompressible from the first. In this paper the eff'ect of 
a distant rigid plane interrupting the fluid in which the sphere is oscillating is 
given to the lowest order of approximation with which the effect is sensible. 
It is shown also that when the ball oscillates in a concentric spiierical enve- 
lope, the eff'ect of the resistance of the fluid is to add to the mass of the 

sphere a mass equal to , — — , where a is the radius of the ball, h that 

of the envelope, and m the mass of the fluid displaced. Poisson, having 
reasoned on the very complicated case of an elastic fluid, had come to the 
conclusion that the envelope would have no effect. 

One other instance of fluid motion contained in this paper will here be 
mentioned, because it seems to afford an accurate means of comparing theory 
and experiment in a class of motions in which they have not hitherto been 
compared, so far as I am aware. When a box of the form of a rectangular 
parallelepiped, filled with fluid and closed on all sides, is made to perform small 
oscillations, it appears that the motion of the box will be the same as if the 
fluid were replaced by a solid having the same mass, centre of gravity, and 
principal axes as the solidified fluid, but different principal moments of in- 
ertia. These moments are given by infinite series, which converge with 
extreme rapidity, so that the numerical calculation is very easy. The oscil- 
lations most convenient to employ would probably be either oscillations by 
torsion, or bifilar oscillations. 

VI. M. Navier was, I believe, the first to give equations for the motion of 
fluids without supposing the pressure equal in all directions. His theory is 
contained in a memoir read before the French Academy in 1822J:. He con- 
siders the case of a homogeneous incompressible fluid. He supposes such a 
fluid to be made up of ultimate molecules, acting on each other by forces 
which, when the molecules are at rest, are functions simply of the distance, 
but which, when the molecules recede from, or approach to each other, are 
modified by this circumstance, so that two molecules repel each other less 
strongly when they are receding, and more strongly when they are approaching, 
* Vol. xviii. New Series, p. 481. t Vol. viii. p. 105. 

X Memoires de I'Academie des Sciences, torn. vi. p. 389. 



ON RECENT RESEARCHES IN HYDRODYNAMICS. 17 

than they do when they are at rest*. The alteration of attraction or re- 
pulsion is supposed to be, for a given distance, proportional to the velocity 
with which the molecules recede from, or approach to each other; so that 
the mutual repulsion of two molecules will be represented by f(r) — V F (r), 
where r is the distance of the molecules, V the velocity with which they recede 
from each other, and/(7-), F (r) two unknown functions of t- depending on 
the molecular force, and as such becoming insensible when r has become 
sensible. This expression does not suppose the molecules to be necessarily 
receding from each other, nor their mutual action to be necessarily repulsive, 
since V and F (r) may be positive or negative. It is not absolutely necessary 
that/(r) and F (r) should always have the same sign. In forming the equa- 
tions of motion M. Navier adopts the hypothesis of a symmetrical arrangement 
of the particles, or at least, which leads to the same result, neglects the irre- 
gular part of the mutual action of neighbouring molecules. The equations 
at which he arrives are those which would be obtained from the common 

equations by wntmg ^ - A (^^ + ^ + J^) m place of ^ m the 

first, and making similar changes in the second and third. A is here an 
unknown constant depending on the nature of the fluid. 

The same subject has been treated on by Poissonf, who has adopted hy- 
potheses which are very different from those of M. Navier. Poisson's theory 
is of this nature. He supposes the time t to be divided into n equal parts, 
each equal to r. In the first of these he supposes the fluid to be displaced 
in the same manner as an elastic solid, so that the pressures in different 
directions are given by the equations which he had previously obtained for 
elastic solids. If the causes producing the displacement were now to cease 
to act, the molecules would very rapidly assume a new arrangement, which 
would render the pressure equal in all directions, and while this re-arrange- 
ment was going on, the pressure would alter in an unknown manner from 
that belonging to a displaced elastic solid to the pressure belonging to the 
fluid in its new state. The causes of displacement are however going on 
during the second interval r; but since these different small motions will 
take place independently, the new displacement which will take place in the 
second interval t will be the same as if the molecules were not undergoing a 
re-arrangement. Supposing now n to become infinite, we pass to the case in 
which the fluid is continually beginning to be displaced like an elastic solid, 
and continually re-arranging itself so as to make the pressure equal in all 
directions. The equations at which Poisson arrived are, in the cases of a 
homogeneous incompressible fluid, and of an elastic^uid in which the change 
of density is small, those which would be derived from the common equations 

by replacing t^ in the first by 

dp _ . fd^u d'^u '^^^N _x> ^ / du dv dio\ 
dx \rf«^2 + f^ "*■ rf^/ dx\di'^in,'^ dzj' 

and making similar changes in the second and third. In these equations A 
and B are two unknown constants. It will be observed that Poisson's equa- 
tions reduce themselves to Navier's in the case of an incompressible fluid. 

The same subject has been considered in a quite different point of view by 
M. Barre de Saint-Venant, in a communication to the French Academy in 

* This idea appears to have been borrowed from Dubuat. See his Principes d'Hydrau- 
lique, torn. ii. p, 60. 

t Journal de I'Ecole Polytechnique, torn. xiii. cah. 20. p. 139. 
1846. C 



18 REPORT— 1846. 

1843, an abstract of which is contained in the < Comptes Rendus*.* The 
principal difficulty is to connect the oblique pressures in different directions 

du du , 

about the same point with the differential coefficients ^. j-, &c., which 

express the relative motion of the fluid particles in the immediate neighbour- 
hood of that point. Tliis the author accomplishes by assuming that the tan- 
gential force on any plane passing through the point in question is in the 
direction of the principal sliding (^glissemeni) along that plane. The sliding 

dw du , 
along the plane xy is measured by j— + -r- m the direction of x, and 

J h 77 in the direction of y. These two slidings may be compounded 

into one, which will form the principal sliding along the plane xy. It is 
then shown, by means of IM. Cauchy's theorems connecting the pressures in 
different directions in any medium, that the tangential force on any plane 
passing through the point considered, resolved in any direction in that plane, 
is proportional to the sliding along tliat plane resolved in the same direction, 
so that if T represents the tangential force, referred to a unit of surface, and 
S the sliding, T=£S. The pressure on a plane in any direction is then 
found. This pressure is compounded of a normal pressure, alike in all di- 
rections, and a variable oblique pressure, the expression for which contains 
the one unknown quantity s. If the fluid be supposed incompressible, and 
£ constant, the equations which would be obtained by the method of M.Barre 
de Saint- Venant agree with those of .M. Navier. It will be observed that 
this method does not require the consideration of ultimate molecules at all. 

When the motion of the fluid is very small, Poisson's equations agree with 
those given by M. Cauchy for the motion of a solid entirely destitute of elas- 
ticity f, except that the latter do not contain the pressure jo. These equations 
have been obtained by M. Cauchy without the consideration of molecules. 
His method would apply, witli very little change, to the case of fluids. 

In a paper read last year before the Cambridge Philosophical Society ;{:, I 
have arrived at the equations of motion in a different manner. The method 
employed in this paper does not necessarily require the consideration of ulti- " 
mate molecules. Its principal feature consists in eliminating from the rela- 
tive motion of the fluid about any particular point the relative motion which 
corresponds to a certain motion of rotation, and examining the nature of the 
relative motion which remains. The equations Anally adopted in the cases 
of a homogeneous incompressible fluid, and of an elastic fluid in which the 
change of density is small, agree with those of Poisson, provided we suppose 
in the latter A = 3 B. It is shown that this relation between A and B may 
be obtained on Poisson's own principles. 

The equations hitherto considered are those which must be satisfied at any 
point in the interior of the fluid mass; but there is hardly any instance of 
the practical application of the equations, in wliich we do not want to know 
also the particular conditions which must be satisfied at the surface of the 
fluid. With respect to a free surface there can be little doubt : the condi- 
tion is simply that there shall be no tangential force on a plane parallel to 
the surface, taken immediately within the fluid. As to the case of a fluid in 
contact with a solid, the condition at which Navier arrived comes to this: 
that if we conceive a small plane drawn within tlie fluid parallel to the sur- 

* Tom. xvii. p. 1240. 

t Exercices dc Mathematiques, torn. iii. p. 187. 

X Transactions of ihe Cambridge Pliilosopliical Society, vol. viii. p. 2S7. s 



ON RECENT RESEARCHES IN HYDRODYNAMICS. 19 

face of the solid, the tangential force on this plane, referred to a unit of 
surface, shall be in the same direction with, and proportional to the velocity 
with which the fluid flows past the surface of the solid. The condition ob- 
tained by Poisson is essentially the same. 

Dubuat stated, as a result of his experiments, that when the velocity of 
water flowing through a pipe is less than a certain quantity, the water adja- 
cent to the surface of the pipe is at rest*. This result agrees very well with 
an experiment of Coulomb's. Coulomb found that when a metallic disc was 
made to oscillate very slowly in water about an axis passing through its 
centre and perpendicular to its plane, the resistance was not altered when 
the disc was smeared with grease ; and even when the grease was covered 
with powdered sandstone the resistance, was hardly increased f. This is just 
what one would expect on the supposition that the water close to the disc is 
carried along with it, since in that case the resistance must depend on the 
internal friction of the fluid ; but the result appears very extraordinary on 
the supposition that the fluid in contact with the disc flows past it with a 
finite velocity. It should be observed, however, that this result is compatible 
with the supposition that a thin film of fluid remains adhering to the disc, in 
consequence of capillary attraction, and becomes as it were solid, and that 
the fluid in contact with this film flows past it with a finite velocity. If we 
consider Dubuat's supposition to be correct, the condition to be assumed in 
the case of a fluid in contact with a solid is that the fluid does not move re- 
latively to the solid. This condition will be included in M. Navier's, if we 
suppose the coefficient of the velocity when M. Navier's condition is ex- 
pressed analytically, which he denotes by E, to become infinite. It seems 
probable from the experiments of M. Girard, that the condition to be satis- 
fied at the surface of fluid in contact with a solid is different according as the 
fluid does or does not moisten the surface of the solid. 

M. Navier has applied his theory to the results of some experiments of 
M. Girard's on the discharge of fluids through capillary tubes. His theory 
shows that if we suppose E to be finite, the discharge through extremely 
small tubes will depend only on E, and not on A. The law of discharge at 
which he arrives agrees with the experiments of M. Girard, at least when the 
tubes are extremely small. M. Navier explained the difference observed by 
M. Girard in the discharge of water through tubes of glass and tubes of 
copper of the same size by supposing the value of E different in the two 
cases. This difference was explained by M. Girard himself by supposing that 
a thin film of fluid remains adherent to the pipe, in consequence of molecular 
action, and that the thickness of this film differs with*the substance of which 
the tube is composed, as well "as with the liquid employed if. If we adopt 
Navier's explanation, we may reconcile it with the experiments of Coulomb 
by supposing that E is very large, so that unless the fluid is confined in a 
very narrow pipe, the results will depend mainly on A, being sensibly the 
same as they would be if E were infinite. 

There is one circumstance connected with the motion of a ball-pendulum 
oscillating in air, wiiich has not yet been accounted for, the explanation of 
which seems to depend on this theory. It is found by experiment that the 
correction for the inertia of the air is greater for small than for large spheres, 
that is to say, the niaj^s which we must suppose added to that of the sphere 
bears a greater ratio to the mass of the fluid displaced in the former, than in 
the latter case. According to the common theory of fluid motion, in which 

* See the Table given in torn. i. of liis Principes d'Hydraulique, p. 93. 

t Meraoires de I'lnstitut, 1801, torn. iii. p. 286. 

X Mfemoires de I'Academie des Sciences, torn. i. pp. 203 and 234. 

c2 



20 REPORT — 1846. 

everytbing is supposed to be perfectly smooth, the ratio ought to be inde- 
pendent of the magnitude of the sphere. In the imperfect theory of friction 
in which the friction of the fluid on the sphere is talcen into account, while 
the equal and opposite friction of the sphere on the fluid is neglected, it is 
shown that the arc of oscillation is diminished, while the time of oscillation 
is sensibly the same as before. But when the tangential action of the sphere 
on the fluid, and the internal friction of the fluid itself are considered, it is 
clear that one consequence will be, to speak in a general way, that a portion 
of the fluid will be dragged along with the sphere. Thus the correction for 
the inertia of the fluid will be increased, since the same moving force has now 
to overcome the inertia of the fluid dragged along with the sphere, and not 
only, as in the former case, the inertia of the sphere itself, and of the fluid 
pushed away from before it, and drawn in behind it. Moreover the addi- 
tional correction for inertia must depend, speaking approximately, on the 
surface of the sphere, whereas the first correction depended on its volume, 
and thus the effect of friction in altering the time of oscillation will be more 
conspicuous in the case of small, than in the case of large spheres, other cir- 
cumstances being the same. The correction for inertia, when friction is 
taken into account, will not, however, depend solely on the magnitude of the 
sphere, but also on the time of oscillation. With a given sphere it will be 
greater for long, than for short oscillations. 



Sixth Report of a Committee, consisting of H.'E. Strickland, Esq., 
Prof. Daubeny, Prof. Henslow, and Prof. Lindley, appointed 
to continue their Experim.ents on the Vitality of Seeds. 

These experiments have again been repeated upon 48 kinds of seeds ga- 
thered in 1843, as well as upon 26 kinds of new seeds added to the general 
collection in 1845. 

Many kinds of old seeds, of various dates from 1812 to 1845 inclusive, 
consisting of 151 packets, have been contributed by Miss Molesworth. These 
were for the most part in small quantities, and were sown only at Oxford, on 
a slight hot-bed. 

A small quantity of soil from the bed of a freshwater lake of the tertiary 
period, at Mundesley, Norfolk, containing scales of fish, elytra of beetles, 
seeds of Ceratophyllum and other plants from Sir W. C. Trevelyan, was sub- 
jected to three tests ; viz. one-third part was placed in a shallow pan, and 
kept moist with distilled water ; the second portion was kept well-saturated 
with the same ; and the third portion, also in a shallow pan, under about 
one inch of distilled water. The whole was kept under a glass case to pre- 
vent the chance of seeds, &c. being deposited in it. No vegetation appeared 
in either case. 

It may be well to remark, that the seeds have been sown under different 
circumstances, and have received different treatment at each of the three 
places they have been experimented upon. At Oxford, as in previous years, 
a selection was made from the whole quantity to be sown, of such as usually 
require the assistance of heat to enable them to germinate ; these were sown 
in pots and placed in slight heat, and the remainder were sown on a small 
bed made in a cold frame, and, Avith the exception of two or three waterings, 
left to nature. 

At Hitcham they were all sown in a border carefully prepared for them, 
and aftervj'ards left to nature. 



ON THE VITALITY OF SEEDS. 



21 



At Chiswick the whole of the seeds were sown in separate pots and placed 
in a pit heated with hot water. 

These several treatments will at once account for the great difference there 
has been in many cases between the length of time the seeds required to 
vegetate, as well as the greater number of seeds which did vegetate at one 
place more than at the other, which will be seen on referring to the following 
statement of the results. 



Name and Date when gathered. 



No. 
sown. 



No. of Seeds of each 
Species which vege- 
tated at 



Ox- „.^ . Chis- 

ford. Hitcham. ^^k. 



Time of vegetating 
in daj's at 



Chis- 
mck 



1843. 

1. Asphodelus luteus 

2. Arctium Lappa 

3. Angelica Archangelica . . , 

4. Ageratum mexicanum ... 

5. Aster tenella , 

6. Allium fragrans 

7. Bidens diversifolia , 

8. Biscutella erigerifolia 

9. Borkhausia rubra , 

10. Bartonia aurea 

1 1 . Callistemma hortensis . . 

12. Campanula Medium 

13. Ceiitaurea depressa 

14. Cladanthus arabicus 

15. Cleome spinosa 

16. Cnicus arvensis , 

17. Convolvulus major , 

18. Dianthus barbatus , 

19. Ecbiuni grandiflorum 

20. Eucharidium concinnum 

21. Euphorbia Lathyris 

22. Gy])sophila elegans 

23. Helenium Douglasii 

24. Hebenstreitia tenuifolia.., 

25. Heliophila araboides 

26. Hesperis matronalis 

27. Hypericum hirsutum 

28. Kaulfussia amelloides .. 

29. Koniga maritima 

30. Leptosiphon androsacea. . 

31. Lunaria biennis 

32. Loasa lateritia 

33. Mathiola annua 

34. Melilotus caerulea 

35. CEnanthe Crocata 

36. Phytolacca decandra 

37. Plantago media 

38. Polemonium caeruleum . . 

39. Rumex obtusifolium 

40. Silene inflata 

41. Smyrnium Olusatrum 

42. Schizanthus pinnatus 

43. Tallinum ciliatum 

44. Tigridia Pavonia 

45. Valeriana officinalis 

46. Viola lutea vars 

47. Xeranthemum annuum .. 

48. Zinnia multiflora 



50 
100 
100 
200 
200 
100 
150 
100 
100 
200 
200 
100 
100 
200 
100 



20 



2 

84 



24 

54 

47 

134 

112 

73 

124 

69 

43 

141 

161 

76 

43 

149 

55 



28 
18 


33 






l** 




13 




18 


50 


■■■4' 

18 


38 

7 

21 


18 
17 

7 


27 
28 


30 





50 
100 
100 
200 

50 
200 
200 
100 
200 
100 
150 
100 
200 
200 
100 
150 
200 
100 
100 

25 
150 
100 
150 

50 
100 
200 
200 
100 
100 
150 
200 
150 



13 




45 


24 


26 


4 


16 












26 


6 


35 


6 


9, 




37 


3 



1 

42 
1 
1 



7 
55 


1 


46 
5 

69 
1"^ 


5 
17 
23 


T 








15 


3 







33 

57 

67 

177 

45 

130 

117 

72 

149 

66 

94 

82 

129 

119 

74 

112 

182 

61 

42 

17 

116 

70 

107 

37 

44 

148 

176 

24 

17 

81 

64 

37 



39 
18 
13 
18 



11 




5 


12 


6 


15 


6 








».„. 


"is" 


5 


12 


'^8 




16 


29 


9 


29 


4 


6 


37 


56 


36 




16 




45 


51 


17 




6 


16 


30 


34 


5 


14 


10 




39 




16 ' 


■■■■36" 







22 



REPORT — 1846. 



Name and Date when gathered. 



No. 
sown. 



No. of Seeds of each 
Species which vege- 
tated at 



Ox- „.^ , Chis- 

ford. Hitcham. „i<.k, 



Time of vegetating 
in days at 



Chis- 
wick. 



1845. 

49. Ailanthus glandulosa .... 

50. Alnus glutiuosa 

51. Alonsoa incisa 

52. Beta vulgaris 

53. Browallia data 

54. Chn'santhemum coronarium 

55. Cjiiisus albus 

56. Eccremocarpus scaber .. 

57. Fagus sylvatica 

58. Fumaria spicata 

59. Gaillanlia aristata 

60. Gleditschia triacanthos .. 

61. Iris, sp 

62. Knautia orientalis 

63. Lopezia racemosa 

64. Lymnanthes Douglasii .. 

65. Petunia odorata 

66. Schizopetalon Walkeri .. 

67. Secale cereale 

68. Spartium Scopaiium 

69. Tagetes lucida 

70. Verbena Aubletia 

71. Viscaria oculata 

72. Xeranthemum annuum.. 

73. ZeaMays 

74. Zinnia grandiflora 



50 

150 

100 

75 

50 

150 

100 

100 

100 

100 

100 

20 

25 

50 

150 

50 

150 

50 

200 

200 

150 

100 

150 

100 

100 

100 



20 



6 
26 
14 
10 
26 
160 
5 



98 



1 

10 



36 
111 
47 
63 
39 
104 
63 
41 
76 
80 
87 



32 



10 



4 
64 
35 



24 

102* 



51 



14 



12 

25 



18 
14 
13 



10 
22' 



From Miss Molesworth : — 



Name and Date. 


g 



3 

1 
> 

1 


8=i 

c. to 

e-i 
H s 


Name and Date. 


i 

6 


> 

6 


fi.S 


1812. 

1. Ricinus communis 

1824. 

2. Catananche caerulea ... 

3. Cucurbita Citriillus ... 

1825. 

4. Swiss Melon 


12 

100 
15 

5 

100 

100 

100 

100 

10 

30 

50 

30 

150 

50 























1828. 


100 

8 

100 

77 
54 

100 
100 
100 
30 

100 

16 

150 

100 






















17. Helianthemum croceum 

1829. 

18. Cucurbita, Cucuzza di 


1827. 
5. Brassica, white Caidi- 
flower 


19. — , Cucuzza Tiascheta 

20. Uorycnium monspeli- 


6. — , Brocoli di Carniyale 

7. ' — , — Primotice 

8. — , Cavoli Feguti 

9. Cucurbita Citrullus ... 
10. Melon 


21. Hvpericum fimbriatum 

22. Milium 


23. Rumex alpinus 


1831. 

24. Cytisus leucanthus 

25. Genista caudicans 

26. Sorghum vulgare 

27. ZeaMays 




12. Water Melon 


13. Hvpericum 


14. Spinacea Oleracea 





• .\t Hitcham, of 3 left, 2 did not flower, and the other produced no seed. 



ON THE VITALITY OF SEEDS. 



23 



Name and Date. 



1832. 

28. Calendula, sp.... 

29. Cucurbita Cucuzza 

30. Green Egyptian Melon 

31. Marari 

32. Gypsophila altissima ... 

33. Linaria genistifolia 

34. Phaseolus compressus 

35. Podalyria tinctoria 

1833. 

36. Cucurbita, Candahar 

Water Melon 

37. Cucuzza Lunga 

38. — di Spagna 

39. Mellone di Acqua 

40. — di Pane Bianca 

41. — della Regina 

42. Orange Gourd 

43. Cucumber, Kabul 

1834. 

44. Catananche crerulea .. 

45. Coix lachryma 

46. Gourd 

47. Valencian Melon 

48. Iris prismatica 

49. Pinus Pinea 

50. Plantago bonariensis .. 

51. Sambucus racemosa .. 

52. Scrophulariagrandiflora 

1835. 

53. Cucurbita 

54. Echinops, sp 

55. Heracleum asperura ., 

56. (Enothera, sp 

57. Orobuslathyroides..... 

58. Podalyria exaltata ... . 

59. Tetragonia expansa . 

60. Verbascura virgatum . 

1836. 

61. Augusta Beans 

62. Cassia Canarina 

63. Melon, early Cantalupe 

64. — , from Lisbon 

I 65. Cassia, sp 

66. Euphorbia arkanocarpa 

67. — Characias 

68. Galega sibirica 

69. Heracleum asperum 

70. Malva, sp 

71. From Malta 

72. Ditto 

73. Pepe 

74. Memoja 

75. CEnothera, sp , 

76. Papaver somniferum .. 

77. Pyrethrum microphyl 

lum 

78. Saponaria, sp 

79. Tragopogon pratensis 



50 

5 

20 
90 
50 
50 
40 

5 

200 
1 
5 

50 
10 
19 
100 
150 
200 

50 
3 

35 
200 

31 

36 
100 

50 



2 
10 
72 
20 
40 
100 
80 

no 

100 

80 

8 

2 

8 

5 

100 

150 

200 
100 





11 
8 
2 







20 

3 















1 

50 
11 



9 

6 



4 









Name and Date. 



30 



1837. 

80. Aubergines. 

81. Melon 

82. Piments .... 

83. Tomates .... 



1838. 

84. Anchusa ochroleuca 

85. Melon from Cassabah ., 

86. Water Melon 

87. Melon from Valparaiso 

88. Cynoglossum, sp , 

89. Papaver somniferum ... 

DO.. Pinus nigricans 30 

91. Tragopogon, sp 22 



92. ? Cassia 

93. ?Dolichos 

1839. 

94. Augusta Beans 

95. Calliopsis 

96. Lapsana communis 

97. Lepidium Draba .. 

98. Pisum, sp 

99. Prunella vulgaris .. 

100. Ricinus communis.. 

101. Salvia verbenaca . 



102. 
103. 
104. 
105. 
106. 
107. 
108. 
109. 

110. 
111. 
112. 
113. 
114. 

115. 
116. 
117. 
118. 



1840. 

Calliopsis tinctoria 

Chenopodium Quinoa 

Canada Beans 

Balsamina hortensis .. 
CEnothera grandiflora. . 
Ornithogalum nutans.. 

Ricinus communis 

Salvia patens 

1841. 
Algoa Bay .• 

Canada Beans 

Ferula, sp 

Papaver somniferum . . . 
Physospermum commu- 

tatum 

Rumex, sp 

Salvia, sp 

• Vicia grandiflora 

Fullard's German Mar- 
row Fat 



PS 



20 
50 

50 

150 

100 
50 
50- 

150 
7 

200 

150 
200 

50 
150 
100 I 

80 

10 

12 

1 

16 
38 
150 

100 
30 
150 
150 



119. 
120. 
121. 
122. 
123. 
134. 
125. 
126. 
127. 
128. 



1842. 

Brassica, Rapa oleifera 

Ervum, sp 

Gossypiimi ? vitifolium 

Malva moschata 

Melilotus macrorhiza.. 
Papaver somniferum ... 
Phacelia tanacetifolia 
Trifolium giganticum... 
— , Alsike Clover .. 
Vicia sativa 




50 


76 



11 






I 



4 
36 

24 




36 




3 



42 

81 



I 




16 


73 


13 


70 



100 
100 
8 
100 
100 
150 
150 
190 
100 
100 



10 



36 



10 

7 



9 
123 

9 
11 

7 

4 
5 
8 
12 
5 



24 



REPORT 1846. 



Name and Date. 


i 
1 


1 
1 


i E. 
ton' 

H 

PS 


Name and Date. 


. 1 

a 

6 
Z 


•a 
S 

1 

> 

6 


E-5 


1843. 


80 
150 
300 

30 
200 
100 
150 
100 
100 
100 
100 

150 


38 
62 

4 
30 
56 
60 
22 
40 
38 


27 

100 


7 
8 
19 
9 
7 
7 

10 

10 

4 

31 

7 


1844. 


5 

27 

30 
15 

4 

1 
2 
4 
3 
5 
1 


5 
12 

27 
14 










8 
7 

8 

7 




130. Diantbus chinensis 

131. Diplotaxis tenuifolia ... 


142. Cobbett's WTieat 

1845. 

143. Augusta Beans 


133. Linumusitatissimum... 

134. Melilotus leucantlia ... 

135. Onopordon tauricum... 


144. Cobbett's Wheat 

About 23 years old. 


ll46. Brazil Nut 


137. Trifolium giganticuin... 

138. —, Alsike Clover 






149. — jEgilops 


140. FuUard's German Mar- 




151. Rhizobolus Pekaea 




1 





W. H. Baxter, Curator. 



On the Colouring Matters of Madder. By Dr. Schunck. 
The organic colouring matters present such a wide tield for inquiry, that it 
would require the labour of years to enable one person fully to elucidate their 
properties, or even to bring this department of organic chemistry into a state 
of development proportionate to the present condition of the science. The 
substances included under the name of colouring matters by no means agree 
in their chemical characteristics ; they merely coincide in being possessed of 
certain vivid colours, or in giving rise to coloured compounds. Strictly con- 
sidered, some of them ought to be classed among the resins and others among 
the extractive matters ; and on the other hand, if we attempt a definition of 
the class according to their chemical characteristics, we shall find it impossible 
to exclude a large number of bodies, which, like tannin and catechin, are 
capable of giving rise under peculiar circumstances to brown substances, 
which in nowise differ in their general properties from the bright red colour- 
ing matters of archil, logwood, &c. Some colouring matters are presented 
to us ready formed in the diiferent parts of plants and animals ; others are 
produced artificially from colourless substances, wliich undergo very complex 
changes during the process ; others arise spontaneously during the first stages 
of oxidation or putrefaction following the extinction of organic life. In the 
investigation of substances thus widely differing in properties and formation, 
it would be vain to expect at present anything approaching to general results 
in regard to the class as a whole. I must therefore content myself on this 
occasion with giving a short account of the results of some experiments 
which I have made on one branch of the subject, at the same time apologising 
for their present vague and undefined nature. 

I have directed my attention in the first instance to madder, partly because 
the colouring matters contained in it are almost unknown, or rather worse 
than unknown, viz. known in such a manner as merely to mislead those who 
wish to inform themselves by the accounts given of them, and partly because 
madder is an article of such an immense importance in the art of dyeing that 
every discovery in relation to it acquires immediately a practical bearing. 

It will be unnecessary for me to allude to the former numerous investiga- 



ON THE COLOURING MATTERS OF MADDER. 25 

tions of madder, except so far as to mention that Robiquet discovered in it 
a crystallized volatile colouring matter, which he called Alizarin, and that 
Runge described five colouring matters which he obtained from it, viz, madder 
purple, madder red, madder orange, madder yellow and madder brown, I 
may here state as one result of my investigation, that I agree with Runge in 
thinking that there is more than one colouring matter in madder, though I 
am of opinion that the substances which he enumerates and describes are 
not pure. Before however entering on this part of the subject, I shall first 
give the results at which I have arrived in regard to alizarin. Alizarin is 
doubtless the most interesting and the most definite in its nature of all the 
substances contained in madder. It also presents itself the most easily to 
the observer even on the most superficial examination. If we heat madder 
spread out in a thin layer on a metal plate without carrying the heat far 
enough to cliar the woody parts of the root, we shall in the course of a few 
hours find its surface covered with small red or orange-coloured crystals, 
which consist of alizarin. In the same way any extract of madder, whether 
with water, alcohol or alkalies, evaporated to dryness and gently heated, gives 
a crystalline sublimate of alizarin, which is variously coloured from a light 
yellow to a dark red or brown. Now one of the first points to be ascertained 
in regard to this body was whether it exists as such in the root, or whether 
it is formed by the process of sublimation. Robiquet, the discoverer, states 
that it pre-exists in the plant. He considered alizarin as the colouring prin- 
ciple of madder, and merely subjected it to sublimation for the purpose of 
purifying it. But his investigation presents us with no convincing proof of 
this opinion, for the extract of madder with water, alcohol, &c., from which 
he prepares his alizarin by sublimation, shows no trace of anything crystalline; 
and many chemists have asserted in consequence that it is a product of de- 
composition, being formed by the action of heat in the same way as pyrogallic, 
pyrotartaric acid, and many other bodies. I have however no hesitation 
in affirming that it exists in the plant as such, having in more than one way 
obtained it in a crystallized state without the intervention of heat. If we 
make an extract of madder with cold water, we obtain a brown fluid which 
produces no reaction on test paper. After being exposed however to the 
action of the atmosphere for some hours, it acquires a distinctly acid reaction ; 
and if it be now examined carefully, there will be found floating about in it 
a number of long hair-like shining crystals : these crystals are alizarin. If 
the fluid be still further exposed to the influence of the' atmosphere, a yellow 
amorphous substance begins to separate, which I shall mention afterwards. 
This is succeeded by a gelatinous substance, and after some days a complete 
state of putrefaction ensues. It seems as if the alizarin in madder, or at all 
events that part which dissolves in the water, exists in combination with lime. 
On exposure to the atmosphere, there is formed, from some constituent 
of the root dissolved in the fluid through the instrumentality of the oxygen, 
some acid, which seizes hold of the lime in the solution and separates the 
bodies which are combined with the lime. Now the alizarin, being a body 
of very slightly acid properties, is separated first, and the other substances 
follow in succession. The fresher the madder is, the purer will be the ali- 
zarin, which separates on exposure to the atmosphere ; in some instances it 
forms on the surface of the fluid a thick light yellow scum ; but in most cases 
it is mixed with brown or red substances, from which it is separated with 
difficulty. It is therefore most advisable to separate the crj^stals which are 
deposited after twelve hours' standing, by filtration. These crystals are then 
washed from the filter and boiled with very dilute nitric acid until they have 
become of a bright yellow colour. They are then dissolved in boiling alcohol, 



26 REPORT — 1846. 

from which they separate on cooling in yellow transparent plates and needles 
having a strong lustre. Alizarin prepared in this way has the following 
properties : — It has a pure yellow colour without any admixture of red. It 
may be volatilized without leaving any residue. The vapour crystallizes on 
cooling in beautiful yellow plates and needles. It suffers hardly any change 
if exposed to the action of the most powerful reagents. It dissolves without 
change in cold concentrated sulphuric acid. Concentrated nitric acid hardly 
affects it even on boiling. It is not changed by chlorine. It is insoluble in 
water, but soluble in alcohol with a yellow colour. It dissolves in alkalies 
with a beautiful purple colour. Its compounds with the alkaline earths are 
red and slightly soluble in water. Its compounds with the earths and metallic 
oxides are insoluble in water and exhibit diff"erent shades of red. It imparts 
no colour to cloth mordanted with acetate of alumina or oxide of iron, on 
account of its insolubility in water. Very little alizarin is obtained in this 
way ; perhaps one 1 gr. from 1 lb. of madder, though there is more of it con- 
tained in the root. 

I shall now shortly describe two other colouring matters which I have 
obtained from madder. If an extract of madder be made with liot or cold 
water, and a strong acid, such as muriatic or sulphuric acid, be added to the 
fluid, a dark reddish-brown flocculent precipitate is produced. This preci- 
pitate was separated by filtration and washed until the acid was removed. 
On being treated with boiling water, apart of it dissolves with a brown colour. 
On adding a few drops of acid to the liltered solution a dark brown pre- 
cipitate is produced, which seems to me to be a peculiar colouring matter 
similar in its properties to orcein, hematin and other soluble colouring 
matters. It dissolves in alkalies with a red colour, and is capable of imparting 
very lively colours to mordanted cloth. As far as I am aware it has not 
been described in the former investigations of tiiis subject, though it seems 
to be the principal substance concerned in the production of the colours for 
which madder is used in the arts. I have however only examined it very 
slightly as yet. The residue left behind by the boiling water was treated 
with dilute boiling nitric acid, by which every trace of the preceding substance 
is destroyed, and the residue itself acquires a bright yellow colour and 
a more powdery consistence. This yellow powder contains alizarin, as is 
shown by its giving crj'stals of that substance on being gently heated ; in 
fact it contains all the alizarin of the root, but mixed with another substance 
of an amorphous nature but very similar properties, from which it is difficult 
to separate it. By crystallising from alcohol no separation can be effected, 
as they are both about equally soluble in that menstruum. They also behave 
in a similar manner towards the alkalies, the earths and most of the metallic 
oxides. I have hitherto only succeeded in discovering one method of se- 
parating them, which is as follows : — The mixture of the two is dissolved in a 
little caustic potash. To the solution is added perchloride of iron, which 
produces a dark reddish-brown precipitate consisting of peroxide of iron in 
combination with the two substances. Now on boiling this precipitate with 
an excess of perchloride of iron, the aljzarate of iron dissolves, forming a dark 
brown solution, wiiile the iron compound of the other substance remains 
behind. On adding muriatic acid to the filtered solution, the alizarin separates 
in yellow flocks and may be purified by crystallization from alcohol. The 
other substance, to which I have not yet given a name, is obtained by de- 
composing its iron compound, which remains behind on treating with per- 
chloride of iron, with muriatic acid, and washing till all the oxide of iron is 
removed. It seems also to be a colouring matter, as it dissolves with a red 
colour in alkalies and gives red compounds with the earths and metallic 



ON THE PHYSIOLOGICAL ACTION OF MEDICINES. 2? 

oxides. It is insoluble in water, but soluble in alcohol with a yellow colour. 
It therefore resembles the resins in its general properties. It cannot be ob- 
tained in a crystallized state. From a hot concentrated solution in alcohol 
it separates on cooling as a yellow powder. It imparts no colour to mor- 
danted cloth. 



On t/ie Physiological Action of Medicines. By J. Blake, M.B.., F.R.C.S. 

The present report is a continuation of those which have already been read 
before this section, and which have been published in the reports of the 
Association. The only additional experiments I now ha<'e to bring forward 
have been instituted to investigate the action of the salts of iridium and 
osmium, and the acids of selenium and sulphur, when introduced directly 
into the blood. 

The experiments that have been made with the salts of iridium and osmium 
prove that these substances closely agree in their physiological action with 
the salts of palladium and platinum. They are, like these salts, very poison- 
ous. A solution containing half a grain of the double chloride of iridium 
and ammonia, was injected into the jugular vein of a dog. Before the injec- 
tion, the action of the heart was regular and strong*; in eight seconds after 
the injection, the action of the heart appeared affected, it being rendered flut- 
tering ; and after a few seconds there was an apparent obstacle to the passage 
of the blood through the systemic capillaries, as the pressure in the arterial 
system became greater. In about a minute the pressure again diminished ; 
the action of the heart was slower, the force it exerted in pi'opelling the blood 
being equal to a column of mercury of but three inches and a half, or little 
more than the half of that under which the circulation is generally carried 
on. The animal appeared to be uncomfortable, owing to the circulation be- 
coming so feeble. On injecting a solution containing a grain of the salt, the 
circulation was arrested in eleven seconds, owing either to the action of the 
heart having ceased, or else that its contractions were so weakened that they 
did not suffice to force the blood through the pulmonary capillaries. The 
pressure exerted by the blood in the arterial system became suddenly dimi- 
nished, so as only to support a column of mercury of an inch and a half, at 
which point the circulation through the capillaries wbuld appear to have 
been suspended, for the pressure remained stationary for more than a minute, 
and then sunk to zero, owing to relaxation of the capillaries taking place. 
Death followed about three minutes and a half after the injection, and the 
eye retained its sensibility to mechanical irritation for three minutes; respi- 
ration and sensibility continuing nearly two minutes longer than would have 
been the case had the injection of the salt totally paralyzed the heart. On 
opening the thorax immediately after death, the heart was found contracting 
rythmically, but very feebly, certainly not with sufficient power to propel its 
contents : both cavities were full of blood ; in the right it was dark, that in 
the left was of a maroon colour, and had evidently been oxygenized, proving 
that the circulation had ceased before respiration was suspended. The blood 
coagulated imperfectly, and this has been noticed after the introduction of 
the salts of palladium and platinum, which are isomorphous with those of 

* The state of the circulation is ascertained by the haemadynamometer, an instrument which 
enables us readily to detect any change in the action of the heart or in the passage of the 
blood through the systemic or pulmonary capillaries. 



28 REPORT — 1846. 

iridium, and which exert, although in such small quantities, a marked effect 
in preventing the perfect coagulation of the blood. Another experiment 
was performed to observe more particularly the general effects following the 
introduction of the salt into the veins ; the liaemadynamometer was not used, 
and the animal was allowed to run about. On injecting a solution containing 
half a grain of the salt, no immediate effects followed, but in about forty 
seconds the animal became unsteady, and there was a tendency to fall back- 
wards : in a minute and a half respiration was longer and deeper ; sensibility 
remained unimpaired ; after a few minutes the animal laid down, and the 
dyspnoea increased, coming on in paroxysms; in six minutes after the injec- 
tion, respiration was suspended for forty seconds, but this was not accompa- 
nied by convulsions, or even by loss of sensibility : this occurred four or five 
times in the coursa of ten minutes ; the animal laid perfectly still, and did not 
appear to be suffering, although sensibility was unimpaired. A grain of the 
salt, on being introduced into the vein, served to increase these symptoms, 
although the animal did not die until some minutes after it had been in- 
jected. These symptoms are such as would result from the gradual weaken- 
ing of the action of the heart, and the consequent diminution of the supply 
of blood to the brain ; they lead to the conclusion that, when injected into the 
veins, this salt does not exert any marked action on the nervous system. 
When introduced directly into the arteries, by being injected through the 
axillary artery so as to mix with the blood as it passes through the aorta, the 
salts of iridium, as those of platinum and palladium, impede the passage of 
the blood through the capillaries, to such an extent as to require the heart to 
exert more than twice the power that is required in the natural state of the 
circulation, to force the blood through them. This sudden increase of the 
pressure in the arterial system is attended by general spasm. When a grain 
of the salt was injected into the artery, in a few seconds the pressure was equal 
to a column of mercury of twelve inches ; violent spasm immediately came on, 
during which respiration was suspended, nor did it again take place regularly. 
Six respiratory movements Mere observed during the next four minutes, after 
which there was no further movement. The action of the heart appeared to be 
arrested by asphyxia ; but even after it had ceased, the pressure in the arteries 
was equal to three inches of mercury, showing that the passage of the blood 
through the systemic capillaries was still impeded, although the animal had 
been dead two or three minutes. When thus brought into direct contact 
with the nervous centres, there can be no doubt but that these substances 
exert a marked action on them ; it is possible that the violent spasm that im- 
mediately followed their injection might be owing to the great pressure the 
brain is subject to from the over- distension of the arterial system, but this will 
not explain the permanent cessation of its functions. 

The salts of osmium are perfectly analogous in their action to those of 
iridium, and the other members of this isomorphous group. The salt used 
was the double chloride of osmium and potassium, for owing to the chlorides 
of both iridium and osmium being decomposed by water, I was forced to use 
them conlbined with another base, although I should have wished to have 
avoided this if possible. 

The effects produced by selenic and sulphuric acids, when introduced into 
the blood, are not very striking, that is, they do not appear to act in a marked 
manner on any one organ. They agree in this respect with other bodies, 
which either are found entering into the composition of the blood, or have 
isomorphous relations with these constituents. The important part which 
sulphur takes in the proteine compounds, might lead us to expect that its in- 
troduction into the blood, as well as selenium, which is so closely isomorphous 



ON THE PHYSIOLOGICAL ACTION OF MEDICINES. 29 

with it, might not produce any very marked effect. In an experiment per- 
formed with selenic acid, the following are the symptoms that presented 
themselves (the acid used was of specific gravity 1'046, containing about five 
and a half per cent, of real acid). On injecting three drachms of the acid, 
mixed with an equal quantity of water, into the jugular vein, no appreciable 
effect was produced ; neither the passage of the blood through the lungs, nor 
through the systemic capillaries, appeared at all impeded ; nor was the action 
of the heart affected ; in about forty-five seconds after the injection its move- 
ments appeared slightly fluttering ; after two minutes the respiration was 
observed to be rather deeper. Immediately after the introduction of half an 
ounce of the acid into the vein, there was a falling off of the quantity of 
blood sent into the arterial system ; and as this took place five seconds after 
the injection, it must have been owing to the passage of the blood through 
the pulmonary capillaries having been impeded, for there had not been time 
for the substance to reach the coronary arteries and act on the heart. After 
thirty seconds the supply of blood to the arteries was restored, and the action 
of the heart was as strong as before. There appeared to be no marked effect 
produced on any organ, although the quantity of acid introduced was very 
considerable (seven drachms) ; after a few minutes the respiratory movements 
became longer and deeper, and the action of the heart decidedly weaker, the 
force with which the blood was propelled into the arteries being only half 
what it is in the natural state of the circulation ; there was no expression of 
pain. Half an ounce of the acid was again injected into the veins. The 
immediate effect was to arrest the passage of the blood through the lungs, no 
blood being sent into the arteries, although the heart could be felt beating 
through the parietes of the thorax. Respiration was stopped at a minute 
and twenty seconds after the arrest of the circulation, sensibility having dis- 
appeared a few seconds earlier. On opening the thorax immediately after the 
cessation of respiratory movements, the heart was found beating rytlimically: 
the right cavities were very much distended with blood, which was dark and 
grumous, and apparently physically incapable of passing through the lungs. 
The left cavities contained a small quantity of scarlet blood, which was co- 
agulated ; the lungs were redder than natural ; the heart retained its irrita- 
bility some time after death. The above symptoms do not suffice to indicate 
any particular organ on which the acid exerts a marked influence; for although 
death was produced by the passage of blood through the pulmonary ca- 
pillaries being arrested, yet this was probably owing to the mechanical 
impediment which the coagulated state of the blood must have opposed to 
its passage through the pulmonary vessels. In other experiments that I have 
made with this substance, I have sometimes seen a serous secretion take place 
in the air passages. There is also some action on the nervous system, as the 
following experiment will show. The animal was small ; it was not confined, 
in order that the effects on the functions of voluntary motion and sensation 
might be more accurately observed. A drachm of the acid was introduced 
into the veins, without giving rise to any marked symptoms. A second in- 
jection, containing a drachm and a half of the acid, did not affect the animal 
in any marked degree : after a few minutes it appeared rather dull, but there 
was no expression of pain, nor was sensibility impaired. On introducing 
two drachms into the veins, the animal fell down in about thirty seconds, and 
respiration was much affected ; it got up again in about a minute, and jumped 
about in a very curious manner, the movements being evidently involuntary, 
as if the animal had chorea : it remained standing quite motionless for a few 
minutes, but gradually became weaker, and death took place ten minutes 
after the last injection ; there were no convulsions, nor was the sensibility 



30 REPORT — 1846. 

destroyed until immediately before death. On opening the thorax, the heart 
was found contracting rythmically. There was a considerable quantity of 
frothy secretion in the bronchial tubes, and this renders it difficult to deter- 
mine if the asphyxia, by which the action of the heart was finally arrested, 
was nervous or pulmonary ; that is, whether the nervous system was aflfected 
by the want of aeration of the blood, or whether the respiratory movements 
ceased in consequence of the action of the acid directly on the nervous 
system. The latter opinion I think the more probable. 

Sulphuric acid, when introduced into the veins, gives rise to exactly the 
same phsenomena, the only organ on which it appears to exert any marked 
effect being the lungs, although slight nervous symptoms are produced when 
a considerable quantity has been introduced into the blood. The action of 
these substances when injected into the arteries, and thus applied directly to 
the brain and over the system, without previously passing through the lungs, 
is evidently on the nervous sj'stem. Two drachms of the diluted acid, mixed 
with four drachms of water, were injected into the left axillary artery, so as 
to pass into the aorta ; in ten seconds all movements ceased ; there was a slight 
spasm, which relaxed in a few seconds. During the continuance of the spasm, 
the pressure in the arterial system was slightly increased, but it rapidly de- 
clined, so that I think the passage of the blood through the systemic capil- 
laries is facilitated, rather than impeded by these substances. All effective 
contractions of the heart ceased a minute after death, probably ouang to the 
shock produced by the sudden annihilation of the functions of the nervous 
system, for it retained its irritability some minutes after death. 

With these experiments I conclude the first part of the series of researches 
which I propose to undertake for the elucidation of this branch of physiology. 
I have been engaged on it for the last six years, but I trust the results ob- 
tained fully repay the labour that has been bestowed on it. The action on 
the animal ceconomy of the compounds of twenty-nine of the most important 
elements lias been experimentally investigated, and the facts which have been 
observed have led to the discovery of a new laiW in vital chemistry which 
had escaped the attention of former observers, viz. tJial the reactions which 
take place between the elements of the livinx/ body a/ul inorganic compounds 
are not governed by tJie ordinary chemical properties of these substances, but 
depend on certain properties they possess connected with. their isomorphous 
relations. The verification of this law enables us to undertake the investi- 
gation of the higher chemical phaenomena of living bodies from an entirely 
new point of view, whilst its existence accounts for the failure that has con- 
stantly attended attempts to explain the chemistry of animal life by analogy 
from ordinary chemical phaenomena. The fact that we now possess the 
means of producing well-marked and definite modifications of some of the 
most important physiological properties of various organs, and this too by 
means of reagents, the laws governing whose action we are acquainted with, 
places in our hands an instrument for discovery which has hitherto been 
wanting in physiological investigations. The enumeration of some of the 
effects that can be produced at pleasure on the more important functions, 
will, I trust, suftice to lead others into this rich field of inquiry. As regards 
the functions of the heart, we can annihilate or increase its irritability, 
quicken or diminish its pulsations, render them regular or irregular, augment 
their force or render them weaker, destroy the contractility of the auricles, 
whilst that of the ventricles remains ; keep up the circulation of the blood 
many minutes after every sign of life has disappeared, and this too more 
actively than when respiration was being carried on ; we can facilitate or 
arrest the passage of the blood through the pulmonary and systemic capil- 



ON THE ACTINOGRAPH. 31 

laries; produce important modifications in the functions of the brain: — in 
short, the injection of various substances into the arteries and veins enables 
us to modify all the most important functions of the body ; and this, as before 
stated, by reagents, the laws of whose action we can fairly hope to discover. 
My reason for having neglected the closer investigation of these interesting 
phaenomena, was a determination fully to establish the law of the analogous 
action of isomorphous substances. This having been accomplished, I shall 
now direct my researches to the elucidation of these secondary questions. 



Report on the Actinograph. By Mr. Robert Hunt. 
It will be remembered that in 1838 Sir John Herschel proposed an instru- 
ment for the purpose of registering the variations of the actinic or chemical 
rays, and published in the Philosophical Transactions a design for what 
he termed an Actinograph, by which it was thought both the chemical action 
of the direct solar rays and of the diffused daylight would be registered. 
Dr. Daubeny, Prof Nichols and Mr. Thomas Jordan have severally designed, 
and I believe used, instruments somewhat similar, but it does not appear that 
any very satisfactory results have been obtained by either of these inquirers. 
At the York meeting I pointed out the importance of some such registration, 
and at the request of the committee I had an instrument constructed, which 
I exhibited to the Association at the Cambridge meeting. The actinograph 
I have been using differs but little from that proposed by Sir John Herschel, 
a modification of Mr. Jordan's being introduced, by which it was thought 
the results could be tabulated for every hour of the day. As the form of 
this instrument is published in the Report for 1845, it will be unnecessary to 
do more than describe such alterations as have suggested themselves during 
the past year. The triangular slit, divided into one hundred parts, has been 
abandoned, it being found in practice almost impossible to discriminate be- 
tween the amount of coloration produced on the paper during an exposure of 
three minutes or six ; consequently it became quite idle to attempt to register 
by this plan to the degree of nicety which it was hoped might be attained. 
A new external cylinder has therefore been constructed, in which are thirteen 
holes, commencing with a mere pin-hole and gradually increasing to a :| 
inch diameter. By this means thirteen bands are marked upon the sensi- 
tive paper, each one separated from the other by an unaltered line, and it 
becomes easy to distinguish with considerable accuracy between the tints 
thus produced. 

Bromide of silver was the material which, from the circumstance that all 
the rays of the prismatic spectrum exert some influence upon it, was em- 
ployed in procuring most of my registrations. It has however been found 
that under all circumstances this preparation is too sensitive, and that although 
in the winter, when the solar radiations are weak, it answers admirably, yet 
in the bright sunshine of summer it assumes too great a degree of darkness, 
in even diffused daylight, and during the shortest exposure to which it is ex- 
posed during the revolution of the cylinder. Many experiments have been 
made with other photographic preparations, and the result has been in favour 
of the general use of the ammonio-nitrate of silver. It is true that this paper 
is not impressed by all the rays of the spectrum, but, as it is acted upon by all 
the rays beyond the yellow ray, and as the influence of the actinic principle 
throughout the entire range of the spectrum is, as it appears, always equally 



32 REPORT — 1846. 

effected by the increased or diminished intensity of the luminous and calo- 
rific rays, and consequently that even the actinism residing in the extreme 
violet ray is relatively as much influenced by an increase of luminous power 
as that which is detected in the yellow ray itself, we may by the use of the 
paper prepared with the ammonio-nitrate of silver arrive at a very close ap- 
proximation to the true result. 

Although I have not been enabled to realize the hope I held forth last year 
of presenting at this meeting a register of actinic influences for the year, (vyhich 
I have been prevented from doing by circumstances which I shall presently 
explain,) yet I have determined, most satisfactorily to my own mind, the prac- 
ticability of procuring, in favourable positions, such a registration as shall 
aflPord much valuable information. 

The circumstances to which I allude as those which have prevented my 
procuring any series of registers, are the impossibility of securing in London 
any position free from the constant interferences of smoke and fog, and the 
difficulty of placing the instrument so as to be free from the reflected radia- 
tions of adjoining buildings. The first alone is a fatal objection ; for instead 
of securing, what is desired, a registration of the relative amount of chemical 
influence as compared with the quantity of light, heat, and the natural at- 
mospheric conditions, we only get a register of the influence of smoke in 
absorbing the actinic rays. 1 therefore propose to hand over the instrument 
to the Association, requesting that it may be placed in a favourable position 
at Kew, under the attention of the excellent observer there, when I do not 
doubt some curious and instructive results may be obtained. 

It is necessary however to state that my experience has pointed out some 
objections to this mode of registration, which indeed militates against the 
use of the actinograph as a philosophical instrument. 

It is a curious fact that upon almost all kinds of photographic paper the 
colour produced by the solar rays at diffierent periods of the day varies 
considerably. It is not a mere difference of tint, but an actual change in the 
colour; thus frequently the light of both morning and evening will give to 
chloride of silver a rose hue, whilst that of noon will change it to a bluish 
variety of brown. There is consequently much difficulty in deciding which 
is the strongest impression. Thus also the rays upon two days, when the sun 
appears equally bright, will in one case produce a red brown, and in the other 
a blue brown. It is left to the eye to decide upon the intensity of the effect 
produced, and with the utmost care it is frequently impossible to say whether 
the actinic influence is greatest on the red brown lines, or those which are 
blue brown. 

The importance of the inquiry has been peculiarly evident during this 
summer ; many peculiarities have been observed in the growth of plants, in- 
fluenced no doubt by the solar radiations. Many of our garden flowers, 
particularly roses, have exhibited an abnormal condition, leaf-buds being 
developed in the centre of the flower, arising from the vegetative functions 
of the plant overpowering its reproductive functions. Again, during the 
intense sunshine and the prevalence of the unclouded skies of the hot wea- 
ther of June and July, practical photographists found the greatest difficulty 
in obtaining portraits by the Daguerreotype process. At this time, although 
the intensity of effect produced on paper in the actinograph under the usual 
circumstances of summer sunlight should have been at a maximum, it was 
found that it was far below this point, the maximum point being repre- 
sented by 120. During several experiments made at the time mentioned, the 
greatest effect indicated was 100 ; whilst the sky still being unclouded and 
the sun shining brightly, it often fell to 90, and sometimes indeed to 80. 



ON THE INFLUENCE OF LIGHT ON THE GROWTH OP PLANTS. 33 

These facts show the importance, amongst many others, of some mode of 
registration by which these ever-varying solar influences may be carefully 
observed. There can be little doubt that they exert an influence, sometimes 
baleful, sometimes beneficial, upon the organized creation, and that we have 
yet to discover, in these emanations or influences, the secrets of many of the 
grand phaenomena of the universe. 



Notices on the Influence of Light on the Gi'owth of Plants. 
By Mr. Robert Hunt. 

The experiments connected with this very interesting inquiry have been 
steadily pursued, and a concluding report would have been made at this 
meeting, but that some experiments, which had been conducted with much 
care, with a view of determining the quantity of solid matter in plants grown 
under different circumstances, were destroyed by the hail-storm which lately 
prevailed over an extensive district of the metropolis, the glasses and troughs 
of coloured fluids being broken, and the plants themselves washed into the 
soil. As it was impossible to repeat this year these experiments, there was no 
alternative but either to present an imperfect report, or to defer the report 
for another year. The latter course has been chosen, and the detail of 
these experiments will be reserved for a future communication ; I have how- 
ever thought it might be attended with some advantages to state a few of the 
leading facts which have been determined. The order of the arrangements 
have been the same as those observed in the former experiments, and nearly 
all the results have been confirmatory of those published six years since. 

The germination of seeds is peculiarly due to the influence of the actinic 
or chemical rays ; and if these are completely isolated whilst the luminous 
rays are permitted to act upon the soil in which the seeds are planted, no 
germination will take place. This influence is exerted and is most necessary 
up to the point at which the first leaves begin to form, when the luminous 
rays are rendered necessary to effect the formation of woody fibre. It must 
be remembered that this was a point upon which I was at issue with some 
other investigators ; and it is due to them that I should state, that the dis- 
crepancies between us appear to have arisen from our not observing with a 
sufficient degree of accuracy the point at which the two influences balance 
each other, previously to the more complete exercise of the exciting force 
of light, as distinguished from actinism. The vegetative process having been 
carried on until the plant arrives at its maturity, a new agency, the calorific, 
is more decidedly necessary to develope the reproductive functions of the 
plant ; and then, again, the chemical rays combined with the calorific be- 
come more active than the luminous rays. In spring we find the chemical 
influences exerting without interference their most decided force : seeds 
then germinate, and young buds and shoots are developed. As soon as this 
is effected, the luminous rays, with the advance of the sun, become more 
active, and the formation of woody fibre proceeds under their particular 
agency ; not that the chemical power becomes dormant, but it is rendered 
proportionally less active by the agency of light. In the late summer and 
the autumn the peculiar properties of the calorific rays are required, and 
under their agency, with diminished powers of light, the ripening of fruits 
and the production of seed are accomplished. 

My experiments have also led me to detect some curious influences which 
appear to be due to dissimilar rays, and which in their action exhibit great 

1846. D 



34 REPORT — 184G. 

inconstancy of effect. One class of rays, the same to which Sir J. Herschel 
has given the name of Parathermic rays, are so subdued by the influences of 
the more refrangible rays, as to be nearly inactive during the spring and early 
summer months; and indeed in the spring they scarcely produce any effect 
upon dead vegetable colouring matter, unless their action is assisted by the 
use of some decomposing agent, such as sulphuric acid. These rays increase 
in power towards the autumn, and to them appears to be due the browning 
of the leaf. 

It is well known that plants will grow in the dark, but that they do not 
then form chlorophylle ; the formation of this colouring-matter has been an 
object of some attention, and I believe I have determined it to result from 
the joint influence of the luminous and actinic rays. Boxes of cress have been 
grown in the dark, and they have then been brought under the influence of 
a large spectrum formed by a water-prism. It has been stated by Dr.Gardner, 
that the plants under those circumstances exhibit a lateral movement, bend- 
ing towards the yellow ray. This appears to be a mistake ; the plants under 
the influence of the red rays bend from the light but along the line of the ray ; 
and those exposed to the most refrangible rays turn towards it, but still in the 
line of the ray. Now the plants which first become green, by careful treatment 
in this way, are those which are exposed to the rays situated between'the mean 
green ray and the extreme blue. The action is continued eventually to the edge 
of the most refrangible violet below the yellow ray. There is not any change 
effected beyond the visible spectrum, notwithstanding the abundance of dark 
chemical rays ; and the change is slow where there is really the largest amount 
of light. I therefore conclude that the luminous rays are fessential in the pro- 
cess, producing the decomposition of the carbonic acid and the deposition of 
the required carbon, which is afterwards, in all probability, combined with 
hydrogen under the influence of purely chemical force as exerted by the ac- 
tinic principle. 

Such are the main results I have obtained. I have several experiments 
now in progress, and I hope to be enable ^ in another year to complete this par* 
ticular branch of investigation so far as to present to the British Association a 
complete report. 



Report on the Recent Progress qf Analysis [Theory of the Comparison of 
Transcendentals). By R. L. Ellis, M.A. 

1. The province of analj'sis, to which the theory of elliptic functions belongs, 
has within the last twenty years assumed a new aspect. A great deal has 
doubtless been effected in other subjects, but in no other I think has our 
knowledge advanced so far beyond the limits to which it was not long since 
confined. 

This circumstance would give a particular interest to a history of the re* 
cent progress of the subject, even did it now appear to have reached its full 
development. But on the contrary, there is now more hope of further pro- 
gress than at the conmiencement of the period of which I have been speaking. 
When, in 1^27, Legendre produced the first two volumes of his ' Theorie 
des Fonctions P^lllptiques,' he had been engaged on the subject for about 
forty years ; he had reduced it to a systematic form ; and had with great 
labour constructed tables to facilitate numerical applications of his results.,; 
But little more, as it seemed, was yet to be done ; nor does the remark of J 



ON THE RECENT PROGRESS OP ANALYSIS. 35 

Bacon, that knowledge, after it has been systematized, is less likely to increase 
than before, seem less applicable to mathematical than to natural science. 
Nevertheless, almost immediately after the publication of Legendre's work, 
the earlier researches of Abel and Jacobi became known, and'it was at once 
seen that what had been already accomplished formed but a part, and not a 
large one, of the whole subject. 

To say this is not to derogate from the merit of Legendre. He created 
the theory of elliptic functions ; and it is impossible not to admire the per- 
severance with which he devoted himself to it. The attention of mathema- 
ticians was given to other things, and though the practical importance of his 
labours was probably acknowledged, yet scarcely any one seems to have 
entered on similar researches*. This kind of indifference was doubtless dis- 
couraging, but not long before his death he had the satisfaction of knowing 
that there were some by whom that which he had done would not willinelv 
be let die. ^ ^ 

The considerations here suggested have led me to select the theory of the 
integrals of algebraical functions as the subject of the report which I have the 
honour to lay before the Association. 

2. The theory of the comparison of transcendental functions appears to 
have originated with Fagnani. In 1714, he proposed, in the ' Giornale de 
Litterati d' Italia,' the following problem : To assign an arc of the parabola 
whose equation is 

such tha.t its difference from a given arc shall be rectifiable. 

Of this problem he gave a solution in the twentieth volume of the same 
j journal. 

, The principle of the solution consists in the transformation of a certain 
, differential expression by means of an algebraical and rational assumption 
I which introduces a new variable. The transformed expression is of the same 
j form as the original one, but is affected with a negative sign. By integrating 
I both we are enabled to compare two integrals, neither of which can be as- 
I signed in a finite form. It is difficult, however, to perceive how Fagnani was 

led to make the assumption in question : a remark which applies more or 

less to his subsequent researches on -similar subjects, 
i The theorem which has made his name familiar to all mathematicians, ap- 
: peared in the twenty-sixth volume of the ' Giornale.' In its application to 

the comparison of hyperbolic arcs we find some indications of a more general 
I method. We have here a symmetrical relation between two variables, x and 

«, such that the differential expression /(a;) rf a; may be written in the form 

z dx. It follows at once that/(2;) dz = x dz, and consequently that 

/f(x) dx +/f(z) dz =/{x dz + zdx}=xz + C. 

The remarkable manner in which the idea of symmetry here presents itself 

suggested to Mr. Fox Talbot his ' Researches in the Integral Calculus.' 

: In applymg bis methods to the division of the arc of the lemniscate, Fao-- 

, iiani obtained some very curious results, and has accordingly taken for the 

' vignette of his collected works a figure of this curve with the singular motto 

Deo ventatis gloria." 

3. In MacLaurin's Fluxions, and in the writings of D'Alembert, instances 
are to be found where the solution of a problem is made to depend on the 

I * Those of M Gauss, which would doubtless have been exceedingly valuable, have not I 
, behave, been pubhshed. They are mentioned in a letter from M. Crelle to Abel Vide the 
I introduction to the coUected works of the latter, p. Tii. ^ 

! ■ D 2 



3G REPORT— 1846. 

rectification of elliptic arcs, or, as we should now express it, is reduced to 
elliptic integrals. But of these instances Legendre has remarked that they 
are isolated results, and form no connected theory. MacLaurin is charged, 
in a letter appended to the works of Fagnani, with taking from the latter, 
without acknowledgement, a portion of his discoveries with respect to the 
lemniscate and the elastic curve. 

4. In 1761, Euler, in the ' Novi Commentarii Petropolitani ' for 1758 and 
1759, published his memorable discovery of the algebraical integral of the 
equation 

m dx n dy 

(A + Ba; + Ca;* + Tioc> + Ea;^)i ~ (A + B?/ + C?/"* + Dz/3 + E?/<)4' 

m and n being any rational numbers. 

He says he had been led to this result by no regular method, " sed id 
potius tentando, vel divinando elicui," and recommends the discovery of a 
direct method to the attention of analysts. In effect his investigations re- 
semble those of Fagnani : he begins by assuming a symmetrical algebraical 
relation between the variables, and hence finds a differential equation which 
it satisfies. In this differential equation the variables are separated, so that 
each term may be considered as the differential of some function. Vv'ith one 
form of assumed relation we are led to the differentials of circular, and with 
another to those of elliptic integrals, and so on. It is in this manner that 
Dr. Gudermann, in the elaborate researches which he has published in Crelle's 
Journal, has commenced the discussion of the theory of elliptic functions. 

5. In the fourth volume of the Turin Memoirs, Lagrange accomplished 
the solution of the problem suggested by Euler. He integrated the general 
differential equation already mentioned by a most ingenious method, which, 
with certain modifications, has remained ever since an essential element of 
the theory of elliptic functions. He proceeded to consider the more general 
equation dx _ dy_ 

where X and Y are any similar functions of a; and rj respectively, and came 
to the conclusion, that if they are rational and integral functions, the equa- 
tion cannot, except in particular cases, be integrated, if they contain higher 
powers than the fourth. He also integrated this equation in a case in which 
X and Y involve circular functions of the variables. It had been already 
pointed out in the summary of Euler's researches, given in the ' Nov. Com. 
Pet.' t. vi., that if X and Y are polynomials of the sixth degree, the last- 
written equation does not in general admit of an algebraical integral, since, 

if so, it would follow that the solution of the equation — = — ^, which 

1 -f- «' 1 + y 
(as the square of 1 -|- j?' is a polynomial of the sixth degree) is a particular 
case of that which we are considering, could be reduced to an algebraical 
form. Now this solution involves both circular functions and logarithms, and 
therefore the required reduction is impossible. This acute remark* showed 
that Euler's result did not admit of generalisation in the manner in which it 
was natural to attempt to generalise it. It was rese^ed for Abel to discover 
the direction in which generalisation is possible. 

6. The discovery of Euler, of which we have been speaking, is in effect 
the foundation of the theory of elliptic functions, as the generalisation of it 
by Abel, or more properly speaking, the theory of which Euler's result is an 

* M. Richelot, in one of his memoirs on Abelian or hyper-elliptic integrals, quotes it, in 
a slightly modified form, from Euler's ' Opuscula.' ' 






ON THE RECENT PROGRESS OF ANALYSIS. 3/ 

isolated fragment, is the foundation of our knowledge of the higher trans- 
cendents. We may therefore conveniently divide the subject of this report 
into two portions, viz. the general theory of the comparison of algebraical 
integrals, and the investigations which are founded on it. Mathematicians 
have been led, by comparing different transcendents, to introduce new func- 
tions into analysis, and the theory of these functions has become an important 
subject of research. 

The second portion may again be divided into two, viz. the theory of 
elliptic functions, and that of the higher transcendents. 

This classification, though not perhaps unexceptionable, will, I think, be 
found convenient. 

7. About sixteen years after the publication of Lagrange's earlier researches 
on the comparison of algebraical integrals, he gave, in the New Turin Me- 
moirs for 1784 and 1785, a method of approximating to the value of any 

integral of the form / tAf, where P is a rational function of x and R the 

*y JR. 
square root of a polynomial of the fourth degree. I shall consider this im- 
portant contribution to the theory of elliptic functions in connexion with the 
writings of Legendre. At present, in order to give a connected view of the 
first division of my subject, it will be necessary to go on at once to the works 
of Abel, and to those of subsequent writers. In the history of any branch 
of science the chronological order must be subordinate to that which is 
founded on the natural connexion of different parts of the subject. 

I shall merely mention in passing, that in 1775, Landen published in the 
Philosophical Transactions a very remarkable theorem with respect to the 
arcs of a hyperbola. He showed that any arc of a hyperbola is equal to the 
difference of two elliptic arcs together with an algebraical quantity. In 1780 
he published his researches on this subject in the first volume of his ' Mathe- 
matical Memoirs,' p. 23. This theorem, as Legendre has remarked, might 
have led him to more important results. It contains the germ of the general 
theory of transformation, the eccentricities of the two ellipses being con- 
nected by the modular equation of transformations of the second order*. It 
is on this account that in a report on M. Jacobi's ' Fundamenta Nova,' con- 
tained in the tenth volume of the Memoirs of the Institute, Poisson speaks 
of Landen's theorem as the first step made in the comparison of dissimilar 
elliptic integrals. Several writers have accordingly given Landen's name to 
the transformation commonly known as Lagrange's. 

8. We have seen that even Lagrange failed in obtaining a result more 
general than that which had been made known by Euler, and yet, as we now 
know, Euler's theorem is but a particular case of a far more general proposi- 
tion. But in oi'der to further progress, it was necessary to introduce a wholly 
new idea. The resources of the integral calculus were apparently exhausted ; 
Abel, however, was enabled to pass on into new fields of research, by bring- 
ing it into intimate connexion with another branch of analysis, namely, the 
theory of equations. The manner in which this was done shows that he w£is 
not unworthy to follow in the path of Euler and of Lagrange. 

I shall attempt to state in a few words the fundamental idea of Abel's 
method. 

Let us suppose that the variable a^ is a root of the algebraical equation 
fx = 0, and that the coefficients of this equation are rational functions of 
certain quantities a, b, . . . c, which we shall henceforth consider independent 
variables. Let us suppose also that in virtue of this equation we can express 

* Vide infra, pp. 50 and 67. 



38 REPORT — 1846. 

certain irrational functions* of x as rational functions of a;, a, b, . . . c. For 
instance, if the equation were x- + ax + —(a- — I) = 0, it follows that 

a/i— a;* = a + x. So that any irrational function of the form F (x Vl —x-) 
can be expressed rationally (F being rational) in x and a. 

From the given equation we deduce by differentiation the following, 
dx = ada + /3«?6 + . . . + ydc, 
where a, jS, . . . y are rational m x, a, b, . . . , c. 

Lety be one of the functions which can be expressed rationally in x, &c., 
it follows that ydx= Ada + Bdb + ... +Cdc, 

where A, B, . . • C are also rational in x, &c. 

The equation fx = will have a number of roots, which we shall call 
«„ x^ .. .x^. It follows that 

y,dx^ + -{-y^dx^zs. 

{A. + .. + A^}rfa + {B, + .. + B^}(Z6 +... + {€,+ .. + C^}(fc, 

where the indices affixed to y, A, &c. correspond to those affixed to x, so 
that yp for instance, is the same function of x^ that y,, is of x„. 

Now A, + . . . + A„ is rational and symmetrical with respect to a?, . . . x^, 
therefore it can be expressed rationally in the coefficients of /(a;) = 0, and 
therefore in a,b..c. We will call this sura Ra, and thus with a similar 
notation for b, &c. we get 

y, dxy + ... -^ yfidXi^= Rarfa + R4 c?Z» + ... + Re rfc. 

The second side of this equation is from the nature of the case a complete 
differential, and it is rational in a, b, c, &c. ; it can therefore be integrated 

ydxhy"^ (x,), we get 
^(x,) + ... + ^(x^) = M, 

M being a logarithmic and algebraic function of a, b, &c., which we may 
suppose to include the constant of integration. 

\j/ (x) is in general a transcendental function, while a, b, &c. are necessarily 
algebraical functions of :r,, . . . , x^, and the result at which we have arrived 
is therefore an exceedingly general formula for the comparison of transcen- 
dental functions. 

The simplicity and generality of these considerations entitle them to espe- 
cial attention : it cannot be doubted that the application thus made of the 
properties of algebraical equations to the comparison of transcendents will 
always be a remarkable point in the history of pure analysis. 

A very simple example may perhaps illustrate what has been said. Let us 
recur to the equation i 

x"-+ax + ^(a"--l) = 0, (1.) 

and suppose that 1 

y = 



Differentiating the first of these equations, we find that 

(^x + a)dx -r (x + a)da = 0. 

* It must be remembered tliat an algebraical function is either explicit or implicit : ex- 
plicit, when it can be expressed by a combination of ordinary algebraical symbols ; implicit, 
when we can only define it by saying that it is a root of an algebraical equation whose co- 
efficients are integral functions of x. Thus y is an implicit function of x ii y^-^xt/-\-l = 0. 
The remarks in the text apply to all algebraical functions, explicit or implicit. 



ON THE RECENT PROGRESS OF ANALYSIS. 39 

Comparing this with the general expression of dx, we perceive that 

2 a; + « 



and as 1 _ 1 

■ a 

da 



V — = (vide ante, p. 38.),* 



ydx =■ 



2a; + « 

80 that ^ 1 

2a; + a' 

Let Xi and a?a be the two roots of our equation, we have thus to find the 
value of 

. , . _ r 1 , 1 1 _ _ 2(3;i+a;o+a) _ q 

^' + ^^-~ Ls^vT^ 2^1+^-1 (2a;,+ a)(2a;,+ «) ' 

since a^i + Xg= — a. 

Hence yidxi + y^d Xc,— 0, 

and ^ x,_-\- '^ Xci— c. 

Since a;,+ «2=— a, 

we see that a^i^ + a:/ = 1, or aj^ = a/1 — x^K 

Hence, as \{/ a; = sin-' x, our result is merely this, that the sum of two arcs 
is constant if the sine of one is equal to the cosine of the other. 

An infinity of analogous results may be obtained either by varying the 
form of y (e. g. by making y = \^l—x-), or by changing the equation (1.). 
A formula applicable to all forms of y, and which, for each, includes all the 
results which can be established with respect to it, is, it will readily be ac- 
knowledged, one of the most general in the whole range of analysis. Abel's 
principal result is a formula of this nature ; he developed at considerable 
length the various consequences which may be deduced from it. 

Generally speaking, the number of independent variables a, b, . . . c will 
be less than that of the different roots, a;,, . , . x/^; hence a certain number, 
say m, of the roots may be looked on as independent (viz. as many as there 
are quantities a, b, . . . c), and the rest will be functions of these. It may 
be shown that it will always be possible to make the difference [/, — m con- 
stant, so that the sum of any number of the transcendents yp is expressible by 
a fixed number of them, together with an algebraical and logarithmic func- 
tion of the arguments, i. e. of a?„ . . . Xm- In the case of elliptic integrals, it 
had long been known that the sum of two may be thus expressed by a third ; 
and Legendre pointed out that the sum of any number may similarly be exr 
pressed by means of one. Accordingly it appears from the general theory, 
that in this ease (x, — m may be made equal to unity. 

9. The history of this important theory is curious. It was developed by 
Abel in an essay which he presented to the Institute in the autumn of 1826, 
when he had scarcely completed his twenty-fourth year. 

In a letter to M. Holmboe, appended to the edition of his collected works, 
Abel writes, " Je viens de finir un grand traite sur une certaine classe de 
fonctions transcendantes pour le presenter a I'lnstitut, ce qui aura lieu lundi 

* The ambiguous sign of the radical is to our purpose immaterial. 



40 KEPORT— 1846. 

prochain. J'ose dire sans ostentation que c'est un traite dent on sera satis- 
fait. Je suis curieux d'entendre I'opinion de I'lnstitut la dessus. Je ne 
manquerai pas de t'en faire part." Long before this memoir was published 
Abel had become " chill to praise or blame." He died at Christiania in the 
spring of 1829. 

M. Jacobi mentions in a note in Crelle's Journal, that while at Paris he 
represented, and as he believed not ineffectually, to Fourier, who was then 
one of the secretaries of the Institute, that the publication of this memoir 
would be verj' acceptable to mathematicians. A long period however was 
still to elapse before the publication took place. It was po.-^sibly retarded by 
the death of Fourier. In 184>1 the memoir appeared in the seventh volume 
of the ' Memoires des Savans Etrangers.' It was prepared for publication 
by M. Libri. 

Thus for about fifteen years Abel's general theory remained unpublished ; 
but in the meenwhile Crelle's Journal was established, and to the third vo- 
lume of this he contributed a paper which contains a theorem much less ge- 
neral than the researches he had communicated to the Institute, but far more 
so than anything previously effected in the theory of the comparison of 
transcendents. This is commonly known as Abel's Theorem. Legendre, in 
a letter to Abel, speaks thus of the memoir in which it appeared : — " Mais le 
memoire . . . ayant pour titre ' Remarques sur quelques proprietes gene- 
rales,' &c., me parait surpasser tout ce que vous avez publie jusqu'a-present 
par la profondeur de I'analyse qui y regne ainsi que par la beaute et la gene- 
ralite des resultats." In a previous letter, with reference I believe to the 
same subject, he had remarked, " Quelle tete que celle d'un jeune Norve- 
gien ! " 

Abel's theorem gives a formula for the comparison of all transcendental 
functions whatever whose differentials are irrational from involving the square 
root of a rational function of x. 

In a very short paper in the fourth volume of Crelle's Journal, which 
must have been the last written of Abel's productions, the chief idea of his 
general theory is stated ; and in the second volume of his collected works we 
find a somewhat fuller development of it, in a paper written before his visit 
to Paris, but not published during his lifetime. 

While Abel's great memoir remained unpublished at Paris, several mathe- 
maticians, developing the ideas which he had made known in his contribu- 
tions to Crelle's Journal, succeeded in establishing results of a greater or 
less degree of generality. Researches of this kind ma}' be presented in a 
variety of forms, because the algebraical function to be integrated, which 
we have called y, may be defined or expressed in different ways. For in- 
stance, if M and N are general symbols denoting any integral functions of x, 

the two suppositions v ^ and y = ._ are precisely equivalent, since 

N V N 

by an obvious reduction, and by changing the signification of M and N, the 
one may be transformed into the other ; and so in more general cases. Thus 
the same function may assume a variety of aspects, and there will be a cor- 
responding variety in the form of our final results. 

In Crelle's Journal we find a good many essays on this part of the sub- 
ject : of these I shall now mention several. 

M. Broch is the author of a paper in the twentieth volume of Crelle's 
Journal, p. 178. It relates to the integration of certain functions irrational 
in consequence of involving a polynomial of any degree raised to a fractional 
power. For tiiese functions he establishes formulae of summation, which of 



ON THE RECENT PROGRESS OF ANALYSIS. 41 

course include Abel's theorem, since the latter relates to cases in which the 
fractional power in question is the (^)th. Subsequently to the publication 
of this paper he presented to the Institute a memoir on the same subject, but 
gave to the functions to be integrated a different but not essentially more 
general form. This memoir, which was ordered to be printed among the 
' Savans Etrangers,' but which will be found in Crelle's Journal (xxiii. 145), 
may be divided into two portions : the first contains results analogous to 
Abel's theorem ; the second relates to the discussion and reduction of the 
transcendents which they involve. In this part of his researches M. Broch 
has followed the method, and occasionally almost adopted the phraseology 
of a memoir of Abel, on the reduction and classification of Elliptic Inte- 
grals (Abel's Works, ii. p. 93). MM. Liouville and Cauchy, in reporting on 
the memoir, conclude by remarking that the author " n'a pas trop presume 
de ses forces en se proposant de marcher sur les traces d'Abel." 

M. Jiirgenson has contributed two papers to Crelle's Journal on the sub- 
ject of which Ave are speaking. The first, which is very short, contains a 
general theorem for the summation of algebraical integrals* when the func- 
tion to be integrated is expressed in a particular form. This paper appears 
in the nineteenth volume, p. 113. In the second (vol. xxiii. p. 126) the au- 
thor reproduces the results he had already obtained, pointing out the equi- 
valence of one of them to tlie theorem established in M. Broch's first essay. 
Besides this, he discusses a question connected with the reduction of alge- 
braical integrals. 

M. Ramus, in the twenty-fourth volume of Crelle's Journal, p. 69, has 
established two general formulae of summation ; from the second he deduces 
with great facility Abel's theorem, and also another result, which Abel men- 
tions in a letter to Legendre, published in the sixth volume of Crelle's 
Journal, but which he left undemonstrated. 

M. Rosenhain's researches (Crelle's Journal, xxviii, p. 249, and xxix. 
p. 1) embrace both the summation and reduction of algebraical integrals. 
His analysis depends on giving the function to be integrated a peculiar form, 
which he conceives leads to a simpler mode of investigation than any other. 

A paper by Poisson will be found in the twelfth volume of Crelle's Jour- 
nal, p. 89. It relates to the comparison of algebraical integrals, ^but is not 
I think so valuable as that great mathematician's writings generally are. 

Beside the memoirs thus briefly noticed, I may mention two or three by 
M. Minding : that which appears in the twenty-third volume of Crelle's 
Journal, p. 255, is the one which is most completely developed. 

There is also a very brief note by M. Jacobi in the eighth volume of 
Crelle's Journal. 

10. To the Philosophical Transactions for 1836 and 1837 Mr. Fox Tal- 
bot contributed two essays, entitled ' Researches in the Integral Calculus.' 
These researches may be said to contain a development and generalisation 
of the methods of Fagnani. They are however far more systematic than the 
writings of the Italian mathematician, and if they had appeared in the last 
century would have placed Mr. Talbot among those by whom the boundaries 
of mathematical science have been enlarged. But it cannot be denied that 
they fall far short of what had been effected at the time they were published, 
nor does it appear that they contain anything of importance not known before. 
I have assuredly no wish to speak disparagingly of Mr. Talbot ; his mathe- 
matical writings bear manifest traces of the ability he has shown in so many 

* I have used the expression " algebraical integrals," though perhaps not correctly, to de- 
note the integrals of algebraical functions. 



42 REPORT— 1846. 

branches of science*. But as in tliis country tliey seem to have been thought, 
and by men not apparently unqualified to judge, to contain great additions to 
our knowledge, I cannot avoid inquiring whether this be true. 

Mr. Talbot points out in the early part of his first paper, that if there are 
n — 1 symmetrical relations among the n variables x, t/ . . .z, then the iden- 
tical equation 

{y . . .z}dx + (x . . .z)d7j +. .,+ {xy .. .}dz — d[xy .. .z) 

will assume the form 

(p{x)dx -\-(p{y)dy -\-...'{- (p(z)dz = d{xy . . .2}, 

and thus give us 

J'(p (x) dx ■\-J^<p (y) dy +,. .+y« (z) dz = xy . . . z + C. 

Precisely the same remark, though expressed in a different notation, is the 
foundation of M. Hill's memoir, published in 1834, on what he calls " func- 
tiones iteratse." It will be found in Crelle's Journal, xi. p. 193. A much 
more general theorem might be established by similar considerations : they 
are of course applicable whether the function <p be algebraical or trans- 
cendent. 

In the course of his researches, Mr. Talbot recognised the important prin- 
ciple, that the existence of w — 1 symmetrical algebraical relations among n 
variables may be expressed by treating them as the roots of an equation, one 
of whose coefficients at least is variable, the others being either constant or 
functions of the variable one. Unfortunately he did not pass from hence to 
the more general view, that the existence of « — p symmetrical relations 
may be expressed in a similar manner if we consider j9 of the coefficients of 
the equation as arbitrary quantities. Had he done so, it is possible, though 
not likely, that he would have rediscovered Abel's theorem; but as it is, he 
has never introduced, except once, and then as it were by accident, more 
than one arbitrary quantity. Thus only one of his variables is independent, 
and consequently, in more than one instance, his results are unnecessarily 
restricted cases of more general theorems. 

The character of his analysis will be perceived from what has been said. 

l( fXdx be the transcendent to be considered, X being an algebraical func- 
tion of X, he makes the following assumption — 

X=f(xv), 

V being a new variable, and /a rational function. From this assumption he 
deduces an algebraical equation in x, the coefficients of which are rational 
functions of w. This equation then is one of those of which we have spoken, 
by means of which the function to be integrated can be expressed in a ra- 
tional form. Taking the sum with respect to the roots of this equation, we 
get 

^(Xdx)=z-2(fCxv)dx). 

It must be remarked that many forms might be assigned to the function f, 
which would give rise to a difficulty, of the means of surmounting which 
Mr. Talbot has given no idea. If x and v are mixed up iny(a; v), it is ma- 
nifest that we cannot integrate f(xv)dx, since 2j is a function of x, which 

* It must be remembered also that Mr. Talbot admits himself to have been anticipated 
to a considerable extent by the publication of Abel's theorem. 



ON THE RECENT PROGRESS OP ANALYSIS. 43 

if we eliminate we merely return to our function X. We must therefore 
express ^f(xv)dx in the form V dv,Y being a function and, as Abel has 
shown, an integrable function of v. Abel has given formulse by means of 
which this reduction may be effected in all possible cases. But there is no- 
thing analogous to this in the writings of Mr. Talbot, and consequently he 
could not, setting aside the defect already noticed, obtain results as general 
as many previously known. In Mr. Talbot's investigations, /(a; v)(i a; is such 
that Sy"(x v)dx may be put in the form — 

V, S {^', xdx} + V.^ ^{f„xdx} + &c., 

(PfX, PdX, &c. (of which (p'l a;, ^'o a:, &c. are the derived functions) being 
rational functions of x. Then S ip a; = a rational function of v by a well- 
known theorem. Let the form of this function be ascertained, and let us 
denote it by % v. Then differentiating, 

ll(p'xdx = ^'v dv, 
and hence 

S X Jx = 'S,f(xv)dx = [V, x'l V + Yo.-xjoj' +. .] rfv, 
and the second side of this equation is of course rational and integrable. 
But the form of the function /(a;?;) is unnecessarily restricted in order that 
this kind of reduction may be possible. Nevertheless, Mr. Talbot's papers, 
from their fulness of illustration and the clear manner in which particular 
cases of the general theory are worked out by independent methods, will be 
found very useful in facilitating our conceptions of the branch of analysis 
which forms as it were the link between the theory of equations and the in- 
tegral calculus. 

In Mr. Talbot's second memoir (Phil. Trans. 1837, part 2. p. 1) he has 
applied his method to certain geometrical theorems. Three of them relate 
to the ellipse, and are proved by the three following assumptions i — 

fl— c2^14 \~ex //^ 

iT-^l =1+^^' ''' = —7T'''' = Tzrx 

These assumptions are all cases of the following — 
r X—e'^x'^ 1 *_ c + c^x 
1 1 — a;2 J ~ a + a' x' 

where a, «', c, c' are arbitrary quantities. The results of this assumption 
are completely worked out by Legendre (Theorie des Fonctions EUiptiques, 
iii. p. 192) in showing how the known formulas of elliptic functions may be 
derived from Abel's theorem. Mr. Talbot's first theorem is a case of" the 
fundamental formula for the comparison of elliptic arcs. This remark has 
reference to an inquiry which Mr. Talbot suggests as to the relation in which 
his theorems stand to the results obtained by Legendre and others. 

In conclusion, it may be well to observe that Mr. Talbot has remarked 
that, apparently, a solution discovered by Fagnani of a certain difierential 
equation cannot be deduced from Abel's theorem ; but as this solution may 
be easily derived from the ordinary formula for the addition of elliptic in- 
tegrals of the first kind it is manifestly included in the theorem in question. 

n. 

II. I now come to the history of researches into the properties of par- 
ticular classes of algebraical transcendents. The earliest, and still perhaps 
the most important class of these researches relates to the transcendents 



44 REPORT — 1846. 

which are commonly called elliptic functions or elliptic integrals. For a reason 
which will be mentioned hereafter the latter name seems preferable, and it is 
sanctioned by the authority of M. Jacobi, though the former was used by Le- 
gendre. Elliptic integrals then may be defined as those whose differentials are 
irrational in consequence of involving a radical of the form //{a + (ioc+y x"^ 
+ Sx3 + £X'^}. But it may perhaps be more correct to say that all such in- 
tegrals may be reduced to three standard integrals, to which the name of 
elliptic integrals has been given. 

In the Turin Memoirs for 1784- and 1785, p. 218, Lagrange considered, 
as has been already mentioned, the theory of these transcendents. He 
showed that the integration of every function irrational in consequence of 
containing a square root may be made to depend on that of a function of 

the form —, P being rational, and R the radical in question ; and that if 

R 
under the sign of the square root x does not rise above the fourth degree, 

"N dx 
it may ultimately be made to depend on that of - /. a~2^7l 4. TT)' 

where N is rational in x-. He thus laid the foundation of that part of the 
theory of elliptic transcendents in which a proposed integral is reduced 
to certain canonical or standard forms, or to the simplest combination of 
such forms of which the case admits. In Legendre's earliest^ writings on 
elliptic functions there is nothing relating to this part of the subject. Having 
thus, in the simple manner which distinguishes his analysis, reduced the ge- 
neral case to that which admits of the application of his method, Lagrange 
proceeded to prove that if we introduce a new variable whose ratio to x is 
the subduplicate of the ratio of 1 + p' a:^ to 1 + g- .r'-, the last written inte- 
gral is made to depend on another of similar form, but in which p and q are 
replaced by new quantities p' and q\ Ifp is greater than q, p^ will be greater 
than^, and g' less than q, and thus by successive similar transformations we 
ultimately come to an integral in which q is so small that the factor 1 + q^ x" 
may be replaced by unity, and the elliptic integral is therefore reduced to a 
circular or logarithmic form. Or by successive transformations in the oppo- 
site direction Ave come to an integral in which p^ and q^ are sensibly equal, 
in which case also the elliptic integral is reduced to a lower transcendent. 
This most ingenious method is the foundation of all that has since been 
effected in the transformation of elliptic integrals, or at least whatever has 
been done has been suggested by it. Thus it is to Lagrange that we owe 
the origin of two great divisions of the theory of these functions. 

In the Memoirs of the French Academy for 1786, p. 616, we find Legen- 
dre's first essay on the subject to which he afterwards gave so much attention. 
We recognise in it what may I think be considered the principal aim of his 
researches in elliptic functions, namely to facilitate, by the tabulation of 
these functions, the numerical solution of mathematical and physical pro- 
blems. 

He begins, not with a general form as Lagrange had done, but with the 

integral /Vl—c-sin^^rf^, which as Ave knoAV represents an elliptic arc, 
and shows how other functions, for instance the value of the hyperbolic arc, 
may be expressed by means of it, and of its differential coefficient with re- 
spect to the eccentricity c. The memoir does not contain Ynuch that is now 
of interest. After Avriting it he became aware of the existence of Landen's 
researches ; and in a second memoir appended to the first gave a demonstra- 
tion of Landen's principal theorem. This demonstration is founded on 



ON THE RECENT PROGRESS OP ANALYSIS. 45 

Legendre's own methods, and he deduces from it the remarkable conclusion, 
that if of a series of ellipses, whose eccentricities are connected by a certain 
law, we could rectify any two, we could deduce from hence the rectification 
of all the rest. The law connecting the eccentricities of the ellipses is that 
which would be obtained by making use of Lagrange s method of transtor- 
mation, with which accordingly this result is closely allied. , , ^ ., 

Legendre's next work was an essay on transcendents*, presented to the 
AcadSmy in 1792 and published separately the year after. It contains the 
same general view as that which is developed in the first volume of the 
'Exercices de Calcul Integral,' which appeared in 1811. 

12. The theory of elliptic functions, as it is presented to us by Legendre, 
may conveniently be considered under the following heads :— 

a. The reduction of the general integral, 

y^ F dx 

in which P is rational to three standard forms, since known as elliptic inte- 
grals of the first, second and third kinds t- 

This classification, though the reduction of the general integral had, as we 
have seen, been already considered by Lagrange, is I beheve entirely due to 
Legendre. If we consider how much it has facilitated all subsequent re- 
searches, we can hardly over-rate the importance of the step thus made. It 
may almost be said that Legendre, in thus showing us the primary forms with 
which the theory of elliptic integrals is conversant, created a new province 
of analysis : he certainly gave unity and a definite form to the whole sub- 
ject. 

For the three species of functions thus recognised Legendre suggested the 
names of nome, epinome and paranome, the name of the first being derived 
from the idea that it involves, so to speak, the law on which the comparison 
of elliptic integrals depends. But these names do not seem felicitous, nor 
have they I believe been ad9pted. To this part of the subject an important 
theorem relating to the reduction of elliptic integrals of the third kind, 
whose parameters are imaginary, seems naturally to belong. 

/3. The comparison of elliptic integrals of the same form differing only 
in the value of the variable, or as it is often called, the amplitude of each. 
This part of the subject divides itself into three heads, corresponding to the 
three classes of integrals. The fundamental results are to be found in the 
memoirs of Euler, of which we have already spoken. By Legendre how- 
ever they were more fully developed. 

It is interesting to observe that Legendre suggested that the discovery of 

* A translation of it appeared in Leybourne's Mathematical Repository, vols. ii. and iii. 
The original I have not seen — it has long been scarce, 
t These three forms are 
x»« d£ /»' /i_e2^2 /»* _££___^___ 

Legendre always replaces x by sin <p, so that the integrals become 

y'*tp d(fi p<p ■ /^<p d(f 

„ -/l-c^sin^^' y„ Vl-c^sin^^rf?; J (1+^,;^=^) Vl-c^-sin^<p- 

The radical v'l — c^ sin^ ip is often denoted by A. 

The constant c is called the modulus; the second constant n (in the third kind) is called 
Vat parameter. The modulus may always be supposed less than unity, and if c=sin t,then 
s is the angle of the modulus. 



46 REPORT — 1846. 

dx dy „ J .i r 

Euler (namely that the differential equation - , — - - 4- —77.,-^ = admits ot 

an algebraical integral,/(x) being the polynomial a.-rpx-Yyx''--\-lx^-\-iX ) 

dx 
might be generalised, if we consider the differential equation ■ /ttt-^ 
(I,. dz ^J \^) 

+ .-^1 — + ... + .--- -= 0. He remarks that this is perhaps the only 

^f(y) </(^) 

way in which it can be generalised. 

y. Theorems relating to the comparison of different kinds of elliptic func- 
tions. One of the most remarkable of these is the relation between the 
complete integrals (those, namely, in which the variable x is unity) of the 
first and second kind, the moduli of which are complementary ; that is, the 
sum of the squares of whose moduli is equal to unity, Legendre's demonstra- 
tion of it is rather indirect, but many others have been since given. Another 
theorem may be mentioned, — that the complete integral of the third kind 
can always be expressed by means of the complete integrals of the first and 
second. A third and most important result shows that in elliptic integrals 
of the third kind we may distinguish two separate species, and that to one 
or other of these any such integral may be reduced. A memorable dis- 
covery of M. Jacobi has greatly increased the importance of this subdivision, 
of which we shall hereafter speak more fully. This part of the subject is, 
I imagine, entirely due to Legendre. 

S. The evaluation of elliptic integrals by means of expansions. 

e. The method of successive transformations. The idea of this method 
originated, as we have seen, with Lagrange. It is developed at great length 
by Legendre, with a special reference to the modifications required in apply- 
ing it to the different species of integrals. As Lagrange had shown, the 
series of transformed integrals extending indefinitely both waj's conducts us, 
in whichever direction we follow it, towards a transcendent of a lower kind 
than an elliptic integral, or in other words, towards a logarithmic or circular 
integral. There are thus two modes of approximation, one of which depends 
on a series of integrals with increasing moduli, and the other on a series 
whose moduli decrease. Thus for the three species of integrals there will 
be in all six approximative processes to be considered. In the case of the 
elliptic integral of the third kind, we have to determine the law of formation 
of the successive parameters ??, n', &c. 

5. Reductions of transcendents not contained in the general formula 



/g. g, I - I to elliptic integrals. 



ij. Lastly, applications to various mechanical and geometrical problems. 

This analysis, however slight, will give an idea of the contents of that part 
of the ' Exercices de Calcul Integral ' which relates to elliptic functions. In 
the third volume there are tables for facilitating the calculation of integrals 
of the first and second kind: they are accompanied with an explanation of 
the manner in which they were constructed. The ninth table is one with 
double entry, the two arguments being the angle of the modulus and the 
amplitude. 

13. In 1825 Legendre presented to the Academic des Sciences the first 
volume of his ' Traite des Fonctions Elliptiques.' A great part of this work 
is precisely the same as the ' Exercices de Calcul Integral.' By far the most, 
important addition to the theory of elliptic functions consists in the disco- 
very of a new system of successive transformations quite distinct from that 
of Lagrange. 



ON THE RECENT PROGRESS OF ANALYSIS. 47 

In the earlier work Legendre had shown that a certain transcendent might 
be expressed in two ways by means of elliptic integrals of the first kind. 
Comparing the two results, he obtained a very simple relation between the 
two elliptic integrals. Their moduli are complementary; while the ratio of 
the A's in the two integrals can be expressed rationally in terms of the sine 
of the amplitude of one. This circumstance seems to have suggested to Le- 
gendre the possibility of generalising the result. He accordingly assumed a 
relation between the amplitudes of two integrals, of which the equation sub- 
sisting in the theorem of which we have been speaking is a particular case ; 
and showed from hence that a simple relation perfectly similar to that wiiich 
he had obtained in the particular instance existed between the two integrals, 
viz. that they bore to each other a ratio independent of their amplitudes. 
Their moduli are connected by an algebraical equation, but are not comple- 
mentary. This circumstance therefore now appeared to be unessential, 
though in the ' Exercices' the investigation is introduced for the sake of ex- 
hibiting a case in which an integral may be transformed into another with a 
complementary modulus. 

Legendre thus obtained a new kind of transformation, which might be re- 
peated any number of times or combined in an infinite variety of ways with 
that of Lagrange. To illustrate this he constructed a kind of table — a " da- 
mier analytique." In the central cell is placed the original modulus c. All the 
moduli contained in the same horizontal row are derivable from one another 
by Lagrange's scale of moduli ; those in each vertical row by the newly- 
discovered scale. He seems to have been very much struck by the infinite 
variety of transformations of which elliptic integrals admit. The integral of 
the first kind is especially remarkable, because of the simplicity of the rela- 
tion which connects it with any of its transformations, viz. that their ratio is 
independent of the amplitudes. 

Legendre's second work was, as we have remarked, presented to the Aca- 
demy in 1825, but it was not published till 1827. In the summer of 1827 
M. Jacobi announced in Schumacher's ' Astronomischen Nachrichten,' No, 
123, that he was in possession of a general method of transformation for 
elliptic integrals of the first kind. He was not acquainted with Legendre's 
discovery of a new scale, and as an illustration of the general theorem gave 
two cases of it, the first being equivalent to Legendre's method of transfor- 
mation. Thus much was announced in a letter to M. Schumacher, dated 
June I3th ; but in one of a later date (August 2nd) he gave a formal 
enunciation of his theorem, but without demonstration. The two commu- 
nications appear consecutively (Ast. Nach. vi. p. 33). 

In No. 127 of the Nachrichten, vi. p. 133, M. Jacobi gave a demonstra* 
tion of his theorem. 

If we can so determine y in the terms of x as to satisfy the difierential 
equation 

V(i-y.)"(l--x^^^)= M V(i-/)0-./t--^'.) ^^ '^^'"S ^""^^^"t>' 

it is manifest that we shall have (F denoting the elliptic integral of the first 
kind) F(Aa;)=MF (A?/), provided that y and x vanish together. The 
question therefore is, how may the differential equation be satisfied, for it is 
clear that by means of a solution of it we transform the elliptic integral 
F(A x) into another, viz. into F(Xy). 

M. Jacobi shows that if y be equal to — , U and V being integral fuuc* 



48 REPORT — 1846. 

tions of X, the difFerential equation will be satisfied, provided U and V fulfil 
two general conditions, the second of which is found to be deducible from 
the first. He then makes an assumption which is equivalent to assigning 
particular forms to U and V, and thence shows, by a most ingenious method, 
that these forms of U and V are such as to fulfil the first of the required condi- 
tions, which, as has been said, implies the other. He thus verifies, a poste- 
riori, the assumed value of the function y. 

In proving that the forms assigned for U and V have the required pro- 
perty, it is necessary to pass from an expression of the value of 1 —y in terms 
of X to one of \— \y in terms of the same quantity. This is done by 
means of a remarkable property of the functions U and V, namely, that if 

in both X be replaced by ~, ~r ox y will (the constants being properly ad- 
hx \ 

iusted) become or -— • Therefore, in any form in which the relation con- 

A U Xy J 

necting y and x can be put, we may replace x by — , provided we at the 

same time replace y by This has been called the principle of double 

substitution, and by means of it we pass from the expression of 1 —y to that 

of 1 — — , and thence obtain that of \ — X y. It is to be observed that 

\y 
this principle is used merely to prove a certain property of the functions 

U and V. Of course, as the change of x into y— implies that of y into — 

ri X '^ y 

in the finite relation between these quantities, the same thing will be true in 
the differential equation by which they are connected, a remark which may 
very easily be verified. But, on the other hand, it by no means follows that 
because it is true in the differential equation therefore any assumed finite 
relation between y and x having this property is the integral required. The 
property in question therefore does not enable us to verify any assumed 
value of y. 

This remark has reference to a communication from Legendre which ap- 
pears in No. 130 of Schumacher's Nachrichten, vi. p. 201. In it he gives 
an account of M. Jacobi's researches, and an outline of the demonstration of 
which we have been speaking. I find it impossible to avoid the conclusion 
that this great mathematician mistook the character of the demonstration in 
question, and that to him it appeared to be in effect a mere verification of 
the assumed value of y by means of the principle of double substitution. 
He remarks that the direct substitution of the value of y in the diff'erential 
equation is impracticable, but that M. Jacobi had avoided this substitution 
by means of " une propriete particuliere de cette equation qui doit etre com- 
mune aux integrales qui la representent." This property is the principle of 
double substitution ; and after showing that it is true of the differential 
equation, the writer proceeds thus: " Ce principe une fois pose, rien n'est 

plus facile que de verifier I'equation trouvee y=- ~, car par la double sub- 
stitution on obtient la meme valeur de y a un coefficient pres qui doit etre 
6gal a I'unite ; " and, after a remark to our present purpose immaterial, con- 

U . . 

eludes, " Ainsi se trouve demontree g^neralement I'equation y = —- ainsi 

que, etc." 

As we have seen, such a verification would be wholly inconclusive, nor is 



ON THE RECENT PROGRESS OF ANALYSIS. 49 

the essential point of M. Jacobi's reasoning, namely, that the assumed forms 
of U and V satisfy the general condition, laid down at the outset of his de- 
monstration, here adverted to. 

In 1828 Legendre published the first supplement to the 'Traite des Fonc- 
tions EUiptiques,' &c. It contains an account of the researches of M. 
Jacobi, and of a memoir by Abel inserted in the third volume of Crelle's 
Journal. The account here given of M. Jacobi's demonstration is fuller and 
more explicit than that already noticed. It leaves, I think, no doubt of the 
error into which Legendre had fallen. No notice whatever is taken of the 
first part of M. Jacobi's reasoning : and after remarking that the differential 
equation is satisfied when the double substitution is made, he goes on, " Tout 

, ^ X \J 

se reduit done a faire cette double substitution dans 1 integrale w = — — et 

IX. V 

a examiner si elle est satisfaite." After showing that it is so, he adds, " Par 

X U 
ce procede tres simple il est constate que I'equation y -■-■ —^ satisfait 

al'equation difi'erentielle dont I'integrale est F (k ^)=/a F (Ji 4'), etc." (Trait, 
des Fonct. Ell., iii. p. 10). 

Legendre remarks, that although M. Jacobi's demonstration rests on " un 
principe incontestable et tres ingenieux," it is still desirable to have another 
verification of so important a theorem. He accordingly gives an original 
demonstration of it, which is however more nearly allied to M. Jacobi's than 
to him it seemed to be. This demonstration had already been hinted at in 
his communication to the Nachrichten. The principal difference is, that 
while M. Jacobi proved generally that if the first of the two required condi- 
tions were satisfied, the second would also be so, and then showed that the 
forms assigned to U and V satisfied the first condition ; Legendre shows the 
assigned forms are such as to satisfy both conditions, on the connection be- 
tween which it is therefore unnecessary for him to dwell. In the third sup- 
plement to the 'Traite des Fonctions EUiptiques,' Legendre has given an- 
other demonstration of M. Jacobi's theorem, remarking that it is both more 
rigorous and more like M. Jacobi's than that which he had first given. I 
have thought it necessary to make these remarks, because it has been said 
that it was in the supplements to Legendre's work that the demonstration of 
this theorem received "le dernier degre de rigueur"*. 

14. In 1829 M. Jacobi's great work on elliptic functions, the ' Fundamenta 
Nova Theorise Functionum EUipticarum,' was published at Kcenigsberg. It 
contains his researches not merely on the theory of transformation, but also 
with respect to other parts of the subject. But the great problem of trans- 
formation is the fundamental idea of the whole work ; the other parts are 
subordinate to it, or at least derived from it. The subject is treated with 
great fulness of illustration and in a manner not unlike that of Euler. 

M. Jacobi begins by considering the possibility of transforming the ge- 
neral transcendent whose differential coefficient is unity divided by the square 
root of a polynomial of the fourth degree. Subsequently, having shown that 
this transcendent may be transformed by introducing a new variable y equal 
to the quotient of two integral functions of x, and also that the general 

transcendent may be reduced to one of the form / . ^ ^., x > 

he proceeds to consider the latter in detail. 

The first step of this reasoning, viz. the possibility of the transformation, 
depends on a comparison of the number of the disposable quantities in the 

* Verhulst, Traite Elementaire des Fonctions EUiptiques. 
1846. E 



50 REPORT — 1846. 

assumed value of y with that of the conditions required, in order that the 
quantity under the radical in the transformed expression may be equal to 
the square of an integral function of x multiplied by four unequal linear 
factors. It is shown that the number of disposable quantities exceeds by 
three that of the required conditions. But, as Poisson has remarked in the 
report already mentioned (^lem. de I'lnstitut. x. p. 87), and as M. Jacobi 
himself intimates, this does nut amount to an absolute a piiori proof of the 
possibility of the transformation ; 7ion constat but that some of these condi- 
tions may be incompatible. 

Granting however the possibility of putting the quantity under the radical 
in the required form, it is shown, as in Schumacher's Journal, that this 
condition is not only necessary but also sufficient, or, in other words, that it 
involves the second condition already mentioned. 

The transcendent / ,, /, mav be transformed bv as- 

suming y = =r, U being composed wholly of odd powers of a*, and V of even 

powers of it. If the degree of U be greater than that of V, the transforma- 
tion is said to be of an odd order, and of an even order in the contrary 
case. 

This being premised, M. Jacobi discusses the particular cases of the trans- 
formations of the third and of the fifth order. The first is the same as that 
of Legendre. It is shown that if we put 

_ (v + 2 u^} V X + u'^ x^ 
^ ~ t,2 + i-s u''- (v + 2 «5) X-* 

where tt and v are constants connected by the following equation — 

n* — V* + 2nv {l — u- v-} = 0, 
we shall get 

d 1/ r + 2?/' dx 

in which k = h* and A = z'*. The equation connecting a and v is called the 
modular equation. 

The "principle of double substitution" may be illustrated by writing 

for X in the expression for y, which then becomes, according to the principle 

1 
m question, 

If we seek to show that the assigned value of y actually satisfies the dif- 
ferential equation just stated, we begin by finding the value of 1 ^y. Re- 
ducing this value by means of the equation between u and r, we can put it in 

the form (1 — a) — -, R being an integral function of a: and V, as heretofore 

the denominator of the expression for y. The value of 1 + y is hence got 
by changing the sign of a', while that of 1 — v*y is obtained by simultane- 
ously replacing x and y respectively by and and reducing. Simi- 

larly for 1 + v^y. Hence it will appear that 

(l-y^)(l-i-3y"-)=(l-a;0(l-"^x^)|^ . . . (a.) 



1 



ON THE RECENT PROGRESS OF ANALYSIS. 51 

where S, like R, is integral. By differentiatiug and reducing, we then show 

and combining these two results obtain the required verification. 

The essence ot M, Jacobi's demonstration consists in showing that if the 
value of y in terms of x is such that an equation of the form (a.) subsists, 
then necessarily ^ g 

rfi^f^W ^^^ 

where ft is a constant ; the existence of the two equations (a.) and (/3.) being 
equivalent to the two conditions of which we have already spoken (p. 48). 
In the particular case we are now considering, 

f«. = — . 

V 

IS. After considering the transformation of the fifth order (in which the 
modular equation is 

m6_^6 ^SvP-v'^^llr — V"-') + 4)M«{1 —ll'^lfi} =0), 

M. Jacobi prepares the way for a more general investigation by introducing 
a new notation. This step is one of the highest importance. We have been 

in the habit of calhng <p the amplitude of the integral / ■ , /.a • a " 

let this integral be called u. The new notation is contained in the equation 

px dx 

a = am «; or II we call sm a, x, so that u:= I ,. . , , . , 

\ ^0 -/(l — a:'^)(l — ^^.r-) 

then X = sin am u. 

A new notation is in itself merely a matter of convenience : what gives it 
importance is its symbolizing a new mode of considering any subject. We 
had hitherto been accustomed to look on the value of the elliptic integral as 
a function of its amplitude, to make the amplitude (if the expression may so 
be used) the independent variable. But in reality a contrary course is on 
many accounts to be preferred. We have in the more advanced part of the 
theory more frequently occasion to consider the value of the amplitude as 
determined by the corresponding value of the integral than vice versa ; and 
it therefore becomes expedient to frame a notation by which the amplitude 
may be expressed as a function of the integral. In a paper in the ninth vo- 
lume of Crelle's Journal by M. Jacobi, which, like many of his writings, 
contains in a short compass a philosophical view of a wide subject, he has 
made use of the analogy between circular and elliptic functions to illustrate 
the importance of the new notation for the latter. When the modulus of an 
elliptic integral of tlie first kind is equal to zero, the integral becomes 
fx dx , 

■ I / -, which, as we know, is equal to the arc whose sine is x, or to 

J <i VI — x'^ 

sin~'a". Now this is a function which we have much less often occa- 
sion to express than its inverse sin x, and we accordingly always look on the 
latter as a direct, and on the former as an inverse function. Yet in the case 
of elliptic functions, the functional dependence for which we had an explicit 
and recognised notation, viz. that of the integral on the amplitude, corre- 
sponds to that which in circular functions has always and almost necessarily 
been treated merely as an inverse function. The origin of this discrepancy 
is obvious ; our knowledge of the nature of circular functions is not derived 

e2 



52 REPORT — 1846. 

from the algebraical integrals connected with them, and therefore these in- 
tegrals are not brought so much into view as in the theory of elliptic func- 
tions the corresponding integrals necessarily are ; but it is certain that while 
the discrepancy continued to exist the subject could never be fully or satis- 
factorily developed. The maxim " verba vestigia mentis" is as true of ma- 
thematical symbols as of the elements of ordinary language. 

We shall see hereafter that Abel took the same step in his first essay on 
elliptic functions. At present I shall only remark, that one of the earliest 
consequences of the new notation was the recognition of a most important 
principle, viz. that the "inverse function" sin am m, that is, the function 
corresponding to sin u in circular functions, is doubly periodic, or that it re- 
tains the same value when u increases by any multiple either of a certain 
real or of a certain imaginary quantity. Now M. Jacobi has shown that no 
function * can be triply periodic, and therefore these inverse functions pos- 
sess the most general kind possible of periodicity, a property which gives 
them great analytical importance. 

Following M. Jacobi, we shall henceforth give the name of elliptic func- 
tions to those which are analogous to circular functions. It is on this ac- 
count better to call Legendre's functions elliptic integrals than, as he has 
done, elliptic functions (vide ante, p. 44). 

By the new notation we are led to consider a great variety of formulae 
analogous to those of ordinary trigonometry. The sine or cosine of the am- 
plitude of the sum of two quantities may be expressed in terms of the sines 
and cosines of the amplitudes of each, &c.t; and we have only to make the 
modulus equal to zero to pass from what has sometimes, though not with 
much propriety, been called elliptic trigonometry to the common properties 
of circular functions. 

M. Jacobi gives a table of formulae relating to the new elliptic functions, 
and proceeds to apply their properties to the problem of transformation. It 
was in this manner that he had treated the problem in the Nachrichten. As 

* %. e. no function of one vaiiable. 
t The fundamental formulae are — 

sin am u cos am n A am « + sin am v cos am uAamu 



sin am {u-{-v) ■■ 



1 — fr sin- am u sin^ am v 



, . cos am M cos am w — sin am u sin am v A am u A am v 

cos am \u + 1') = :; ,o ■ ., r-5 j 

^ ' 1 — «■' sm- am « sin^ am » 

A am M A am w — ^'^ sin am u sin am v cos am n cos am » 

A am (M -4- «;) = :; ,9 ■ 9 :— 5 ; 

^ ' ^ \ — k^ war am u %\Vi.^ am v 

k being the modulus, and A am m = a^I — A^ sin^ am u. If 

K = f^ ^ ^^ and K' = Z* ^ , ^'^ , 

J ^ Vl - F sin« ^ J ^ VI - A'2 sin2 ^ 

where A" + ^'3 = i^ then it may be shown that 

sin am (m -|- 4 K) = sin am m, 

and 

sin am (w -|- 2 K' v — 1) = sin am u, 

so that 4 K is the real and 2 K' v — 1 the imaginary period of sin am w. Heuce it is ob- 
vious that we shall have generally 

sin am (m + 4 m K -}- 2 w K' v — 1) = sin am «, 
m and n being any integers. 



ON THE RECENT PROGRESS OF ANALYSIS. 53 

in his earlier essay, he assumes y equal to a rational function of x, whose 
coefficients are elliptic functions, and shows that this assumption satisfies the 
difi"erential equation already mentioned. It may be asked what is gained by 
the introduction of elliptic functions into a problem of which, as we have 
seen, particular cases (e.^. the transformations of the third and fifth order) 
can be solved by algebraical considerations. The answer is, that the pro- 
perties of these functions enable us to transform the assumed relation between 
y and a? in a manner which would otherwise be impracticable. It is con- 
ceivable that any particular case might be solved by mere algebra, but it 
does not seem possible to discover in this way a general theorem for trans- 
formations of all orders, and practically the labour of obtaining the formulae 
for the transformation of any high order would be intolerable. 

Having proved the theorem for transformation in nearly the same manner 
as he had already done, M. Jacobi developes the demonstration which, as 
we have said, Legendre hinted at in No. 130 of Schumacher's Journal. 

He then proceeds to consider the various transformations of any given 
order. We have seen that the modular equation for those of the third order 
rises to the fourth degree, that is to say, for a given value of the modulus of 
the original integral four new moduli exist, corresponding to four new in- 
tegrals, into which the given one may be transformed. These four trans- 
formations are all included in the general formula for the third order ; but 
it is to be remarked that in general only two of the roots of the modular 
equation are real. Thus there are two real transformations and no more. 
The same thing is true, mutatis mutandis, of the transformations of any 
prime order (to which M. Jacobi's attention is chiefly directed), that is to 
say, there will be » + 1 transformations of the nth order, w — 1 of which 
are imaginary. The two real transformations are called the first and the se- 
cond ; the second is sometimes called the impossible transformation, because 
it presents itself in an imaginary form*. Of the formulae connected with 
these two transformations M. Jacobi gives copious tables. 

He next shows, in a very remarkable manner, that, corresponding to a 
transformation in which we pass from a modulus ^ to a modulus A, there 
exists another, whose formulae are derivable from those of the former, in 
which we pass from a modulus \^\ — A* to a modulus \/\ — X% or which 
connects moduli complementary to A. and k. The latter is accordingly called, 
with reference to the former, the complementary transformation. The first 
real transformation of k corresponds to the second real transformation 
V\ — k% and vice versa. 

The next theorem which M. Jacobi demonstrates is not less remarkable. 
It is that the combination of the first and second real transformations gives 
a formula for the multiplication of the original integral, or, in other words, 
that the modulus of the integral which results from this double transforma- 
tion is the same as that of the original integral, so that the two integrals 
diflTer only in their amplitudes. Of this theorem he had in the earlier part 
of the work proved some particular cases f. 

* Mr. Bronwin, in the Cambridge Mathematical Journal and in the Phil. Mag., has 
made some objections to this transformation ; but from a correspondence which I have re- 
cently had with him, I believe I am justified in stating that he does not object either to M. 
Jacobi's result or to the logical correctness of his reasoning, but only to the form in which 
the result is exhibited. 

t It may be shown that if we pass from A to X by the first traiisfonuation, we can pass 
from Vl — >.2 to Vl — F also by the first transformation. Also, as has beeujaid, we 
derive from the transformation -fito x\ a transformation •{ vl — A^to vl — x^|, and 



54 REPORT— 1846. 

After fully developing this part of the subject, he next treats of the nature 
of the modular equation, and shows that it possesses several remarkable 
properties. One is, that all modular equations, of whatever order, are par- 
ticular integrals of a differential equation of the third order, of which the 
general integral can be expressed by means of elliptic transcendents. 

16. We now enter on the second great division of M. Jacobi's researches, 
the evolution of elliptic functions. 

The evolution of elliptic functions into continued products with an infinite 
number of factors presents itself as the limit towards which M. Jacobi's 
theorem for the transformation of the nth order tends as n increases sine 
limite. It is for this reason that we may look on the problem of transforma- 
tion as the leading idea in M. Jacobi's researches. 

We may in some degree illustrate these evolutions by a reference to cir- 
cular functions. A sine is, as we know, an elliptic function whose modulus 
is zero. Now if k is zero, X is also zero. Thus if we apply a formula of 
transformation to a sine, we shall be led to another sine either of the same 
or of a multiple arc. Accordingly the first real transformation degenerates 
in the case in question into the known formula for the sine of a multiple 
arc ; while the second, leading us merely to the sine of the same arc, becomes 
illusory. Thus in the case of a sine, transformation is merely multiplication ; 
but from the formula for multiplication, viz. 

sin(2?»+l)9=(2m + l)sin9/l - 



/l — sin"^ 9 \ /, _ sin"- 9 \ 

I ^~c Tt l""B • » 2m7r I 



we at once deduce, by making (2m + l)9 = and 2 m + \ infinite, the 
common formula 



^^"^=K'-S)0-5> 



This then is a formula of evolution deduced from the first real transfor- 
mation. It is however only when h, is zero that the first transformation will 
give such a formula. In all other cases it is, for a reason which we cannot 
here enter on, impossible to derive from it a formula of this kind. M. Jacobi's 
formulae are accordingly derived from the second real transformation, and 
therefore are illusory when k is zero, or for the case of the sine. There is 
nothing therefore strictly analogous to them in the theory of angular sections. 
By means of them we express the function sin am x in terms of sin m x, m 
being a certain constant. 

From the fundamental expressions in continued products, of which there 
are three, many important theorems may be derived. This part of the sub- 
ject seems to admit of almost infinite increase, and it is difficult to give any 
General view of it. I may, however, mention a remarkable transcendental 

similarly from { V\ — x^ to Vl — F} a transformation {x to A}. The first and last of 
these transformations correspond respectively to the differential equations — 

dy _ 1 dx^ 

•v/(l-y2)(i_A.2y2) ~ M V(l - ar2) (1 - K^)' 
daf 1 dy 



V(l-y2)(l-/l2/r'2) M' ^/(l _ yi) (1 _ X2y2)' 

Hence, combining these equations and integrating, 
and it may also be shown that tjtt; is an integer. 



ON THE RECENT PROGRESS OF ANALYSIS. 55 

function of tlie modulus k which is usually denoted by q, and which occurs 
perpetually in this part of the theory of elliptic functions. If for the moment 
we denote this function by Y k, so that (/ = F k, then if for k we write k^, 
which we suppose to represent the modulus of the first real transformation 
of the nth order, we find that q" = F k,^, so that if q^ is the same function of 
^„ that g' is of A 

g^ = g"- 

This singular property, and others of an analogous character, are of great 
use in establishing various formulae *. 

Before discussing the evolution of integrals of the third kind, M. Jacobi 
has premised some important theorems. He proves that the elliptic integral 
of the third kind, though it involves three elements, viz. the amplitude, the 
modulus and the parameter, can yet be expressed in terms of other quantities 
severally involving but two. In order to this we introduce either a new trans- 
cendent t or a definite elliptic integral of the third kind, whose amplitude is a 
certain function of its modulus and parameter. It is almost impossible to 
tabulate the values of a function of three elements, on account of the enormous 
bulk of a table with triple entry ; we therefore see the importance of the step 
thus made. M. Jacobi announced this discovery as generally true of elliptic 
integrals of the third kind, but his demonstration applies to that subdivision 
already mentioned, which was designated by Legendre " Fonctions du troi- 
sieme ordre ^ parametre logarithmique," and not to functions " a parametre 
circulaire X'' It is probable that this limitation was in M. Jacobi's mind, but 
lie does not seem to have expressed it. Further on, in the ' Fundamenta 
Nova,' we find another mode of expressing integrals of the third kind in 
terras of functions of two elements, but this method also applies only to 
" fonctions du troisieme ordre a parametre logarithmique," the two methods 
being in fact closely allied. 

Legendre appreciated the importance of this discovery of M. Jacobi. He 
speaks of it in a letter to Abel, as a " decouverte majeure," but adds that 
bis attempts to extend M. Jacobi's demonstration to the other class of inte- 
grals of the third kind had been unsuccessful. The same remarks occur in 
his second supplement (Traite des Fonct. Ell., iii. p. 141). The distinction 
thus made between the two classes of integrals of the third kind appeared 
to Legendre sufficient to make it desirable to recognise in all four classes of 
elliptic integrals, so as to make the division between the two species of the 
third class coordinate with that between either and the first or second. 
Legendre says explicitly that M. Jacobi had announced, in making known 
his discovery, that it applied to functions " a parametre circulaire." This 

* A method of calculating elliptic integrals by means of q was suggested by Legendre. 
Vide Verhulst, p. 252, and M. Jacobi in Crelle. 
t This transcendent is denoted by T, and is defined by the equation 



=yE («??); 



A (e ^)' 

where E (c <f) is the elliptic integral of the second kind. If we introduce the inverse nota- 
tion, and make ^ = am M,^we can readily establish the following result, 

T = —'ifi — c^fr&in- z.xaud u^. 

The function T, which is the logarithm of n(vide infra, ip. 66), has many remarkable pro- 
perties. 

J In the former species (1 + «) ( ^ H — ) is negative, and in the latter positive (vide 

ante, p. 45). The specific names are derived from the circumstance that for the former the 
fundamental formula of addition involves a logarithm, for the latter a circular arc. 



56 REPORT — 1846. 

however possibly arose from some misconception of M. Jacobi's meaning. 
Dr. Gudermann, in the iburteenth volume of Crelle's Journal, has given it 
as his opinion that the circular species of integrals of the third kind does not 
admit of the reduction in question ; and remarks, that it occurs much more 
frequently than the other species in the applications of mathematics to na- 
tural philosophy. 

After having discussed at some length, and by new methods, the proper- 
ties of elliptic integrals of the third kind, M. Jacobi concludes his work by 
investigating the nature of two new transcendents which present themselves 
in immediate coimexion with the numerator and denominator of the con- 
tinued product by which sin am u is expressed. One of them however 
M. Jacobi had already recognised by a distinctive symbol, in consequence of 
its intimate connexion with the theory of integrals of the third kind. 

Such is the outline of this remarkable work : before it appeared M. Jacobi 
gave in the third and fourth volumes of Crelle's Journal (iii. pp. 192, 303, 
403, iv. p. 185) notices, mostly without demonstrations, of the progress of 
his researches. Almost everything in the first and second of these notices 
is found in the 'Fundamenta.' In the third we find a remarkable algebraical 
formula for the multiplication of the elliptic integral of the first kind. The 
fourth and last relates to ulterior investigations, which it was the intention 
of the author to develope in a second part of his work. It contains an indi- 
cation of a method of transformation depending on a partial differential 
equation * ; values of the elliptic functions of multiple arguments ; a method 
of transforming integrals of the second and third kinds ; a most important 
simplification of the method of Abel for the division of any integral of the 
first kind, &c. Of this simplification he had already given some idea in a 
note in the preceding volume of the same Journal, p. 86. 

17. It may not be improper in this place to observe, that in 1818, and 
thus in the interval between Legendre's first and second systematic works on 
the theory of elliptic functions, M. Gauss published the tract entitled ' De- 
terminatio Attractionis,' &c. The illustrious author begins by i-emarking 
that the secular inequalities due to the action of one planet on another 
are the same as if the mass of the disturbing planet were diffused according 
to a certain law along its orbit, so that the latter becomes an elliptic ring of 
variable but infinitesimal thickness. The problem then presents itself of 
determining the attraction exerted by such a ring on any external point. 
In the solution of tliis problem M. Gauss arrives at two definite integrals ; 
they can readily be reduced to elliptic integrals of the first and second kinds. 
For the evaluation of the integrals to which he reduces those of his problem, 
M. Gauss gives a method of successive transformation, analogous in some 
measure to that of Lagrange. But the transformation of which he makes 
use is a rational one, and is in fact the rational transformation of the second 
order. The discovery of this transformation appears therefore to be due to 
M. Gauss. He has remarked, though merely in passing, that his method is 
applicable to the indefinite as well as to the definite integral. The rational 
transformation in question leads to a continually increasing series of moduli, 
or is, to use an expression of M. Jacobi a transformation "minoris in 
majorem." The law connecting two consecutive moduli is the same as in 
Lagrange's, which is, as we have seen, an irrational transformation ; so that 
M. Gauss's method does not afford us a new scale of moduli. Nevertheless, 
as no rational transformation had I believe been noticed when his tract ap- 

* Mr. Cayley, to whose kindness I have been, while engaged on the present report, greatly 
indebted, has communicated to me a demonstration of the truth of this equation. 



ON THE RECENT PROGRESS OF ANALYSIS. 57 

peared *, his method is, in a historical point of view, of considerable in- 
terest. 

18. In the second volume of Crelle's Journal, p. 101, we find Abel's first 
memoir on elliptic functions. It was published in the spring of 1827, and 
therefore before M. Jacobi's announcement in No. 123 of Schumacher's 
Journal. But it contains nothing which interferes with M. Jacobi's disco- 
very of the general theory of transformation. Abel's researches on this part 
of the subject appeared in the third volume of Crelle's Journal, p. 160. 
This second communication is dated, as we are informed by an editorial 
note, the 12th of February, 1828, and though it is announced as a continu- 
ation of the former memoir, it is yet in effect distinct from it, as its contents 
are not mentioned in the general summary prefixed to the first communica- 
tion. 

These details may not be without interest, though it is not often that ques- 
tions of priority deserve the importance sometimes given to them. There 
is no doubt that Abel's researches were wholly independent of those of 
M. Jacobi ; and though the coincidence of some of their results is therefore 
interesting, yet the general view which they respectively took of the theory of 
elliptic functions is essentially different, as different as the style and manner 
of their writings. 

With M. Jacobi the problem of transformation occupied the first place ; 
with Abel that of the division of elliptic integrals. Both introduced a nota- 
tion inverse to that which had previously been used, and as an immediate 
consequence recognised the double periodicity of elliptic functions. Ex- 
pressions of these functions in continued products and series were given by 
both, but those of Abel were deduced by considering the limiting case of 
the multiplication of elliptic integrals, those of M. Jacobi, as we have seen, 
from the limiting case of their transformation. Hence Abel's fundamental 
expressions depend on doubly infinite continued products, corresponding to 
the double periodicity of elliptic functions. On the other hand, M. Jacobi's 
continued products are all singly infinite. 

Other differences might of course be pointed out, but the most remarkable 
is that which we find in the character and style of their writings. Nothing 
can be more distinct. In M. Jacobi's we meet perpetually with the traces 
of patient and philosophical induction ; we observe a frequent reference to 
particular cases and a most just and accurate perception of analogy. Abel's 
are distinguished by great facility of manner, which seems to result from 
his power of bringing different classes of mathematical ideas into relation 
with each other, and by the scientific character of his method. We meet in 
his works with nothing tentative, with but little even that seems liiie artifice. 
He delights in setting out with the most general conception of a problem, 
and in introducing successively the various conditions and limitations which 
it may require. The principle which he has laid down in a remarkable pas- 
sage of an unfinished essay on equations seems always to have guided him — 
that a question should be so stated that it may be possible to answer it. 
When so stated it contains, he remarks, the germ of its solution f . 

* The fundamental formula of his transformation is incidentaDy mentioned in Legendre's 
second work (Traite des Fouct., i. 61). 

t For instance, Is it possible to trisect an angle by the rule and compass ? The ques- 
tion thus stated leads us to consider the general character of all problems soluble by the 
methods of elementary geometry ; and following the suggestion thus given, we find that it 
is to be answered in the negative. But if the last clause be omitted or neglected, we can 
only proceed, as many persons have done, tentatively, i. e. by attempting actually to solve 
the problem. If we fail, the question remains unanswered ; if we succeed, we do answer 
it, but as it were only by accident. 



58 REPORT — 1846. 

I do not presume to compare the merits of these two mathematicians. 
The writings of both are admirable, and may serve to show that if ever the 
modern method of analysis seems to be an en-n-eipia lather than a re^vq, it 
does so, either because it has not been rightly used, or because it is not duly 
understood. 

To obtain a general view of Abel's writings it may be remarked, that his 
earliest researches related to the theory of equations. Of the ideas with 
which he was then conversant he has made two principal applications. The 
one is to the comparison of transcendents in the manner already described ; 
the other to the solution of the equations presented by the problem of the 
division of elliptic integrals. The second of these applications is contained 
in the memoir published in the second volume of Crelle's Journal. 

He begins by introducing an inverse notation (u) corresponding to the 
function denoted in the ' Fundamenta Nova' by sin am «, while/ (i<) and 
F (m) correspond respectively to cos am u and A am u. This notation 
has the defect of appropriating three symbols which we cannot well spare. 
On the other hand it is certainly more concise than M. Jacobi's. 

He then verifies the fundamental formulae for the addition of the new 
functions, and goes on to show that they are doubly periodic *, He next 
considers the expressions of <pna,, &c. in (p a, &c., and proceeds to prove 
the important proposition that the equation of the problem of the division 
of elliptic integrals of the first kind is always algebraically soluble. 

In order to illustrate this, which is one of the most remarkable theorems 
in the whole subject, it may be observed, that as any circular function of a 
multiple arc can be algebraically expressed in terms of circular functions 
of the simple arc, so may <pna,fna., Fwa be algebraically expressed by 
means of (p a, fa, F a. 

Conversely, as the determination (to take a particular function) of sin a in 
terms of sin w a requires the solution of an algebraical equation, so does that 
of ^ a in terms of ^ wa. The equation which presents itself in the former 
case is, as we know, of the ni\\ or of the (2/i)th degree as n is odd or even. 
But the equation for determining <p a, rises to the {n")th degree in the former 
case, and in the latter to the (2 n-)i\\. We may however confine ourselves 
to the case in which n is a prime number ; since if it be composite the ar- 
gument of the circular or elliptic function may first be divided by one of 
the factors of n, and the result thus got by another, and so on. Thus setting 
aside the particular case of n = 2, we shall have to consider, in order to 
determine sin a or (p a, an algebraical equation of the nth or (n-)th degree 
respectively. 

In consequence of the periodicity of sin at, the roots of the equation in 
sin a admit of being expressed in a transcendental form ; they are all in- 
cluded in the formula sin | a + — — ), in which p is integral, and which 
therefore admits of only n different values. 

* The formulae in question diifer from those already given, only because Abel's form of 

the elliptic integral is / — / =. which becomes the same as Legendre's on 

J V(l-c2>2-)(I + e-a-2) 
making e^ = — l. The double periodicity of the functions is expressed by the formula 

with similar formulae for/ and F. The quantities m and n are integral, and 



ON THE RECENT PROGRESS OF ANALYSIS. 59 

But elliptic functions are doubly periodic, and therefore the roots of the 
equation in ip a are expressible by a formula analogous to the one just written, 
but which involves two indeterminate integers corresponding to the two 
periodicities of the function, just as p does to the single periodicity 2 iz*. 
Giving all possible values to these integers, we get nP^ different values for 
the formula. 

The question now is, how are we to pass from the transcendental repre- 
sentation of these roots to their algebraical expression ? Or, in other words, 
how are the relations among the roots deducible from the circumstance of 
their being all included in the same formula, to be made available in effect- 
ing the solution of the algebraical equation ? 

The answer to this question is to be found in the following principle : that 
if ;^ M be such a rational function of u that 

X, y, ' -z being the roots of an algebraical equation, then any of these quan- 
tities may be expressed in terms of the coefficients of the equation. This 
follows at once from the consideration that we shall have 

%« = — {%^+%y + ••• +X^)' 

jw, being the number of the roots x,y,...z. For the sum within the bracket 
being a rational and symmetrical function of the roots, is necessarily expres- 
sible in the coefficients of the equation, and the same is therefore of course 
true of % X, or of any of the other quantities to which it is equal. 

If, therefore, by means of the relations which -we know to exist among the 
roots of the equation to be solved we can establish the existence of a system 
of such functions, %, %', %", &c., each of which retains the same value of 
whichever root we suppose it to be a function ; and if by combining these 
functions we can ultimately express x in terms of them, the equation is solved, 
since each of these functions may be considered a known quantity. 

Such is the general idea of Abel's method of solution. The principle on 
which it depends, namely, the expressibility of any unchangeable function %, 
is one which is frequently met with in investigations similar to that of which 
we are speaking. M. Gauss's solution of the binomial equation is founded 
upon it. 

I have already remarked that an important simplification of Abel's process 
was given by M. Jacobi. The result which M. Jacobi has stated without 
demonstration may be proved by means of a theorem established by Abel in 
the fourth volume of Crelle's Journal, p. 194. 

M. Jacobi shows the existence of a system of li^ functions v, yj, &c., by 
combining which we can immediately express the values of the roots. In the 
last of his ' Notices ' on elliptic functions we find, as has been said, the ex- 
plicit determination of all the roots. The formula given for this purpose is, 
like the former, undemonstrated, and I do not know whether any demonstra- 
tion of it has as yet been published ; but from a note of M. Liouville, in a 
recent volume of the ' Comptes Rendus,' we find that both he and M. Her- 
mite have succeeded in proving it. 

But in whatever manner the solution is effected it will always involve cer- 
tain transcendental quantities, which are introduced in the expressions of the 
relation subsisting between the different roots. The solution can therefore 
be looked on as complete, only if we consider these to be known quantities. 
They are the roots of a particular case of the equation to be solved. They 
relate to the division of what are called the complete integrals. We may 
therefore say that the general case is reduced to this particular one. But 



60 REPORT — 1846. 

the latter is not, except under certain circumstances, soluble, though the 
solution of the equation on which it depends can be reduced to the solution 
of certain other equations of lower degrees. 

But for an infinity of particular values of the modulus, the case in ques- 
tion is soluble by a method closely analogous to that used by M. Gauss for 
the solution of binomial equations. Thus for all such values the problem of 
the division of elliptic integrals is completely solved. 

The most remarkable of these cases corresponds to the geometrical pro- 
blem of the division of the perimeter of the lemniscate. Abel discovered 
that this division can always be effected by means of radicals, and further, 
that it can be constructed by the rule and compass in the same cases (that 
is for the same values of the divisor) as the division of the circumference of 
a circle. Of this discovery we find Abel writing to M. Holmboe, " Ah qu'il 
est magnifique! tu verras*." 

In order to form an idea of the nature of the difficulty which disappears 
in the case of which we are speaking, let us suppose that we have to solve 
the algebraical equation which is represented by the transcendental one 
(p (3 9) ^ 0, in the same manner as the equation 'ix^ — 3a? = is represented 
by sin (3 fl) = 0. 

The roots of 4- a;' — 3 .r = 0, are, setting aside zero, 

.2* . 47r 
sm — , sm — . 
3 3 

Those of the former algebraical equation, which, as we know, is of the ninth 

degree, are, beside zero, 

^2w ^iOJ 

^3 3 

2 cr t ^ 4 •zir e 

^-3-' ^-3- 

2 (w -F w i) 4<(a} + ■uxi) 



2( w + 2v:i) ^4(w + 2ari) 



3^3 
3 



where i = V* — 1 . 

To satisfy ourselves that these are the roots required, we observe that 
<p(muj + n'uji) = for all integral values of jn and n. Hence the general 

form of the roots of our equation is <p ^'^ "^ ^^^ ; but it will be found that 

3 

if we give any values not included in the above table to m and n, the resulting 

expression can be reduced to one or other of the forms we have specified in 

virtue of the formula ip(9) = (p{( — l)'"+'*9 + TO«;-|-«ari}. E.ff. Thenon- 

tabulated root ^ "^ — — is equal to our sixth root <p ^^ ^'s since the 

3 3 

sum of their arguments is Sui + 2nii, and the sum of 3 and 2 is an odd 
number. 

* It is right to mention that M. Libri has disputed Abel's title to the theory of the di- 
vision of the lemniscate. I shall, however, not enter on the merits of the controversy which 
arose on this point between him and M. Liouville. The reader will find it in the seventeenth 
volume of the ' Comptes Rendus.' It appears that M. Gauss had himself recognised the 
applicability of his method to the equation arising out of the problem of tlie division of the 
perimeter of the lemniscate (vide Recherches Arithmetiques, § vii. p. 429. I quote from 
the translation published at Pai-is in 1809). 



ON THE RECENT PROGRESS OF ANALYSIS. 61 

On considering our table, we observe that it consists of 3 + 1 horizontal 
rows, each containing 3 — 1 terms, and that the arguments of the terms in 
each row are connected by a simple relation ; that of the second being double 
that of the first. If we were to replace 3 by any odd number p, we should 
get an equation of thep'^ degree, whose roots, setting aside zero, might simi- 
larly be arranged in j» -f 1 rows, each of jo — 1 terms, the arguments of the 
terms in each row being as 1, 2, 3, &c. 

Moreover, sin — is rarionally expressible in sin — , and generally sin — ^ — ■ 

is so in sin , n and p being any integers we please. So too are all 

the terms in each horizontal row of our table, whether for the particular case 

we have written down, or for that of any odd number, rationally expressible 

in the first term. 

Hence it may be shown that when the divisor 2 w + 1 is a prime number, 

an equation whose roots were the terms in any horizontal row could be solved 

algebraically, by a method essentially the same as that of Gauss, just as we 

2 o tt 

can solve the equation the type of whose roots is sin „ ^ . But to con- 

^ •'^ 2w + 1 

struct this equation, i. e. to determine its coefficients, requires the solution of 

an equation of the same degree as the number of horizontal rows, i. e. of the 

degree 2 « + 2. And this equation is in general insoluble. The difficulty 

we here encounter may be expressed in general language, by saying that 

although we can pass from one root to another along each horizontal row, 

yet we cannot pass from row to row. 

Our table, however, has the remarkable property, that supposing, as we 
may always do, 2 n + 1 to be a prime number, all the roots are rationally 
expressible in terms of any two not lying in the same row. This depends on 
a property of the function <p, which it is very easy to demonstrate, and it is 
intimately connected with the relations which exist among the terms of the 
same row. 

If, then, which is the case for an infinite variety of values of the modulus, 
we can express any root rationally in terras of another of a different row, 

2 w 2 w 

say in a , all the roots become rational in terms of a . Moreover, 

2n+l 2ra+l 

it appears that not only are the roots all expressible in one, but they are so 
in such a manner that the functional dependencies among them fulfil a cer- 
tain simple condition, which, as Abel shows in a separate memoir (Crelle, iv. 
p. 131 ; or Abel's works, i. p. 114), renders every equation, all whose roots 
are rationally expressible in terms of one, algebraically soluble. 

To take the simplest case, the arc of the lemniscate may be represented by 

the integral / . If ^ be the function inverse to this integral, we have 

»/ V 1 X ' Q ' O 

the simple relation between roots of different rows, a ^ r =i0 ^ -, 

^ ^2^4-1^ 2m + 1 

w being in this case equal to ar. 

To apply what has been said to the solution of the general equation for 
determining ^ a in terms of ^ (2 « + 1) a, it is sufficient to remark that the 
transcendents introduced in considering the relations among the roots of this 

equation, are simply <p - — -— - and (p - — -— -, or at least may be algebraically 

expressed in terms of these two quantities. 
The remainder of the first memoir contains developments of the functions 



62 REPORT — 1846. 

^,y and F in doubly and singly infinite continued products and series. They 
are derived from the expressions of (p a, Sec. in terms of <p _, &c., by supposing 

n to increase sine Ibnite, and are therefore analogous to the expression of 
sin (p in terms of <p which we have already mentioned. 

The second contains the development of what had already been pointed 
out Avith respect to the lemniscate, so far as relates to the division of its peri- 
meter by any prime number of the form 4wi + !• In an interesting note 
which M. Liouville communicated to the Institute in 1844, and which is 
published in the eighth volume of his Journal, p. 507, he has proved gene- 
rally that the division of the perimeter of this curve can always be effected 
whether the divisor be a composite or prime number, real or complex (that 
is, of the formp -t- v — q, p and q being integers). In order to do this, it 
was only requisite to follow m.m., the reasoning by which Abel has shown 
that the equation which presents itself in the problem of the division of the 
circumference of the circle is always resoluble. Thus, as M. Liouville has 
remarked, his analysis is implicitly contained in Abel's. 

This memoir also contains Abel's theorem for the transformation of elliptic 
integrals of the first kind. It is equivalent to that of M. Jacobi ; nor is the 
demonstration, though presented in quite a different form, altogether unlike 
M. Jacobi's. 

Abel begins by considering the sum of a certain series of (p functions whose 
arguments are in arithmetical progression. He shows that the sum of this 
series is a rational function of its first term. If we call this sum (multiplied 
by a certain constant) y, and the first term x, then y is such a function of x 
as to satisfy the differential equation already mentioned, viz. 

dy 1 dx r . -> 

^(l-r/2)(l-A*i/°-j "" M V{l-x-){\-h'-x'')' 
or rather an equation of equivalent form. In fact y is m .m . the same func- 
tion of X that it is in M. Jacobi's theorem. Thus the sum of the series of 
elliiJtic functions is itself, when multiplied by a constant, a new elliptic func- 
tion, having a new modulus, and whose argument bears a constant ratio to 
that of the first term of the series. It appears also that for the sum of the 
elliptic functions we may, duly altering the constant factor, substitute their 
continued product. Thus, beside the algebraical expression of y, there are 
two transcendental expressions of it, both of which are given by M. Jacobi 
in the ' Fundamenta Nova.' At the close of the memoir Abel compares his 
result with the one in Schumacher's Journal, No. 123, and mentions that he 
had not met with the latter until his own paper was terminated. 

19. In the 138th number of this journal Abel resumed the problem of 
transformation, and treated it in a more general and direct manner than had 
yet been done. This memoir appeared in June 1828. M. Jacobi, in a letter 
to Legendre, has spoken in the highest terms of Abel's demonstration of the 
formulee of transformation : he says, " EUe est au-dessus de mes eloges, 
comme elle est au-dessus de mes travaux." An addition to this memoir, 
establishing the real transformations by an independent method, appeared in 
Number 148 of the same journal. These two papers are printed consecu- 
tively in the first volume of Abel's Works, pp. 253, 275. 

In the first of these two remarkable essays Abel makes use of the perio- 
dicity of the function f 9, or, as he here denotes it, A 9, to determine a priori 
what rational function of x, y must be in order that the differential equation 
dy _ dx 



ON THE RECENT PROGRESS OF ANALYSIS. 63 

may be satisfied. [I have altered his notation for the sake of uniformity.] 
Le rj/a; be the function sought, then considering ?/ = \|/ .r as an equation de- 
termining X in terms of y, he shows that certain relations necessarily exist 
among its roots. Let A 6 be one of them and X 9' another, it Avill readily be 
seen that we may put 

dS' = dQ, 
since each is equal to 

dy 

Hence 

a. being the constant of integration, or, which is the same thing, being inde- 
pendent of ?/. Hence A 6 being one root, every other root is necessarily of 
the form A (9 + «)• Again, we see from hence that 

■which is to be true for all values of 9, and which therefore implies the exist- 
ence of a series of equations, of which the type is 



if/(A(9 + A - 1 «)) = T^(A(9 + ka.)), 

where k is an integer. Hence A (9 + ^ a) is a root, whatever integral value 
we may give to k. But the equation ij ^=^fx has but a finite number of 
roots, and therefore the values of the general expression A (8 + A a) must 
recur again and again. This consideration throws light on the nature of the 
quantity a ; it must in all cases be an aliquot part of a period (simple or 
compound) of the function A 9. 

All the values of A (9 + A a) got by giving different values to k are roots ; 
but the converse is not necessarily true ; all the roots are not necessarily 
included in this expression. But it is not difficult to perceive that all the 
roots are included in a more general expression, viz. A(9 +A, a,^ +^2 ^i • • K «*«)» 
and conversely, that all the values of this expression are roots. The number 
« is indeterminate : we may have formulas of the form y = \|/ a:, in which n 
is unity, others in which it is two, &c. ; but in all cases a is an aliquot part 
of some period of A 9, and k is integral. 

It is easy when the roots of y = ^ a? are known, to express y in terras of 9. 

For let vj/ a; = p— , / and F being integral functions. Then 

yY a -fx = {yp - q) {{x-\^){x - K{^ + a)) ....} 

is (jyp — q being the coefficient of the highest power of .r in y F jr —/a?) an 
identically true equation ; whence, to determine y in 9, we have only to assign 
a particular value to x, or to compare the coefficients of similar powers of it*« 

This then determines the form which the function y must necessarily be 
of: the question which Abel goes on to discuss is this: Under what circum- 
stances will a function of the form thus determined a priori be such a fuuc^ 
tion as we require? The character of the reasoning by which this question 
is treated is similar to that of the method by which Abel had, in his second 
tnemoir on elliptic functions, verified the form which, without assigning any 
reason, he had there assumed for the function y. 

The second essay is singularly elegant. If (p^ denote the function inverse 

* I have trot noticed an ambiguity of sign at the outset of this reasoning, as given by 
Abel, as for the purposes of illustration it is immaterial. 



64 REPORT — 1846. 

to the integral / . =^, and ffl^ the corresponding function for 

•y V (1— M-) (1— A^'m*) 

the modulus c, then, on introducing the inverse notation, the differential 

equation 

dy dx 

becomes of course dQ' = ad6, with j = ^<,9 and y = ^^9'. Hence for a 
given increment a of 9, that of 9' is a a. 

Let us take the simplest case, and suppose y to he a rational function of 
X ; then, as a? or 0^ 9 remains unchanged when 9 increases by a period of the 
function (p^, y does so too; that is 0;t9' remains unchanged when 9' increases 
by a times a period of <p^, or in other words, a times a period of p^ is neces- 
sarily one of (p^. 

Suppose now k and c to be both real and less than unity ; then (p^ and ^^ 
have each a real period, here denoted by 2wi, and Sw^ respectively, and each 
an imaginary period ■ar^i and ■nx^i respectively, nr^ and rs^ being both real. 
Let 9 receive first the increment 2Wf, and secondly the increment vj^i, then, 
by what has been said, 

2 a Wj = 2 m Wi + « ot^ « 

a CTj e = 2 j9 Wi + y ra-j z*, 

»w, w, /?, q being certain integers. But can these two equations subsist simul- 
taneously ? Not generally, since if we eliminate a and equate possible and 
impossible parts, we get two relations among ca^ vs^ uif. -us^, which are con- 
tinuous functions of the tivo quantities k and c. Hence both are determinate ; 
and if we wish c to remain indeterminate, we must either make m and q 
equal to zero, in which case a is impossible, or, making n and/) equal to zero, 
assign a real value to it. When a is real we have 

Wt I'll 

and hence the remarkable conclusion, that 

— : — ',: q\m, 

m and q being integers. 

The commensurability of the transcendental functions — -i — ~ is therefore 

a necessary condition, in order that an integral with modulus c can be trans- 
formed into one with modulus k, the regulator a being real and c indeterminate. 
And it may be shown that this condition is not only necessary but sufficient. 
Similar considerations apply to the case in which a is impossible. 

Simple as this view is, it leads to many consequences of great interest. 

The function q, of which we have already spoken (p. 55), is merely e~''V> 
and as we know for the first real transformation of the wth order, it becomes 

nisr _ _ r ^ "1 r ^ 1 

e '"ir. Hence in this case we have — = « — according to the 
general law. It may be well to remark, that if /e = c we have a^=-m^=^m 

* -nr here is in M. Jacobi's notation 2 K', so that ^^ = (p(^+2»j« + «'Bri). m and n 
being any integers. j 



ON THE RECENT PROGRESS OF ANALYSIS. 65 

(an integer). Hence in multiplying an integral, the multiplier must be an 
integer, if y is rational in x, except for particular values of c. 

In the paper of which we are speaking Abel has applied precisely similar 
considerations to the case in which x and y are connected by any algebraical 
equation. 

Passing over one or two shorter papers, one of which has been already 
referred to at p. 59, we come to a ' Precis ' of the theory of elliptic func- 
tions, published in the fourth volume of Crelle's Journal, p. 236. The 
work of which it was designed to be an extract was never written, and the 
' Precis ' itself is left unfinished. A general summary was prefixed to it, from 
which we learn that the work was to be divided into two parts. In the first 
elliptic integrals are considered irrespectively of the limits of integration, and 
their moduli may have any values, real or imaginary. Abel proposes the 
general problem of determining all the cases in which a linear relation may 
exist among elliptic integrals and logarithmic and algebraical functions in 
virtue of algebraical relations existing among the variables*. 

His first step is to apply his general method for the comparison of trans- 
cendents to elliptic integrals, which may be done by what is called Abel's 
theorem, in at least two different ways: the one, that of which he now makes 
use ; the other, that which we have seen is applied to the case of four func- 
tions by Legendre in his third Supplement. 

He next determines the most general form of which the integral of an al- 
gebraical differential expression of any number of variables is capable, pro- 
vided it can be expressed linearly by elliptic integrals and logarithmic and 
algebraical functions. The result at which he arrives admits of many im- 
portant applications. It is, that the integral in question may be expressed in 
a form in which the sine of the amplitude of each elliptic integral and the 
corresponding A, and also the algebraical and each logarithmic function are 
all rational functions of the variables and of the differential coefficients of the 
integral with respect to each. 

He proceeds by an interesting train of reasoning to establish the remark- 
able conclusion, that the general problem which we are considering may 
ultimately be reduced to that of the transformation of elliptic integrals of the 
first kind. The problem of this transformation is then discussed, and by a 
method essentially the same as that of which he had made use in his paper in 
Schumacher's Journal. The appearance however of the two investigations is 
dissimilar, because no reference is made to elliptic functions (as distinguished 
from elliptic integrals) in the first part of the ' Precis.' The relations there- 
fore which exist among the roots of 3/=4'a? are established by considerations 
independent of the periodicity of elliptic functions ; though it is not difficult 
to perceive that they were suggested by the results previously obtained by 
means of that fundamental property. It is shown, that if the equation 
y=.'ifx, where ■i^x is a rational function, satisfy the differential equation (A.), 
then this equation, considered as determining x in terms of y, is always alge- 
braically soluble. As the multiplication of elliptic integrals may be consi- 
dered a case of transformation (that, namely, in which the modulus of the 
transformed integral remains unchanged), this theorem may be looked on as 
an extension of that which we have spoken of (p. 58) in giving an account of 
Abel's first memoir on elliptic functions. The two theorems are proved by 
the same kind of reasoning. 

The second part of the memoir was to have related to cases in which the 
moduli are real and less than unity ; of this however only the summary exists. 

* In the assumed relation, the amplitude, or rather the sine of the amplitude of each 
elliptic integral, is to be one of the variables, and not a function of one or more of them. 
1846. F 



66 REPORT — 1846. 

Abel proposed to introduce three new functions, the first corresponding to 
that which lie had previously designated by <p9*. He now denotes it by A 9. 
The second and third functions are apparently what the second and third 
kind of elliptic integrals respectively become, when, instead of x, we intro- 
duce the new variable 9 ; x and 9 being of course connected by the equation 
x = XS. The double periodicity of the function X and its other fundamental 
properties having been established, it was his intention to proceed to more 
profound researches. Some of his principal results are briefly stated. I may 
mention one, that all the roots of the modular equation may be expressed 
rationally in terms of two of themf. 

One of the last paragraphs of the summary relates to functions very 
nearly identical with those which M. Jacobi discusses at the close of the 
' Fundamenta Nova,' and which he has designated by the symbols H and 0. 

The second volume of Abel's collected works consists of papers not pub- 
lished during his life. Two or three of these relate to elliptic functions. 
The longest contains a new and very general investigation for the reduction 

p 
of the general transcendent, whose differential is of the form -=, P being, 

as usual, rational and R a polynomial of the fourth degree ; together 
with transformations with respect to the parameter of integrals of the third 
kind. 

20. Having now given some account of the revolution which the disco- 
veries of Abel and Jacobi produced in the theory of elliptic functions, I shall 
mention some of the principal contributions Avhich have been made towards 
the further development of the subject since the publication of the ' Funda- 
menta Nova.' In Crelle's Journal, iv. p. 371, we find a paper by M. Jacobi, 
entitled ' De Functionibus EUipticis Commentatio.' It contains, in the first 
place, a development of the method of transforming elliptic integrals of the 
second and third kind, and introduces a new transcendent Q,, which takes the 
place of 0, with whicli it is closely connected. M. Jacobi proves that the 
numerator and denominator of the value of 7/, mentioned above, and which 
have been denoted by U and V, satisfy a single differential equation of the 
third order. The remainder of the paper relates to the properties of H (vide 
ante, note, p. 55). When this function is multiplied by a certain exponen- 
tial factor it becomes a singly periodic function, and, which is very remark- 
able, its period is equal to one of the single or composite periods of the el- 
liptic function inverse to the integral of the first kind. By composite period 
I mean the sum of multiples of the fundamental periods. The exponential 
factor being properly determined, its product by H is equal to multiplied 
by a constant. In considering this subject M. Jacobi is led to introduce the 
idea of conjugate periods. These are periods by the combination of which 
all the composite periods may be produced. It is obvious that the funda- 
mental periods are conjugate periods; and there are, as may easily be 
shown, an infinity of others. 

In the sixth volume of the same journal we find a second part of the 
' Commentatio.' It contains a remarkable demonstration of the fundamental 

* In the 'Precis' Abel has adopted the canonical form of the integral of the first 
kind made use of by Legendve and M. Jacobi ; so that the quantity under tlie radical is 
{l — x") (1 — c-a-). It is worth remarking, that in Ids first paper in Schumacher's Nachrichten 
this quantity is ( 1 — e^ a;") (1 — e" J'-) , while in the second it is the same as m the ' Precis.' To 
this form he appears latterly to have adhered. 

t It is not clear whether by roots of the modular equation we are to imderstand the trans- 
formed moduli themselves, or their foiu-th roots, i. e. in M. Jacobi's notation X or v. Vide 
sttpra, p, 50. 



ON THE RECENT PROGRESS OP ANALYSIS. 67 

formulae of transformation of the odd orders founded on elementary proper- 
ties of elliptic functions. 

In a historical point of view a notice by M. Jacobi in the eighth volume 
of Crelle (p. 413) of the third volume of Legendre's 'Traite des Fonctions 
EUiptiques' is interesting. It was here, I believe, that M. Jacobi first pro- 
posed the name of Abelian integrals for the higher transcendents, which we 
shall shortly have occasion to consider. After some account of the contents 
of Legendre's supplements, the first two of which contain the greater part of 
M. Jacobi's earlier researches, he goes on to generalise a remarkable reduc- 
tion given by Legendre at the close of his work. 

21. I turn to one of the very few contributions which English mathema- 
ticians have made to the subject of this report, namely, to a paper by Mr. 
Ivory, which appeared in the Phil. Trans, for 1831. His design is to give 
in a simple form M. Jacobi's theorem for transformation. The demonstra- 
tion is essentially the same as that in the ' Fundamenta Nova.' But Mr. 

Ivory does not set out with assuming y = — ., U and V being integral func- 
tions of a;, but with assuming it equal to the continued product of a number 
of elliptic functions (whose arguments are in arithmetical progression), mul- 
tiplied by a constant factoi". This is one of M. Jacobi's transcendental ex- 
pressions for y, and the two assumptions are therefore perfectly equivalent 
in the transformations of odd orders ; but in those of even orders, or where 
the continued product consists of an even number of factors, Mr. Ivory's 
amounts to making y equal to an irrational function of x. Transformations 
by irrational substitutions, though long the only kind known (since Lagrange's 
belongs to this class), had not of late been considered in detail. Abel 
indeed remarked in the beginning of the general investigation contained in 
Schumacher's Journal (No. 138), that the existence of an irrational trans- 
formation implied that of a rational one leading to an integral with the same 
modulus as the other. He was, therefore, in seeking for the most general 
modular transformation, exempted from considering irrational substitutions ; 
but in a historical point of view it is interesting to see the connection between 
Lagrange's transformation and those which have been more recently disco- 
vered*. 

* Ify=(l+J)jr . /J— ^, where fi=+c2=l, then 






V(l-/) (1-/^2 y2) - ^ + -' V'(l_«2)(l_c8j^) 

where i._l — * 

This is Lagrange's direct transformation. The corresponding rational transformation is 

^ I -(1+5) ^2 

which satisfies the same differential equation as before. 

Again, dy_ 1+c dx 

V(l-y2)(l-A2y2) 2 V(l-^2)(i.J^^' 

where _ 2 Vc 

is satisfied by 2y2=i+ca^_-v/(l-^2) (i_e2x2), 

which may be called Lagrange's inverse transformation, k being now the same function of 
c, that c was before of ft. The corresponding rational transformation is 

_ a+c)x 
^~ 1-He*2' 

v2 



68 REPORT— 1846. 

The question presents itself, what is the connection between the irrational 
transformation (that of which Lagrange's is a particular case) and the rational 
transformation of even orders ? Perhaps the simplest answer to it (though 
every question of the kind is included in the general investigations contained 
in Abel's ' Precis') is found in a paper by M. Sanio in the fourteenth volume 
of Crelle's Journal, p. 1. The aim of this paper is to develope more fully 
than Mr. Ivory has done the theory of transformations of even orders, and 
particularly of the irrational transformations, which M. Sanio considers more 
truly analogous to the rational transformations of odd orders than the rational 
transformations of even orders ; and also to discuss the multiplication of 
elliptic integrals by even numbers, a subject intimately connected with the 
other. We have already mentioned the existence of what are called com- 
plementary transformations, each of which may be derived from the other 
by an irrational substitution, by which two new variables are introduced. In 
the case of transformations of odd orders, the original transformation and the 
complementary one are both rational, and are both included in the general 
formula given by M. Jacobi's theorem ; but to the rational transformation of 
any even order corresponds as its complement the irrational transformation 
of the same order. This remark, which, as far as I am aware, had not before 
been made, sets the subject in a clear light*. 

22. In the twelfth volume of Crelle's Journal (p. 173), Dr. Guetzlaff has 
investigated the modular equation of transformations of the seventh order : it 
is, as we know from the general theory, of the eighth degree, and presents 
itself in a very remarkable foi'm, which closely resembles that in which 
M. Jacobi, at p. 68 of the ' Fundamenta Nova,' has put the modular equa- 
tion for the third order. Dr. Sohncke has given, at p. 178 of the same vo- 
lume, modular equations of the eleventh, thirteenth and seventeenth orders, 
none of which apparently can be reduced to so elegant a form as those of 
the third and seventh. Possibly the transformation of the thirty-first order 
might admit of a corresponding reduction. The whole subject of modular 
equations is full of interest. Dr. Sohncke has demonstrated his results in a 
subsequent volume of the Journal (xvi. 97). 

In the fourteenth volume of Crelle's Journal there is a paper by Dr. Gu- 
dermann on methods of calculating and reducing integrals of the third kind. 
I have already quoted from this paper the expression of the opinion of its 
learned author, that it is impossible to express the value of integrals of the 
circular species in terms of functions of two arguments. If this be so, it is 

which is M. Gauss's, and is termed in M. Jacobi's nomenclature the rational transformation 
of the second order. It satisfies the equation 

^y =(i+c) ^^ ^ 



where, as before, , 2 v^e 

k = . 

\+c 

* Lagrange's transformation being 

, = (l+i)... /;i^.lety=^p£andx=424. 
then we find that , _ {\+b)x' 

while the differential equation becomes 

dy' . dx' 

where h-=\~k'. 



ON THE RECENT PROGRESS OF ANALYSIS. 69 

impossible to tabulate such integrals, and therefore our course is to devise 
series more or less convenient for determining their values when any pro- 
blem, e.g. that of the motion of a rigid body, to which Dr. Gudermann espe- 
cially refers, requires us to do so. The formation of such series is accordingly 
the aim of this memoir, which contains some remarkably elegant formulae ; 
one of which connects three integrals of the third kind with three of the 
second. 

In the sixteenth and seventeenth volumes of the same Journal, Dr. Guder- 
mann has given some series for the development of elliptic integrals ; and he 
has since published in the same Journal a systematic treatise on the theory 
of modular functions and modular integrals, these designations being used to 
denote the transcendents more generally called elliptic. The point of view 
from which he considers the subject has been already indicated (vide supra, 
p. 36). In a systematic treatise there is of course a great deal that does not 
profess to be original, and it is not always easy to discover the portions 
which are so. Dr. Gudermann's earlier researches are embodied and deve- 
loped in his larger work ; and in some of the latter chapters (XXIII. 329, &c.) 
we find some interesting remarks on the forms assumed by the general trans- 
cendent when the biquadratic polynomial in the denominator has four real 
roots. Dr. Gudermann points out the existence of a species of correlation 
between pairs of values of the variable. 

23. The development of the elliptic function (p in the form of a continued 
product may be applied to establish formulae of transformation. This mode 
of investigating such formulae was made use of by Abel in his second paper 
in Schumacher's Journal, No. 148, which we have already noticed ; and a 
corresponding method is mentioned by M. Jacobi in one of the cursory no- 
tices of his researches which he inserted in the early volumes of Crelle's 
Journal. Mr. Cayley, in the Philosophical Magazine for 1843, has pur- 
sued a similar course. Another and very remarkable application of the same 
kind of development consists in taking it as the definition of the function ^, 
and deducing from hence its other properties. It has been remarked that 
the continued products of Abel and M. Jacobi are derived from considera- 
tions which, although cognate, are yet distinct ; those of the latter being 
singly infinite, while Abel's fundamental developments consist of the product 
of an infinite number of factors, each of which in its turn consists of an in- 
finite number of simple factors. Thus w^e can have two very dissimilar de- 
finitions of the function (p by means of continued products. M. Cauchy, 
who has investigated the theory of what he has termed reciprocal factorials, 
that is, of continued products of the form 

{{l+x){\+tx) ){{\ + tx-^)(\ ^f^x-^) }, 

which is immediately connected with M. Jacobi's developments, has accord- 
ingly set out from the singly infinite system of products, and has deduced 
from hence the fundamental properties of elliptic functions (Comptes Rendus, 
xvii. p. 825). 

Mr. Cayley, on the other hand, has made use of Abel's doubly infinite 
products, and has shown that the functions defined by means of them satisfy 
the fundamental formulae mentioned in the note at page 52, which, as these 
equations furnish a suflScient definition of the elliptic functions, is equivalent 
to showing that the continued products are in reality elliptic functions. He 
has therefore effected for Abel's developments that which M. Cauchy had 
done for M. Jacobi's. Mr. Cayley 's paper appeared in the fourth volume of 
the Cambridge Mathematical Journal, but he has since published a trans- 
lation of it with modifications in the tenth volume of Liouville's. On the 



70 REPORT — 1846. 

same subject we may mention a paper by M. Eisenstein (Crelle's Journal, 
xxvii. 285). 

24. M. Liouville has in several memoirs investigated the conditions under 
which the integral of an algebraical function can be expressed in an alge- 
braical, or, more generally, in a finite form. This investigation is of the 
same character as that which occurs in the beginning of Abel's last published 
memoir on elliptic functions (vide supra, p. 65). But while Abel's re- 
searches are more general than M. Liouville's, the latter has arrived at a 
result more fundamental, if such an expression may be used, than any of 
which Abel has left a demonstration. 

He has shown that if ?/ be an algebraical function of x, such thatjydx 
may be expressed as an explicit finite function of x, we must have 
ri/dx = t -\- Alogu + B\ogv +...+ Clog 10, 

A, B, . . . C being constant, and t, u,v, . . . w algebraical functions of x. 
The theorem established by Abel in the memoir referred to includes as a 
particular case the following proposition, that if 

Cydx — t-\- A log M + B log V + . . . + C log w, 

then t,u.,v, . . .w may all be reduced to rational functions of x and y. 

Combining these two results, it appears that \ijydx be expressible as au 
explicit finite function of x, its expression must be of the form 

< + AlogM -f- Blogzj -f . . .+ Clogw, 
where t, u,v,...w are rational functions of x and y, or rather that its ex- 
pression must be reducible to this form*. 

After establishing these results in the memoir (that on elliptic transcen- 
dents of the first and second kinds), which will be found in the twenty-third 
cahier of the ' Journal de I'Ecole Polytechnique,' p. 37, M. Liouville sup- 

p 
poses y to be of the form — — , where P and R are integral polynomials, and 

'vR ^p 

hence deduces the general form in which the integral / d x may neces- 
sarily be put, provided it admit of expression as an explicit finite function of j:. 

/»p 

He shows from hence that if / - d x cannot be expressed by an alge- 
braical function of x, it cannot be expressed by any explicit finite function 
of it, and finally demonstrates that an elliptic integral, either of the first or 
second kind, is not expressible as an explicit finite function of its variable. 

In a previous memoir inserted in the preceding cahier, M. Liouville 
proved the simpler proposition, that elliptic integrals of the first and second 
kinds are not expressible as explicit algebraical functions of their variable 
(Journal de I'Ecole Polytechnique, t. xiv. p. 137). His attention appears to 
have been directed to this class of reseai'ches by a passage of Laplace's 
' Theory of Probabilities,' in which the illustrious author, after indicating the 
fundamental, and, so to speak, ineffaceable distinctions between different 
classes of functions, states that he had succeeded in showing that the inte- 
gral / , — — is not expressible as a finite function, explicit or 

implicit, of x. Laplace however did not publish his demonstration. 

* An equivalent theorem is stated by Abel in his letter to Legendre for implicit as well 
as explicit functions (Crelle's Journal, vi.). 



ON THE RECENT PROGRESS OF ANALYSIS. /I 

In his own Journal (v. 34 and 441), M. Liouville has since shown that 
elliptic integrals of the first and second kinds, considered as functions of the 
modulus, cannot be expressed in finite terms. 

25. In the eighteenth volume of the 'ComptesRendus'(Liouville's Journal, 
ix. 333), we find in a communication from M. Hermite, of which we shall 
shortly have occasion to speak more fully, a remarkable demonstration of 
Jacobi's theorem. It is stated for the case of the first real transformation, 
but might of course be rendered general. This demonstration depends es- 
sentially on the principle already mentioned (p. 59), that any rational func- 
tion of a root of an algebraical equation which has the same value for every 
root of the equation is rationally expressible in the coefficients. The equa- 
tion to which this principle is applied is that to which we have so often re- 
ferred, viz. y = — , considered as an equation to determine x in terms of y, 

and by means of it, M. Hermite shows at once that a certain rational func- 
tion of X is also a rational function of y, the form of which is subsequently 
determined. 

M. Hermite goes on to prove other theorems relating to elliptic functions. 

As elliptic functions are doubly periodic, we may determine certain of 
their properties by considering to what conditions doubly periodic functions 
must be subject. This view is mentioned by M. Liouville in a verbal com- 
munication to the Institute (Comptes Rendus, t. xix.). He states that he 
had found that a doubly periodic function which is not an absolute constant 
and has but one value for each value of its variable must be, for certain va- 
lues of it, infinite ; that from hence the known properties of elliptic func- 
tions are easily deduced ; and that by means of this principle he had suc- 
ceeded in proving the expressions of the roots of the equation for the division 
of an elliptic integral of the first kind, which M. Jacobi had given without 
demonstration in Crelle's Journal*. I am not aware that any development 
of M. Liouville's view has as yet appeared. 

In the recent numbers of Crelle's Journal there are many papers by M. 
Eisenstein on different points in the theory of elliptic functions. Among 
these I may mention one which contains a very ingenious proof of the fun- 
damental formula for the addition of two functions, derived from the differ- 
ential equation of the second order, which each function must satisfy. 

Other contributions to the theory of elliptic functions might be mentioned ; 
some of these, not here noticed, are referred to in the index which will be 
found at the end of this report. But in general it may be remarked that the 
form which the subject has assumed, in consequence of the discoveries of 
Abel and M. Jacobi, is that which it will probably always retain, however 
our knowledge of particular parts of it may increase. What has since been 
effected relates for the most part to matters of detail, of which, however im- 
portant they may be, it is difficult or impossible to give an intelligible ac- 
count. 

26. It does not fall within the design of this report to consider the various 
applications which have been made of the theory of elliptic functions ; but 
I shall briefly mention some of the geometrical interpretations, if the expres- 
sion may so be used, which mathematicians have given to the analytical re- 
sults of the theory. 

The lemniscate has, as is Avell known, the property that its arcs may be 
represented by an elliptic integral of the first kind, the modulus of which is 

* M. Liouville has mentioned that M. Hermite had demonstrated the formiilBe in question 
in a different manner. « 



72 REPORT — 1846. 

—7=.. The problem of the division of its perimeter is accordingly a geome- 
trical interpretation of that of the division of the complete integral, and was 
considered by mathematicians at a time when the theory of elliptic func- 
tions was almost wholly undeveloped. Besides Fagnani, whose researches 
with respect to the lemniscate have been already noticed, we may mention 
those of Euler, who however did not succeed in obtaining a solution of the 
problem. Legendre, who seems to have attached considerable importance 
to geometrical illustrations of his analytical results, assigned the equation of 
a curve of the sixth order, whose arcs measured from a fixed point represent 
the sum of any elliptic integral of the first kind and an algebraical expression. 
He showed also that an arc of the curve might be assigned equal to the el- 
liptic integral, but in order to this both extremities of the arc must be con- 
sidered variable, so that in effect the integral is represented by the difference 
of two arcs measured from a fixed point (Traite des Fonctions Elliptiques, 
i. p. 36). 

M. Serret, in a note presented to the Institute in 1843 (Liouville's Journal, 
viii. 14-5), has proved a beautiful theorem, viz. that the sum and difference of 
the two unequal arcs, intercepted by lines drawn from the centre of Cassini's 
ellipse to cut the curve, are each equal to an elliptic integral of the first 
kind, and that the moduli of the two integrals are complementary. In the 
lemniscate, which is a case of Cassini's ellipse, one of these arcs disappears, 
and the moduli of the two integrals are equal, each being the sine of half a 
right angle. So that M. Serret's theorem is an extension of the known pro- 
perty of the lemniscate. 

M. Serret has since considered the subject of the representation of elliptic 
and hyper-elliptic arcs in a very general manner. His memoir, which was 
presented to the Institute and ordered to be published in the ' Savans 
Etrangers,' appears in Liouville's Journal, x. 257. He had remarked that 
the rectangular coordinates of the lemniscate are rationally expressible in 
terms of the argument of the elliptic integral which represents the arc, 

.— z -\- z^ .— z — z^ 

for if we assume x = v2 a y^T^i ^"^ ^ ~ ^^ " i 4- -4 ' ^^'^ ^^^'^ ^^^® 

dz 

ds = V {dx'^ + dy^} =: 2a . — =, and if between the first two of these 

equations we eliminate z, we arrive at the known equation of the lemniscate*. 
So that if we state the indeterminate equation 

dx'^ ^d7f--=Z .dz\ 

{x, y and Z being real and rational functions of 3), the lemniscate will afford 
us one solution of it ; and every other solution will correspond to some curve 
whose arc is expressible by an elliptic or hyper-elliptic integral. Of this in- 
determinate equation M. Serret discusses a particular case. He succeeds in 
solving it by a most ingenious method, which is applicable to the general 
equation, and shows from hence that there are an infinity of curves, the arcs 
of which represent elliptic integrals of the first kind. M. Serret's researches 
however have not led him to a geometrical representation by means of an 
algebraical curve of any integral of the first kind, though his results are ge- 
neralised in a note appended to his memoir by M. Liouville. In order that 

* On reducing; the integral / — -, to the standard form of elliptic integi-als, we 

.7 VI -i-i-' 
find that it is an elliptic integral of the first kind, of which the modulus is the sine of 45°. 



ON THE RECENT PROGRESS OF ANALYSIS. 73 

the curve may be algebraical, it is necessary and sufficient, as M. Liouville 
has remarked, that the square of the modulus of the integral should be ra- 
tional, and less than unity. 

In a subsequent memoir (Liouville's Journal, x. 351) he has very much 
simplified the analytical part of his researches, and in the same Journal 
(x.421) has proved some remarkable properties of one class of what maybe 
called elliptic curves. In the fourth number of the Cambridge and Dublin 
Mathematical Journal (p. 187), M. Serret has developed this part of the sub- 
ject, and has also given a general sketch of his previous papers. M. Liou- 
ville (Comptes Rendus, xxi. 1255, or his Journal, x. 456) has given a very 
elegant investigation of an analytical theorem established by M. Serret. 

In the fourteentli volume of Crelle's Journal (p. 217), M. Gudermann has 
considered the rectification of the curve called the spherical ellipse, which is 
one of a class of curves formed by the intersection of a cone of the second 
order with a sphere. He has shown that its arcs represent an elliptic integral 
of the third kind. 

In the ninth volume of Liouville's Journal (p. 155), Mr. W. Roberts proves 
that a cone of the second order, whose vertex lies on the surface of a sphere, 
and one of whose external axes passes through the centre, intersects the 
sphere in a curve whose arcs will, according to circumstances, represent any 
elliptic integral of the third kind and of the circular species ; or any elliptic 
integral of the same kind and of the logarithmic species, provided the angle 
of the modulus is less than half a right angle ; or (subject to the same con- 
dition) any elliptic integral of the first kind ; or lastly, by a suitable modifi- 
cation, any elliptic integral of the second kind. The cases here excepted 
may be avoided by introducing known transformations. The cases in which 
the arcs represent elliptic integrals of the first kind, Mr. Roberts has pre- 
viously mentioned in the eighth volume of Liouville's Journal (p. 263). He 
has since given in the same Journal (x. 297), a general investigation of the 
subject, in which it is supposed that the vertex of the cone may have any 
position we please. M. Verhulst has represented the three kinds of elliptic 
integrals by means of sectorial areas of certain curves, and the function T by 
the volume of a certain solid. It is manifest, however, that it is incom- 
parably easier to do this than to represent these transcendents by means of 
the arcs of curves. 

Beside one or two other papers I may mention a tract by the Abbe Tor- 
tolini, on the geometrical representation of elliptic integrals of the second and 
third kinds. This tract, however, I have not seen. 

Lagrange long since proved (vide Theorie des Fonctions Analytiques, 
p. 85), that by means of a spherical triangle a geometrical representation of 
the addition of elliptic integrals of the first kind may easily be obtained, and 
that hence by a series of such triangles we are enabled to represent the mul- 
tiplication as well as the addition of these integrals. 

M.Jacobi has given (Crelle's Journal, iii. p. 376, or vide Liouville's Journal, 
X. p. 435) a geometrical construction for the addition and multiplication of 
elliptic integrals of the first kind. It is founded on the properties of an irre- 
gular polygon inscribed in a circle, and the sides of which touch one or more 
other circles. It is to be remarked that Legendre, in giving an account cJf 
this construction in one of the supplements to his last work, has only con- 
sidered its application to multiplication and not to addition, and has been 
followed in this respect by M. Verhulst, whose treatise on elliptic functions 
has been already mentioned. In consequence of this, M. Chasles was led to 
believe that until the publication of his own researches, no construction for 
addition excepting that of Lagrange was known. But he has recently 



74 REPORT — 1846. 

(Comptes Rendus, January 1846) pointed out the error into which he had 
fallen. 

27. In the Transactions of the Royal Irish Academy (ix. p. 151), Dr. 
Brinkley gave a geometrical demonstration of Fagnani's theorem with respect 
to elliptic arcs, and in the sixteenth volume of the same Transactions (p. 76), 
we find Landen's theorem proved geometrically by Professor MacCullagh. 

M. Chasles has considered the subject of the comparison of elliptic arcs 
by geometrical methods, and with great success. His fundam^tal propo- 
sition may be said to be, that if from any two points of an ellipse we draw 
two pairs of tangents to any confocal ellipse, the difference of the two arcs of 
the latter respectively intercepted by each pair of tangents is rectifiable. 
Or, what in effect is the same thing, if we fasten a string at two points in 
the circumference of an ellipse, and suppose a ring to move along the string, 
keeping it stretched, and winding it on and off the arc which lies between its 
two extremities, the ring will trace out a portion of an ellipse confocal to the 
former. If for the first ellipse we substitute an hyperbola confocal with the 
second, the sum of the arcs will be constant. From hence a series of theo- 
rems is deduced, remarkable not only for their elegance, but also for the 
facility with which they are obtained. They furnish constructions for the 
addition and multiplication of elliptic integrals. The whole of this investi- 
gation, of which an account is given in the 'Comptes Rendus' (vol. xvii. 
p. 838, and vol. xix. p. 1239), shows, like others of M. Chasles's, how much 
is lost in treating geometrical questions by an exclusive adherence to what 
may be called the method of co-ordination. Invaluable as this method is, 
it yet often introduces considerations foreign to the problem to which it is 
applied *. 

III. 

28. The first outline of a detailed theory of the higher transcendents was 
given by Legendre in the third supplement to his ' Traite des Fonctions 
Elliptiques.' He proposes to classify the transcendents comprised in the 
general formula 

f{x)dx 



J(x 



ix — a) Vi^ X 

according to the degree of the polynomial (f x, the first class being that in 
which the index of this degree is three or four ; the second that in which it 
is five or six, and so on. The first class therefore consists of elliptic inte- 
grals ; all the others may be designated as ultra-elliptic. This epithet, how- 
ever, which was proposed by Legendre, has not been so generally used as 
hyper-elliptic, which was, I believe, first used by M. Jacobi. M. Jacobi, how- 
ever, has proposed to call the higher transcendents Abelian integrals. 

The principle of Legendre's classification is to be found in the minimum 
number of integrals to which the sum of any number of them can be reduced. 
As A\e know, this number is unity in the case of elliptic integrals, and by 
Abel's theorem we find that it is two in the first class of the higher trans- 
cendents, three in the next, and so on. 

Following the analogy of elliptic integrals, Legendre proposed to recognise 
three canonical forms in each class of hyper-elliptic integrals, and thus to 
divide it into three orders. The sura of any number of functions of the first 

* M. Chasles has also considered the subject of spherical conies, as well as that of the 
lines of curvature and shortest lines on an ellipsoid. The latter has recently engaged the 
attention of several distinguished mathematicians — MM. Jacobi, Joachimsthal, Liouville, 
MacCullagh and M. Roberts may be particularly mentioned. 



ON THE RECENT PROGRESS OP ANALYSIS. 75 

kind will, when the required conditions are satisfied, be equal to a constant ; 
that of any number of the second and third kinds respectively will, under 
similar conditions, be equal to an algebraical or logarithmic function. 

Much the greater part of the remainder of the supplement consists of a 
discussion of the particular transcendents 

y-» dx p dx 

V\—x^ ^" J Vl+x^' 

It contains a multitude of numerical calculations, and if the writer's age be 
considered (he was then almost eighty), is a very remarkable production. 
By means of the numerical calculations he recognised, as it were empirically, 
the values to be assigned in different cases to the above-mentioned constant : 
what these values ought to be, he did not attempt to determine a priori. 

At the close of the supplement we find a remarkable reduction of an inte- 
gral, apparently of a higher order to elliptic integrals. The method em- 
ployed has been generalised by M. Jacobi, in a notice of Legendre's ' Sup- 
plements,' inserted in the eighth volume of Crelle's Journal (p. 413). 

29. In the ninth volume of Crelle's Journal (p. 394'), we find a most im- 
portant paper by M. Jacobi (Considerationes Generales, &c.), which may 
be said to have determined the direction in which the researches of analysts 
in the theory of algebraical integrals were to proceed. 

The writer proposes two questions, both suggested by the cases of trigo- 
nometrical and elliptic functions. First, as in these cases we consider certain 
functions to which circular and elliptic integrals are respectively inverse, and 
which are such that functions of the sum of two arguments are algebraically 
expressible in terms of functions of the simple arguments, what are the cor- 
responding functions to which the hyper-elliptic or Abelian integrals are 
inverse, and how by means of them can Abel's theorem be stated ? 

Secondly, as in the same cases we obtain algebraical integrals of differen- 
tial equations, whose variables are separated, but which nevertheless can only 
be directly integrated by means of transcendents *, what are the differential 
equations of which Abel's theorem gives us algebraical integrals? These 
two questions are, it is obvious, intimately connected. 

M. Jacobi first takes the particular case in which the polynomial under the 
radical is of the fifth or sixth degree. If we call this polynomial X, it follows 
from Abel's theorem, that if 

, pxdx 

and^,a:=/;^7=, 

we shall have the equations 

(p a -|- ^ 6 = 9 a; + ? 3/ + ^ a;' -f- ^ y', 

where a and h are given as algebraical functions of the independent quan- 
tities x, y, x^,y^. 

Let fx-\-(py=.u ^x' + (py' = M' 

^ dx dy / . 

* E.g. / „ + ■ - — — ^ = 0, of which the algebraical integral is a"v 1 — y*-|-y v 1 — ar^ = C. 

Each term of tliis differential equation is a differential of a transcendent function sin~'x or 
8in~V 



76 REPORT — 1846. 

Then x and y are both given as functions of u and v. We may therefore 
put 

x = k(ti v), y = >^ (m ^') ; 

and similarly, 

x' = X (m' ?;'), y^ = X, (m' i>') ; 

and as ^ « + (p 6 = m + ?«' 

^, a + ^ 6 = z; + ^'j 

M'e shall have a = X (?^ + m', « + «•) 

Hence the functions X (m + ?f ', v •{- v^) and X, (2^ + u^, v + ?;') are expres- 
sible as algebraical functions oi \{uv), X, {uv), X (m" w'), X, (m' r'). 

These then are functions to which the integrals are in a certain sense in- 
verse, and which have the same fundamental property as circular and elliptic 
functions. 

In the general case of Abel's theorem, we introduce (when the degree of 
the polynomial is 2 m or 2 m — 1 ), w — 1 functions analogous to X, each 
being a function of m — 1 variables. These functions will, it may easily be 
shown, have the fundamental property just pointed out for the case in which 
m is equal to three. 

Again, the diiferential equations of which Abel's theorem gives us alge- 
braical integrals, are, if the degree of the polynomial X be five or six, the 
following : 

dx dy dz 

X dx ydy z dz 

and generally, if the degree of the polynomial be 2 m or 2 m — 1, there are 
m — 1 such equations, the numerators of the last containing the (>» — 2)th 
power of the variables. 

M. Jacobi concludes by suggesting as a problem the direct integration of 
these differential equations, so as to obtain a proof of Abel's theorem cor- 
responding to that which Lagrange gave of Euler's (vide ante, p. 36). 

30. Another important paper by M. Jacobi is that which is entitled ' De 
Functionibus duarum Variabilium quadrupliciter periodicis, etc' (Crelle, xiii. 
p. 55). It is here shown that a periodic function of one variable cannot 
have two distinct real periods. In the case of a circular function, though we 
have for all values of x 

sin a; = sin (a; -I- 2 w ir) 
= sin (a; -|- 2 « ir), 

m and n being any integers, yet 2 m if and 2 71 it do not constitute two 
distinct periods, since each is merely a multiple of 2 if, which is the funda- 
mental period of the function. But if we had for all the values of x 

f(x)=f{x + a}=f{x+^}, 
we should also have 

/ (^) =/(* -f- m a + « /3), 
where m and n may be any integers, positive or negative. Hence »J a + n j3 



ON THE RECENT PROGRESS OF ANALYSIS. 77 

may, provided a and j3 are incommensurable, which is implied in their being 
distinct periods, be made less than any assignable quantity, so that we may 
put 

fx =f{x + £), 

where £ is indefinitely small, and this manifestly is an inadmissible result. 
Accordingly we see that one at least of the periods of elliptic functions is 
necessarily imaginary. 

Again, similar reasoning shows that in a triply periodic function, that is 
in one in which we have 

/(^)=/{a.+»i(a+/3A/::n") + ?«'(a' + /3'V'^) + »»"(a" + ^"'^~l)} 
for every value of x, m, m', m" being any integers, and in which the three 
periods a+ ^V^—i, &c. are distinct, we can make 

f{x)=f{x + e) 
by assigning suitable values to m, m', m" ; s being as before less than any 
assignable quantity. Hence as this result is inadmissible, it follows that 
there is no such thing as a triply periodic function. Whenever therefore a 
function appears to have three periods they are in reality not distinct, and 
so a fortiori when it appears to have more than three. But now we come to 
a difficulty. For M. Jacobi proceeds to show that if we consider a function 

of one variable inverse to the Abelian integral / ^ — -^ , X being of 

the sixth degree in x, this function has four distinct and irreducible periods. 
His conclusion is that we cannot consider the amplitude of this integral as 
an analytical function of the integral itself. In the present state of our 
knowledge, this conclusion, though seemingly forced on us by[the impossibi- 
lity of recognising the existence of a quadruply periodic function of one va- 
riable, is not, 1 think, at all satisfactory. The functional dependence, the 
existence of which we are obliged to deny, may be expressed by a differen- 
tial equation of the second order; and therefore it would seem that the 
commonly received opinion that every differential equation of two variables 
has a primitive, or expresses a functional relation between its variables, must 
be abandoned, unless some other mode of escaping from the difficulty is dis- 
covered. It is probable that some simple consideration, rather of a metaphy- 
sical than an analytical character, may hereafter enable us to form a con- 
sistent and satisfactory view of the question, and this I believe I may say is 
the opinion of M. Jacobi himself. The same difficulty meets us in all the 
Abelian integrals : as in the case of those of Legendre's first class, namely 
where X is of the fifth or sixth degree, so also generally, the inverse function 
has more than its due number of periodicities. 

Abel, in a short paper in the second volume of his works, p. 51, has in 
effect proved the multiple periodicity of the functions which are inverse to 
the integrals to which his theorem relates. The difficulty to which this gives 
rise did not strike him, or was perhaps reserved for another occasion. 

M. Jacobi next proves that his inverse functions of two variables are 
quadruply periodic, but that quadruple periodicity for functions of two va- 
riables is nowise inadmissible. 

A difficulty however seems to present itself, which is suggested by M. 

Eisenstein in Crelle's Journal, viz. that if for each value of the amplitude 

/*d X . f> • J 

the integral ^ a? or / -/= (vide supra, 1^.15), has an mfimty of magnitudes 

real and imaginary, and the same is the case for (py, it is by no means easy to 



78 REPORT — 1846. 

attach a definite sense to the equation m = ip a? + f> y, or to see how the value 
of u is determined by it *. 

31. Two divisions of the theory of the higher transcendents here suggest 
themselves, which are apparently less intimately connected than the corre- 
sponding divisions in the theory of elliptic functions, viz. the reduction and 
transformation of the integrals themselves, and the theory of the inverse 
functions. 

But before considering these I shall give some account of what has been 
done in fulfilment of the suggestion made by M. Jacobi at the close of 
the * Considerationes Generales.' Mathematicians have succeeded in effect- 
ing the integration of the system of differential equations to the consideration 
of which we are led by Abel's theorem, and which is commonly designated 
by German mathematicians as the " Jacobische system ; " its existence and 
its integrability having been first pointed out by M. Jacobi. 

In Crelle's Journal (xxiii. 354), M. Richelot, after modifying the form in 
which Lagrange's celebrated integration of the differential equation of 
elliptic integrals is generally presented, extended a similar method to the 
system of two differential equations which occurs when we consider the 
Abelian transcendents of the first class. He thus obtains one algebraical 
integral of the system. In the case of Lagrange's equation one integral is 
all we want ; but in that which M. Richelot here discusses we require 
two. Now if in the former case we replace each of the variables by its reci- 
procal, we obtain a new differential equation of the same form as the original 
one, and integrable therefore in the same manner ; and if in its integral we 
again replace each new variable by its reciprocal, that is by the original va- 
riable, we thus, as it is not difficult to see, get the integral of the original 
equation in a different form. That the two forms are in effect coincident 
may be verified a postenori. But the same substitutions being made in M. 
Richelot's equations, which are of course those we have already mentioned 
at p. 76, the first of them becomes similar in form to the second, and vice 
versd the second to the first. Thus the system remains similar to itself; and 
if in the algebraical integral we obtain of it we again replace the new vari- 
ables by their reciprocals, Ave fall on a new algebraical integral of the original 
system ; this integral being, which is remarkable, independent of that pre- 
viously got. Thus the system of two equations is completely integrated. 
Extending his method to the general system of any number of equations, 
M. Richelot obtains for each two integrals, but of course these are not all 
that we want. At the conclusion of his memoir M. Richelot derives from 
Abel's theorem the algebraical integrals of the " Jacobische system." 

Though in this memoir M. Richelot only obtained by direct integration 
two of the m — 1 algebraical integrals of the " Jacobische system," yet he put 
the problem of its complete integration into a convenient and symmetrical 
form. As there are m variables and m — 1 relations among them, we may 
suppose each to be a function of an independent variable t, Lagrange, as 
we know, in integrating the equation 

dx dy 

* The difficulty here mentioned may perhaps be met by saying that the value of ^.r de- 
r^ dx 
termined by the integral / ~'7^ is necessarily determinate, and so likewise is that of «. 

That considerations connected vrith the conception of a function inverse to ^ a? make the 
latter quantity appear indeterminate is undoubtedly a difficulty ; but it is, so to speak, a 
difficulty collateral to M. Jacobi's theory, and therefore need not prevent our accepting it. 



ON THE RECENT PROGRESS OF ANALYSIS. 7^ 

introduced such an independent variable by the assumption — = -/x, which 

of course implied that -^ = — Vy, This assumption is unsymmetrical, 

and it is therefore difficult to see how to genejralise it. But if we assume 

~= ■ ^. we shall of course have-^f = , and therefore t is symme- 

dt x—y "■* y—so 

trically related to x and y. Let F m = be an equation whose roots are 
X and y, then, as we know, when u = x,F'u = x—y and, when u = y, 
V ti =^ y — X, so that using an abbreviated notation 

— =^^and^ = J^ 
dt p(a?) dt p(3,)- 

Nothing is easier than to generalise this result. For instance, the " Ja- 
cobische system " of two equations is 

dx dy dz 

xdx ydy zdz 

Now if F M =s have x, y, z for its roots, the two preceding equations 
may, in virtue of a very well-known theorem, be replaced by the three follow- 
ing. 

dx_s/X ^_ Vy dz^_^'L 
dt~'Y~x^ dt~Wy' dt~'Wz^ 

which introduce an independent variable t, symmetrically related to x, y and 
z ; and so in all cases. 

M. Richelot * then takes a symmetrical function oi x, y, , . . z, viz. their 
sum, and by means of the last written equations arrives at an integrable dif- 
ferential equation of the second order, the principal variable being the said 

* As M. Riclielot's method of demonstrating Euler's theorem is more symmetrical and 
far more easily remembered than Lagrange's, it ought, I think, to be introduced into all 
elementary works on elliptic functions. The equation to be integrated being 

' dx ^ dy ____ -0, 



assume 



dx^^y/a + lix + yx^ + lx^ + tx* 



dt 



y -X 



Then iy__ V'g-HSy-i-yy'-t-Sy^ + ty^ 

dt a? — y 

Let p = X ■{■ y. Then after a few obvious reductions 



Hence dp- 



di^-'^P + 'P'' 



« {Vx + ^x + Yo.'^ + ix^ + ^a:*- V* + j8y + 7 y^ + 3y'+ «y'}^ 

the algebraical integral sought. It may easily be expressed ia other forms. 



80 REPORT— 1846. 

sum, and the independent variable being t. From the first integral of tiiis 
equation it is easy to eliminate the differentials, and we have thus an alge- 
braical relation m x,y, . . . z, from which, in the manner already mentioned, 
M. Richelot deduces another. We now see that if we could find any other 
symmetrical function which would lead to an integrable equation we should 
get a finite relation among the variables. 

In the next volume of Crelle's Journal M. Jacobi took the following func- 
tion as his principal variable, 

\/ {(f- - ^) (f* -y) • • • (p - z))> 

fj. being a root of X = or fx = if we suppose X =fa;. Calling this 
function v, we get a simple differential equation in v and t, and a correspond- 
ing integral of the system. Novv/j = has 2 m or 2 m — 1 roots, and we 
only want m — 1 integrals. The integrals therefore which we get by making 
ju, the first, second, &c. root of/.r = are not all independent. 

In the twenty-fifth volume of Crelle's Journal M. Richelot resumed the 
subject of his former paper, and discussed it in a very interesting memoir. 
This fundamental or principal result may be said to be a generalisation of 
M. Jacobi's. It is that in the function 

^ {(ix, - oc) (i^ - y) . . . (fx - z)} 

/* may have any value whatever. The resulting differential equation, 
though rather more complicated than when, with M. Jacobi, we suppose ju< 
a root of fee = 0, is still very readily integrable. We have thus an indefi- 
nite number of algebraical integrals, since the quantity jw, is arbitrary, but 
of course not more than m — 1 of them are independent. 

In the same volume of Crelle's Journal, p. 178, there is a curious paper by 
Dr. Hoedenkamp, in which the algebraical integrals of Jacobi's system are 
for the case of a polynomial of the fifth degree under the radical deduced 
from geometrical considerations. It is shown that in a system of curvilinear 
coordinates (those of which MM. Lame and Liouville have made so much 
use), the equations of the system are the differential equations given by the 
Calculus of Variations for the shortest line between two points. Conse- 
quently the finite equations of a straight line are the integrals sought. This 
very ingenious consideration is aft-^rwards generalised. 

32. In the twelfth volume of Crelle's Journal, p. 181, M. Richelot has 
considered the Abelian integral of the first class. The principal result at 
which he arrives is, that the only rational transformation by means of which 
such an integral may be changed into one of similar form is linear in both 
the variables which it involves. By means of this substitution, he transforms, 
under certain conditions, the integral in question into a form analogous to 
the standard forms of elliptic integrals. The subject of the division of hyper- 
elliptic integrals of each class into three genera is also considered, and the 
same principle of classification as Legendre's is made use of. The paper 
concludes by pointing out an error which Legendre committed in the appli- 
cation of his principle. Legendre had thought that the formula of summa- 
tion given by Abel's theorem for integrals of the form / , — could not 

involve a logarithmic function. Thus these integrals would belong to the 
first or second kind, according to the value of the index e, and of X the degree 
of <p X. But in reality, though the integrals in question are of the first kind 
(that is, they admit of summation without introducing either an algebraical 
or logaritiimic function) if e be less than a certain limit, yet if it be not so 
their formula of summation will in general involve both algebraical and loga- 



ON THE RECENT PROGRESS OF ANALYSIS. 81 

lithmic functions. Either may, under certain conditions as to the form of 
(px, disappear, but while (p j? is merely known as the polynomial of the Ath 
degree, we cannot decide whether the integral is to be referred to the second 
or third kind. 

I may mention here a very elegant result due to M. Jacobi. It appears in 
the thirtieth volume of Crelle's Journal, p. 121, and is a generalisation of the 
fundamental formula for the addition of elliptic arcs. With a slight modifi- 
cation it may be thus stated. li (p x involve only even powers of x, the 

highest being j?^", then the sum of the integrals / , is equal to 

the product of their arguments, that is of the different quantities denoted by 
the symbol x. In this case then the logarithmic function disappears, and 
the integral belongs to the second kind. 

In the twenty-ninth volume of Crelle's Journal there is a paper by M. 
Richelot on a question connected with hyper-elliptic integrals. The reader 
will find in it a good many fully-developed results, which may be considered 
as particular cases of Abel's theorem. They illustrate the learned author's 
criticism of Legendre's classification of hyper-elliptic integrals, though they 
are not adduced for that purpose. 

The function M (vide ante, p. 38) is a function of the arbitrary quantities 
a, b, ... c, which, as has been remarked, may themselves be considered func- 
tions of the arguments Xi, x^, ... x^. "To determine M as a function of the 
last-written quantities is a necessary ulterior step in almost any special appli- 
cation of Abel's theorem, and this M. Richelot has done in several interesting 
cases, establishing at the same time the relations which exist among the 
quantities m question. His investigations, however, have an ulterior purpose, 
and are not to be considered merely as corollaries from Abel's theorem. 

Another paper of M. Richelot, on the subject of the Abelian integrals, is 
found in the sixteenth volume of Crelle's Journal, p. 221. The aim of it is 
to furnish the means of actually calculating the value of the Abelian integral 
of the first class by a method of successive transformation, that is, by a method 
analogous to that used for elliptic integrals. M. Richelot's process depends 
essentially on an irrational substitution, by means of which we can replace 
the proposed integrals by two others which differ only with respect to their 
limits. In the development of this idea the author confines himself to the 
first kind of the Abelian integrals of the first class, though the same method 
may m .m .he more generally applied*. From the formula which expresses 
the proposed integral as the aggregate of two others is deduced another, in 
which it is expressed by means of four integrals, the inferior limits of all being 
zero. The first and second of these integrals differ only in their amplitude, 
and the same is true of the third and fourth. There are two principal trans- 
' formations, either of which may be repeated as often as we please ; and though 
it might seem that the number of integrals would in the successive trans- 
formations increase in a geometrical progression, yet by the application of 
Abel's theorem we can always reduce them to the same number. But the 
development of this part of the subject M. Richelot has reserved for another 
occasion f. 

* The integral to be transformed is — ; ^^ -, M and 

\/ {z{l-~z) {1-k^z) {l-}>?z) {1-fifiz)} 
N, &c. being constant. 

t His transformations ultimately reduce the hyper-elliptic integral to elliptic integrals ; 
the latter may be considered known quantities, " vel per paucas adjectas transformationes 
directe computentur." 

1846. G 



82 REPORT--1846. 

At the close of his memoir, M. Richelot has given some numerical exam- 
ples of his method for the case of a complete hyper-elliptic integral. The 
third example he had previously given in a brief notice of his researches, 
published in No. 311 of Schumacher's Journal. 

33. For many years after the death of Legendre the subject of the com- 
parison of transcendents was studied principally by German and Scandi- 
navian writers* : a young French mathematician, M. Hermite, has recently 
made important discoveries in this theory ; but as the principal part of what 
he has done is as yet not published, a very imperfect outline is all that can 
be given. 

In the seventeenth volume of the ' Comptes Rendus ' we find the report 
of a commission, consisting of MM. Lame and Liouville, on a memoir pre- 
sented to the Institute by M. Hermite. This report is reprinted in the eighth 
volume of Liouville's Journal, p. 502. A remark which incidentally occurs 
in it, namely, that Abel was the first to give the general theory of the divi- 
sion of elliptic integrals, led to a very warm discussion between MM. Liou- 
ville and Libri, on the subject of the claims which, as I have already remarked, 
the latter had made with reference to this theory. 

It appears from the report, that M. Hermite has succeeded in solving the 
problem of the division of hyper-elliptic integrals. The division of elliptic 
integrals depends on the solution of an algebraical equation ; that of the 
hyper-elliptic integrals (as the functions inverse to them involve, as we have 
seen, more than one variable), on the solution of a system of simultaneous 
algebraical equations. This solution can, M. Hermite has shown, be effected 
by means of radicals asuming, as in the analogous case of elliptic functions, 
the division of the complete integrals. M. Hermite's method depends for 
the most part on the periodicity of the functions considered. A transcen- 
dental expression of the roots of the equation of the problem having been 
obtained, their algebraical values are deduced from it. 

These researches, in themselves of great interest, are yet more interesting, 
when we consider how completely they justify the views of M. Jacobi as to 
the manner in which Abel's theorem ought to be interpreted, by showing 
that his theory of the higher transcendents is no barren or artificial gene- 
ralisation. 

At page 505 of the volume of Liouville's Journal already mentioned, we 
find an extract from a letter of M. Jacobi to M. Hermite, in which, after 
congratulating him on the important discovery he had made, he points out 
that the transcendental functions X {u v), A, (?< v') (vide ante, p. 76) are alge- 
braical functions of transcendental functions which involve but one variable. 

M. Hermite's subsequent researches have embraced a much more general 
theory than that of the Abelian integrals, namely, that of the integrals of 
any algebraical function whatever. Thus his views bear the same relation 
to Abel's general theory, developed in the ' Savans Etrangers,' that those of 
M. Jacobi in the ' Considerationes Generales ' do to Abel's theorem. 

All that has yet been published with respect to them is contained in the 
* Comptes Rendus,' xviii. p. 1133, in the form of an extract of a letter from 
M. Hermite to M. Liouville. This extract is reprinted in Liouville's Journal, 
ix. p. 353. It was communicated to the Institute in June 184'4?. 

Following the course of M. Jacobi's inquiries, M. Hermite proposed to 
determine what are the differential equations of which Abel's investigations 
give the complete algebraical integrals. When this is done it suggests the 

* The papers of M. Liouville, already noticed, may be said to be an exception to this 
remark. 



ON THE RECENT PROGRESS OP ANALYSIS. 83 

nature of the inverse functions which are to be introduced. The number 
of these functions will of course vary in diflPerent cases, just as in M. Jacobi's 
less general theory. Let us suppose this number to be denoted by y, then 
each function will involve y variables. And if each of these variables be 
replaced by the sum of two new variables, then all the functions are given 
as the roots of an equation of the yth degree, whose coefficients are rational 
in terms of the corresponding functions of each of the new variables and of 
certain known algebraical functions. From hence is derived the theory of 
the periodicity of these functions. 

After some other remarks on the theory of the higher transcendents, M. 
Hermite states that the method of division of which he made use in the 
problem of the division of Abelian integrals extends also to the new trans- 
cendents now considered, but that in the theory of transformation he had 
not as yet been successful. The greater part of the remainder of this re- 
markable communication relates to elliptic functions, and has been already 
noticed. The remark just mentioned as having been made by M. Jacobi for 
the functions which are inverse to the Abelian integrals, extends, M. Hermite 
observes, to the functions which he considers. 

In conclusion, M. Hermite remarks that the method of differentiation with 
respect to the modulus of which Legendre made so much use in the theory 
of elliptic functions, may be applied to all functions of the form 

where y is given by the equation 

In concluding this report, it may be remarked that the subject of it is still 
incomplete, and that there is yet much to be done which we may hope it 
will not be found impossible to do. It is however difficult to predict the 
direction in which progress will hereafter be made. Yet I think we may 
reasonably suppose that the question of multiple periodicity, from the para- 
doxical aspect in which it has presented itself, and from its connexion with 
the general principles of the science of symbols, will sooner or later attract the 
attention of all philosophical analysts. M. Liouville's idea of considering the 
conditions to which a doubly periodic function must as such be subject, can 
scarcely be developed or extended to the higher transcendents without leading 
to results of great generality and interest. 

The detailed discussion of different classes of algebraical integrals, their 
transformations and reductions, form an endless subject of inquiry. But in 
this, as in other cases, the increasing extent of our knowledge will of itself 
tend to diminish the interest attached to the full development of particular 
portions of it; and with reference to analytical problems arising out of 
questions of physical science, the theory of the higher transcendents will it 
is probable never become of so much importance as the theory of elliptic 
functions. We have occasion to make use of circular much more frequently 
than of elliptic functions, and similarly we shall, it may be presumed, have 
less frequently to introduce the higher transcendents than elliptic functions. 
Numerical calculations of the values of the higher transcendents are therefore 
less important than similar calculations in the case of elliptic functions*. 

* The Academy of Sciences has proposed as the subject of the great mathematical prize 
for 1846 the following question : — " Perfectionner dans quelque point essentiel la theorie 
des fonctions abeliennes ou plus generalement des transcendantes qui resultent de la con- 
sideration des integrales de quantites algebriques." The memoirs are to be sent in before 
the 1st of October. 

g2 



84 REPORT — 1846. 

The following index is intended to contain references to all the papers in 
the first thirty-one volumes of Crelle's Journal, and in the first ten volumes 
of Liouville's Journal, more or less connected with the subject of this report, 
together with a considerable number of others. 

In the following Index Crelle's Journal is denoted by C, Liouville's hy L., 
and the present Report hy R. 

Abel. Ueber die Integration der differential Formel .— wenn R and p 

V R 

ganze Functionen sind C. i. 185. This paper contains formulae of re- 
duction. It is mentioned by M. Liouville, 'Journal de I'Ecole Polytech- 
nique,' 23d cah. p. 38. It appears in French in Abel's collected works, 
torn. i. 33. 

. Recherches sur les Fonctions Elliptiques — C. ii. 101, and iii. 160 

[1827]. V. Abel's works, i. HI ; also R. p. 57. 

Remarques sur quelques Proprietes Geuerales d'une certaine sorte 

de Fonctions Transcendantes. — C. iii. 313. This paper contains the 
theorem commonly known as ' Abel's Theorem.' V. Abel's works, 
i. 268 ; also R. p. 40. 

. Sur le Nombre de Transformations Differentes qu'on peut faire subir 

a une Fonction EUiptique par la Substitution d'une Fonction donnee 
du premier desse. — C. iii. 394. V. Abel's works, i. 309. 

. Theoreme General sur la Transformation des Fonctions Elliptiques 

de la seconde et de la troisieme espece. — C. iii. 402. The theorem is 
stated without demonstration. V. Abel's works, i. 317. 

. Note sur quelques Formules Elliptiques. — C. iv. 85. This paper con- 
tains developments of ellijitic functions, &c. V. Abel's works, i. 299. 

. Theoremes sur les Fonctions Elliptiques. — C. iv. 194. They relate 

to the demonstration of the theorem stated by M. Jacobi in the third 
volume of Crelle's Journal) p. 86, by means of which Abel's method 
for the division of elliptic integrals is greatly simplified. V. Abel's 
works, i. 318 ; also R. p. 59. 

. Demonstration d'une Propriete Generale d'une certaine Classe de 

Fonctions Transcendantes. — C. iv. 200. This short paper contains the 
fundamental idea of the memoir presented to the Institute in 1826. 
V. Abel's works, i. 324 ; also R. p. 40. 

. Precis d'une Theorie des Fonctions Elliptiques. — C. iv. 236 and 309. 

This jDre'm was left unfinished. V. Abel's works, i. 326 ; also R. p. 65. 

— — . Extracts from Letters to M. Crelle, one of which relates to the com- 
parison of Transcendents. — C. v. 336. V. Abel's works, ii. 253. 

. A Letter to M. Legendre. — C. vi. 73. Works, ii. 256. It contains a 

theorem proved by M. Ramus in the twenty-fourth volume of Crelle's 
Journal, p. 78 ; and another, proved by M. Liouville in the twenty-third 
cahier of the 'Journal de I'Ecole Polytechnique.' V. R. p. 41 and p. 70. 

. Memoire sur une certaine classe de Fonctions Transcendantes. Pre- 
sented to the Institute, Oct. 30, 1826, published in 1841 in the ' Memoires 
des Savans Etrangers,' vii. 176. It is the only memoir of Abel's not 
contained in the collected edition of his works published in 1 839 ; the 
editor, M. Holmboe, not having been able to procure a copy of it. 
V. R. p. 39. 

■ . Solution d'un Probleme General concernant la Transformation des 

Fonctions Elliptiques. — Schumacher's Astronomische Nachrichten, No. 
138, vi. 365. V. Abel's works, i. 253 ; also R. p. 62. 



ON THE RECENT PROGRESS OF ANALYSIS. 



85 



Abel. Addition au Memoire Precedent.— Schumacher's Astronomische 

Nach. No. 147, vii. 33. V. Abel's works, i. 275 ; also R. ubi supra. 

The following papers were published for the first time in Abel's col- 
lected works. The references are to the second volume : — 
Proprietes reraarquables de la Fonction y=(px, etc., p. 51. The 

multiple periodicity of a function inverse to a hyper-elliptic integral is 

here mentioned. V. R. p. 77. 
Sur une Propriete remarquable d'une Classe tres etendue de Fonc- 

tions Transcendantes, p. 54.. This paper contains a generalisation of a 

theorem relating to elliptic functions. 

Extension de la Theorie Precedente, p. 58. 

. Sur la Comparaison des Fonctions Transcendantes, p. 66. This paper 

contains a somewhat fuller development of his general theory than that 

which is inserted in the fourth volume of Crelle's Journal, p. 200. 

V. R. p. 40. 

Theorie des Transcendantes Elliptiques, p. 93. V. R. p. 66. 

. Demonstration de quelques Formules Elliptiques, p. 210. 

Broch. Sur quelques Proprietes d'une certaine classe des Fonctions Trans- 
cendantes.— C. XX. 178. An extension of Abel's theorem. V. R. p. 40. 

. Memoire sur les Fonctions de la forme 

s 

rx'-yP-^Y(xP)(R(xP))~''''dx, etc.— C. xxiii. 145 and 201. 

This memoir, of which the first part may be considered a generalisation 
of the preceding, is accompanied by a report of MM. Liouville and 
Cauchy. V. R. p.41. 
Bronwin. On Elliptic Functions.— Camb. Mathematical Journal, ui. 123. 
Mr. Bronwin puts the transcendental formula of transformation m a 
very neat form. 

. On M. Jacobi's Theory of Elliptic Functions — Lond., Ed. and Dub. 

Phil. Mag. xxii. 258. V. R. p. 53. 

Reply to Mr. Cayley's Remarks.— L., E. & D. Phil. Mag. xxiu. 89. 

V. R. ubi supra. 
' Catalan. Sur la Reduction d'une Classe d'Integrales Multiples.— L. iv. 323. 

. Sur les Transformations des Variables dans les Integrales Multiples. 

Memoires Couronnes par I'Academie Royale de Bruxelles, xiv. 2de 
partie, p. 1. The third part contains a transformation of a multiple 
integral leading to properties of hyper-elliptic integrals analogous to 
known properties of elliptic integrals. 
Cauchy. Comptes Rendus, xvii. 825. — V. R. p. 69. 

Cayley. Memoire sur les Fonctions doublement Periodiques. — L. x. 385. 
An enlargement of his paper on the inverse elliptic functions, published 
in the fourth volume of the Cambridge Mathematical Journal. V. R. 
p. 69. 

Remarks on the Rev. B. Bronwin's paper.— L., E. and D. Phil. Mag. 

xxii. 358. 

. Investigation of the Transformation of certain Elliptic Functions — 

L., E. and D. Phil. Mag. xxv. 352. V. R. p. 69. 

. On the Inverse Elliptic Functions — Camb. Math. Journal, iv. 257. 

V. R. p. 69. _ _ 

Chasles. Comptes Rendus de I'Institut, xvii. 838, and xix. 1239. M. 
Chasles in these two communications presents to the Institute notices 
of his geometrical researches illustrative of the theory of elliptic func- 
tions. V. R. p. 74. 



86 REPORT — 1846. 

Clausen. Schumacher's Nachrichten, xix. 178. On a particular Integral 

mentioned by Legendre. 
. Schumacher's Nachrichten, xix, 181. It is shown that the arcs of 

one of the curves, known as the Spirica of Perseus, may be rectified 

by means of an elliptic integral. 
EiSENSTEiN, Theoremes sur les Formes Cubiques. — C. xxvii. 75. At the 

end of this paper we find some developments of elliptic functions in con- 
tinued fractions. This subject is continued in the following paper of 

M. Eisenstein's. 
. Transformations remarquables de quelque Series— C. xxvii. 193. 

and xxviii. 36. See also Theorema, C. xxix. 96. 
. Bemerkungen zu den elliptischen und Abelschen Transcendenten. — 

C. xxvii. 185. M. Jacobi has criticised this paper (of which a trans- 
lation appears in Liouville's Journal, x. 445) in the thirtieth volume of 

Crelle's Journal. 
. Eleraentare Ableitung einer merkwiirdiger Relation zwischen zwei 

unendlichen Producten C. xxvii. 285. V. R. p. 70. 

— — . Beitrage zur Theorie der Elliptischen Functionen. — C. xxx. 185, 

211. This paper contains a demonstration of the fundamental formula 

of elliptic functions. V. R. p. 71. 
GuDERMANN. Integralia Elliptica Tertiae Speciei Reducendi Methodus 

Simplicior, &c.— C. xiv. 159, 185. V. R. p. 68. 
— — . Einige Bemerkungen iiber EUiptische Functionen. — C. xvi. 78. 
. Series novae quarum ope Integralia Elliptica Primse et Secundse 

Speciei computantur, &c. — C. xvi. 366. and xvii. 382. 
. Theorie der Modular Functionen und der Modular Integrale. — 

C. xviii. 1, 142, 220, 303; xix. 46, 119, 244; xx. 62, 103; xxi. 240; 

xxiii. 301 ; xxv. 281. A systematic treatise on elliptic functions. V. 

R. p. 69. 
GiJTZLAFF. ^quatio Modularis pro Transforraatione Functionum Ellipti- 

carum Septimi Ordinis. — C. xii. 173. V. R. p. 68. 

Haedenkamp. De Transformatione Integralis / / .. . ". ; rr* 

*' ^^ -\/(sm9 V — sm* f> cos^ \J/) 

— C. XX. 97. It is shown to be the product of two elliptic integrals. 

. Uber Transformation vielfacher Integrale C. xxii. 184. Analo- 
gous to the researches of M. Catalan in the ' Memoires de Bruxelles,' 
which appears to have been previously published. 

. tJber Abelsche Integrale. — C. xxv. 178. V. R. p. 80. 

Hermite. Sur la Theorie des Transcendantes a DifFerentielles Algebriques. 
— L. ix. 353. Extracted from the ' Comptes Rendus' [June 1844]. 
This note, which contains scarcely more than an indication of M. Her- 
mite's results, may be said to mark the furthest advance yet made in 
the theory of the comparison of transcendents. V. R. p. 82. 

Hill. Exemplum usus Functionum Iteratarum, &c. — C. xi. 193. This 
paper contains some interesting applications of the calculus of functions 
to the comparison of transcendents. V. R. p. 42. 

Jacobi. Addition au M6moire de M. Abel sur les Fonctions Elliptiques. 
— C. iii. 86. A short note, containing an important simplification of 
Abel's method of solving the equation of the problem of division. 
V.R. p.59. 

. Note sur la Decomposition d'un Nombre donne en quatre quarr^s. — 

C. iii. 191. The demonstration referred to is founded on elliptic 
functions. 

'. Note sur les Fonctions Elliptiques. — C. iii. 192. 



I 



ON THE RECENT PROGRESS OF ANALYSIS. 8/ 

Jacobi. Suite des Notices sur les Fonctions Elliptiques. — C. iii. 303. 

. Suite des Notices, «fec. — C. iii. 403. 

. Suite des Notices, etc. — C. iv. 185. These notes contain theorems 

stated for the most part without demonstration. V. R. p. 56. 

— — . Ueber die Anwendung der elliptischen Transcendenten auf ein 
bekanntes Problem der Elementar-geometric, u. s. w. — C. iii. 376. 
This paper contains a geometrical construction for the addition and 
multiplication of elliptic integrals of the first kind. A translation of 
the most important part appears in Liouville's Journal, x. 435. V. R. 
p. 73. 

> De Functionibus Ellipticis Commentatio. — Civ. 371. Transforma- 
tions of integrals of the second and third kinds, &c. V. R. p. 66. 

"— -r. De Functionibus Ellipticis Commentatio altera — C. vi. 397. We 
find here an elementary demonstration of M. Jacobi's theorem. V. R. 
p. 66. 

■ Note sur une nouvelle application de 1' Analyse des Fonctions Ellip- 

tiques a I'Algebre. — C. vii. 41. It relates to the development in con- 
tinued fractions of a function of the fourth degree. 

» ' • ■■ ' . Notiz zu Theorie des Fonctions Elliptiques de Legendre, Troisieme 
Supplement.— C. viii. 413. V. R. p. 67. 

—— . De Theoremate Abeliano — C. ix. 99. V. R. p. 41. 

-> . Considerationes Generales de Transcendentibus Abelianis. — C. ix. 

394 [1832]. This memoir lays the foundation of the theory of the 
higher transcendents. V. R. p. 75. 

. De Functionibus Duarum Variabilium quadrupliciter Periodicis, &c. 

— C. xiii. 55. M. Jacobi here proves the impossibility of a function of 
one variable being triply periodic. V. R. p. 76. 

. De usu Theoriae Integralium Ellipticorum et Integralium Abeliano- 

rum in Analysi Diophantea. — C. xiii. 353. It is here pointed out that 
a problem of indeterminate analysis, discussed by Euler in the posthu- 
mous memoirs recently published by the Academy of St. Petersburg, 
is in effect that of the multiplication and addition of elliptic integrals. 
Suggestions are made as to the corresponding application that might be 
made of the Abelian integrals. 

. Formulae novas in Theoria Transcendentium Fundamentales. — C. xv. 
199. Elegant elementary formulae. 

. Note von der Geodatischen Linie auf einera Ellipsoid, u. s. w. — 

C. xix. 309. M. Jacobi has here announced the important discovery 
that the equation to the shortest line on an ellipsoid is expressible by 
means of Abelian integrals of the first class. As this is perhaps the first 
application made of Abelian integrals since their recognition as elements 
of analysis, I have thought it well to mention it in this place. A trans- 
lation of the note is found in Liouville's Journal, vi. 267. 

Deraonstratio nova Theorematis Abeliani. — C. xxiv. 28. V. R. 
p. 80. 

Zur Theorie der elliptischen Functionen. — C. xxvi. 93. This paper 
contains series for the calculation of elliptic functions, and a table of 
the function q. 

Ueber die Additions-theoreme der Abelschen Integrale zweiter und 
dritter Gattung. — C. xxx. 121. We find here some remarkable formulae. 
V. R. p.81. 

Note sur les Fonctions Abeliennes. — C. xxx. 183. This note relates 
principally to the fact announced in M. Jacobi's letter to M. Hermite, 
V. L. viii. 505. 



88 REPORT— 1846. 

Jacobi. Ueber einige die Elliptischen Functionen betreffenden Formeln. — 

C. XXX. 269. 

Extrait d'une Lettre a M. Hermite. — L. viii. 505. V. R. p. 82. 

. Extraits de deux Lettres de M. Jacobi, &c — Schumacher's Nach- 

richteu, vi. 33 [Sept. 1827]. They contain the first announcement of 

his theorem. 
— — . Demonstratio Theorematis ad Theoriam Functionum EUipticarum 

Spectantis. — Schuraaclier, vi. 133. The first published demonstration 

of his theorem. See also Legendre at p. 201 of the same volume. 

V. R. pp. 47 and 48. 
JiJRGENSEN. Sur la Sommation des Transcendantes a DifFerentielles Alge- 

briques. — C. xix. 113. 

■ Remarques Generales sur les Transcendantes a Differentielles Alge- 
briques. — C. xxiii. 126. V. R. p. 41. 

Ivory. On the Theory of the Elliptic Transcendents. — Phil. Trans., 1831, 

p. 349. V. R. p. 67. 
LiBRi. Sur la Theorie des Nombres. — Memoires des Savans Etrangers, v. 1 . 
. Sur la Resolution des Equations Algebriques, &c C. x. 167. These 

memoirs are referred to in the controversy between MM. Liouville and 

Libri. 
Liouville. Sur les Integrales de Valeur Algebrique. — Journal de I'Ecole 

Poly technique, cah. xxii. 124 and 149. These two memoirs are printed 

also in the fifth volume of the 'Memoires des Savans Etrangers,' pp. 

76, 105. Poisson's report on them is inserted in the tenth volume of 

Crelle's Journal, v. infra. 
— — . Sur les Transcendantes EUiptiques de Premiere et de Seconde 

Espece. — Journ. de I'Ecole Polytech., cah. xxiii. 37. V. R. p. 70. 

■ Note sur la Determination des Integrales dont la Valeur est Alge- 
brique. — C. X. 347. This note is appended to Poisson's report. 

— — . Sur ITntegration d'une Classe de Fonctions Transcendantes. — C. xiii. 
93. On the same general subject as the preceding memoirs. 

. Sur la Classification des Transcendantes. — L. ii. 56, and iii. 523. 

These papers contain an exposition of the principles on which this clas- 
sification is to be effected. 

■ Sur les Transcendantes EUiptiques de Premiere et de Seconde Espece 
considerees comme Fonctions de leurs Modules. — L. v. 34 and 441. It 
is proved that these transcendents so considered cannot be reduced to 
algebraical functions. 

Rapport fait a I'Academie des Sciences, &c. — L. viii. 502. Report 

on M. Hermite's memoir. V. R. p. 82. 
— — . Sur la Division du Perimetre de la Lemniscate. — L. viii. 507. 

V. R. p. 62. 
. Rapport sur le Memoire de M. Serret sur la Representation Geo- 

metrique des Fonctions EUiptiques et Ultra-elliptiques. — L. x. 290. A 

note is appended to this report generalising M. Serret's theory. V. R. 

p. 72. 
Sur un Memoire de M. Serret, &c.— L. x. 456. V. R. p. 73. 

LoBATTO. Sur rintegration de la Difi'erentielle — . — „ 

— C. X. 280. \^w^ + ax^+^j:' + y ic+S 

LucHTERHANDT. De Transformatione Expressionis 

^ &c. 

— C. xvii. 248. 



ON THE RECENT PROGRESS OF ANALYSIS. 89 

MacCullagh. Transactions of the Royal Irish Academy, xvi. 76. An ele- 
gant geometrical proof of Landen's theorem. 

Minding. Theoreme relatif a une certaine Fonction Transcendante. — 
C. ix. 295. The function in question was shown by M. Richelot to be 
reducible to elliptic integrals. _ 

Sur les Integrales de la forme f^^^ ^ , &c.— C. x. 195. An ad- 

dition to this memoir is found at p. 292 of the same volume. 

. Recherches sur la Sommation d'un certain nombre de Fouctions 

Transcendantes, &c. — C. xi. 373. These researches relate to an exten- 
sion of Abel's theorem. 

Propositiones quasdam de Integralibus Functionum Algebraicarum 

unius variabilis e principiis Abelianis derivatse. — C. xxiii. 255. This, 
memoir is mentioned by M. Hermite. 

PoissoN. Rapport sur deux Memoires de M. J. Liouville, &c. — C. x. 342. 
V. supra, Liouville. 

Theoremes relatifs aux Integrales des Fonctions Algebriques — 

C. xii. 89. V. R. p. 41. 

Raabe. Bemerkungen zum Principe der doppelten Substitution, u. s. w. — 
C. XV. 191. 

Ramus. De Integralibus DifFerentialium Algebraicarum. — C. xxiv. 69. 
V. R. p. 41. 

Richelot. Note sur le Theoreme, &c. — C. ix. 407. V. supra, Minding. 

. De Integralibus Abelianis Primi Ordinis Commentatio Prima. — 

C. xii. 181. V. R. p. 80. 

. De Transformatione Integralium Abelianorum Primi Ordinis Com- 
mentatio.— C. xvi, 221 and 285. V. R. p. 81. 

. Ueber die Integration eines merkwiirdigen Systems Differential- 

gleichungen. — C. xxiii. 354. These equations are those known as the 
" Jacobische System." V. R. p. 78. 

Einige neue Integral-gleichungen des Jacobischen Systems DilTeren- 

tial-gleichungen. — C. xxv. 97- The results contained in this paper are 
much more general than those of the preceding one. V. R. p. 80. 

Nova Theoremata de Functionum Abelianorum cujusque ordinis 

Valoribus, &c.— C. xxix. 281. V. R. p. 81. 

. Ueber die auf wiederholten Transformationen beruhende Berech- 

nung der ultra-elliptischen Transcendenten, — Schumacher Astr. Nach. 
xiii. 361 [July, 1836]. V. R. p. 82. 

Roberts. Sur une Representation Geometrique des Fonctions Elliptiques 
de Premiere Espece.- — L. viii. 263. 

. Sur une Representation Geometrique des Trois Fonctions Ellip- 
tiques. — L. ix. 155. Mr. Roberts's papers relate to curves formed by 
the intersection of a cone of the second order with a sphere. The fol- 
lowing paper contains a more general exposition of his views. 

. Memoire sur quelques Proprietes Geometriques relatives aux Fonc- 
tions Elliptiques.— L, x, 297. V. R. p. 73. 

RosENHAiN. Exercitationes Analyticaa in Theorema Abelianum de Inte- 
gralibus Functionum Algebraicarum. — C. xxviii. 249, and xxix. I. 
V. R. p. 41. 
Sanio. De Functionum Ellipticarum Multiplicatione et Transformatione 
quae ad numei'um parem pertinet Commentatio. — C. xiv. 1. V. R. p. 68. 
Serret. Note sur les Fonctions Elliptiques de Premiere Espece. — L. viii. 
145. V. R. p.72. 



90 REPORT — 1846. 

Serret. Proprietes Geometriques relatives a la Theorie des Fonctions 
EUiptiques. — L. viii. 495. 

Note a I'occasion du Memoire de M. William Roberts, &c. — L. ix. 

160. 

Memoire sur la Representation Geometrique des Fonctions EUip- 
tiques et Ultra-elliptiques. Addition au memoire precedent. — L. x. 
257 and 286. It was on this memoir that M. Liouville made so favour- 
able a report to the Institute. V. R. p. 72. 

. Developpemens sur une Classe d'Equations relatives a la Represen- 
tation des Fonctions EUiptiques. — L. x. 351. 

. Note sur les Courbes EUiptiques de la Premiere Classe. — L. x. 421. 

. Sur la Representation des Fonctions EUiptiques de Premiere Espece. 

— Camb. and Dublin Math. Journ. i. p. 187. 

SoHNCKE. jSlquationes Modulares pro Transformatione Functionum EUip- 
ticarum et undecimi et decimi tertii et decimi septimi ordinis. — C. xii. 
178. M. Sohncke here gives the results which he investigates by a ge- 
neral method in the following paper. 

. ^quationes Modulares, &c.— C. xvi. 97. V. R. p. 68. 

Talbot. Researches in the Integral Calculus. — Phil. Trans. 1836, p. 177 ; 
1837, p. 1. V. R. p. 41. 



On Comparative Analytical Researches on Sea Water. 

By Prof. FORCHHAMMER. 

In a paper read today in the Chemical Section, I have tried to show that 
in the ocean between Europe and America, the greatest quantity of saline 
matter is found in the tropical region far from any land ; in such places 1000 
parts of sea water contain 36'6 parts of salt. This quantity diminishes in 
approaching the coast, on account of the masses of fresh water which the 
rivers throw into the sea; it diminishes likewise in the westernmost part of 
the Gulf-stream, where I only found it to be 35"9 in 1000 parts of the water. 
By the evaporation of the water of this warm current, its quantity of saline 
matter increases towards the east, and reaches in N. lat. 39° 39' and W. long. 
55° 16', its former height of 36*5. From thence it decreases slowly towards 
the north-east, and sea water, at a distance of sixty to eighty miles from the 
western shores of England, contains only 35*7 parts of solid substances ; and 
the same quantity of salt is found all over the north-eastern part of the At- 
lantic as far to the north as Iceland, always at such a distance from the land 
that the influence of fresh water from the land is avoided. From numerous 
observations made on the shores of Iceland and the Faroe islands, it is evi- 
dent that the water of the Gulf-stream spreads over this part of the Atlantic 
Ocean, and thus we see that water of tropical currents will keep its character 
even in high northern latitudes. 

Besides the southerly direction which any current flowing from the north- 
ern polar regions must take, it will, according to well-known physical laws 
depending upon the rotation of the earth, always take a direction towards the 
west, and thus be driven towards the eastern shores of the continents, while 
any tropical current flowing towards the north will, according to the same law 
of rotation, take a direction towards the western shores of the continents. 
This is at present the case in the Atlantic Ocean, and its effects upon the 
shores of Europe, which by a branch of a tropical current are surrounded by 
warm water, produce a mild and moist climate. 



ANALYSIS OF SEA-WATER, 91 

The water of the different seas is much more uniform in its composition 
than is generally believed. In that respect my analyses agree with the newer 
analyses of atmospheric air, in showing that the differences are very slight 
indeed. Sea water may contain more or less salt, from a very small quantity, 
as in the interior part of the Baltic, to an amount of 37*1 parts in 1000 parts, 
which I found in water from Malta, and which is the greatest quantity I ever 
observed ; but the relative proportion of its constituent saline parts changes 
very little. 

In order to get rid of those differences which might arise from the dif- 
ferent quantity of saline matter in sea water, I have compared sulphuric acid 
and lime with chlorine, and the following results are the mean of many ana- 
lyses : — 

In the Atlantic, the proportion between chlorine and sulphuric acid is 
10,000 : 1188; this is the mean of twenty analyses, which differ very little 
from each other. 

In the sea between the Faroe islands, Iceland and Greenland, the same 
proportion, according to the mean of seventeen analyses, is 10,000 : 1193. 

In the German Ocean, according to ten analyses, it is 10,000 : 1191. 

In Davis's Straits, according to the mean of five analyses, it is 10,000 : 1220. 

In the Kattegat, according to the mean of four analyses, it is 10,000 : 1240. 

Thus it appears that the proportion of sulphuric acid increases near the 
shores, a fact which evidently depends upon the rivers carrying sulphate of 
lime into the sea. 

The proportion between chlorine and lime in the Atlantic Ocean, according 
to the mean result of seventeen analyses, is 1 0,000 : 297 ; and in the sea 
between Faroe and Greenland, according to the mean of eighteen analyses, 
it is 10,000 : 300. 

In the longitude of Greenland, and more than 100 miles to the south of 
the southernmost point of that large tract of land, sea water contains only 
35*0 in 1000 parts. In going from this point towards the north-west it de- 
creases constantly, and in Davis's Straits, at a distance of about forty miles 
from the land, it only contains 32*5 parts of salt in 1000 parts of sea water. 
This character seems to remain in the current which runs parallel to the 
shores of North America ; and at N. lat. 43^-° and W. long. 46| the sea water 
contained only 33*8 parts of salt. Thus tropical and polar currents seem not 
only to be different in respect to their temperature, but also in the quantity 
of salt which they contain ; from which it appears, that while the quantity 
of water carried away from the tropical sea by evaporation is greater than 
that which rain and the rivers give back to that sea, the reverse takes place 
in the polar seas, where evaporation is very small and the condensation of 
vapour very great. The circulation must on that account be such, that a 
part of the vapour which rises in tropical zones will be condensed in polar 
regions, and in the form of polar currents flow back again to warmer climates. 
Although my analyses are only made on water from the ocean between Eu- 
rope and America, yet little doubt can be entertained that that part of the 
ocean which separates America from Asia is constituted in a similar manner, 
and that currents flowing from the poles are the rule, and currents flowing 
towards the poles the exception. 

Lime is rather rare in the sea around the West Indian islands, where mil- 
lions of coralline animals constantly absorb it, the proportion according to 
five analyses being 10,000 : 247 ; and it is rather copious in the Kattegat, 
where the numerous rivers of the Baltic carry a great quantity of it into the 
ocean. The proportion is there, according to four analyses, 10,000 : 371. 



92 REPORT — 1846. 

On the Calculation of the Gaussian Constants for 1829. By A. Erman. 

As purely theoretical speculations on natural phEenomena remain highly- 
unsatisfactory until they can be founded on a sufficient number of observa- 
tions, in the same manner collections of the most careful observations must 
be almost useless before they are thoroughly elaborated according to a given 
theory. Nay, the accumulation of observed numbers, notwithstanding the 
value they possess when viewed by themselves, may even become injurious 
to science, by retarding its progress. Indeed the aspect of progressively in- 
creasing, but not duly elaborated, materials, must at last give rise to the ap- 
prehension, both on the part of those engaged in furnishing them, and of 
every one interested in the results to be gathered from them, that the means 
may be wanting to bring such a stock of matter to bear for their proper 
purpose. The loss of the whole, that is to say, of data which have not been 
acquired without the exertion of considerable scientific labour, and which 
seemed pregnant with beautiful germs, would then be a most discouraging 
consequence. 

The British Association for the Advancement of Science has many times 
proved itself convinced of the truth of this principle. A resolution adopted by 
the Association in 1833, during its first meeting at Cambridge, warded oflT the 
peril just mentioned, even from a department of science whose long-established 
rate of progress had not been able to protect it sufficiently against such a 
risk. The reduction of the Greenwich observations of planets, undertaken in 
consequence of this resolution, and now published by order of the Lords 
Commissioners of the Admiralty, has been fully appreciated by all astrono- 
mers, and particularly by the late M. Besscl, who in the last moments of his 
life welcomed it as the beginning of a new period in astronomy. Moreover, 
the condition that a uniform progress of observation and calculation is equally 
indispensable in less-developed or only nascent branches of physical science, 
has been expressed by the British Association at its second meeting at 
Cambridge in 1845 ; first, by several of the members being inclined to 
raise the question, whether the continuation of magnetic and meteorological 
observations were desirable, as long as a great part of the materials collected 
by them are still waiting their first employment ; and, secondly, by including 
the calculation of the Gaussian constants of terrestrial magnetism for 1829 
within the sphere of their own operations, being pleased at the same time to 
entrust me M'ith the superintendence of the same, and to place at my disposal 
the sum of £50, granted for this purpose for the year 1845 to 1846. I shall 
endeavour to point out in a few words the fruits this arrangement seems to 
promise, and the results it has already obtained. 

I think we are authorised to suppose that all those phaenomena which we 
have learned to express by numbers, with the help of remarkably accurate 
instruments, will at length lead us to a theory of the forces which produce 
them ; and that, in consequence, the intrinsic value of observations on such 
phaenomena — a value which hitherto could not be demonstrated — will then 
at once become most evident. It was this expectation alone which often 
encouraged observers to persevere in labours apparently rather tedious, and 
the zeal with which the meteorological and part of the magnetic variations 
are pursued by your members in British and colonial observatories, is, I think, 
attributable to the same cause. In the branches of physics which they cul- 
tivate, these observers, it is true, have still to look to futurity for both kinds 
of progress, viz. the discovery of an abstract theory, and the true establishment 
of the same by means of observed numbers. As to the first and most import- 



ON THE GAUSSIAN CONSTANTS FOB 1829. 93 

ant of these steps, they have a consolation in the fate of their predecessors 
in most similar labours : I allude to the long series of philosophers who de- 
voted themselves during the first thirty years of this century to ascertaining 
the mean values of magnetic elements for as many points on the surface of 
the globe as possible, and whose undertakings are so carefully recorded by 
one of them — I mean Col. Sabine, in his admirable report on magnetic in- 
tensity. They too were long enough under the necessity of restricting the 
immediate application of their operations to refuting some evidently super- 
ficial or erroneous theoretical views, and then, after detaching from their re- 
sults every accidental influence, to register them in the annals of science, as 
contributions to a theory which they only hoped might be attained. But M. 
Gauss's admirable theorem, that any terrestro-magnetical element, that is 
to say, any observable part of the intensity of magnetic force at a given point 
of the earth, or any angle formed by this force with a given plane or line, can 
be represented by combining with given functions of the latitude and longi- 
tude of this point a limited number (probably twenty-four) of constant quan- 
tities, and the way pointed out by him for deriving these constants from a 
sufficient number of observed mean values of magnetic elements, have in a 
short time so completely realized these hopes, that a great encouragement 
was held out, both to former observers of mean magnetic elements, and 
to those who were then, and are still employed in less-advanced branches of 
physics: nevertheless this encouragement was but an imperfect one. To 
complete it, the possibility of applying those former observations had to be 
changed to a reality. On this account I am inclined to think that the com- 
mittee appointed to conduct the cooperation of the British Association in the 
system of combined magnetic and meteorological observations, have parti- 
cularly contributed to the satisfaction of their own observers, by encouraging 
the calculation of the Gaussian constants for 1829; for, by so doing, they in the 
first place have confirmed their adhesion to the general principle, that no set of 
observations whatever must remain longer than is indispensably necessary 
without reduction to theory; and secondly, they have made the immediate ap- 
plication of the mean magnetic values for 184-5, that may be furnished by the 
combined British and Russian observatories, the more probable, as it will then 
be already preceded by a similar application of the analogous values for 1829. 
Besides this, to prove the influence of your resolution on the dej)artment of sci- 
ence most directly connected with it, I may remark that a more and more exact 
determination of magnetic constants (the Gaussian) is equally indispensable 
at the present moment (and for the same reason), as the obtaining of the con- 
stants for planetary orbits was formerly, from the moment in which Kepler's 
and Newton's discoveries opened a possibility of arriving at them. Whatever 
may be the analogies once to be found in magnetism for the secular varia- 
tions and other perturbations of planetary orbits, the entrance into these 
untouched fields of science cannot fail to be effected by fixing the actual 
values of Gaussian constants. 

It was under these circumstances that I long ago felt it to be a debt I had 
contracted towards science, that the magnetic elements which I observed 
from 1828 to 1830, at about 650 equidistant stations, on a line encircling 
the globe, between latitudes 67° north and 60° south, conjointly, perhaps, 
with the magnetic elements that had been observed in Europe during the 
same years, should be fully applied to the development of the now existing 
theory. For the undertaking of such a work, however, it was evidently ne- 
cessary to have more time at my disposal than I have ever enjoyed. M. 
Henry Petersen, too, a most industrious and talented young mathematician, 
who in 1842 had undertaken and performed at my request a small part of 



94 REPORT — 1846. 

this comprehensive task, found his leisure hours unequal to its completion. 
Now, on the contrary, the support of the British Association has enabled and 
induced this gentleman to suspend his other official duties for the year just 
expired, and to devote himself entirely to the prosecution of the work in 
question, in which his success will, I think, be appreciated by the Association, 
from the results which I have the pleasure of laying before you, accompanied 
by some remarks on the means employed to obtain them. What proportions 
these one year's results bear to the final term of the whole labour, and how 
far they deserve to be continued, is a subject which I shall take the liberty 
of touching upon at the conclusion of this paper. 

The object of the calculations committed to my superintendence may be 
stated to consist in finding, by a sufficiently large series of observations, 
twenty-four corrections, — 

^g*fi, A^*'i, AA*.', A^r*.* A^'.'j AA'.', 

to be singly applied to the twenty-four preliminary values, 

5r*.«, Sr4,l, ^4,1, /j4,2 gUX^ ^i,,, 

which M. Gauss assigned to the constants of terrestrial magnetism, and in 
calculating at the same time the probable errors of the so-corrected constants. 
To this effect (preserving the literal denominations used in M. Gauss's theory 
of terrestrial magnetism, and marking by AX, AY, AZ, Aw, wAJ, ^j/Ai, the 
differences (theoretical value — actual (or observed) value)), the following 
expressions have been derived : — 

0=AX + Ay''OsinM— cosM(A5''''.cosX+AA'>'.sinA.)-f-2cosM.sinM.A5'®'° 
—cos 2 u . (A^2,i.cos x-i- AA^.i.sin A)— sin 2 u (^g^'^.cos 2 X 

+ Ah^>^.sin2X) + 3.Ag^fi.{cos'^u—-\.smu—f3cos^u——cosu\ 

(Ag'S'' . cos X -f AA3'' . sin X)— sin u.(3 cos* u—l)(Ag3fi^ cos 2 X 
-f- Ah^fi.sin 2 X)— 3 cos u. sin^ u(Ag^'3 . cos 3 X+ AA^.s . sin 3 X) 

+4[ cos^Ji— -coswJsinM.A^'"— [4cosm4— — - cos*m + - J. 

(A(/^''.cosX + AA4.' .sin X)— 2( 2 cos' «— -cosm JsinM.(A^'2.cos2X 

-i- A/i*- . sin 2 X) — (4 cos2i« — 1 )sin2 m( A^f*.' . cos 3 X -t- AA4'3 . sin 3 X) 
—4 cos u . sin' u (Ag*'* . cos 4 X+ AA*'* . sin 4 X). 

0=AY-l-A5'''i.sinX — AAi'' . cos X + cos m (A^r^.i .sinX— AA^.' . cos X) 
+ 2sin7i(A5ra.2.sin2X— AA2.2.cos2x) + (A^3.'.sinX— AA'-'cosX), 

fcos^ «<— -] + sin 2 u . (A<73.« , sin 2 X— AA^.^ . cos 2 X) 

-1-3 sin-?<(Ap3.3 .sin 3 X— AA^.^ . cos 3X)-f | cos'm— -cosm j. 

( A^r*.' .sin X— AA*-' .cos X)+ [2 cos^ m— 1 Jsin u (Ag*'^ . sin 2 X 

— AA4'2 cos 2 X)-t-3 cos M . sin«7^ (A^*.' . sin 3 X— AA*-' cos. 3 X) 
-1-4 sin^.M (A^'-*"* . sin 4 /— AA^.* . cos 4 X). 



(!•) 



(2.) 



ON THE GAUSSIAN CONSTANTS FOR 1829. 95 

0=AZ+2 cosM . A5'''° + 2sin?i . (A^'>i . cos A + A/t''' . sin X) 

+ (3 cos2 M— 1) . A^-^'O + S cos ?< . sin «< (A^^.i . cos X + A/t^.i . sin X) 
+38in9M(A^«>2.cos2X + AA«'«.sin2X) + ('4!COs3«— — cosMJ.A^3,o 

+ ('4 cosStt— - jsin « (A^-s-'cos X+ A/i3>> sin X)+4 cos u sin-u. 
(A^3,2.cos2X + AA3.a.sin2X)+4sin3M.(A5'3>'.co83X+AA3.3.sin3X) ("(S.) 
+ ^5 cos «* -— cos^M + 1) . Atr*'" + (5 cos3 M - y cos M V 

sinM(A5r4.i cosX + AA*'' sin X) + [5cos*m— - jsin3M(A^>".cos2X 

+ AA*''* sin 2 X) + 5 cos 4 . sin^ u {^g*'^ . cos 3 X + AA^-^ . sin 3 X) 
+ 5 sin* z< . ( A^'* . cos 4 X + AA*-* . sin 4 X) ; 
and denoting by AX, AY, AZ their just- mentioned developments according 
to the corrections of constants, 

0=Am; —cos J. ax — sinJ.AY (4.) 

0=M>.AJ. + sin5.AX — cosJ.AY (5.) 

0=»I'. Ai+sin2(cos5.AX+sinJ.AY)— cosi.AZ . . (6.) 

and then 283 numerical primary equations, relating to magnetic elements, 
observed on a line from Berlin to the east coast of North Asia, at the port 
of St. Peter and St. Paul, have been formed, by alternately i-ecurring to one 
or the other of these six formulae. For the sake of uniformity, their first term, 
which always meant the value of the magnetic element calculated with M. 
Gauss s numbers — the observed value of the same element, — has always been 
marked by the letter n, independently of its having been derived by the 1st, 

the 2nd, the 6th of these formulae ; and also the whole numerical primary 

equation has been represented by 

0=n + coeff. A5r*'°-( A^'O") + coefF.A^^-^CA^^.i) + coeffiA/t*.'. 

(AA*.i) +coeff.A/ji''.(AA'.>), 

independently of their origin from formulae (1) (2) or (6). 

In five of the accompanying tables you will find, according to these de- 
nominations, — 1. the numerical values for 

log. w, log. coeff. A^"", log. coeff. A^*-', log. coeff. AA*-' log. coefi^.A/i'-'fj 

furnished by each single element ; 2. the name, the latitude and the lon- 
gitude of a station to mark the place of observation ; and 3. one of the letters 
X, Y, Z, V), $ or i, which respectively indicate that the observed element has 
been the northern or the western component of horizontal force, the vertical 
force, the whole horizontal force, a declination or an inclination*. In the 
three last cases in which therefore n denotes the value of Aw, of toAS, or of 
\[/At|, it must still be noticed that the values of the declination (S) in the 
first, and respectively of the horizontal force (w), or of the total force (4/) in 

* It k understood that, according to M. Gauss's memoir, the meaning of the letters em- 
ployed is, — 



X Northern horizontal force. 
Y Western horizontal force. 
w Total horizontal force. 
Z Vertical force. 
'</' Total force. 



S Declination. 
i Inclination. 

M North polar distance of the station. 
X Longitude east from Greenwich of the 
station. 



t Instead of log. coefF. the further abbreviation I.e. being usually employed. 

t The arcs A^ and Ai being previously changed to the ratio of their sines to unity. 



96 KEPORT — 1846. 

the two other, being only approximately re<iuired for this purpose, have been 
merely calculated by theory, viz. by those values of constants which we are 
about to correct. 

The correctness of the numbers in those 283 primary equations for the 
twenty-four unknown, has then been controlled by determining the theoretic 
values of the X, theY, the Z, or of the two or three of them that were required 
for the composition of to, of tang S or tang i, a second time in a somewhat 
different way. It consisted in calculating according to 

X=F(m) + F' (u) cos X+F" (u) sin A + F"' (u) cos 2 K 

+ + F™(M)cos4A + F""(«)sin4X, 

or to a quite analogous expression for Y and Z, for which the numerical values 
contained in F(m), F'(m) F''"'(m) are given as resulting from the pre- 
liminary values of the twenty-four constants in M. Gauss's theory of terres- 
trial magnetism, § 27. 

This part of the task being completed, the second, and by far the most 
laborious one, consisted in forming out of the coefficients in each primary 

equation ^^^^" =325, and therefore altogether 325 x 283 = 91975 products 

of two factors, each according to this form, — 

un, w.(c.A5'*>°), w(c.A5'*''), ^.(c.A^r".!), «(c.A/^''>), 

(c. A^^.o).( cA^^.o), (c. A5r*.o).(c. A^*.> ), . . . (c. A5r*.o).(c. A^r' , ' ), (c. A^+.o).(c. AA' -• ), 
(c.A</*.')-(c.A5r4,'), ... (c.A5r4.i).(c.A^'.'), (c.A^*.').(c.A/t'.'> 



(c.A^>.')-(c.A^'''). (c.A5r'.>).(c.A/i'.'), 
(c.A/i'.').(c.A/<i.'). 

The 283 products, assembled under each of these 325 titles, were then 
separately summed up, and by this means (marking by [] a sum of analogous 
terms) the twenty-four final equations of the following form were obtained: — 

-[M.(c.A^*.o)] = [(c.A5r4,o).(c.A^4,o)].A^4,o+[;(c.A^*.o)(c.A5'*.')]-A^*'' + " 
-f-[(c.A5r4,o)(c.A^i,i)].A^'.i + C(c.A^.o)(c.A/ii.')]-^^'''' 

+ [(c.A5?4.')(c.A^'.')].A^'.' + [(c.A5'''.i)(c.AA'.')].AA'.', 

— [w.(c.AA4.')] = [(c.A/t*.').(c.A5't.o)].A</4.<'+[(c.A/«4.')(c.A^-*,ij].^^4,i + ,. 

-l-[(c.A/i4.i)(c.A^'.')].A^'.'+[(c.AA4.')(c.A/t'.')].A/*''', 



— [w.(c.AA''')] = [(c.AA'.')(c.A5r4.o)].A^*.o+[(c.A/i''')(c.A<74.')].A^.' + .. 
-f-[(c.AA'.')-(c-A5'''')]-^^''' + [(c-A/i''')(c.AA>.i)].AA'.'. 

The numerical expressions of these equations will be found in the table 
marked VI. Hitherto they have been controlled by the calculation leading 
to them from the primary equations, being repeated a second time in the 
same manner as the first, but with the suppression of one decimal figure in 
the products and in their sums. To obviate the danger, arising from the ad- 
dition of such extensive rows of numbers, lest the compensation of opposite 
errors might produce an illusory agreement, M. Petersen, besides the forma- 
tion of new primary equations, has proceeded to subject these final equations 
to another kind of control, — I mean the process usually employed in similar 



ON THE GAUSSIAN CONSTANTS FOR 1829. 



97 



cases, and which consists in the formation for each primary equation of a 
supplementary term s, equal to the sura of the other, viz. in one case, in the 
calculation of 

s=coeff. A^'+.o + coeff. A^^-' +coefF. AA*-' + + coeff. A^i-' + coeff. AA'.>, 

whereby we obtain as controlling equations, 

[«»] = [w.c.A^'-'.o] + [w.c.A^'*''] + [?i.c.AA*.'] + ... + [7i.cAg^'^2 + [«-c.A/i'.'], 

[«.c.A^*.o] = licAg^fiXc.Ag'fi)} + [(c.A^*.o)(c.Ai/*.0] +..+ [(c.A5r4,o)(c,A/ii>')], 



[5.c.AA'.i] = [(c.AA'.i)(c.A^*''')] + + [(c.A/t''')(c.A/i'.0]' 

independently of the extension which is given to the sums marked by []. 

If we now consider, in the first place, the linear primary equations con- 
tained in the Tables annexed, we shall find that the values of the Gaus- 
sian constants hitherto accepted sufficiently approximate to the truth to 
authorise the supposition from which we start, that the powers of their cor- 
rections superior to the first can be neglected ; but, on the other hand, that 
these values are still so erroneous, that the elements calculated by them differ 
from the empirical ones by far more than can be ascribed to errors of obser- 
vation, and in quite another manner than would arise from local irregularities 
of terrestrio-magnetic power. Indeed we see that when the places of obser- 
vation are in similar parts of the globe, the values of n belonging to them 
remain nearly enough equal to each other ; whereas on the longitudes of the 
places increasing, these values of n are gradually lessened, and at length 
replaced by a series of values with opposite signs. Of course, to observe this 
regularity of progress, we must only compare such values of )i as relate to 
magnetic elements of a similar character ; as for instance, all to x, or all to 
w, and so on. 

The value [ww] =233423, marked in the Table of Final Equations as re- 
sulting from 283 equations, shows that for the part of the earth on which 
the observations hitherto considered have taken place, the mean difference 
between an observed magnetic element and the corresponding calculated one 
amounts to 29, the intensity of the whole force at London being =1372 ; and 
in agreement with this result, we find, for example, by immediate inspection 
of the Table of Primary Equations, — 



The mean difference 


Lat. 56. 


56. 


67. 


54. 


59. 


56. 


1^ 

Is 


Long. 43. 


60. 


67. 


100. 


145. 


222. 


In northern horizontal force 
In western horizontal force 
In perpendicular force 


+4.a 

+ 18 
+ 11 


+ 39 
+ 19 
+ 13 


+ 23 
+ 40 
+ 35 


- 6 



-15 


-28 
-25 
+ 30 


j +35 
+ 38 



The concluding table contains besides, as already mentioned, twenty- four 
final equations for the twenty-four unknown quantities, by which, mathema- 
tically speaking, the whole problem in question would appear to be ready for 
a definitive and now most easy solution. In practice, however, this is evi- 
dently far from being the case. Thus indeed it is plain, even at a first 
glance, that each observation, hitherto registered, has already contributed 
to the solubility of the problem all that it will ever be able to do ; there 
is not in this circumstance alone a sufficient reason for that solubility being 
reached, and then that, on the contrary, the probability of the value to 

1846. H 



98 REPORT— 1846. 

be obtained for any one of the twenty-four unknown quantities, for ex- 
ample, A^f''' depends entirely on its weight, that is to say, on the magni- 
tude of the coefficient with which this quantity remains in the equation 

containing at the origin the terms + [(c.A^''').(c.A^''')].A5'''', after 

the elimination of the twenty-three others ; and that these weights, as is easily 
shown, will only become sufficiently extensive when there are neither two 
nor more of the unknown quantities whose coefficients remain in a constant, 
or in a nearly constant relation in the whole series of primary equations, tri- 
butary to the final ones. Hence the examination in this same respect of the 

above expressions for Aa;, Ay, ^^i, will easily show that by reason 

of the similarity of the latitudes in which by far the greatest part of the ob- 
servations till now calculated have been effected, the solution of the final 
equations as hitherto obtained would give but a very trifling weight to almost 
all the corrections we are seeking for, and therefore be still without interest. 

Even the seventy elements that, according to date of observation, follow 
next to the 283 finished ones, and for which M. Petersen has also nearly 
accomplished the primary equations, will, by their contributing to the final 
ones, most sensibly improve them in respect of solubility. Indeed in full 
opposition to the now finished ones, these latter elements relate to points of 
a line rather northern than eastern in its direction, and extending from 57° 
latitude north to about the equator. It is therefore precisely those unknown 
quantities whose coefficients have hitherto exhibited the least variations, or 
followed in their varying a course parallel with that of the coefficients of 
other unknown ones, that will vary the most, both relatively and absolutely 
speaking, in the set of primary equations next to be formed, and thus will 
add to the final equations just what is requisite for increasing the weight of 
each of the quantities sought for and thereby preparing their due separation. 

M. Petersen will at all events subjoin, in the course of the ensuing months, 
this next continuation of his present labour, which I shall then forward to the 
Association. 

It is plain, notwithstanding, that even then we shall not have reached the 
most favourable state which our fund of observations for the year 1829 would 
allow of attaining in the knowledge of the Gaussian constants for the same 
year. Indeed even then, by substituting, as could be effected through your 
further patronage, a full execution of the task to a but partial one, the places 
of observation contributing to the final equations may be trebled, and what 
is still more important, a considerable improvement be attained in their re- 
petition over the globe. 

I thought it my duty to submit to the Association my opinion of the 
benefits to be derived from the continuation of M. Petersen's labours, leaving 
it to their decision whether they will consider it advisable to grant him their 
further support in devoting himself entirely to the prosecution of his task. 




IAN CONSTANTS 



;. coef. 


Log. coef. 


Log. coef. 


,p3'l. 


Afi^-K 


A^3,2. 


4.8 3 5w 


8-31498W 


1 
9-4867771 


6233W 


8-32996w 


9-485297? 


2229 


7-90650 


9-452857} 


9225 


8-87940 


9-1765771 


953^ 


8-87372 


8-4709971 


6564 


8-84974 


8-339867* 


8701 


8-89717 


8-14350 


io6i 


8-96857 


8-81254 


0364 


8-96640 


8-92877 


3439 


8-90906 


9-00657 


4.361 


8-59045 


9-29898 


9088 


8-46569 


9-37007 


6679 


8-64810 


9-38158 


4250 


9-6758in 


9-87814 < 


4368 


9-676o3» 


9-87835 


3785 


9'65364W 


9-89819 


7004 


9-58289« 


9-95416 


2844 


9-55oo4» 


9-96567 


I3380 


9-5497CM 


9-96487 


4384 


9-53368M 


9-96639 


5606 


9-508 lOM 


9-96613 


6346 


9-50o69« 


9-96388 


7454 


9-49987» 


9-95945 


2208 


9-47524^ 


993149 


3596 


9-460557} 


9-91924 


3126 


9-4499571 


9-92003 


7767 


9-74430 


9-62987 


7739 


9-74504 


9-62756 


7376 


9-75721 


9-61558 


'7357 


975777 


9-63108 


6384 


9-78061 


9-58067 


15622 


9-79616 


9-53372 


4801 


9-80841 


9-49290 


3526 


9-82241 


9-42509 


2890 


9-83297 


9-35665 


2110 


9-84630 


9-24762 


9792 


9-87632 


5-70909 


9568 


9-87978 


8-57363 


8210 


9-89227 


8-3828472 


6070 


9-90866 


9-06 i6on 


5259 


9"9i535 


9-1761771 


4705 


9-92172 


9-24306?^ 


4151 


9-92757 


9-29531M 


3533 


9"93353 


9-3411971 


197s 


9-94380 


9-4317671 


0513 


9'95i97 


9-4976571 


8657 


9-96138 


9-5606971 


8148 


9-96279 


9-5837371 


1270 


8-89625 


9-5048371 


2997 


8-87698 


9-281227: 


4663 


8-83713W 


9-262437; 


2560 


8-63757n 


9-221607 


0837 


i-44862n 


9-173437 


5229 


8-1292671 


9-11094J 















TABLE OP PRIMAHY EQUATIONS FOR THE GAUSSIAN 


CONSTANTS IN 1829. 










Sl..i«.,..J.b,er,cJ.leme.l.. 


L.ti.udc 


Long. EiUt. 1 
Grcctiwkli. 1 ^^- "■ 


Loj.coet 


Log. CO, 


Log-rarf 


l^g.co. 


Log.eoef 


Log.cot 


Log^«<= 


Log. coef. Log. coe 


L«e.raef.'Lo5.totf 


Logjjoe 


Log. <Mcf.!Log. coi'f.'Log. coef.lLog. coe 


■^/^"'■■-l^r' 


LogcortW^cf 


Log. coef. Log. coef. Log. cncf 


■-Sr'- 


pf;!':::::::;:::::::-^ 


5? S6 19 

P*5 \ 


« » !« 


iiii 

1-80693 
174068 
166651 


9-74494 

9-67578 
,-68414 
9-68767 


916I64 
9-34813 
9-41938 
9-37795 
9-36818 


9-03619 


7-, .174" 


9-418790 
9-34893" 


7-8,158 
7-969S5 
8-55486 
,-31916 


9'699iio 

9-6I65.., 


9;+4:98 
975673 


9- 568 501 

9-598430 


9-9,65, 
,■9164, 

,■'"67 
,■,.1,6 


8-5483S" 

8-5613 3„ 

3-S95J: 
3-86564 


i-3,4,8" 

1-87,40 
8-84974 


8-4^1^' 

8'3„St. ■ ... 




1 




,)-64o,-o 


ir ::::::. ■'■-■: 


Sfi 6 34 


Jl *9-4 " 


r667i7 


,-68,.8 
,■67519 


,-36191 

n|56j 


'-40S 


7-70689 

S.,s.49 


9; 3 3 78 3" 


9;6o,47 

966464 
9-67177 
,■70646 


9-57097., l97;97S 
9-5038611 19-7601; 


9:t::;; 


r9.i58 


8-8B701 
8-9106. 


'S;; 


8-8,154 I9-76S641I ;9-So593 1,-65, 66n 




.■4,.o...V„,45 






H''- :;:::;:::::::::::i: 


S6 .J 
S8 . 14 


s" 1°* S6 


1-67614 


,-68556 


9-31364 


9-3983. 
,-366,7 
,-36053 


8;s7487 


9^87980 


9-439380 


9-60891 
9-5 5M 


ViW 


fl'.lY, 


8'B3+39 
8-44361 
8-19088 


8-90,06 
8-59045 
8-46569 


'■^8^1 


9767330 


9;S'974 

9-84155 
9-84505 


9-625980 
9-58988., 

9-i4S85" 


,■96139 

99658, 
,■95585 


9„058" 
9-39617" 
9-3«743" 


'■Si?l 


9^.969 1 

9-46879 


9-9633^., „ ■ 




Polcrslmrgh. i y. 

Doskino'. '. .... ij. 




'o f '* 
43 34 3* 




9-71067 


9-17478 


9-33769" 

9-2i«6n 
9-196380 


9-76961 
9-78415 


9-5698KO 
S'+Ssi" 


9^!.]06 
,■84167 
9-85936 
976767 


9-48541 


9;667;8 
8-83605 


9-8,737 


,■9.6,, 


8-46679 

9-44368 
9;4378s 

9-51R44 


8-648,0 
9-6758,0 

,-676„,n 
9-65,640 
9-5818,. 


,■,8.58 

9-87835 
,■898.9 


9-74S39.1 


9-85558 

9-87655 
9-87564 

9-94IC6 
9-5483. 


9-107110' 
8-06945 
8-14627 
875116 
9-56812 


9-95683 


9;3«!4." 

'Si 


9-873491 

9'87338'i 
9;36i46n 


9-94087 

9-96734 
0-03685 


9-69183™ 
9-6898 in 
9^68i33n 

9-456440 


9-71947 


9-666760 

9-70361 
9-7.581 
9^7SSS, 


9-S4Sc7n 
9-898660 


Nijnci Nowgorod ,j. 


,6 ,, ,0 


43 57 4 


1-S9I70 


-« 


9-18308 
9-18936 
9-19+83 


9-198980 


978+74 


s'^^ir 


9-757.4 


9-70944 


8-698.0 


9-83165 


-« 


9-53380 


9-54970" 




8-518680 




9-78913 


-« 
















Angikowo y, 




4« 9'4 


;:?;js: 


-„ 


9-14687™ 


978H1 




965966 


9-S0593 


9-'9333" 


,-84305 


ZZ 


v^zt 


9-5„6«» 
9-508,0,, 


9966,9 


9-o"o3 


9-7°iS 




-" 


9-77361 


9-76346" 




8-4.6n 


-» 


^■»i4]o 


9-84414" 


















9615,8 






9-8„i9 




9-56346 




9-96388 


























57 4. 


?4 30-4 ' 


ri968S 


-„ 


9-19391 


5-i«o*,; 


9-76.60 


9-29890 




l^ 




1-68, ,1 


-Z 


9-61208 


;4,,87" 




IX, 


9-67010 
9-38561 


990358 
9-91456 


ZZ 


9;8o31i 


9 728640 


o-ooJso 


9 54180 


i; 


;8836j 


lUliT 


" 




S! 1 >4 

li 31 4 
57 i6 


56 13 56 
S6 37 3« 
30 17 5. 


i-.i,6, 
'•■«S54 
i'6i4o8 

.•3832! 


8-379" 

8 -848 son 


9-31110 

9-30088 
9-77807 

,•75453 
,-736,6 


9-54587 

,;54958 


9-75164 
9-57007 


9-368.0 
9;38i33 

9-8iI*9 

9-83817 
,■.49.7 


9;|458S 

7-9183, B 

!-O04g4n 
!-57,g3., 


9-7848 




9-61,96 
,-6181, 

9-41^36 


)-68i3i 
9-63763 


9-63596 

9-63.16 
9-97767 
9-97739 
9-97376 


|46o55n 
■74504 


9-'i?56 


9-S7S91 
9-89997 


9-18079 

8-548^00 
8-59396" 


9-9165J 

9;69983 
975461 


0-0,651 


j-84827 
9-84807 

0-05744 


9-8169S 
9-8.7+5 


9-56766 
9-56S10 

9-55606 


9 60718 


— 


"": 


, ,4,40 






,8.670 ,0 3 30 




98404; b In 




UM.I 
















9'413°4" 


































" 'S ',= 










•)'46oi4 




,-,9631,, 


979890 






9-5813. 




-7,6,6 






9-772830 






0-04761 


9-88757 


9-481.9 




























































37 jS i« 






9-67.35 






































99631^ 




DiLiiiri'HBk " 


56 o 




,■54444 


O-igosSn 


,-66,64 


9-5868, 


^'■\lTl 


9-91,63 


9-56i,=n 


9-79657 


9-6558;.. 
9-663,90 


-i;"° 


9-46:67 




■84630 


9-35665 
9-14761 


&-C0899 


9-496490 

9-547^4" 


9-80505 
9-78109 


rs; 


0-03676 
0-01694 


9-94084 
9-93114 




996658 
996583 












Nijnci NowgoroJ , x. 


56 ,,5 


43 34 36 


a-48»87 


9-i7io5n 


,-641.7 


9-61957 


3-61458 


9-,.So, 


9^670l7n 




9-68oion 


-67878 








8-70909 






9-71907 


0-01914 


0-00131 








0,9 


99068 




« ^ 






(■150690 


9-64187 


961597 


.-49041 








,-67348'- 




1-48845 


1-89568 






D-00981 










9-98264 


1-52848 


9646 








Anjri'k owo' ' '* 




:r;.4" 


0-71418 


9-»os8n 


9-61453 
9-59638 


9-64434 


8-97379" 


,-,iI,o 


,-77831.. 


9'690S7 
9-63105 




■35416" 

■04078" 




9-86070 


■89117 
■,0866 


8-381840 


0-0.378 




9-67484 


o-oiIh 


9-98694 


o-o?j6o 


8-338780 
9-0.9460 


.972 


VI 


' ;/, 


,'' 




Miijetij'ko !..."'; 






■34791 












1-79.41,, 






























1 97116 










50 38-4 
stir 56 


0-94988 

-11385 
■S5.36 

-.6458 
-41813 


9-1641371 


5-SD.43 




9.59030 


9,1.57 


9-79013" 


9-55685 


9-653520 


-losijn 


9-47985 


9-84705 


•91.71 


9-Z4306.. 


0-00500 


977364" 




0-0,0,7 


9-95088 


001 555 


9-198460 


9 96098 


110 


855 


9.97s 




J', ,4 


1-014670 

HE 


9-57387 
9-,6.i. 


9-6,440 
9-7.013 


i-i64S7n 

.-4.14670 


'If" 
9-«7,54 


':*E 


9-46147 
9-36406 

9-09176 


9;5S883n 

9-506650 
9-44483" 


■39988" 
■44376" 


9'SS4i9 

9-57793 
9-58774 
-60798 


9-83533 

9-81975 


•93,5, 
-,4380 

-PI38 






9-767310 


9-34088 


0-04908 


9-89581 
9-87454 


0-01854 












9-43.760 9-97007 
9-497650 9-96035 
9-560690 y-94413 


Will 










7 J3 45 






94099" 




9 71931 


S0974H 






905844 




-47749" 


9-57868 


•78.48 




9-58373" 


9-95053 


9783660 




0-05571 






9 53245" 






rierl.r. 


- J» S 


13 14 iS 


-66304 


-56685 








-„5ion 9-3S5oon 


9-74667" 




-85440„ 


■87,60 


■411-0 








9-6ii99n 




,-,6i70 








,87,6 n 














•73SIS 
















■9.189 




■816,8 
















9-46758" 
























■471,9,1 5'o8i37 








■91330 8-34663 1 










1-8.778" 










, 910760 


, 70,90 9 8104,0 














-38064 


.■OS601 


7008.0 9-443290 |9'i56i6 


|-704i6n 










9-77704" Ir^Soii 




1-94767 






























,■70.560 










,■835,3" 






1-50*54" 


,-390660 


9 ,,849" 






IT. 


7 37 


34 404 


-66,6, 


■70947 


-3966. 




-5.1410 9-4iSS6«|9-i9870 9'69483'> J9'6j!07 | 


-46900" 9-9,398 885119 




9-iiC94n 9-778300,9-41338 


,■83184" 


9-95414 


9-4779," 


9-49i87'i 


9-3359 in 


9-94,540 


9-7166, 9-80,97" 


9-74B67" 



5TANTS IN 1829 (Continuj 



}g. coef. 



Log. coef. 



XcJ24iii 

7*2785 
N<l67579 
PI 71054 
1^7270^ 

MI85699 
AI82420 

P(l8724S 
£^88993 
T<t9iioi 
-•089549 

KI81405 
8,176890 

81^92132 
P()6i562ra 
Kr64097>i 
Mf58870» 

Bf58386« 
T<'56954ra 

OrsS7S4» 

Pf22I38 

Or35744 
Br82868 

•84524 
TJ85125 
Q'32966 
/(• 10809 
S|-83075 
179845 

'I1'7oo93 

T*4226o 

B'43i2in 

K'4io23w 

Tc39ii2re 

Ii'42i73M 

^)"4i 52671 

"'•34775" 
i"32256w 
i*29907M 
i"2863ira 
1-96555 

1-97325 
1-97972 
1-97810 
1-98288 

(■97915 
(-98514 

1-99273 
,•99923 
1-00538 
1-00864 
1-00974 
-2690071 



9-08 164W 
9-03459n 
8^88487n 
8^S2752n 



8^71 ii9« {9-77295ra 9-6390) 
8-48266M 9*76976« 9-6853( 



8-0942771 



Log. coef. 

AA3'2. 



Log. coi 



9'77567« 9'4437^ 
9-77574W 9-4856^ 
9'76969» l9'58o6c 
9^77038ra 9-6046^ 



8-48301 
8-62500 
8-94633 

8-99942 
9-09554 
9-27335 
9-74748 
9-84935 
9-95818 

9-96029 
9-96437 
9^96659 
9^85329 
8-74915 
9*40464 

9-45490 

9'49397 
9*48065 
9-51054 
9-56563 
9-59872 

9-62843 
9-64960 
9-91922 
9-90689 
9-89421 
9-89153 

9-87756 
9-86037 
9-83803 
9-81243 
9-78978 
9-62401*1 

9-6753371 
9-7I4347J 
9-6646771 
9-6799771 
9-641827* 
9-7463171 

9-7821172 
9-8110571 
9-8313971 
9-8361071 
9-8359971 
9-37412 



9-7743771 19-7105^ 

977395«|9"74ii! 
9-7746271 19-7677(1 
9-7739671 :9-78i3< 
9-774607* 9-81531 



9-82485 
9-97926 
9-7383571 



9-8136 
9-8226J 
9-8612! 



9-7773271 
9-776557* 
9-761357* 
9-872577*19-9492! 
9-775677* J9-97713 
9-284897* 9-9288: 

9-234657* 9-9206; 
9-022697* [9-8873^ 
8-796567*19-8650' 



9-8235 
9-7969 
9-8711 



9-725447* 9-8753 
9-712797*, 9-8756 
9-729277* 9-84831 
9-722617* i9-836i 
81077* ,9-8646 
9-660977* i9-852oi 



9-64036?* 9-8322! 
9-626317* 9-8082 

9'5855i 
9-63635 
9-67539 
9-64291 



9-0779 
8-6242 
8-4279 
8-0932 



9-66549 

974493 
9-77578 
9-80050 
9-81307 
9-95772 



8-5082 
9-2381 
9"3953 
9'4974 
9'543^ 
9-810J 



9-94587 9-815.! 

9-93316 9-8i5( 

9-91329 19-7645 

9-89204 [9-742^ 

9-88073 '9-709* 

9-92093 9-814 

9-89755 I9-802 

9-87330 9-786 

9-84332 ,9-760 

9-82174 ,9-736 

9-81270 9-725 
9-763727* 9-816 



TABLE OF PRIMARY EQUATIONS FOR THE GAUSSIAN CONSTANTS IN 1829 (Continued). 



Suiinos nnil obtcrved elcmenti. 



Long. East, j^ Log.cocf. Log.coef.Log.coef.Log. coeClLog.c 



Tjumcu-.,. 
Tnjnkowo . 



965643 

9-67487 
9'67S4S 



g-S6s3:n 9 



i'63S58n 
1-65437 
i-S8i6j 
1-68646 









r6«44! 




-66008 9 


,-7.!j. 






,•8.6,. 






,18, i6 






5-84»07 


■514^9 


9-+9180 9 


,8,„8 


■S6SS9 






■64467 


























9-48.51 


m>7« 






,■7185. 




I-+0766 


t'llllt 


8-3.696 


9-3J086 
9-14860 


9 7"47 


9-I791I 


8-88154 




918680 


8'i5675 
















)'87558 








)-76i90(i 


80.300 


9'84830 


9-69669n 


'rZlZ 


9-8,6.5 
9-86+63 


9-69i57n 




9-85103 


9-800340 














9-830471 


8-96031 


9'4S907" 


g-gjs.e- 


8-50684 




9-8JS38. 


8-31059^ 




9-79o89n 


g-p6o?« 


9>9S!!» 
9loS74n 






910776,1 


9-83+15" 


8-8j..3n 


9-.3!'7» 


9-8ii6in 


9-ii96en 


,-osl09» 


9-8c8.» 










8-05733" 


9-761071 




8-,.6« 


975>9S" 


8-80969T. 


!-3>s67 

9-4.076 





'S1167 






9■87^;7» 


-94918 


9-77567'i|9*977'» 1 
9-18+89-. 9-9»88. 


.■.14*." 








3-79656" 






rsijso 










9-71 544B 


9-875JS 




,■87560 




,■84830 




.-83615 






9-66097" 


9-8510J 




9-85111 












J6a+2i 






9-64191 


S093J4 




8-5081911 




9138 .on 






9-80050 




9-81307 


9-5+366« 




9-8105411 


llVul 


I'lllZ 





-73344 9 


94'4i" 








9S46." 


-.,.64, . 






-3364,,. 


■659'7" 9 










■8347I" 




■94876 


■97658 . 


-087.7 




9'9».» 


9-73747 9 


-,0689. 


9-73795 9 


■8,4.,~ 




■8,,5,„ 










1-73615 


■83803, 








■789,8n 
















9-7')'S 


-» 


9-73301 


-so 


9l..]6 




1-8514. 








1-88375 




9-90976 


o^.i40S 


,-8,8,. 




,-88540 


o^..3«s 


)-85»59 




9-8346. 








9-87330 


0-0.37. 



. 9-58.07«!r54" 



coef. 



NSTANTS IN 18: 



Log. coef. 



;i4« 
13 



T'29 
Si92 

K'49 
K570 



B'9^ 
K?83 

\^73 

(.568 

I23 



pz66 o"o2595 
.T676 p'ozSgS 
T 633n 8"304.o6 
-j7i4re y75282w 
0596/1 |8"34i24ra 
g6o5re 8'6oo86?i 



Log. coef 



8-06623 
8-99939 
8-99933 
8-92808 
8-76821 
0-01147 

0-01443 
0-01289 
0-00946 
0-00440 
9-99764 
9-98914 

9*98557 
9-97767 
9-96918 
9-97251 
0-01429 
0*02284 



j832re 
j,948« 
J366W 
j,632W 
5697/1 
^S44» 

j^443W 

206»l 

ji207n 

(985/1 

766 

1j75a 

J 063 
(007 
1351 
T935 
T57I 
t 858/1 

51148/1 

J 403/1 

^916/1 
J 991/1 

T329 

1^745 

1521 

r '604 
II752 
j)2l6 

Ii97a 



8-84061/1 
8-97649/1 
9-08991/1 
9-25890/1 

9"34i75' 
9-36275/1 

9-01908 

9'i3354 

9-25472/1 

9-43568/1 

3227 
9-16974 

9-40798/1 

9-40424/1 

9-13890/1 

8-857i3« 

8-97705 

9-05572 

9-39411 
9-49295 
9-46289 
9-44572 
9-15066 
8-62195/1 

8-40031 
9-02258 
9-04691 
9-05866 
9-07560 
9-09897 



9*37773 

9"38749 

9"4°777 : 

9"5ii34 

9'55Sii 

9-82550/1 

9-82625 
9-81150! 

9-78779/i 
9'753M 

9-66864/^ 

9-65050?! 
9-61463/j 
9-58552^ 
9-60227/i 
9-89788/! 
9-92881/1 

9-97800/! 

0-01043/1 
9"57523 
9'S9833! 
9"59367j 
9'57944| 

9-56277! 

9"53745! 
9-50744: 

9'49574| 
9-49088 
9-50613 

9-59468 
9-67881 
■60465! 
9-62027: 
9-70990 
974453 

9-78072 
9-72517 
9-68382 
9-45109 
9-77148 
9-77 14( 

9-6727 
9-62oo( 
9-6315, 
9-63251 
9-7001 
9-1906 



9-9797 
9-6667 
9-7278 
9-7262 

97354 
9-680- 




TABLE OF PRIMARY EQUATIONS FOR THE GAUSSIAN CONSTANTS IN 1829 (Continued). 



9-38960 

j-375J^ 

9-387S4 

9-43689 



9-3Sjasii 5 

9-45378" 

9-SS8s5« 






8-87938 



J7S376 

9-69874 



1-6687 6n 
'65759" 



8;734io 
8-37986 



8-46849-1 
7-S48o4n g 



9'84997" 



9'*7345 

9-65779 
9-64556 



9-78779-1 
9-75384" 
9-7146611 
9-66864/1 9 



9-556440 9 

9-678780 

9-64458" 



9;6545J 
9-67665 



9-74809" 
9-6845an '• 



9-59647 
9-73656 



9-68468 



ANTS IN 1829 (Continued j 


)g. coef 


Log. coef. 


Log. coef. 


Log, coe< 


Ah^>\ 


A^3.2, 


AAW. 


A^S'S. 


33184 


0'0142IM 


9-40731 


9-41789/8 


33731 


o'ooo8i« 


9-23363 


9-22800/8 


33830 


o'oo8oow 


8-84385 


8-84221/8 


33688 


o*oi3i4« 


9-02819/8 


9-03203 


33430 


o'oo927» 


9-34462/1 


9-34884 


32475 


0'0i3O7« 


9-57680/8 


9"594S9 


31637 


0*Ol6l2W 


9-66700/1 


9-69570 


50395 


o-oi6jyn 


9-75448/1 


9-79584 


)93o5 


9'9905ore 


9-84332/1 


9-87880 


b8298 


o'022i6n 


9-83378/1 


9*89723 


,8875 


o'o 144771 


9-82642/1 


9-88029 


?9°47 


o"oo6o4W 


9-83275/1 


9-88003 


^9688 


9*9963 iw 


9*82169/1 


9-85710 


30046 


o"oi5ii»i 


9-77620/1 


9*81982 


30801 


o"ooo48ra 


9-76157// 


9*78871 


31443 


9'97925» 


9-75194/8 


9*75857 


31857 


9"964ioji 


9-73849/8 


9-73024 


32Il6 


9-93992M 


9-73425/1 


9*70589 


31984 


9*902 59n 


9-75232/8 


9*69875 c 


16302 


972820 


9-40843/8 


9-60002 k 


30343 


9*76698 


9-18166/8 9-27549 '5 


36595 


977337 


8-62980 9-21546/8 ^ 


11237 


9-7+834 


9"i339S 


9"547S3»'g 


19957 


9-72513 


9-25186 


9*65725/8 5J 

1 


(.0821 


9-64720 


9-45661 


9"8475S« i9J 


39580 


9-65793 


9'4343i 


9*82632/8 19 


(.0289 


9-67973 


9-36636 


9*77950/8 19I 


33^43 


9-69547 


9-3S146 


9*76391/819! 


13067 


9-70799 


9-40826 


9*75359/819 


37287 


9-72264 


9-44746 


9*73430/8 i9| 


70973 


9-71365 


9-50672 


9*75134/8 9! 


D1937 


9-890i6« 


9-75376/8 


9*69154 9! 


31608 


9-858301 


9-76946/8 


9*68834 9' 


31065 


9-82009?! 


9-79046/8 


9*68873 9' 


30193 


9-76370™ 


9-81201/8 


9*68177 9* 


99342 


9-71 150W 


9*82673/8 


9*67203 8" 


98819 


9-69409?i 


9*86042/8 


9-70667 sJ 


98032 


9-63967^ 


9*86285/8 


9-68403 8* 


95980 


9-5009IW 


9*88328/8 


9-65516 8i 


93574 


9-29653/8 


9-88533/8 


9-59404 9*: 


92253 


9-15124^ 


9-88326/8 


9'55657 9" 


90894 


8-9336571 


9-88807/8 


9-52678 9* 


89525 


8-5428IM 


9-88612/8 


9*48588 9*; 


88686 


7-15099W 


9-89254/8 


9-46986 9*i 


87829 


8-48978 


9-89615/8 


9-44847 9- 


87708 


8-58613 


9-90230/8 


9-45286 9- 


87213 


874775 


9-90239/8 


9-43712 9- 


8633s 


8-95506 


9-90886/8 


9-41633 9* 


84255 


9-20483 


9-90895/8 


9-33763 9* 


82729 


9'339i7 


9-92092/8 


9-28653 9-1 


81943 


9-40173 


9-93331/8 


9-26666 9-1 


819*5 


9-40297 


9-93301/8 


9-26463 9-1 


78853 


9-54107 


9-93052/8 


9-05462 9-' 


71601 


9-71700 


9-89399/8 


8-74271/8 9- 











TABLE OF PRDIARY EQUATIONS FOR THE GAUSSIAN CONSTANTS IN 1829 (Continued). 




9'S6565 



'67676 l9-6S76gR ,9-5604.9 



ir 


r747S8n 
9^68s79n 


|ii 


1-48 5,sn 

9^578o4n 
9SJI69" 
9444!3» 


■+987411 
■39639-1 


,■08913 
1-87619 


■00636™ 


S8aj88» 


:?'i" 


9'07 14311 
9-346 son 


■4"»5 


9-37357" 



B57 jg-gei: 



K" 


Log, coef. 


■4=73. 


94i78,i, 


■=!363 




■8418, 










,34884 




9S94S9 


■6670011 








■8433!» 




■83378« 








8317511 




S.,6,,1 


,■857.0 




18 198! 


7'i57» 




7519411 


9'75i57 


iB'i« 


9'7os39 


7513m 


9-69875 


4084311 


9'6oooi 



9^MS5 
964059 



9-78917 
9-77876 



8-97569 

965139" 
96855811 



•ANTS IN 1829 (Contin 



og. coef. 



Log. coef. 
Ap3.2. 



Sea 63010 
Mag6io57 
Tel^55440 
Cha<5S76i 
KosyS7989 
Masl5898s 

NaWs92''9 

St. 158365 

Paci|6o664ra 

Iteii76743« 
ItenfS 5067*1 

Itenf87834»» 



Iterf88676n 8-18409 
Kir<6a534 9-69368 



Log. coef. 
AA3.2. 



9-81858 
9-83894 
9-89522 
9-89984 
9-89674 
9-90145 

9-92248 
9-92975 
9-92511 
9-64949 
9-19250 
8*69271 



Pari'896ii» 
Terf743S^« 
01er7°377« 



9-65684 
9-58125 
9-50716 



Talif97744»* 9*05894 



For "96 315" 

Leb"9730o» 
Nocj-93o6oM 
Tsci"65444W 
Gai1"84454M 
OcH"o2732n 

Man 1396 
Teir4i4a4 
CW'49a92 
Kof 62064 
Ma|-733oo 
Na<"92046 

St. 1-91982 

Kiij'2042i 
Paij'29420 
TeB-39i67 
Ol|43788 

Tafs69i8 

Po-58741 
LeV6o457 
No|-6ii63 
Tsi-61760 
Gal-62171 

Oc '63602 
Mi-68355 
Te 1-67592 
Ch -67013 
Kc -65768 
Mi 1-64406 

Nj 1*6 II 06 
St, -61059 
Pa -56993 
Its -95317 
Ka -6io33« 
Be •488o7» 



'°l 



8-84123 

8-45452 

7-05679 

8-48 i6o» 

8-64254M 

9-24430W 

9-648 3 7w 
9-67624/1 
9-67ii3« 
9-65120*1 
9-63498M 
9-60236W 

9-6o677« 
9-5446o« 
9-72596B 
9-77912W 
9-83291M 
9-86779» 



9-82768» 
9-8i634» 
9-79077W 
9-8oi27« 
9-83376» 
9-85824» 

9'89942M 

9-89423W 

9-95242 

0-01840 

0-01860 

9-99912 

9-99716 
9-55728 
9-62721 
9-69648 

973034 
9-79096 

9-79565 
9-79783 
9-79834 
9-79777 
9-79718 
9-77588 

9-62577 
9-57179 
9-57256 
9-58822 
9-59177 
9-57930 

9-57125 

9-5719IW 

9-86236 

9-80875 

9-71768 

9-64461 



9'33 
9"39l 

9"54 

9"S5 
9*53 
9"Sa 



9-91098M 
9-91571W 
9-91619M 
9-9191371 
9-92289W 
9-92260M 

9-92500W 
9-79i38n 
9-75245M 
975856« 
9-783697* 
97994o« 

9-8200IW 

9-8i38i« 

9-85296 

9-69714 

9-63397 

9"55a'9 



9-17896 

8-96129 

8-57288 

7"i7758 

8-6o672« 

87679578 

9-3934171 
9-8139871 
9-8569071 
9-8571371 
9-8466771 
9-8426171 

9-8430871 

9-8493371 

9-82565*1 

9-56263 

9-65585 

9-70928 




TABLE OF PlilMAUY EQUATIONS FOR THE GAUSSIAN CONSTANTS IN 1829 (Continued). 



c.,(iAn. n„H ftlisprvcd elements. Latitude hP^^' ^^}' Lou ti t^K-***^-''*'8-™«f' L"?- wef.Log.coef. U)g.coef, Ug.coef.,Log.coef. Log.coefMx>g.coef.Log.co€f.Log.coef. I«g.coef.Ug.coef.UB.coef.iI^^^ 

Statiom aua Oliseirea Clements. ■^.>iiui.e.| ce„„,,|| >^- n. j ^^ ^, ^j, , ^^^ ^^.j ^^^ ^,j ^^, ^^, | ^^ ^^ ,_ | ^i ^^ ^^^„ ^^^ | ^ | 2^^_ ^, ^^, , ^_, | ^„^ ^ ^^,,, j ^,., | ^j,., 






Takiiik 

I'orotowsk . . 
Lebcginc . . . 
Nocliinsk . . . 
Tschcrnoljcs 
Gamastacli 



Telowka 

Cliarlscliinsk . 
Kosuircwsk . 
Mascbura . . . 



Pacific Ocean 

lucliora 

Kantinsk 

Beresowskjt Oftrow . 



S' !! Si 
56 ji 6 
SS S» S 



60 54 
60 4! 
59 !■» 
58 55 



53 J* 37" 
55 33 IS" 
5' 54 ■ 



61 56 45 1 
31 ijt 
J9 361 



!)'43 



5« 53 5! 
56 3 
55 5' 5 



158 H 4! 
160 54 4 
160 43 1 

159 34 1 
158 5i 1 

157 »S 3 

158 40 1 
,03 14 

08 4'4 



56 45 
3" '3 
19 36 



129 44 51 

131 49 s: 

<n 4« I 

134 5' S3 



160 S4 
160 43 
■59 34 
■58 55 



IS8 40 1 
03 24 
109 3S'4 



■36418 
■88547 
■79449" 
■6S31In 
■66087" 
■Ii866n 

■37438" 
■30384^ 
■4i304n 
■472610 
■4 "497" 
■34«4"" 

I '470481 

■62941 
■13577' 
■25iJin 
•23147" 

"■"9S"3" ! 
'■0775! 
0S97"" 



■ ■3382: 

i*66469n 
fS7s6«ri 
1-2656911 
l-28983n 
'■49'33" 
I-5ir 



'■704' S 
o'79449" S 



9^o67267i 
9^iz45&n 
. 057S" 
9^28556n 

9-4265 m 
1-4324111 
9'S»4S9» 
9'398»9" 
9'«39S5" 
9-06633" 

9-04434" 
9'7>45" 
9-73460 
75085 
974985 
,■76518 

. 765" 
9-76636 
9-76454 
9-76108 
9-76082 
9-73770 



5-69891 

9-67825 
9-66035 
9-60783 



9-77 I26n 
9'76347n 
9^74695" 
r)-727l8li 

9-6727011 
9-6708311 
9-5969111 
9-6399"" 
9'64765n 
965582™ 

965ii4n 
8^943i6)i 
8^94884n 
8^94477" 
S^9988on 
8^97l84Ji 

8^99o63i 
!^9883Si 
9-0227711 
9-07373" 
9-0808 ir 
9-274»6i 

9-4085711 
9-464581 
9-478691 
9^49864i 
9-5223011 
9-572201 

9575651 
960733" 
9- 3637 " 
9' 39084 
9-40871 
9-39321 

)'37763 
,■36390 
9-35449 
9-33998 
931939 
9^3I37S 

0586 
8-96135 
8-87413 
S-86656 
8-87005 
8-85632 



.■40738 
938425 
31039 
9-30721 
9- 3 1806 
931310 

9-27340 
9-26244 
33'3" 
9-4594411 
9-58307" 
9^63438" 



,■03877 

902356 
9-0 5»77 



. '0""43 
9-06975 

,-17190 

9'"43S5" 

-'87738 
■98676 



9'"9759 
9-3 "576 
9-33479 
9'339'9 
9-34035 
9'34»'S 

9-31180 
9-36032 
,-33500 
,-32282 
15894 
9-37040 



9-74960 
9-76826 
9-81638 
9-81813 



9-81894 
979578 
54375 
. 10327 
8-61388 

8-10630 
9-32305 
9-3"776 



8-58130 
8-19797 
6-75540 



566950 
9'719S6" 9-57148" 
9-7470in 9-54557" 
5-7651211 553573" 

>-55943n 
1 5-5815311 
9-51491" 



99»937 

9-91,37 
9-18665 
9-29813 
9-40245 
9-43409 
953114 



3695 
5-52763 
9-52637 
9'45337 

9-26303 

9-16849 

5385 

4063 



9-64863 



9-75881 
9-77282 

9-8369X 



9-89315 
9-95858" 
9183711 
. 79998" 
,-7i3i6n 



5827911 9-62280 
.59410" '9-60814 
9-63569" i9^S3"70 
,'6546011 9^5040] 
,■6,05811 9-45035 
9-728071 



9-6891711 • 
1-49075 
m 5-34274 
.-6843511 8-98312 
9-6938311 8-05964 
9-60955" 935775" 



5^6327 in 
5^68974n 
■645811. 

■641650 
■I5286n 
•49996" 



5-811830 
_ - S08n 
5-85183 
9-65355 
9-'744S 
8^64I49 



8-12971 9-54353 



9-57963" 
5-543150 
1-530750 
9-4937111 



-46070 
9-43038 
9-39678 



8-973350 
8-64,531 
9-556691 
,■619111 
9-681630 
715930 

,'769150 

9-773850 

9-775310 

9-776840 

9-778090 

9'777'3" 

,-766550 c 

9-63379" . 

9-576680 t 

.•S7994'i i 

9-600040 



-93773 
-676830 
9-69309 
9-64874 
,-56640 
9'49"7S 



3,160 
,-37871 
9-634980 

-464330 



7-0352, 
8-461920 ( 
8-633990 



,-45461 
953567 
9-57948 



S-79606 
8-78408 
8-73573" 
8-983580 
8-974431 



9-19536 
,-19347 
9-0,951 
9-38596 
9-2,240 
9-3,259 



9-613311 
9-606091 
,'62670 
,-35866 



5652550 

.■641610 
15,9410 
919,71 
,■34261 
53,564 



9-79967 
5-80519 
583613 
585726 



9-431770 
9-483910 
9-518430 
9-555800 
9-568000 
9-678840 

9-679800 
9-656410 
1-661040 
^-681831 
9-693,70 
9-70751 

9-7031: 
9-71231 
9-76714" 
9^787 
793850 

9^706411 
9-676510 
9-638711 
9-61S260 ; 
9-593530 ! 
9-577390 ! 

9-36911 
9'47514 

9-607,7 



9-55'34 
955369 
,•56491 
9-57916 
. 57795 
9-58331 

8-418440 

098! 

1969: 

115480 

8-7856( 

8-5650. 



9-10311 
9-08350 



-C06480 9-63010 
.-C09540 9-61057 
-015170 9-55440 
1-013870 ,-55761 
'-008780 ,9-57989 
i-003930 19-58985 



9-94791019-767430 
9-915350 , ■850670 
9-89,780 ,-878340 



,-81858 
5-838,4 
5-8,531 



'688700 
'6656. 
,'6048' 
9-5685; 



8-503660 

24400 
8-56503 



9'9i64i 
5-91710 
9'9"577 
9-9t593 
9-51175 

9-91178 



7-4,103 
"-43663 
8-45743 
-■89740 



■S17680 
. -790770 



5-330970 ,-73338 
9-3,5480 5-7330S 
9-546950 9-73881 
955337" ,-7547> 
9-530760 9-79111 
539070 ,-83636 



9-91148 
9-9"975 
9-915" 

64,49 
9^19350 

'6,371 

8-18409 
,-6,368 
9-65684 



9-899410 
894230 
9-9S141 



S-65444n 
8-844540 
8-017310 



9-70179 
9-71458 
9'7373' 
.'76353 
9-78710 
9-83951 



9^91668 
1-51485 
9'9'399 
,-91105 



8-945180 
9049530 
5-147081 
9-31,761 



8-754160,-8. 
8-68603 ,-887280 ( 
,-7612, 9-165450 
9-731,8 8-566340 
,-66532 ,-09418 
9-61537 9-35897 

9-58596 
5^60533 
,■60700 

1 ,^6351' 
1 ,'63408 



9-6,828 
,'69857 
,-68413 

,-64,33 
9-63555 
,■62427 



-73300 
8-,3046 

S-,1,81 
,-076011 

,-19410 



8-45453 

5679 

8-481600 

8-641540 

144300 

,■6483. 

,■676140 

,■671131 



979096 

9-79565 
9'79783 
79834 
9-79777 
9-797"i' 
977588 

9'6i577 
9-57179 
9-57356 



'517110 9-970880 

■7395 9-9'S85" 

'58709 9-780800 

9-61861 9-693140 

5-64845 9-668 5c 
9-767340 9-61498 
- 1 9-46433 
,■793850 . 

9-706411 



1-06818 

,-06395 
1-04346 



9^963i8 
,•96086 
9^9 1078 
9'974"S 



1091440 

/097SCO 
)-| 12940 
>-| 14940 



,-69853 
9-65207 
965868 



-077810 
99974 



I 9-51,670 
■ 9-579481 
1 5'6 15500 
1 9-656611 
V 9-669071 



9-94496 
9'93337 
9-9341 ■ 
9^5l831 

,•91837 
,•,1664 
,■,■,■3 
'91340 
'91366 



•11956 



976397 9'77307" 0^1307o 

9^78S47 

,•84719 

9'8S370 

985390 

9^86176 



,•761870 i 
9'74174" '' 
,•755130 011113 
9'7909»" o^il893 
,•819550 ©•H478 



" 95939' 
» 955189 
o ,56139 
" 9'59197 
9^6i469 



11 y89457 
>9 990141 
, , ,150 990938 
9897340 5-6l894 
•960710 9^I5H9 
■97597" 864466 



9-643650 
,■663710 . 
,'648040 5 
9-633950 



/87.510 
,-866900 
9-93669 
9-98787 
9'97739 
995107 



)-1034i 



,■,,6121 
,-,31941 
9-89811, 



9-698410 
9-815640 
9-868360 
1-876770 



9-83191 
,•86779 
99109S 



,•6,27, 
,•69403 
,67125 



■89191 
.•93S301 
9-635891 
9-675080 

-699300 



9'S59i' 
9^58029 
94,463 



9-45735" 
,•773260 
,'8014,0 
,•816010 
9'844630 
9'8733S" 

9-93655" 
9-93878" 
,■9,649 



8^78616 9^19920 



5-64864 
8-489911 
9-186790 
9-175711 



8-79514 

9-67409" S 

9-63843" 5 
,-616060 5 



1-56918 
,•58741 
,•60457 
,•61163 

,•63601 
,•68355 
,-675,3 
9-67013 
,-65768 



1-603360 ,-57,30 

.1-606770 9-57115 
9-S446< . _ . 
,-715,60 9-86136 
9-77,110 ,-80875 
9-831510 ,-71768 
,•8677,0 9-64461 

,-910980 9-178,6 
.'9157 ".' 
,-9161,0 8-57288 
7758 

8-6067; 

8-767950 



9'47514 

,-607,7 
9-57468 

9-55911 

9-49463 
,•6,7570 
,•5306111 



9^50858 

58366 

9^63179 



,-758560 
9-783690 
9-799400 5 



;o6i5o s 

8-3599,0 ; 

7-70824 ■ 
'07397 



,'813811 
,85256 
,•69714 
1 9-63397 



,•843080 
9'84933" 
,•835651 
,•56163 
,•65585 
,•70918 



99191 
98951 
047510 
764680 
9^790000 

rsisiio 



9^836130 
9^857260 
,•8,4880 

,•8,1,10 
993830 . 
-839830 
9-85348" 
9-854300 
,■865060 

9-760380 

9^7305" 
,■6,1030 
,•671540 
,-649540 
9-653420 

9-4344!" 
,-54661 
,-68471 
9-68675 
9-65678 
9-64631 

9-65614 
9-67790 
9-60384 
,-58709 
9'3313' 
8-89816 



9-95349 
9-96141 
9-96391 
9-96797 
9'97159 
198137 



,■610220 

,■596130 
957533" 
,•56,700 
,•459170 

11531 



9'5'747 
,■54038 
9^50699 
r4i389 



,■18390 
550543 
,•,0123 



,•49815 
958536 
■63335 



,•,4961 
9,3830 
,■,3983 



983187 
9^81813 
,80580 
,■79560 
978174 
9-77761 

9-71119 
9-49748 
9-43766 
9-43988 
,-46080 
5-46960 

9-47163 
9-46318 
9-489740 
9-05185 

935"03 



,•79481 

9-80801 

9-87813 
989645 
989853 
,■8,614 
9-88969 
9-88363 

9-87093 
9-87157 
985351 
9-641000 
,-663980 
,-641150 



8-57188 
7-17758 
8-606710 
8-76795" 
9'3934 

, 5-813580 
1-856,00 
/8571!" 
,-846670 
5-841610 
)-843o8o 

,-84,330 
1-815650 
(-79*55" 
7-845500 
,-853360 
5-538640 

,•964,50 
,•,6,700 
,•969500 
9973410 
9^97850D 
9-97873" 

7-990310 
9-863850 
9-825360 
5-837360 
9-865790 
5-885650 

9^917050 



9^52289 
,•92360 
y92500 



9-75856 
.■78369 
,•79,40 
^•82001 

,•81381 
,•852,60 
9^93495 
9^87513 
977813 
,■70546 

9'1!"93 



,■458730 
9885450 
y9338lo 
993593" 
,■918770 
,■918860 

,•,40110 
,•946940 
,•534860 
9'71755 5 
,•80068 5 
,•85448 i 



9-4,815 
958536 
963115 
975183 

9-77009 
9^78610 
979481 
,■80370 
,■80801 
,•83813 

,•89645 
989853 
,'8,614 
9'88,6, 
,•88368 
,•870,3 

9-87157 
9-85353 
,-,7802 
,-56861 
,■95271 
993945 



9-818230 
5-8058C0 
9-7956. 



I 



9-84988 
9-83875 
9'83375 

97775' 
9^568,5 
9-51457 
9^51868 
,•541,0 
9555«5 



,•56079 
959895" 
,•40127 
951159 
959,08 



,•463180 
948974 
9^4916, 
,56453 
9^64581 
,69310 

9^80580 
,■81408 
983941 
,■84909 
9^85971 
9^86415 

9-90345 
,■,67,2 
997 544 
9'97494 
557179 
996993 

9-96796 
,■96,18 
9-96173 
9-910950 
,-,03300 
9-884750 



:ONSTAN 



lOg. coef. 



6487s 
83417 
•91189 
•81852 

77173 
54797 

10508 
•96477M 
50222M 

57S79« 
75085*1 
71926 



[^•05340 
JJC08657 



Log. 
Ah 



8-8o( 

■931 
■90c 

■77< 
•72] 
•50c 

8"i2a 

7"379 
7767 
8-377 
9-41 il 
9'38i 

9-342 
9'33S 



, DERIVEJ 



AA3.2. 



- 11-5592 

+ '8-6579 

— 0-2149 

- 1-9325 
+ 73-9182 

- 55-4073 

+ 2-5941 

- 17-3423 

— 13-2269 

- 10-1757 
+ 46-2468 
+ 3-2973 

2-1125 

+ 106-0129 

— 61-8273 

+ 3-5400 

— 2-2458 

+ 74-7282 

+ 8-4568 

— I-8413 
+ 117-3487 

+ 12-4076 

+ 79-1037 
+ 11-2201 




TABLE OF PRIMARY EQUATIONS FOR THE GAUSSIAN CONSTANTS IN 1829 (Continued). 



Stations and obseired elements. 



Lattudea™!;.,!.!'' 



Log. coef. Log, coef. Log. coef. Log. coef.jLog. cocf. Log. coef. Log. coef. Log. coef. Log. coef. Log. coef. Log. coef. 



>g. eoef.jLog. corf. Log. corf. Log. coef. Log. coet. Log. cocf. Log. coef. Log. cocf. Log. coef. Log. coef. Log. cocf. Log. coet.JLog. coef. Log. cocf. Log. coef. Log. coef.lLog. cocf. Log. coef. Log. cocf. Log. coef. Log. 
V"- I '">'-■ V-*. I ■!''"■=■ V''. as'.i. Af. V'. I i'l'"- V-- i*'". <iy'». All"- J/-". aj-i. Ik". irr*». iA=-. is'''. aji'i. i/ 



Sarajuchlatsk 

Tojon.tain 

Tliree wenls from Bjehkji Perewse 

AlUcfajooa 

Tuiioinsk 

.\tki 

Rank of Kiishtin near Ochozk 

Sea of Ochork 

Pacific Ocean 

Item 

Item (harbour of Silctia) 



60 6 



141 : 



"3836411 
•a7944Ji 
■327971 
•385781 
■54357" 
■43949" 



143 13 
58 45 
58 16 1 
58 IS 54157 : 
S3 34 37]i 

55 33 152 

56 54 i 



975516 
9^76324 
9^7577i 
9^75533 
9-74680 

9-73780 
9-71936 
9-7»i9S 



9-275891 
9-32748" 
9-38191" 
9-404461 
9-412101 
9-235011 



9-06414 
9-10552 



■942S3 
9-410341 
9-434021 



9^°9a35 9^47967 
"■91997 ^■51774 
.-09981" 9-53333 
8-S5Sii"l9'5i7i5 
8^77788ll9'5o645 
8-876121 '9-48013 



7061 9-37894 

190W 9-3494S 

9-44504" 8-3963111 

9-474291 9-]4ioon 



9-678071 
9-637001 
9-495951 
9^44734" 
9-36707" 
9-297441 

.-0576011 
■35547 



9^27576i 

5456711 

9-593081 

974" 



■702451 
■75 = 14 
9-78248 



958442 
957136 
9^S7S04 



70794 
9-61699 



9-45358" 
■25741" 
■4565S 



9-31776 

9-38875 
9-51658 
.65071 
9-69275 
9-833701 
■670331 



9-91496 
9-91293 
991 = 5! 
9143 s 
991492 
,1631 



S7S79" 
750851 
3-71926 



234251 
34754" 
■494091 
585271 
749601 
■69378" 



9-78270 
9-77676 



9-68340 
9-497601 
9-659941 



■784181 9-087281 
9-736971 9'37Sion 
561 ,9-6445211 



9-374601 [9-797211 

9-178301] 9-828401 

8-54648 I9-84821H 

31242 l9-S3oiin 



9-95055 
9^94002 
9-92033 



9-46834 
9-52961 
9-59958 
959936 



9-58230 

9-57612 

57765 



9-640221 9-51010 
963633" 9-31377 

58442" 18-503221 
,55278118-965441 
9-52943W 9-202461 
9-493661 9-3282411 



9-464781 



35695" 
9-107301 
8-858S41 



9-92047 
9-92438 

■9=157 
9-92305 



9-68351 
9-686S9 
9-69763 



'S7796 
9-92360 
9-93050 



9786551 
9'767i2i 
9-74749" 











FIN.AL EQUATIONS FOK 


THE CORRECriONS OF GAUSSIAN CONSTANTS, 


DERIVED FROM 283 OBSERVED 


ELEMENTS. 










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- 6-2885 


+ 7-7=6. 


- 4-0981 


+ 8-89. 8 




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- 0-9548 


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- =9-4357 + 14-8789 - 15-1839 + ■5-01C6 
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- 5-1817 


- 45-5888 


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- 167-706-1+ 7-4550 






— 6-217. 




+ 0-5697 


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+ 4-0848 




+ 18-6579 


- 4-0145 








+ 21 4=71 


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-20-0157 


- 0-2.49 


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-1274-333 = ;+ 5-9292 


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-.9-5229 
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- 2-1662 


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-.C-.2.6 


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- 9-1377 

+38-8037 


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- ,-,283 
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- 28-979= 


- 54-5557 
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- 16-2783 


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- 6-2.7. 


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-37-"4=S 


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-16-8569 


- 8-1499 


+ 0-7102 


- SS-4073 


+65-2013 






28 0272 




- 647-3==- 

- 4o6-3>3 = 
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- 3-0981 
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- 3-4168 

- 0-9765 

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- 8-7596 
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+ 0-6655 

— 10-12.6 

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- 6-6373 
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+ 56-74=3 
-30-4774 
- 4-8492 


-30-4774 
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- 3-8575 
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- 0-7064 


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- 7-0914 

- s-3843 

- 4-3685 


+46-73.2 
-.3-8286 
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- 17-34=3 

- 13-2269 


- 8-0472 
+ 9-451= 
+ 41-4770 


+65-7055 
-36-8534 
- 5-7606 


+ 4-679S 
- 2-2930 


+ 1.-580. 
- 0-5B45 


- 10-1565'+ 53-2076 

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- 9-5804 '1+ 4-8506 


+ 4-1294 

- =14145 

- 15-1690 


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+ 13-8777 
- 11-9=49 
+ 0-9705 


- 31-0184 

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-■564-564- 
- 5="'7S6= 
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— 3-8640 

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- 1-12.5 

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+ 2-530, 
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- 10-17S7 
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+ 9-970= 
I '"060 


- 5-9819 
+ 4-59=9 

- 7-9634 


+ 134-4645 
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- 5-=934 
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- 162021 
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+ =-47=5 


+.20-0916 
- 4-9200 
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- 8-1970 
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- 30-1954 
+ ^^ 


— 1650-280— 
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- 642-674- 


+ .1-2025 
-11-5592 
+ 9-7937 


- 2-3692 
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- 4-0145 


-20-0.57 

- 0-2149 

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-40-.238 


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-55-4073 
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+46-7312 
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-.3-8286 
-.734=3 
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+ 3-8279 
-13-2269 

+41-4770 


+ =-5301 
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- s-1738 
+46-2468 
-15-5200 


- 47-7003 
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- 7-9060 


+85-5.86 
- 2-. 115 
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_ 2-1.25 
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- 61-8273 


+ 1-0551 
-6.-8273 
+77-5180 


+52-8321 
+ 2-5400 
- 8-7316 


- 19-0963 

- 3-=43= 
+ 6-5774 


- 7-0457 
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- 28-96.5 


- 77-!i=9 + 95-495= 
+ 8-4568 1- 1-8413 

- 1S-=43S + ■■5=91 


- 13396 


- 39-9794 
+ ..-4076 


- 5-73=3 
+ 79-1037 

- 3=-7=50 


- 79-8057 
+ 11-1201 

- 17-6483 


- 384-743 = 
—2061-483 — 
+ 347-481- 


- 3-9394 
+43-1971 

- 3-4183 


- 0-6867 
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- 3-6575 
+76-8295 
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—28-0272 


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-36-8534 


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+52-832. 
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- 8-7326 
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-28-96.5 


+78-5268 

- 5;8354 


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+ = 33=571 

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- 8-9338+ 62-5119 
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+ 5-9868 
- 9-7735 
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- .3-2836 
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-=9-4357 


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-56-4565 


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-14-094. 


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— 1796-4C0 — 
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— .-2624 1 —.6-4226 

+.2-3.84 ! - .-998. 


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— 1-1283 


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-21-4145 


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-39-9794 


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— 1-6217 


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+=57-3495 


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1 +i?6l-88i- 


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-54-5557 


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-11-9=49 
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-79-8057 


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+ 11-7835 
















































(6) 



ON THE IRON MANUFACTURE IN GREAT BRITAIN. 99 

On the Progress, present Amount, and probable future Condition of 
the Iron Manufacture in Great Britain. By G. R. Porter, F.R.S. 

In obedience to the request of the Council of the British Association, made 
at its meeting in June 1845 at Cambridge, — a request from that body being 
equivalent to a command, — I avail myself of the first moment of leisure that 
has since presented itself, to investigate the condition of the iron manufac- 
ture in Great Britain. 

The incessant claims upon my time, of public duties, which have called in 
their performance for the most anxious and unremitting labour, throughout 
all of the present year that has hitherto elapsed, may perhaps be allowed to 
plead in excuse for the imperfect manner in which I am able to perform my 
task. I wish, most sincerely, that it had been otherwise, and that it had 
been possible to devote to its accomplishment an amount of time and a de- 
gree of research that might have enabled me to present a work more worthy 
of the acceptance of this body, and better proportioned to the importance of 
the subject. 

It was, doubtless, a conviction of the great and growing influence which 
the progress of the iron manufacture must exercise upon other important 
branches of our national industry, that led the Council of our body to desire 
information concerning it, and all that has since arisen in the course of our 
legislation has given additional interest to the subject, so that it has become 
piore than ever of consequence to know the actual condition of this great 
branch of our industry, and of the capabilities which present themselves for 
its increase. The enormous demand for iron caused by the general and 
simultaneous construction of railways all over this kingdom, and not here 
only, but in various parts of Europe and in the United States of America, 
and also by their promised extension to India, is calculated to produce much 
of anxious inquiry into the subject, in order to ascertain, in the first place, 
whether, and in what way, that enormous demand can be met, and then to 
satisfy ourselves that through the cessation of that demand, which from its 
nature must be in a chief degree temporary, we may not be exposing to 
ruinous depreciation establishments for the formation of which vast capitals 
have been and will be sunk, in which many skilled workmen are trained, who 
during the continuance of the existing great demand will be receiving high 
wages, but who when it ceases may, many of them, be thrown out of em- 
ployment, and who must be so, unless some new and permanent uses can be 
found for the produce of their industry. 

The object of the present inquiry does not call for any research into the 
remote history of the iron manufacture. It will not assist us in the solution 
of the questions now pressing upon our attention, to ascertain whether, in 
centuries preceding the Christian sera, when the Phoenicians traded with our 
ancestors for tin, the Britons did, as some writers have assumed, know and 
practise the manufacture of iron. Certain it is, that the rise of that manu- 
facture upon any scale deserving of notice as a national object, dates from a 
time within the memory of persons now living. In 1788 the whole quantity 
of pig-iron made in England and Wales is said to have amounted to no more 
than 61,300 tons, of which quantity 48,200 tons were made with coke of 
pit-coal, and the remaining 13,100 tons were still made with charcoal (see 
Appendix No. 1). In the same year the production in Scotland did not 
exceed 7000 tons. In Ireland charcoal-iron was made on a moderate scale 
during the seventeenth century. Sir William Petty tells us in his ' Political 
Anatomy of Ireland,' that in 1672 the quantity of iron made there was about 
1000 tons, giving employment to about 2000 persons of both sexes. Works 

H 2 



100 REPORT — 1846. 

established by Sir William Petty in the county of Kerry in 1660, continued 
to be carried on until tlie exhaustion of the timber in the neighbourhood 
brought them to a stand, and in 1788 there does not appear to have been 
any ii-on-work in existence in Ireland. 

About this time the iron-masters in Great Britain began to avail them- 
selves of Mr. Watt's improvements of the steam-engine, and were thus en- 
abled greatly and rapidly to increase the productive power of their works, 
so that in eight years from 1788 the quantity of British-made iron was nearly 
doubled. An inquiry made in 1796, consequent upon the proposal of Mr. 
Pitt, which was afterwards abandoned, to place a tax upon coal at the pit's 
mouth, showed the make of British iron to be then — 

In England and Wales... 108,993 tons. 
In Scotland 16,086 tons. 



Together 125,079 tons. (See App. No. 2.) 

Ten years later, in 1806, it was proposed to tax the production of iron, 
and again on that occasion an account was taken of the number of furnaces 
and the quantity of iron produced, which was found to have been more than 
doubled in ten years ; the production being 

In England and Wales... 234,966 tons. 
In Scotland 23,240 tons. 

Together 258,206 tons. (See App. No. 3.) 

Of this quantity it was stated that about 95,000 tons were converted into 
bars and plates, and that the capital engaged in the manufacture amounted 
to £5,000,000. The proposed tax was so powerfully opposed in the House 
of Commons, tliat the bill was carried through the Committee by a majority 
of only ten, and the measure was abandoned. 

The next account of this manufacture which has been given, was prepared 
by Mr. Francis Finch, formerly member for Walsall, and had reference to 
the year 1823. From that account (see App. No. 4) it appeared that in 
seventeen years the make of iron in Great Britain had been increased from 
258,206 tons to 452,066 tons. Between 1823 and 1830 there were erected 
ninety-six new furnaces; and in the latter year it was found, on a further ex- 
amination by Mr. Finch, that the quantity of pig-iron made in Great Bi'itain 
amounted to 678,417 tons (see App. No. 5). Our confidence in the cor- 
rectness of the quantities here stated should be confirmed by their having 
been adopted in his evidence before the Committee on Import Duties in 1840 
by Sir John Guest, whose authority upon this subject is conclusive. 

From this time (1830) a series of improvements has been introduced 
into the processes of making iron, which has had the effect of improving 
the quality of the metal and of materially ceconomising the cost of its pro- 
duction. One of thq most important of these improvements was made the 
subject of a patent in 1829 by Mr. Neilson of Glasgow, and consisted in the 
artificial heating of the air previously to its being passed into the furnaces. 
The eff"ect of this plan in saving fuel has been most remarkable. In 1829, 
at the Clyde Iron Works, where Mr. Neilson's experiments were made, and 
in which his patent M'as first adopted, it required more than 8 tons of coal, 
when converted into coke, to produce 1 ton of cast iron. This was when 
the air was forced into the furnace at its natural temperature. By heating 
the air to 300° Fahrenheit preparatory to its introduction, it became neces- 
sary to consume for each ton of iron produced only 5 tons 3^ cwt. of coal 
converted into coke ; but in heating the air to the required degree, nearly 



ON THE IRON MANUFACTURE IN GREAT BRITAIN. 101 

8 cwt. of coal was consumed. The saving was thus found to be 2^ tons of 
coal for each ton of iron. Thus encouraged, further experiments were made. 
The previous heating of the air was raised to 600° Fahrenheit, and it was 
then found, not only that a further great ceconomy was produced in the fuel, 
but that coal could be used for smelting in its raw or uncoived condition. It 
was further discovered that the same blast-machinery, when the air was thus 
heated, sufficed for a greater number of furnaces, so that the power neces- 
sary for three furnaces, when cold air was employed, became ample for four 
furnaces of equal size when the air was previously heated. The result may 
be thus stated : — 

In 1829, using coiie and cold air, each ton of iron required for its produc- 
tion 8 tons I cwt. 1 qr. of coal. 

In 1830, using coiie and heated air, each ton of iron was made with 5 tons 
3 cwt. 1 qr. of coal. 

In 1833, using raw coal and heated air, each ton of iron was made with 
2 tons 5 cwt. 1 qr. of coal. 

The saving in fuel is thus seen to amount to 72 per cent. 

The effect of the hot-blast upon the quality of the iron produced has 
been the object of many experiments to determine. As those experiments 
were in great part undertaken at the instance of the British Association, and 
as their results have been published from time to time in its Transactions, it 
cannot be necessary to notice them further here. Mr. Neilson's invention 
was for a long time greatly decried, and to this day it is the practice with 
some few of our leading engineers, when drawing specifications for works, to 
forbid the use of hot-blast iron. Under these circumstances, the introduc- 
tion of this plan has been by no means universal in the iron-works of England 
and Wales, although it is otherwise in Scotland, where the increased make 
of iron, from 37,500 tons in 1830, to nearly 500,000 tons in the past twelve 
months, maj' be in great part, if not altogether, ascribed to the ceconomy 
which Mr. Neilson's plan has occasioned. But for the introduction of that 
plan, we should in all likelihood not have witnessed the unequalled develop- 
ment exhibited during the past fifteen years in this, which has now become 
one of the greatest branches of our national industry. Without this discovery 
our railroad system could not have marched forward with such giant strides, 
and in all probability the application of iron to the building of ships, — an 
application from the extension of which, in future years, so many advantages 
may be made to arise, — might have continued unthought of. 

In a letter which has reached me while writing, from a most intelligent 
iron-master in the North of England *, the subject is thus noticed : — 

" Previously to this invention, metal was made with such coal only as was 
easily destructible before the blast, thereby admitting a greater quantity of 
air into the furnace. Air is the food of fire. Coals of a stronger or more 
bituminous character were not serviceable ; the current of cold air at the 
Tuyeres had the effect of caking the coal and choking the admission of air, 
by which the process of reduction was stopped. But when Mr. Neilson intro- 
duced his method the difficulty was conquered. By heating the air up to 600° 
Fahrenheit, the caking at the Tuyeres no longer took place ; the air entered 
freely into the furnace, and coal hitherto unserviceable was enlisted into the 
service of mankind, and applied to the great improvement of their condition. 

" It was pretended that the metal made with hot-blast was not so good ; 

that it was weaker ; and for a long time it was tabooed in all contracts ; but 

this delusion is gradually giving way to truth. There was no foundation for 

such prejudice. It is known that air does not burn until it reaches 3000° 

* Charles Perkins, Esq. 



102 REPORT 1846. 

Fahrenheit ; the raising of it to 600° before admission to the furnace was 
nothing, nor did it destroy any of its elementary qualities ; it only secured 
its admission and ensured its regularity of action in the process of reduction. 
This was an increase of man's power over eletnentary matter : it is by the 
additions to and the increase of this power that men will in time accomplish 
a greater and more powerful condition." 

The disinclination to adopt an innovation, which as we have seen in this 
case of the hot-blast, has not been entirely overcome by more than fifteen 
years' experience of its advantages, has not been confined to that instance, 
but has been allowed for a much longer period to influence, in another case, 
the proceedings of our iron-masters. It was as long ago as 1801, that Mr. 
David Mushet, to whom the world is greatly indebted for his scientific re- 
searches and his practical exertions in this important branch of metallurgy, 
discovered when crossing the river Calder, in the parish of Old Monkland, a 
description of ironstone, to which the name of black-band, or Mushet-stone, 
has been given. For many years following this discovery the black-band was 
used only in the Calder Iron Works, which were established in 1800 by Mr. 
Mushet, and it was not even there employed alone, but was used in combi- 
nation with other iron ores of the argillaceous class. It was not until 1825 
that it was first used alone by the Monkland Company, whose success in the 
experiment led gradually to its adoption by other establishments, and to the 
erection of additional works. 

Mr. Mushet, in his ' Papers on Iron and Steel,' p. 128, thus describes the 
advantages of this kind of ironstone : — 

" Instead of 20, 25 or 30 cwt. of limestone formerly used to make a ton of 
iron, the black-band now requires only 6, 7 or 8 cwt. to the production of a 
ton. This arises from the extreme richness of the ore when roasted, and 
from the small quantity of earthy matter it contains, which renders the ope- 
ration of smelting the black-band with hot-blast more like the melting of 
iron than the smelting of an ore. When properly roasted, its richness ranges 
from 60 to 70 per cent., so that little more than a ton and a half is required 
to make a ton of pig-iron ; and as one ton of coal will smelt one ton of 
roasted ore, it is evident that when the black-band is used alone, 35 cwt. of 
raw coal will suffice to the production of one ton of good gray pig-iron." 

This calculation is strongly corroborated by a statement which was pro- 
duced by Dr. Watt to the Statistical Section of the Association at Cambridge, 
from which it appeared, that to make 400,400 tons of iron in the counties of 
Lanark, Ayr, Stirling and Clackmannan, the quantity of coal consumed was 
934,266 tons, or 2 tons 6 cwt. 2 qrs. 18 lbs. for each ton of iron, part of which 
is the produce of argillaceous ores. 

The statement of these discoveries appears necessary in order to account 
for the great and rapid extension given since 1830 to the production of iron 
in this kingdom, and especially in Scotland. 

In 1836 every iron-Avork in Great Britain was visited, and an account taken 
of its produce, by a highlj^-gifted gentleman, M. F. Le Play, " Ingenieur en 
chef," employed in the Ministry ot^ Public Works at Paris, under whose di- 
rection are made the yearly reports describing the progress of mining indus- 
try in France, of which I have on former occasions availed myself in pre- 
paring papers read before this Section of the Association. The result of his 
inquiries showed that in that year the quantity of iron made reached to 
1,000,000 tons, an amount then deemed almost incredible, but which in the 
years immediately following was greatly exceeded. In his ' Papers on Iron 
and Steel,' to which reference has already been had, Mr. Mushet states 
(p. 421) that the quantity of British iron made in 1839, was 1,248,781 tons 
(See App. No. 6). 



ON THE IRON MANUFACTURE IN GREAT BRITAIN. 



103 



In the following year a very elaborate inquiry into this subject was made 
by Mr. William Jessop of the Butterley Works in Derbyshire, and the result 
of his inquiries was printed by him for private distribution. His statement 
embraces every iron-work in Great Britain, and gives the number of furnaces 
in blast and out of blast, with the weekly produce of each establishment. 
From Mr. Jessop's tables it was shown that the number of furnaces in blast 
in that year (1840) was 402, and the number out of blast 88; the weekly 
produce of the 402 furnaces being 27,928 tons, and consequently the yearly 
produce, taken at 50 weeks' working, 1,396,400 tons. In the production of 
this quantity Mr. Jessop states that there were consumed 4,877,000 tons of 
coal, being at the rate of 3 tons in Scotland, and 3 tons 12 cwt. in England 
and Wales, for each ton of iron. This was exclusive of the coal used in 
converting pig-iron into wrought iron, and which he sets down at 2,000,000 
tons additional (see App. No. 7). At the time Mr. Jessop's account was 
taken, it appeared that out of 420 furnaces erected in England and Wales, 
there were 82, or 1 in 5, out of blast, and that of 70 furnaces in Scotland, 
6, or I in 11, were in that condition. The rapid increase of this manufac- 
ture during the preceding ten years forbids the belief that this large number 
of furnaces could have been idle through dilapidation. In fact, the country 
was then suffering under an amount of commercial depression of no ordinary 
character, and which continued to press heavily upon almost every branch of 
its industry, until the abundant harvest of 1844, joined to the effect of the 
fiscal reforms introduced in 1842, caused the return of healthiness to our 
trading interests. The continuance of the depression, which had no doubt 
extinguished so many of the furnace fires in 1840, caused still more of them 
to be put out of blast in the years immediately following, and it was shown 
by a statement drawn out under the direction of an association of the iron- 
masters of Yorkshire and Derbyshire, that the quantity of iron made in the 
first six months of 1842 in Yorkshire, Derbyshire, Staffordshire, Shropshire, 
South Wales and Scotland did not exceed 523,214 tons, or at the rate of 
1,046,428 tons per annum. The quantity of iron made in those divisions of 
the kingdom in 1840 was, according to Mr. Jessop's statement, 1,343,400 
tons, so that the diminution of production was at the rate of more than 22 
per cent., which rate was probably experienced throughout the kingdom; 
and in this case the whole quantity of iron made in 1842 did not much ex- 
ceed one million of tons, the quantity ascertained by M. Le Play to have 
been made in 1836. 

A great impulse had been given at that time to this branch of industry by 
the demand arising from the construction of railways. This impulse, and 
the subsequent depression, may be easily inferred from the following state- 
ment of the number of railway acts passed in each year from 1831 to 1843, 
distinguishing such as were for new lines from those which authorized ex- 
tensions or amendments in former acts, and giving the amount of capital 
authorized by Parliament to be raised under those acts. 



Years. 


Acts passed for 


Capital authorized. 


New Lines. 


Extensions,&c. 


1831. 
1832. 
1833. 
1834. 


5 
5 
5 
5 


4 
4 
6 
9 


1,799,873 

567,685 

5,525,333 

2,312,053 



104 



REPORT — 1846. 

Table (continued). 





Acts passed for 




Years. 




Capital authorized. 








New Lines. 


Extensions,&c. 




1835. 


8 


11 


£ 
4,812,833 


1836. 


29 


6 


22,874,998 


1837. 


15 


27 


13,521,799 


1838. 


2 


17 


2,096,198 


1839. 


3 


24 


6,455,797 


1840. 


— 


24 


2,495,032 


1841. 


1 


18 


3,410,686 


1842. 


4 


18 


5,311,642 


1843. 


5 


19 


3,861,350 



We see from these figures, that in the two years 1836 and 1837, Parliament 
passed 7 7 railway bills, of which 4;4 were for new lines, and that the capital thus 
authorized to be raised, amounted to more than 36 millions of money. The 
length of the lines then sanctioned amounted in the aggregate to nearly 
1200 miles, and would call for the production of more than 500,000 tons of 
iron. The price of bar-iron, which in 1834 had been 6/. 10s. per ton, and 
in 1835 was Tl. lOs., advanced in 1836 to 11/., and this gave a powerful 
stimulus to the extension of the manufacture. So great a rise in the market 
value of the metal checked its use however for a great variety of purposes, 
and when, in the years following 1837, the railway speculation so far sub- 
sided, that only 15 acts were passed for the construction of new lines in the 
six years from 1858 to 1843, the price of iron fell as rapidly as it had previ- 
ously risen, and it could with difficulty be sold at less than half the price 
which it commanded in 1836. In this state of things, the iron-masters 
sought to lighten their loss by limiting the production, rather than by forcing 
their goods into use by lowering the price. This appears to have been done 
to a greater extent in England and Wales than in Scotland, where for reasons 
already explained, the cost of production had been so lessened as to enable 
the iron-masters to work to a profit at prices by which their English compe- 
titors were losing on every ton they brought to market. 

Since Mr. Jessop made his statement in October 184'0, not any attempt 
has been made to ascertain the progress of the iron manufacture throughout 
England and Wales, from which any result can be confidently given. In 
Scotland, where the principal extension has occurred, several statements of 
the kind have been put forward. One of these, the correctness of which has 
been generally admitted by those whose knowledge upon the subject should 
give weight to their opinion, states the number of furnaces in blast in March 
1845, and the weekly and yearly produce from the same to have been 76 
furnaces, yielding 8250 tons of iron weekly, or in the year of 50 weeks,412,500 
tons. At that time there were, according to this statement, 22 more fur- 
naces built and building, and the whole of these it was expected would be 
in blast in the course of 12 months from that time. It is stated by a respect- 
able firm in Glasgow, that in December 1845, there were 87 furnaces in 
blast in Scotland, the number at the end of 1844 having been 69; the in- 
creased make of pig-iron in 1845 as compared with 1844 is stated at 60,000 
tons. The lowest price at Glasgow in January 1844 was 40*. per ton, and 
the highest price for the year, caused in great part by the purchases of spe- 



ON THE IRON MANUFACTURE IN GREAT BRITAIN. 105 

culators tempted by that extremely low price, occurred in April, when the 
quotation stands at 655. per ton : in September the price was again reduced 
to 50s., and the average price of the year was 55s. 6d. per ton. In 1845, the 
lowest price, which also occurred in January, was 60s., and in March the 
price had advanced to 100s. ; in May purchases were freely made at 110s. ; 
and we cannot wonder that with a rise in price equal to 175 per cent., so great a 
stimulus should be given to the extension of iron-works. On the authority 
of the same firm, it is stated, that the number of furnaces in blast, which at 
the end of 1845 was 87, was on the 30th of June 1846 increased to 97, and 
that the computed make of pig-iron in Scotland, in the first six months of 
the present year, is 260,000 tons, equal to 520,000 tons in the year, showing 
that the production has been more than doubled in the six years since 1840. 
I have before me a detailed account of the iron-works of Scotland in 
August 1846, which gives 105 as the number of furnaces in blast, 21 as those 
out of blast, in addition to 11 more building. The weekly make of pig-iron 
at the 105 furnaces is said to be 11,010 tons, equal to 550,050 tons per 
annum ; estimating that each furnace is in action during 50 weeks. This ac- 
count is in part corroborated by a table kindly sent to me by Dr. Watt, of the 
works in Lanarkshire, and which places the yearly produce of that county at 
390,000 tons. It is further stated, that notwithstanding the great increase in 
the quantity made, according to the concurrent testimony of all parties, the 
stock of iron in the hands of the makers and dealers has materially decreased. 
The stock in Glasgow at the end ot 1845 was 210,000 tons 
and on the 30th of June 1846, only 140,000 

Decrease 70,000 tons. 

It may be assumed that this increase of production, although it may have 
been at first called forth by speculation, has not been sustained by those 
means, since the stock has thus diminished in the face of that increase, while 
the price has been declining. In January it was 80s. and in June 68s. per ton. 

A statement which appeared in the ' Glamorgan Gazette' computed the 
make of iron in 1843 at 1,210,550 tons, of which quantity 238,750 tons were 
assigned to Scotland. The entire quantity was stated to have been the pro- 
duce of 339 furnaces in blast, while there were said to be 1 90 furnaces out 
of blast in different parts of the kingdom. Another statement, communicated 
to me by Mr. Buckley (Member of Parliament for Newcastle), differs but 
slightly from that which was inserted in the ' Glamorgan Gazette,' the total 
quantity made being given as 1,215,350 tons (see App. No. 10). 

In the absence of any authentic statement of the make of iron in England 
and Wales at this present time, an attempt has been made by correspondence 
to ascertain the facts as they exist in different localities. The result of in- 
quiries thus conducted cannot have the same value as the investigations made 
by Mr. Jessop in 1840, but is offered here as the best which it has been in 
my power to produce. 

It is given as the opinion of several most intelligent iron-masters whom I 
have consulted, that nearly all the increased production of iron in this king- 
dom since 1840, has been drawn from Scotland. It is true that in some of the 
Scotch works there is already experienced a short supply of materials, but on 
the other hand, new fields are discovered and brought into working. Mr. 
Jessop states, that " a new field of coal and iron has been opened out in 
Ayrshire, but not so favourable as the Airdrie and Coalbridge district." The 
great demand at present experienced, and that which is sure to follow from 
the extent of the railway projects which have received legislative sanction in 
1845 and 1846, have naturally stimulated every establishment to its utmost 



106 REPORT — 1846. 

point of production. But in order to add materially to the make of iron, a 
great variety of circumstances must be brouglit to concur. One of the 
greatest difficulties with which the manufacturers liave to contend in such 
circumstances is offered by the workmen, who naturally enough, perhaps, 
strive to obtain for themselves the largest possible share of the increased value 
of that which they produce. To be of much use in any branch of this ma- 
nufacture a man must have undergone a season of instruction, and as the 
number of skilled workmen is limited, these, whenever any great or unwonted 
demand arises, hardly know how to set limits to their demands. On this 
subject, a recent number of the ' Merthyr Guardian' contains the following 
paragraph. 

" Prosperity in the Iron Trade. — We believe the iron trade in this district 
was rarely ever known to be in a more thriving state than at the present 
time. Forgemen and puddiers realize from 10/. to 18/. per month. But 
this state of prosperity brings its attendant snare, inasmuch as the surplus 
which should be accumulating in the Savings' Banks, is in too many cases 
squandered in debauchery or lavished in vice. The state of things calls 
loudly for a remedy." 

This complaint will not surprise us when we call to mind the fearful de- 
scription of the state of the population of Merthyr, read before this section 
of the Association at Cambridge, by Mr. Kenrick. But the same complaint 
is made in other quarters, and there is but too much reason to fear that it 
might be universally preferred. In a letter now before me, an extensive 
iron-master in the north of England writes on the 15th of August last, — " The 
cost of making iron from the recent spirt of prosperity has increased so 
enormously that the ' prosperity ' has well-nigh ruined many makers ! wages 
are so ruinously raised." 

A gentleman writes from Scotland in June 1845, — "This is the present 
position of the trade. The speculators were first bitten by the mania of an- 
ticipated consumption ; then the masters took the fever, and, as was to be 
expected, the workmen follow, and say they must have a rise of wages equi- 
valent to the 1 105. price. It is nothing to them that iron has again fallen ; 
they say, first put us up equivalent to the high price before you can ask us 
to conform to the present. The iron-master will therefore find that he must 
give the wages corresponding to 110s., although he may sell at 70s. or less." 
It is understood that chiefly I'rom this cause the cost price of iron in Scotland 
has been increased about 15s. per ton. 

May we not reasonably allow ourselves to indulge the hope, that at some 
future, and perhaps not very distant day, the two classes of employers and 
workmen may come to the better understanding of their mutual interest, so 
that the sun of prosperity, whenever it arises, may shine equally for both ? 

Statements were inserted in the course of last year in the ' Mining Journal,' 
and were made the subject of remark by different persons without impugn- 
ing their accuracy, to the effect that the make of iron in Great Britain during 
184'5 would amount to about 1,330,000 tons. If the statement concerning 
the production in Scotland already mentioned should be correct, this would 
leave for the make in England and Wales 917,500 tons, being 134',000 tons 
less than the quantity stated by Mr. Mushet as made in 1839, and 238,000 
less than the produce of IS^O, as given by Mr. Jessop. The increase in 
Scotland during the five years was, on the other hand, 171,500 tons, thus 
leaving the whole produce of 1845 less than that of 1840 by 66,500 tons. 

It will doubtless appear extraordinary, that with so much cause for increa- 
sing the quantity as had arisen last year out ot the actual and anticipated 
demand for railway purposes, the produce in England and Wales should not 



ON THE IRON MANUFACTURE IN GREAT BRITAIN. 107 

at least have overtaken that obtained in 1 840, and that it had not done so 
calls for explanation. In the endeavour to obtain this I have been met by- 
statements which might appear to be in some respects somewhat contradic- 
tory of each other, the different writers representing matters as presented to 
their own views and experience, and without possessing that general acquaint- 
ance with the facts existing in other districts which it is so desirable to attain. 
One most highly intelligent iron-master whom I have consulted writes, " I 
consider now," that is, since the discovery of the hot-blast system, " that all 
the ironstone and coal of this country is applicable to the production of iron. 
I fear however that the deposits of ironstone exceed very much those of coal, 
and that the increasing demand upon this latter article will before many years 
show its effects. I think in Staffordshire they already feel a want in the high 
price of coal, and the iron trade seems migrating northward, where coal is 
more abundant and different deposits of ironstone are continually discovered." 
The gentleman who thus writes is interested in iron-works in the county of 
Northumberland, where the make of iron has increased and is increasing 
greatly and rapidly, but still not sufficiently to compensate for the falling off 
of production elsewhere. 

Another gentleman, from whom I have received great assistance in my 
inquiries, writes, " In some of the localities in Scotland there is beginning to 
be a great scarcity of ironstone, several furnaces being recently put out iu 
consequence ; and in Staffordshire still more so. People's ideas about increase 
of make of iron travel much faster than the reality. In fact, during 1845 
great numbers of the furnaces in Staffordshire were going on half-quantity, 
simply from want of materials ; this I knoio." It is corroborative of this re- 
presentation, that a powerful iron company, having works in Staffordshire, has 
for some time had two new furnaces completed without putting them in action. 

From a third correspondent, whose interest is in the great iron district of 
South Wales, I hear of so great a number of new works building in Durham, 
Cumberland, Northumberland and Scotland, that any account taken of the 
produce in those districts, even so recently as last April, must necessarily be 
very imperfect. He adds, " They are progressing so rapidly, and can pro- 
duce iron at such a cheap rate in these new iron districts, as to lead to the 
conclusion that ultimately the principal seat of the iron manufacture will be 
removed from South Wales to the North of England and Scotland." 

On the other hand, Mr. Mushet, whose acquaintance with the subject is 
probably of a more general nature than that of my correspondents previously 
quoted, writes so recently as the 16th of August in the present year, — "The 
principal object in the iron trade which now attracts attention is the recent 
discovery of an extensive district of black-band ironstone, ranging from 
beyond Cwm Avon, through Maesteg, towards the valley of the Taffe. The 
two principal beds or veins of black-band lie high in the coal series, and in 
this respect differ from the Beaufort black-band, which wafe found over the 
lowest coal, and from the Scotch, which was found descending in the coal- 
series at various depths. These beds measure fifteen inches each in thick- 
ness, and will each yield fully 3000 tons per acre. The lower bed contains 
40 per cent, of iron, and is put raw into the furnace ; the other is previously 
roasted, as it contains more shale, and when so roasted yields the same quan- 
tity of iron as the other when raw. As this range of minerals occupies both 
sides of the line from Cwm Avon to Cwm Taffe, large tracts of black-band 
ironstone must unavoidably be found, and it may not be hazardous to pro- 
nounce that this in time may rival Merthyr, and become an extensive ii'on- 
making district, probably the Lanarkshire of South Wales." Mr. Mushet 
furnishes a list of fourteen furuaces in which this black-band ironstone is 



108 REPORT— 1846. 

partly used, and mentions three other furnaces now building in which it will 
be employed. 

Referring to the counties of Durham and Northumberland, Mr. Mushet 
gives a list of thirty-five furnaces Avhere twenty years ago only one blast- 
furnace, at Chester-le-street, was known to exist ; and he mentions, but not 
as of his own knowledge, another source of supply as about being brought 
forward into notice from the spoil and waste of the lead-mines in Weardale, 
" which are now worked and have been so for ages." He says, " The rider 
of the lead-ore is a true carbonate of iron, some of it yielding from 25 
to 40 per cent. A small blast-furnace has been erected at Stanhope, where 
a very important and interesting experiment has been made, and a suc- 
cessful result obtained, in which this rider ironstone has been smelted, and 
pig-iron of a strong and excellent quality produced. This ore, even after 
being ground and washed, still contains some particles of galena, and which 
in smelting gives out at the furnace-top a heavy cloud of sulphurous smoke, 
of a forbidding aspect. The pig-iron, however, when remelted, yields no 
smoke from its surface, which would be the case if a small quantity of me- 
tallic lead were thrown in, from which it may be inferred that the lead is in 
the process of smelting entirely dissipated and driven off. What effect may 
be produced upon the conversion of this iron into bar-iron remains to be de- 
termined. The result of this experiment has been deemed so satisfactory as 
to induce the company to erect large smelting-works about three miles from 
Wolsingham. These works consist of two powerful blast-engines and six 
large blast-furnaces. In this enterprise we shall by and by behold the spoil 
of ancient mines, which has reposed for ages, brought to light, no longer as a 
useless, but as a useful material for the production of the common and ordi- 
nary sorts of pig-iron. Great and beneficial results are calculated upon, and 
should they be realized, will no doubt contribute greatly to the produce of 
our iron manufacture." 

Other authorities do not speak so hopefully of this discovery, and certain 
it is, that of the six blast-furnaces of which Mr. Mushet speaks, only three 
have hitherto been erected, and only one of these is lighted. A great part 
of the furnaces now existing in Durham are chiefly employed in reducing 
ores procured from Whitby and from Scotland, and occasionally small quan- 
tities of hematite ore are procured from Devonshire and from Cumberland. 
There is a considerable quantity of ironstone of the argillaceous kind in the 
eastern division of Durham, but it is for the most part found at inaccessible 
depths, or in such positions of dislocation as to render the cost of working it 
too great. At a place called Shottley-bridge, about fifteen miles west of 
Newcastle, where the ore is plentiful and very accessible, there have been 
eight furnaces at work for some time, and six others are now about to be 
lighted. The argillaceous ironstone found at that place resembles in quality 
the ironstone of Staffordshire. This is the only portion of the county of 
Durham in which it has hitherto been found practicable to make iron in any 
large quantity with materials wholly found on the spot. I have a list of 
twenty-two furnaces now in blast in the two counties of Durham and North- 
umberland yielding weekly 1895 tons of pig-iron, equal to a yearly produc- 
tion of 94',750 tons, the quantity made in 1843 having been estimated at 
25,750 tons, and in 1844 to only 21,250 tons. In various localities in North- 
umberland there is abundance of clay-ironstone in the immediate vicinity of 
plenty of excellent coal and limestone, and in the course of years large quan- 
tities of iron may be made in that district. The great obstacle to any sudden 
increase there and elsewhere is offered, as already mentioned, by the difficulty 
of procuring skilled labour. Any addition to the number of coal-miners can 



ON THE IRON MANUFACTURE IN GREAT BRITAIN. 109 

be made only by slow degrees, and the same condition applies to all other 
classes of persons whose labour is required for the manufacture of iron. It 
is hopeless to stimulate the exertions of the persons already employed. They 
are naturally ready enough to exact higher rates of wages when the demand 
for their labour becomes more urgent, but succeeding in this they prefer to 
obtain the same amount of earnings, with higher rates of wages, to the secu- 
ring of greater gains by the exertion of even the same amount of toil, so that 
a greater urgency in the demand may be, and frequently is, accompanied by 
a lessened production. 

Under these circumstances, how the enormous demand existing and to arise 
from carrying out the railway schemes already sanctioned is to be met, it 
would be most difficult to say. The laying down of these lines and providing 
them with the needful working stock of carriages, &c. would absorb all the 
iron which it is reasonable to expect will be made in Great Britain during 
the next three years, and it affords no satisfactory solution of this difficulty to 
say that the quantity required will only be called for progressively, and that 
the demand will be spread over the same three years. To render this circum- 
stance effective, we should be assured that no further projects will be sanc- 
tioned during the time spent in their construction, an assurance for which 
we can hardly look, and even then we should be left without a ton of iron 
applicable to the thousand other purposes for which this metal is so indis- 
pensable. If the difficulty presented by the want of larbour could be sur- 
mounted, there appears no rational ground for supposing that we should, for 
a very long time to come, experience any deficiency in the means for making 
iron. In the anthracite coal district of South Wales, where clay-ironstone is 
thickly interstratified with the coal seams, there appears to be no reason to 
doubt that if the means employed in the anthracite district of America for 
smelting the clay-ironstone Avere adopted, it would prove equally successful. 
The difficulty consists in the sluggish nature of anthracite, which requires a 
more rapid draught than can be provided by the ordinary means of bellows 
and tall chimneys; and this is overcome in America by obtaining a great 
volume of air by means oi fanners. The iron made with raw anthracite coals 
has been found by Mr. Mushet to be much stronger than iron made with 
coke, and after a variety of experiments, the earlier of which afforded but 
small encouragement, this fuel has been adopted by the proprietors of twenty- 
three furnaces, who avail themselves of only the ordinary means for providing 
a blast. The manner in which the railway demand has already limited other 
uses of iron, may be gathered from the following extract from a letter of 
recent date written to me by Mr. Mushet : — 

" At the above period (1840) merchant bar-iron, boiler-plate, sheet-iron 
and rod- iron, principally occupied our mills ; but these of late, particularly in 
South Wales, have given way in a great measure to the manufacture of 
railway bars, so as to eclipse in a striking manner the varied and extensive 
assortments required by the merchants' demands." The long period of dul- 
ness that intervened between 1839 and the beginning of 1845, accompanied 
as it was by a continued fall in the market price of iron, caused this metal to 
be applied, most advantageously, to a variety of new purposes, from which it 
will be prejudicial henceforth to withdraw it. In a well-known mercantile 
circular letter issued in February ISiS by Messrs. Jevons of Liverpool, it is 
stated that there had arisen " a new and increasing demand for iron roofs, 
iron houses, and fire-proof buildings in Liverpool," and that during the year 
then just passed upwards of 20,000 tons of cast and wrought iron had been 
so used iu that town. These gentlemen further stated that preparations were 
going forward for the erection of still more extensive ranges of buildings of 



1 10 REPORT — 1846. 

similar construction during 184-5, and that the sailing-ships and steam-vessels 
then under construction in that port would require 25,000 tons of plate-iron 
and angle-iron. 

The employment of iron for the purpose last mentioned, that of ship- 
building, has already been an object of very great national importance. The 
extent to which this use for the metal may be carried in future years it is 
not possible to foresee, but we may base upon even our present limited ex- 
perience the hope that by this means our furnaces and forges may be pro- 
vided with some employment when our system of railways shall be completed. 
The tonnage of mercantile shipping belonging to the British empire in 1845 
was 3,7]4',061 tons, and exceeded the amount in existence in 1814 by 
1,097,096 tons, but during that interval there were built and registered ships 
amounting in their measurement to. 5,476,957 tons, so that there were required 
to be built ships of the aggregate burthen of 4,379,861 tons, in order to re- 
pair the waste occasioned by wear and tear and by losses : altogether the 
building of ships has gone forward at the average rate of 176,676 tons 
yearly. Assuming for the moment that this same rate of building will be 
called for in future years, and that the whole of the mercantile shipping con- 
structed would be built of iron, this would prove a very insufficient substitute 
for the demand now existing for railway purposes. I have before me a 
statement of the weight of iron used in building eight large sea-going steam- 
vessels, the aggregate measurement of which was 5922 tons, by which it is 
shown that the metal used was 2877 tons weight, or 9 cwt. 2 qrs. 24 lbs. for 
each ton of measurement, and at this rate the construction of 176,676 tons of 
shipping in each year would provide a market for no more than 85,814 tons 
of wrought iron, equal to 115,849 tons of pig-iron. We cannot suppose, 
however great may be the advantages attendant upon the substitution of iron 
for timber in ship-building, that this use of the latter material will be all at 
once, or indeed for many years, abandoned. There are many existing inter- 
ests opposed to the change, and there is much of prejudice still to be over- 
come before all our merchant-ships will be built of iron. We must likewise 
bear in mind the now well-established fact, that iron ships are far more durable 
than those built of timber, that they require much less repair, and that they 
are less subject to accident and to loss. It cannot be necessary, however, to 
enlarge upon this subject, since the Association has already been favoured 
by Mr. Fairbairn at one of its former meetings — that held at Glasgow in 
1840 — with a valuable paper upon the subject. 

Placing this subject in another point of view, may we not however feel 
justified in believing, that when opposing interests shall have been silenced, 
and existing prejudices shall be overcome, and the fast increasing commerce 
of this country shall have experienced some degree of that development 
which is expected to spring from late changes in our commercial legislation, 
the rate of increase hitherto sufficient to supply the waste of our mercantile 
marine, and to provide what has been necessary for its inci'ease, will no longer 
suffice to that end, and that although our iron ships may outlast by three or 
four times the less durable vessels now constructed, and through all their 
existence may call for little or no materials to be used for their repair, that 
the necessity for additional shipping may in great part prove an equivalent 
for the lessened demand otherwise arising? 

The building of iron ships is at this time proceeding at a greater rate than 
at any previous moment since their first introduction, although the price of 
iron has so materially advanced, and this should give us the assurance that 
when, as we may expect it will happen, the falling off of railway demand, or 
the exertions of our iron-masters, shall have restored the equilibrium between 



ON THE IRON MANUFACTURE IN GREAT BRITAIN. Ill 

aupply and demand, and the price shall again have become more moderate, 
an impetus will be given to the production of shipping, not alone for the 
uses of our own merchants, but for carrying on the trade and navigation of 
other countries. The cost of our shipping will then be so materially reduced, 
both their first cost and the expense of their maintenance, that the objection 
so often and unfortunately so successfully offered by our shipowners to any 
relaxations in our commercial code affecting their business, that the greater 
cheapness with which shipping can be produced in foreign countries prevents 
their successfully competing with ships of those countries, can no longer be 
urged with any plausibility ; but on the contrary, tliat ships of English con- 
struction will then be the cheapest in the world. It has been said that fluc- 
tuations in the price of iron do not cause any considerable difference in the 
cost of iron vessels, so large a proportion of their whole cost consisting in 
labour. A reduction of 20s. per ton in the price of the material will, how- 
ever, cause an oeconomy of 10s. per measurement ton in the cost of the ship, 
and it will hardly be said that the very possible rise or fall of 31. or 4:1. per 
ton in the price of iron plates is an immaterial circumstance to the ship- 
builder. But the cheapness here spoken of will no doubt be principally 
found in the greater durability and the insignificant cost of repairs of metal 
ships. 

A statement was inserted a few months ago in a Scotch newspaper, giving 
the particulars of the iron ships then under construction in the Clyde ; they 
amounted to twenty-four in number, and were of the aggregate burthen of 
14,032 tons (see Appendix No. 8). These were all steam -vessels, to which 
class of shipping iron has hitherto been principally applied, although there is 
no reason for supposing that it is not equally applicable to every description 
of naval architecture. The reason for this circumstance may probably be 
found in the fact, that the construction and employment of steam-vessels has, 
for the most part, been undertaken by persons not previously interested in 
shipping, and who consequently had no prejudice or habit to overcome in 
their choice of material. 

This statement, imperfect as it necessarily is, would be more glaringly so 
if it did not present some particulars of our external iron trade. 

So recently as the beginning of the present century more than two-fifths 
of all the iron used in this kingdom was imported from the north of Europe. 
Foreign metal was then used for very many of the purposes to which iron 
was at that time generally applied in England, and it was so used indiscrimi- 
nately with British iron. In 1806 the use of foreign iron had been lessened 
by nearly one-third, while the home production was so increased as to form 
seven-eighths of the quantity used. In a few years after our make was be- 
yond our own wants, and foreign iron ceased to be imported for any pur- 
poses to which the produce of our own forges could be applied. Thence- 
forward our demands have been confined to metal of the qualities from which 
alone steel can be made. Our exports of British iron have, on the contrary, 
increased progressively, and have now become an object of great national 
importance. The statement given in the Appendix, No. 9 shows the yearly 
progress of the trade since 1827 up to the year 1845 inclusive. It will be 
seen on consulting this statement, that the quantity had increased from 92,313 
tons in 1827 to 351,978 tons in 18-t5, and that the declared value of the 
shipments advanced in that interval from £1,215,561 to £3,501,895. A 
column has been added to the table, exhibiting the average value per ton of 
all forms of iron exported in each year, from which it will be seen how great 
an influence price has, in its advance and its diminution, upon the lessening 
or increase of our exports. In 1840, when the average value appears to have 



112 REPORT 1846. 

been 9/. 85. 2d., the quantity of British iron exported was 268,328 tons. The 
price in the following years fell rapidly, and the demands from other countries 
increased as rapidly. In ISIS, when the average price is represented by 
51. 15s. 5d. per ton, the exports were 448,925 tons. In 1844 the quantity was 
slightly increased, viz. to 458,745 tons, although the price had advanced to 
6/. 195. 2c?. per ton ; but in 1845, the further advance in the average declared 
value to 9^. 18s. llrf. per ton, reduced our foreign shipments to 351,978 tons, 
or by more than 23 per cent. 

It is worthy of remark, that we now export largely, more largely than in 
former periods we ever imported from the same quarter, iron in its crude 
state, and articles manufactured with the same, to the countries whence we 
once drew the largest proportion of what was used by us. In 1844 our ship- 
ments of iron, in its various forms, to the north of Europe amounted to 
178,635 tons, equal probably to 200,000 tons of pig-iron ; and in 1845, not- 
withstanding the great speculative demand and rise in price at home, our 
shipments amounted to 140,006 tons, equal probably to 160,000 of pig-iron. 
In those two years the whole of our colonies and dependencies took from us, 
in 1844, 78,594 tons; and in 1845, 60,683 tons. 

Our largest customers are found in the United States of America, and it 
is probable that they will long continue to be so, unless the citizens of those 
states in which materials for producing iron are found should be unduly 
stimulated to increase their home production through the existence of high 
prices in this countiy. An increased demand from that quarter is expected 
when the more liberal tariff recently passed at Washington shall come into 
operation, but it is clear that the realising of that expectation must depend 
greatly upon the state of markets in this country (see Appendix, No. 11). 

A writer of the protectionist school, in an article inserted in the ' National 
Magazine,' published in New York in July 1845, states that the make of iron 
in the United States in that year from 540 blast-furnaces would amount to 
486,000 tons, and that the domestic supply would ere long be brought to 
meet the entire wants of the country. New furnaces and rolling-mills are, 
according to this writer, being erected in every direction, and those works 
that had been inopei'ative and unproductive, from the low prices of iron in 
1843 and 1844, were again at work, so that it might soon be unnecessary to 
import a ton of the metal from Europe. With a moderate price in England 
we need not put much faith in this assertion, which was put forth as an in- 
ducement to Congress to add to the high protection then afforded by the 
tariff, but which is now reduced. 

France, notwithstanding the exorbitant duties charged on importation, 
takes from us a considerable and constantly increasing quantity of this metal 
(see Appendix, No. 12), and although the production of pig-iron in that 
country has increased from about 220,000 tons in 1831 to about 420,000 
tons in 1843, and is still increasing, the want of a sufficient supply of this 
all-important metal is severely felt in that country, and the high price is 
found to weigh grievously upon various branches of industry. In particular 
a cry has been raised, which it is expected may be successful, in favour of 
the admission, free of duty, of plate-iron suitable for ship-building, but tiie 
eagerness now shown to obtain this concession will be much abated should 
the price of the material advance in any great degree in England. At this 
time we are certainly not in any condition to meet the demand that might 
come upon us should that concession be made by the French Chambers. 

With the exception of England, Sweden appears to be the only country 
which has or can be expected to have any disposable quantity of iron for export- 
ation, and it does not seem likely that we shall be a customer for any, except 



ON THE IRON MANUFACTURE IN GREAT BRITAIN. 113 

that which we need for converting into steel, to which use Swedish iron is 
peculiarly applicable, and for which its high price causes it to be reserved. To 
Russia our shipments of this metal are fully equal to the quantity imported 
thence. 

After much consideration given to the circumstances in which our iron 
manufacture is now placed, and to its prospects for the future, I venture, 
with some hesitation, to oifer the following opinion. 

Legislative sanction has been given in this and the two preceding years 
to the construction of many thousand miles of new railways, in the comple- 
tion of which so many interests are engaged, that we must not expect any 
considerable portion of them to be abandoned by their projectors. We must 
for this reason expect that for some few years to come, during which these 
works will be going forward, the price of iron will be high. The tendency 
of this high price will be, to give an impetus to the manufacture, and to cause 
much new capital to be invested for its extension, for which ample opportu- 
nity presents itself in different localities, although in other places, as in 
Staffordshire, where the manufacture has hitherto flourished, there is more 
reason to expect diminution than increase, owing to a failure in the supply 
of materials. The great obstacle to the forming of new establishments, and 
to the extension of those already in operation, consists in the difficulty of 
procuring the necessary amount of labour, miners, furnace- men and others. 
This obstacle will, however, be gradually and progressively lessened, and 
when the present exaggerated railway demand shall have ceased, as it must 
necessarily do through the completion of the lines which alone can be pro- 
fitably opened, and the demand thence arising for iron shall be limited to 
the quantity — still, however, considerable — which will be needed for keep- 
ing the lines in repair (see App. No. 13), we shall find ourselves in posses- 
sion of means for making iron much beyond what have at any previous time 
existed, and very greatly beyond any probable demand to arise from other 
and existing channels of employment at home, or from foreign countries. 
The price will consequently fall, as it has done at former times and under 
analogous circumstances. We shall then find that this metal will again be 
employed in uses from which it may have been excluded by the previous 
high price. From the improvements already made, and from others which 
we may expect will be introduced into the processes of manufacture, we may 
even find that the market price will fall to a lower point than has hitherto 
been witnessed, and new uses may in consequence be discovered whereto to 
apply this metal. All this, however, must be the work of time, and it seems 
but too probable that in the meanwhile our iron-masters will have to undergo 
a somewhat lengthened season of adversity, for the enduring of which they 
are in a measure prepared by former experience. 



1846. 



114 BBPORT — 1846. 

Appendix No. 1. 

Manufacture of Pig-iron in England and Wales, 1788. 
Made with coke of pit-coal. 

Counties. No. of Furnaces. Tons of Iron. 

Shropshire 21 23,100 

Staffordshire 9 6,900 

Derbyshire 7 4,200 

Yorkshire 6 4,500 

Cumberland 1 700 

Cheshire I 600 

Glamorganshire 6 6,600 

Brecknockshire 2 1,600 



Made with wood-charcoal. 

Gloucestershire 4 2,600 

Monmouthshire 3 2,100 

Glamorganshire 3 1,800 

Carmarthenshire 1 400 

Merionethshire 1 400 

Shropshire 3 1,800 

Derbyshire 1 300 

Yorkshire 1 000 

Westmoreland 1 400 

Cumberland 1 300 

Lancashire 3 2,100 

Sussex 2 300 



■ 48,200 



13,100 



Total 61,300 



Appendix No. 2. 
Manufacture of Iron in Great Britain in 1796. 

Counties. No. of Furnaces. Tons of Iron. 

Chester 2 '. 1,958 

Cumberland 4 2,034 

Derbyshire 3 2,107 

Gloucestershire 2 380 

Herefordshire 5 2,529 

Lincolnshire 2 705 

Shropshire 23 32,969 

Sussex 1 173 

South Wales 25 34,251 

North Wales 3 1,434 

Staffordshire 14 13,211 

Yorkshire 20 17,242 



Total— Great Britain ... 121 125,079 



1 



ON THE IRON MANUFACTURE IN GREAT BRITAIN. 115 

Appendix No. 3. 
Production of Iron in 1806 in Great Britain. 

Counties. No. of Furnaces. Tons of Pig-iron. 

Cumberland 4 1,491 

Derbyshire 12 10,329 

Gloucestershire 2 1,629 

Lancashire 2 2,500 

Monmouthshire 3 2,444 

Shropshire 28 54,966 

Staffordshire 31 49,460 

Yorkshire 23 26,671 

South Wales 36 75,601 

North Wales 3 2,075 

Old charcoal furnaces in dif- "1 , , ^ „,. « 

ferent counties /^^ ^'^^^ 



Total— England and Wales 155 234,966 

Scotland 18 23,240 



Total— Great Britain 173 258,206 



Appendix No. 4. 
Production of Iron in Great Britain in 1823. 

Counties. No. of Furnaces. Tons of Pig-iron. 

Staffordshire 84 133,590 

Shropshire 38 57,923 

Yorkshire 26 27,311 

Derbyshire 15 14,038 

Northumberland and Durham 2 2,379 

South Wales 72 182,325 

North Wales (estimated^ 10,000 



237 427,566 

Scotland 22 24,500 



Total— Great Britain 259 452,066 



Appendix No. 5. 
Production of Iron in Great Britain in 1830. 

Counties. No. of Furnaces. Tons of Pig-iron. 

Staffordshire 123 212,604 

Shropshire 48 73,418 

Yorkshire 27 28,926 

Derbyshire 18 17,999 

Northumberland and Durham 4 5,327 

South Wales 113 277,643 

North Wales (estimated) , 25,000 



333 640,917 

Scotland 27 37,500 



Total— Great Britain 360 678,417 

i2 



116 



REPORT — 1846. 



Appendix No. 6. 

Quantity of Iron made in Great Britain in 1839, as stated by David 
Mushet, Esq. 

Districts. No. of Furnaces. Tons of Pig-iron, 

South Wales 122 453,880 

Forest of Dean 5 18,200 

Shropshire 29 80,940 

Staffordshire (South) 106 346,213 

Staffordshire (North) 7 18,200 

North Wales 13 33,800 

Derbyshire 14 34,372 

Yorkshire 22 '. 52^416 

Northumberland and Durham 5 13,000 

Scotland 54 196,960 



Lancashire (charcoal-iron) 



377 1,247,981 

800 



Total of pig-iron in Great Britain 1,248,781 



Appendix No. 7. 

Production of Iron in Great Britain in the year 184'0, as ascertained by 
Mr. William Jessop, of the Butterley Iron Works, Derbyshire. 



Districts. 


Number of Furnaces 


Iron made. 


Coal used. 


In blast. 


Out of blast. 


Forest of Dean 


4 

132 

12 

5 

25 

13 

7 

116 

24 

64 


31* 
3 
1 
7 
5 
9 

19 
7 
6 


tons. 
15,500 

505.000 
26,500 
11,000 
56,000 
31,000 
20,500 

407,150 
82,750 

241,000 


tons. 

60,000 

1,436.000 

110,000 

38,500 
306,500 
129,000 

83,000 

1,582,000 

409,000 

723,000 


South Wales 


North Wales 


Northumberland 


Yorkshire 


Derbyshire 


North Staffordshire 

South Staffordshire 

Shropshire 


Scotland 


Coal used in converting tc 


402 
wrought- 


88 
iron 


1,396,400 


4,877,000 
2,000,000 






6,877,000 



Of the above 402 furnaces, there were using hot air, 162; cold air, 240. 



ON THE IRON MANUFACTURE IN GREAT BRITAIN. 11? 



Appendix No. 8. 
Iron Steam-Vessels being built in the Clyde during the Spring of 184-6. 
of 2,000 tons burthen and 750 horse-power. 
... 1,300 



1,225 
1,100 
900 
800 
700 
600 
600 
600 
500 
450 
450 
450 
346 
346 
300 
275 
275 
275 
150 
150 
120 
120 



24 14,032 



520 

450 

400 

400 

400 

280 

300 

220 

300 

200 

250 

250 

150 

50 

50 

160 

35 

35 

35 

120 

100 

65 

60 



5,580 



Appendix No. 9. 
Quantities of British Iron exported, with the declared value of the same, and 
the average value of each ton exported in each year from 1827 to 1845 
inclusive, stated in tons. 















Average de- 


Yeata. 


Bar-iron. 


Pig-iroji. 


Castings. 


AU kinds*. 


value. 


clared value 
per ton. 












£ 


£ s. d. 


1827. 


45,284 


7,095 


6,292 


92,313 


1,215,561 


13 3 5 


1828. 


51,108 


7,826 


6,205 


100,403 


1,226,617 


12 4 4 


1829. 


56,178 


8,931 


8,219 


108,275 


1,162,931 


10 14 9 


1830. 


59,885 


12,036 


8,854 


117,420 


1,078,523 


9 3 8 


1831. 


64,012 


12,444 


10,361 


124,312 


1,123,372 


9 8 


1832. 


74,024 


17,566 


12,495 


147,636 


1,190,749 


8 1 3 


1833. 


75,333 


22,988 


14,763 


162,815 


1,405,035 


8 12 7 


1834. 


70,809 


21,788 


13,870 


158,166 


1,406,872 


8 17 10 


1835. 


107,715 


33,073 


12,604 


199,007 


1,643,741 


8 5 2 


1836. 


97,762 


33,880 


19,891 


192,352 


2,342,674 


12 3 7 


1837. 


95,663 


44,387 


12,373 


194,292 


2,009,259 


10 6 10 


1838. 


141,923 


48,554 


14,942 


256,017 


2,535,692 


9 18 1 


1839. 


136,452 


43,460 


10,836 


247,912 


2,719,824 


10 19 6 


1840. 


144,719 


49,801 


9,886 


268,328 


2,524.859 


9 8 2 


1841. 


189,249 


85,866 


14,077 


360,875 


2,877,278 


7 19 5 


1842. 


191,301 


93,851 


15,934 


369,398 


2,457,717 


6 13 


1843. 


198,774 


154.770 


16,500 


448,925 


2,590,833 


5 15 5 


1844. 


249,915 


99,960 


18,969 


458,745 


3,193,368 


6 19 2 


1845. 


153,813 


77,362 


22,036 


351,978 


3,501,895 


9 18 11 



• Including the kinds stated in the previous columns, together with bolt- and rod-iron, iron- 
wire, anchors, grapnels, &c., hoops, nails, and all other sorts not included in the foregoing. 



118 



REPORT— 1846. 



Appendix No. 10. 
Make of Iron in 184-3 compared with 1840. 



Decrease. Increase. 



Forest of Dean 

South Wales 

North Wales 

Northumberland .... 

Yorkshire 

Derbyshire 

North Staffordshire 
South Staffordshire 

Shropshire 

Scotland 



Less in 1843 



tons. 

15,500 

505,000 

26,500 

11,000 

56,000 

31,000 

20,500 

407,150 

82,750 

241,000 



tons. 

8,000 

457,355 

19,750 

25,750 

42,000 

25,750 

21,750 

300,250 

76,200 

238,550 



tons. 

7,500 

47,650 

6,750 



14,000 
5,250 



106,900 
6,550 
2,450 



1,396,400 
1,215,350 



1,215,350 



197,050 
16,000 



181,050 



181,050 



14,750 
1,250 



16,000 



Appendix No. 11. 

Quantity and declared Value of Iron, wrought and unwrought, exported to 
the United States of America in each year from 1831 to 1844. 

Years. Tons. Value. 

1838 71,235 £634,395 

1839 74,772 801,198 

1840 38,603 355,534 

1841 79.186 626,532 

1842 68,418 394,854 

1843 31,909 223,668 

1844 107,379 696,937 



XOOl, 

1832. ,. 


... 37,565 ... 


... 284,502 


1833. .. 


... 54,124 ... 


... 412,515 


1834. .. 


... 40,625 ... 


... 322,156 


1835. .. 


... 51,951 ... 


... 408,368 


1836. .. 


... 79,330 ... 


... 912,387 


1837. .. 


... 49,204 ... 


... 489,309 



Appendix No. 12. 

Quantity and declared Value of Iron exported to France in each year from 
1831 to 1844. 

Years. Tons. Value. 

1838 15,723 £103,026 

1839 14,288 93,356 

1840 16,804 88,631 

1841 19,0.99 95,943 

1842 23,428 105,172 

1843 29,626 120,220 

1844 21,362 100,982 



Years. 


Tons. 


Value. 


1831. ... 


... 2,721 .. 


...£ 21,416 


1832. ... 


... 5,657 .. 


.... 32,768 


1833. ... 


... 7,424 .. 


.... 41,696 


1834. ... 


... 8,306 .. 


.... 55,060 


1835. ... 


... 14,863 .. 


.... 82,302 


1836. ... 


... 14,016 .. 


.... 115,718 


1837. .. 


... 15,015 .. 


.... 96,415 



ON ATMOSPHERIC WAVES. 119 

Appendix No. 13. 

Estimated quantity of Iron required for the construction and putting into 
operation each mile of Railway. 

Tons per mile. Tons of Pig-iron. 

Rails, 75 lbs. per yard 235 equal to 3l7i 

Chairs, 40 lbs. each 125 125 

Locomotive engines, 1 per mile 25 33| 

Wagons and carriages, iron-work 25 33| 

Tanks, &c 5 5 

Turntables, points and sidings 100 110 

Workshops 30 40i 

Coke, ovens and sundries 5 5 

Bridges, roofs, stations, &c 30 40^ 

711 
Required to maintain the above, each year — 

Rails, chairs, locomotives, turntables, &c., 50 tons of wrought- and cast-iron, 
equal, each year, to 61 tons of pig-iron. 

N.B. The above estimate has been furnished by an experienced railway engineer 
to the chairman of a railway company. The quantities are greater than are com- 
monly assigned, but an abatement of 25 per cent, would not disturb the calculation 
made by me (page 109) ; and when provision is made for maintaining in repair the 
railways now open, it would absorb all the iron which will probably be made in the 
next four years, to construct, at that abatement of 25 per cent., the lines now sanc- 
tioned by Parliament. 



Third Report on Atmospheric Waves. 
By William Radcliff Birt. 

The two former Reports which I have had the honour to present to the Asso- 
ciation necessarily possessed a fragmentary character. Sir John Herschel, 
in his Report on Meteorological Reductions (1843), distinctly traced two 
well-defined atmospheric waves which passed over the British Isles and the 
west of Europe, one in September 1836, the other in December 1837. These 
may be regarded as the earliest instances of our detecting and clearly appre- 
hending the character of the atmospheric undulations constantly traversing 
our oceans and continents, and mark the commencement of that sera in atmo- 
spheric research to which Mr. Forbes alluded in his Report on the Recent 
Progress and Present State of Meteorology, presented to the Association in 
1832, when he said, " The great extent of country over which the accidental 
variations of the barometer take place, is one of their most striking features ; 
and in a future and more advanced state of meteorology we may be able to 
draw the most interesting and important conclusions from the great atmo- 
spheric tidal waves which are thus perpetually traversing oceans and conti- 
nents." 

Sir John Herschel, in the conclusion of the report to which allusion has 
been made, noticed the larger fluctuations which I had observed in the 
autumn of 1842, especially the symmetrical wave which occupied thirteen 
days in November for its complete rise and fall. The curves representing 
these larger undulations were appended to Sir John Herschel's report ; and 
the Association, under the direction of the Magnetical Committee and the 
immediate superintendence of Sir John, entrusted me with the further in- 



120 REPORT— 1846. 

vestigation of these waves, especially that of November. The mode of in- 
vestigation and the partial results arrived at during the period between the 
sittings of the Association in 1843 and 184'4' form the subject of my first re- 
port, which, as before stated, must be regarded only as a fragment. 

During the further investigation of the wave of November various obser- 
vations came to hand, which appeared to throw considerable light on the 
general character of atmospheric undulations. The publication of the Green- 
wich and Toronto observations afforded an interesting comparison of the 
passages of certain maxima at these distant stations, and by extending this 
comparison to Prague and Munich, several interesting features of certain 
secondary waves during the transit of a supposed normal wave appeared so 
clearly to be made out, that it was deemed desirable to include the whole of 
this comparison in the succeeding report, rather than run the risk of its being 
lost by deferring it until after the examination of the great wave should be 
completed. Another most interesting result arrived at about this time, was 
the recurrence of the great Avave of November. The return of this interest- 
ing phaanomenon appeared so strikingly distinct in 1843 and 184'4, that to 
have omitted noticing it in the Report would have greatly contributed to re- 
tard the inquiry. It accordingly forms the second section of the Report of 
1845. These circumstances, with the further investigation of the great wave 
of November 1842, give to the second report a more fragmentary character. 

Previous to entering on the immediate subject of the present report, it will 
be desirable to review the steps that have been taken for observing the great 
symmetrical wave on its return in 1845; and also to notice any other cir- 
cumstance that may have transpired during the past year at all calculated to 
throw any light on the subject of our investigations. With regard to the first 
point, certain instructions were drawn up, w^hich were forwarded to gentle- 
men interested in meteorological research, and otherwise circulated, in con- 
sequence of Avhich a number of interesting and valuable observations were 
obtained. The results of the examination of these observations, as far as it 
has yet proceeded, will form the first part of the present report. In the Phi- 
losophical Magazine for April in the present year Mr. Brown published a 
voluminous paper on the oscillations of the barometer, with particular refer- 
ence to the meteorological phasnomena of November 1842, the month in 
which I first observed the great symmetrical wave. This paper is accom- 
panied by diagrams representing the direction of the wind in England, Scot- 
land and Ireland every day, from the 1st to the 26th inclusive. Upon a very 
careful perusal of it, I found that the observations, as given in the diagrams, 
very beautifully illustrated Prof. Dove's theory of parallel currents or alter- 
nately disposed beds of oppositely directed winds, and appeared to throw so 
clear a light on the real character of the atmospheric undulations, that I was 
induced to enter upon a very careful examination of the barometric obser- 
vations in connexion with the diagrams of the wind. The result of this ex- 
amination has been to give the inquiry a completeness which it was before 
destitute of. It was previously difficult to define the real notion we formed 
of an atmospheric wave ; not so much from the distribution of pressure over 
a tract of country gradually decreasing on each side a line of maxima, as 
from the relation of the aerial currents or winds to this distribution of 
pressure of which we were to a certain extent ignorant. The examination 
of these observations has exhibited very clearly tlie distribution of the aerial 
currents in relation to the distribution of pressure, and enabled us to define 
the nature of an atmospheric wave both as regards its undulatory and mole- 
cular motion. This definition, with the examination of the observations, 
forms the second part of this report. 



ON ATMOSPHERIC WAVES. 121 

Part I. — Recurrence of Symmetrical Wave. 

The following were the instructions drawn up for observing the Great 
Symmetrical Wave on its return in IS^S. 

" The recurrence of the great November Wave observed in ISi^ (an en- 
graving of which is inserted in the Report of the Thirteenth Meeting of the 
British Association for the Advancement of Science), during the autumns of 
1843 and 1844, renders the barometric movements of the months of October 
and November highly interesting. It is accordingly proposed that meteoi'o- 
logical observations, on a similar plan, should be made as extensively as pos- 
sible, with a view to observe this particular wave; and meteorologists are 
invited to direct their particular attention to the oscillations of the barometer 
during the months above-named. 

" Times of Observation. 

" The following hours are the most suitable for the object now in view : 
3 A.M., 9 a.m., 3 P.M. and 9 p.m.; these hours divide the day into four equal 
parts ; they have been recommended by the Royal Society as meteorological 
hours, and are the hours at which observations are made daily, by direction 
and under the superintendence of the Honourable the Corporation of the 
Trinity House, which have been most advantageously used in the examina- 
tion of atmospheric waves. 

" In cases, however, in which the observation at 3 a.m. may be inconve- 
nient or impracticable, it will be important to substitute for it two observa- 
tions, one at midnight and the other at 6 in the morning, so that the hours of 
observation will in such cases be 6 a.m., 9 a.m., 3 p.m., 9 p.m. and midnight. 

" To individuals who cannot command these hours, it is recommended 
that observations should be made as near them as possible ; these will still be 
valuable, although not to so [great an extent as those made at the regular 
hours. In these cases, however, it will be absolutely necessary to substitute 
two readings for every one of the regular hours omitted — one previous to, 
the other succeeding the hour so omitted ; and these should, if possible, in- 
clude an equal interval both before and after such hour. In all cases the 
exact hour and minute of mean time at the place of observation should be 
inserted in its appropriate column in the form sent herewith. 

" At the regular hours of observation, or any others that the observer may 
fix upon, in accordance with the foregoing instructions, it will be necessary 
to observe, 

" 1st. The barometer, with its attached thermometer, and enter in the 
form the aclmil height observed with the temperature of the mercury. 

" 2nd. The external and dry thermometer. 

" 3rd. The wet bulb thermometer. 

" [These observations are particularly essential in order to separate the 
pressure of the vapour from the aggregate pressure, as measured by the mer- . 
curial column.] 

" 4th. The direction and force of the wind. 

" [These are important to determine the connexion between the undula- 
tory and molecular motion of the wave.] 

" 5th. The character of the weather at the times of observation ; which 
may be recorded by Capt. Beaufort's symbols. 

" It is proposed to commence the observations on the 1st of October next, 
and continue them daily until the end of November, unless it should be found 
that at that time the Wave is not completed, in which case it will be requisite 
to continue them a few days longer. 

" It will be necessary, on returning the form when filled, to accompany it 



122 



REPORT — 1846. 



with the following data for reduction. A blank is left for this purpose on 
the back of the form. 

" The geographical co-ordinates of the place of observation, viz. latitude 
and longitude. 

" The altitude of the cistern of the barometer above the level of the sea, 
exactly ; if not, as near as it can be obtained. 

" The internal diameter of the tube of the barometer. 

" The capacity, neutral point, and temperature. 

" [These are usually engraved on the instrument.] 

" If the co-efRcients of the diurnal and annual oscillations have been de- 
termined for the place of observation, include them. 

" Those sets of observations which may be reduced by the observers, should 
be accompanied with the original observations, and a reference to the tables 
used in their reduction, also the data above-mentioned. 

" All observations that may be made in accordance with these instructions 
and forwarded to me, will be carefully examined and reported on at the next 
meeting of the British Association. " W. R. Birt." 

" 2 Sidney Place, Cambridge Road, Bethnal Green." 

In accordance with these instructions observations were received from the 
following stations and observers. 

Table I. 



I 



station. 



Sandwick Manse, Orkneys 

West coast of Scotland 

East coast of Great Britain ... 

Firth of Forth 

Longstone, Northumberland .. 

Newcastle-upon-Tyne 

Belfast, Ireland 

Stokesley , Yorkshire 

Markree, Ireland 

York 

Heligoland 

Galway, Ireland 

Lough Corrib and Galway ... 

Portarlington, Ireland 

DubMn, Ireland 

Limerick, Ireland 

Bai-dsey Island off Wales 

Birmingham 

Haisboro', Norfolk 

Coast of Suffolk 

South of Ireland 

South of Ireland 

South Bishop off Wales 

Pembroke 

Gloucester 

Harwich 

London 

Ramsgate 

Scilly Isles 

Helston, Cornwall 

Portsmouth 

St .Catherine's Point, I. of Wight 

Hastings 

St. Helier's, Jersey 



Vessel or Establishment. 



H.M.S.V. "Shearwater" 
H.M. Ketch " Sparrow ' 
H.M.S.V. "Mastiff" .. 

Lighthouse 

PhdosopMcal Society .. 



Lighthouse. 



Ordnance Survey OflSce 



Lighthouse. 

Philosophical Institution 
Lighthouse. 

H.M.S.V. "Blazer" 

H.M.S.V. "Lucifer" ... 
H.M.S.V. "Tartarus"... 
Lighthouse. 
H.M.S.V. "Firefly" 



H.M.S.V. " Porcupine " 



H.M.S.V. "Porcupine" 
Lighthouse , 



Lighthouse. 



Observer or Authority. 



Rev. Charles Clouston. 

Commander C. G. Robinson. 

WiDiam Turton, R.N. 

The Officers. 

WiUiam Darling. 

George Muras, Esq. 

Dr. Stevelly. 

John Call, Esq. 

Edward J. Cooper, Esq., M.P. 

John Phillips, Esq., F.R.S. 

Lieut. Sidney, R.N. 
Lieut. Beechey, R.N. 
M. Ilanlon, M.B. 
Capt. Larcora, R.E. 
R. T. MaunseU, Esq. 

William Onion, Esq. 

Capt. Owen Stanly, R.N. 
Commander G. A. Frazer. 
Commander James Wolfe. 

Capt. Beechey, R.N. 
John Jones, Esq. 
The Officers. 
W. R. Birt. 
The Officers. 
E. L. Davis. 
M. P. Moyle, Esq. 
Commander W. L. Sheringham, 
[R.N. 
Sir Howard Elphinstone, Bart. 
Capt. Childers. 



N.B. I am indebted to the Honorable the Corporation of the Trinity House for the Light- 



ON ATMOSPHERIC WAVES. 123 

house observations, and to Rear-Admiral Beaufort, R.N., for the observations made on board 
the surveying vessels. — W. R. B. 

These observations, which were principally undertaken with a view to ob- 
serve the return of the great wave, have been attended with highly interesting 
results. I shall first notice the result of the comparison of the observations at 
this station (Cambridge Heath) with those made at Leicester Square in IS-tS, 
as fully establishing not only the return of the great wave, but also that of 
other extensive undulations. 

Section I. — Comparison of observations made at Cambridge Heath (north- 
east of London) from Oct. 1, lSi5 to Nov. 21, 1845, with observations 
made at Leicester Square from Sept. 14, 1842 to Nov. 25, 1842. 
The observations of 1842 are projected in curves and appended to Sir John 
Herschel's 'Report on Meteorological Reductions,' 1843. 

1842. I. Plate I. fig. 1 (Report 1843) exhibits an undulation consisting of 
a gentle barometric fall and almost as gentle a rise during seventeen days, 
namely from Sept. 14 to Oct. 1, interrupted only by the diurnal oscillations, 
which are in general well-developed. 

1845. In 1845 this undulation of seventeen days' interval returned. It 
was observed from Oct. 1 (the commencement of the observations) to the 
19th, but instead of exhibiting the gentle fall and ascent noticed in 1842, it 
was interrupted by two most remarkable superposed waves. The first oc- 
curred on the 4th, 5th, and 6th of October, and the second from the 11th to 
the 16th. 

When these waves (the commencement and termination of each being well- 
marked) are abstracted from the general curve, the resemblance between the 
curves of 1842 and 1845 is very apparent. 

1842. September 14 to October 1. 
1845. October 1 to October 19. 

II. The curves of the succeeding four days in the two periods 1842 and 
1845 exhibit similar barometric fluctuations, so that the movements during 
the four days succeeding the seventeen-day wave in 1845 are identical with 
those of the four days succeeding the same wave in 1842. 

1842. October 2 to 5l , ., . , . 

lOA e r\iU irvi. nnf both incluSlVe. 

1845. October 19 to 22 J 

III. The exact identity between the curves of 1842 and 1845 breaks off 
on Oct. 23, 1845. The barometer maintains an elevation above thirty inches 
during the period in 1845 that the movements are not in accordance with 
those of 1842, 

IV. Plate I, fig. 3 (Report 1843). The identity between the curves of 
1842 and 1845 again commences on Oct. 27, 1845, and is maintained in a 
very close manner until midnight of Nov. 6. 

1842. October 31, midnight to November 11, noon. 
1845. October 27, noon to November 6, midnight. 

V. In consequence of the movements from Oct. 27 to Nov. 6, 1845, ex- 
hibiting so close a similarity to those between Oct. 31 and Nov. 11 in 1842 
which immediately preceded the great wave in that year, considerable ex- 
pectation was raised that the great wave M'ould set in on the morning of the 
7th. At midnight of the 6th, the similarity between the curves that had been 
so closely maintained during ten days and a half began to fail, and rendered 
it difficult to determine for some days if the preceding movements had really 
been followed by the great wave. This question was set at rest as the obser- 
vations proceeded; for on comparing the curve from the 6th to the 21st with 
that of the great wave of Nov. 1842 (Plate II., Report 1843), there was every 



Nov. 


18. 


Nov. 


14. 


Oct. 


27. 


Nov. 


14. 



124 REPORT — 1846. 

reason to believe that it had again returned and that its fourth transit had been 
observed. Between these epochs, Nov. 6 and 21, all its essential features were 
exhibited. The large central undulation, also forming the crown of the great 
wave and occupying in this instance five days, was very distinctly marked ; 
and the two smaller undulations on each side the central wave, making with 
it the five of which the great wave is composed, were also well-developed. 
These smaller waves did not appear to co-ordinate with those of former 
transits. The great wave culminated on the 14th. 

Corresponding barometric movements. 

Crest. 

1842. November 11 to November 25. 

1843. November 6 to November 21 . 

1844. October 20 to November 4. 

1845. November 6 to November 21. 

VI. The movements between the above epochs in each year were more or 
less symmetrical, the axes occurring on the dates indicating the pjissage of 
the crests. In the year 1845 the symmetrical movements appeared to extend 
greatly beyond the limits noticed above, for not only did the central undula- 
tion which culminated on the 14th form the axis of the great wave (properly 
so called), but also of a system at least double its extent, namely from Oct. 29 
to Nov. 28. Observations received from Hastings appear to indicate that 
the barometric oscillations during October, November and December were 
symmetrical, the axis occurring about the middle of November. 

VII. In my last report (Report, 1845, page 116) I stated that the mini- 
mum of the 16th of Feb. and that of the 5th of Oct. in the year 1841, formed 
the limits of the period of least range for that year. It is well known that 
the barometric oscillations are divisible into two classes ; those of small range, 
confined to the summer half year ; and those of great range, the period of 
their development being the winter. These greater oscillations begin to appear 
in October. Fig. 2, Plate I. (Report, 1843) exhibits a similar undulation 
to that of Sept. 14 to Oct. 1, 1842, of seventeen days' interval with two sub- 
ordinate m,axima interposed. The depression of the 23rd was very considera- 
ble, and rendered memorable by the inundation of the Madeiras. We ac- 
cordingly find that the larger undulations began to appear in these latitudes 
in 1842, about the 16th of Oct. The seventeen-day undulation, Sept. 14 to 
Oct. 1, occurring about a fortnight later in 1845, brought it within the period 
of the commencement of the greater secondary undulations, and we find it 
interrupted by two very remarkable waves, in both cases rising above the 
general surface of the normal wave. On comparing the curve from Oct. 1 to 
17, 1845, with that of Oct. 15 to 31, 1842, and bringing the minima in both 
cases on the same vertical line, but little if any resemblance can be traced 
between them. There are however these interesting exceptions. During 
the first seven and a half days the descent in each case is interrupted by a 
superposed wave, the co-ordinates of that of 1842, being about double those of 
the superposed wave of 1845. The ascent during the succeeding seven and 
a half days is also interrupted in each case by superposed waves, but the 
characters of them are reversed, the largest occurring in 1845 and the smallest 
in 18*2 ; the relations are nearly similar to those characterizing the super- 
posed waves of the descent, that of 1845 being nearly double that of 1842. 
Another most remarkable circumstance is also apparent on the comparison 
of these curves, the displacement of the maxima of these superposed waves, 
or the interval between their crests. It is probable that three waves transited 
during the seventeen-day undulation, Oct. 14 to 31, 1842, having their re- 
spective maxima on the 21st, 25th, and 27th ; there are also traces of three 



ON ATMOSPHERIC WAVES. 125 

during the seventeen-day undulation of Oct. 1 to 19, 18^5, having their re- 
spective maxima on the 5th, 10th, and 14th. Taking the same vertical or- 
dinates in each curve, we have the epochs of the troughs of the first superposed 
waves nearly similar but separated by an interval of civil reckoning of four- 
teen and a half days ; that is, the gentle undulation of the last half of September 
occurred a fortnight later, and the superposed waves indicating the disturbed 
state of the atmosphere, and characterizing the period of greater barometric os- 
cillation, came rolling on a fortnight earlier ; the two coinciding and producing 
the compound curve really observed. The first of these superposed waves 
being about half the size of the corresponding wave in 184<2, passed its maxi' 
mum about a day and a half earlier, and a small wave succeeding it brought 
the minimum on the same vertical line as that of 1842. In a similar manner 
the largest superposed wave in 1 845 culminated at a later period of the normal 
wave than the smaller wave of 1842. In consequence of these different rela- 
tions of the superposed waves of 1842 and 1845, the two apices were much 
nearer in 1842 than in 1845. 

From these remarks it appears that, taking the barometric movements from 
Sept. 14, 1842 to Nov. 25 of the same year, containing two undulations of 
seventeen days' interval, and comparing them with those from Oct. 1 to Nov. 
21 of 1845, only one undulation of seventeen days' interval was observed in 
the latter year, namely from Oct. 1 to 19 ; that this undulation was not of the 
gentle flowing character manifested by that from Sept. 14 to Oct. 1, 1842, 
but was interrupted by the same number of superposed waves as that from 
Oct. 14 to 31, 1842; and that this state of things was brought about by the 
later occurrence of the normal wave, and the earlier occurrence of the super- 
posed waves. Of the two seventeen-day undulations of 1842 the first (Sept. 
14 to Oct. 1) returned in 1845. 

VIII. In addition to the absence of the second seventeen-day interval, Oct. 
14 to 31, 1842, in the observations of 1845 ; the preceding movements, Oct. 6 
to 13, 1842 (the barometer attaining a considerable altitude), were not ob- 
served in 1845. 

IX. During the period from Oct. 1 to Nov. 21 in 1845, the barometric 
movements of Oct. 23 to 26 were the only oscillations that appeared to have 
no corresponding movements in 1842. 

X. The distinctness with which the great wave commenced in 1842 and 
1845, and the breaking off of the exact similarity between the curve of the 
preceding ten and a half days which had been so closely maintained just as 
the wave commenced in 1845, exhibit this interesting phaenomenon in all its 
individuality, and completely separate it from all the preceding barometric 
movements. 

XI. The individuality which is thus given to the great wave, the distinct- 
ness of its essential features, the close resemblance of its curves in 1842, 1843 
and 1845, and the closer relations existing between those of 1842 and 1845, 
induce the strong belief that we have obtained the ti/pe of the barometric 
oscillations during the middle portion of November. This type I propose to 
express in the following language. 

"That during fourteen days in November more or less equally disposed about 
the middle of the month, the oscillations of the barometer exhibit a remarkably 
symmetrical character, that is to say, the fall succeeding the transit of the 
maximum or highest reading, is to a great extent similar to the preceding 
rise. This rise and fall is not continuous or unbroken ; in three out of four 
of the occasions on which it has been observed, it has been found to consist of 
Jive distinct elevations. The complete rise and fall has been termed the great 
symmetrical barometric wave of November, and as such has been considered 



126 REPORT— 1846. 

to result from the transit of a large wave ; but there is great reason to believe 
that while it may be due to the transit of a normal wave of about fourteen 
days' amplitude, it also exhibits the transits of Jive secondary superposed waves 
of a similar character to those riding on the wave of seventeen days' interval, 
Oct. 1 to 19 (VII.). The great November wave consequently possesses a com- 
pound character : at its setting-in the barometer is generally loio, sometimes 
below twenty-nine inches. This depression is succeeded by two well-marked 
undulations, varying from one to two days in duration. Thecentral undulation, 
which also forms the apex of the gi'eat wave, is of larger extent, occupying 
from three to five days ; when this has passed, two smaller undulations, cor- 
responding to those at the commencement of the wave, make their appearance, 
and at the close of the last the wave terminates." This was the order of 
things in 1842, 1843 and 1845. The smaller undulations in these instances 
were not identical, that is, they did not occur on the same points of the wave 
in each case ; but the two preceding and the two succeeding undulations to 
the larger or central one were well-marked ; the physiognomy of the wave was 
readily recognized. 

The wave of 1844 exhibited a striking departure from this type in two re- 
markable particulars ; the epoch of transit and compound form of wave. The 
epoch was considerably earlier than in 1842, 1843 or 1845, namely Oct. 27; 
and the compound form consisted only of thee instead of Jive undulations. 
The symmetry however was very apparent. This departure from]the November 
type may probably be connected with the earlier occurrence of the wave ; 
future observations will doubtless make us acquainted with its cause. 

XII. Capt. Larcom of the Royal Engineers has most obligingly forwarded 
me, in addition to the observations made during the months of Oct., Nov. and 
Dec. 1845, curves of the barometric undulations observed at Dublin during 
the Novembers of 1829 to 1845 inclusive. These curves are so admirably cal- 
culated to confirm or disprove the views advanced in XI., that I avail myself of 
his permission to lay them before you ; and I beg to acknowledge the great 
obligations I am under to that oflicer for the valuable assistance he has rendered 
me inthisinquiry, both with respect to the immediate subject now under discus- 
sion (the great wave), and the return of the other extensive undulations before 
alluded to, which are admirably illustrated by the curves with which he has 
furnished me, and which I have much pleasure in submitting to the Association. 

Review of the essential features of the Great Symmetrical Barometric Wave, 
as exhibited in a series of Curves representing the Barometric Undulations 
as observed at Dublin (Mount Joy, Ordnance Survey Office, Phoenix Park) 
during the Novembers of 1829 to 1845 inclusive. 

[It may be well to notice, previous to proceeding with this review, that 
Dublin is not constantly situated in the line of greatest symmetry. In 1842 
it appeared to form one of the points in the line, but the observations of Nov. 
1845 have shown that this line is 7iot stable. The line of greatest symmetry 
appeared on that occasion to coincide to a great extent with the southern 
coast of England, so that Dublin was thrown to the north of it. It is pro- 
bable this line has a sort of oscillatory motion, and this may to a great ex- 
tent explain the nodal character of Brussels as a barometric station, the fixed 
point in the line being not far removed from that city. The departure from 
symmetry on many of the returns of the great wave at Dublin, may readily 
be accounted for by the line of greatest symmetry being considerably removed 
from thence.] 

1829. The great wave very distinct; the two anterior undulations well- 
marked, the two posterior not so distinct. 



ON ATMOSPHERIC WAVES. 127 

Transit of anterior trough, Nov. 9. 

Transit of apex „ 16. 

Transit of posterior trough „ 23. 
Amplitude in time, fourteen days. 

1830. The symmetry of the great wave not so apparent; the subordinate 
undulations strongly marked. 

Transit of anterior trough, Nov. 15. 
Transit of apex „ 23. 

1831. The symmetry of the great wave clearly observable; its amplitude 
much smaller ; two subordinate waves, one on the anterior, the other on the 
posterior slope, well-developed. 

Transit of anterior trough, Nov. 6. 

Transit of apex „ 12. 

Transit of posterior trough „ 15. rf|Lw 

Amplitude in time, nine days. ^. 

1832. The symmetry very distinct on this occasion ; the curve somewhat 
resembled that of 184?2 ; the two anterior and two posterior undulations well- 
developed. 

Transit of anterior trough, Nov. 9. 

Transit of apex „ 16. 

Transit of posterior trough „ 23. 
Amplitude in time, fourteen days. 
The trough succeeding one of the posterior undulations is deeper than the 
posterior trough of the great wave, a circumstance that occurred in the year 
1845 at this station (Dublin). The similarity between the transits of the 
great wave in 1829 and 1832, especially as to time, is highly interesting. 
The remaining movements in this month were also strikingly symmetrical. 

1833. The great wave very difficult to recognize; taking the well-marked 
depressions of the seventh and twenty-first as the anterior and posterior troughs, 
and regarding the movements between these epochs as due to the great wave, 
although greatly concealed by the strongly developed subordinate waves, we 
may regard the whole as a transit of the great wave at a station considerably 
removed from the line of greatest symmetry ; the two anterior subordinate 
waves are strongly marked ; the posterior appear to be broken into a numbel- 
of smaller undulations. 

Transit of anterior trough, Nov. 7. 

Transit of apex „ 13. 

Transit of posterior trough „ 21. 
Amplitude in time, fourteen days. 

1834. The great wave very distinct, the subordinate waves but slightly 
developed. 

Transit of anterior trough, Nov. 7. 

Transit of apex , 14. 

Transit of posterior trough „ 21. 
Amplitude in time, fourteen days. 

1835. The great wave very distinct and considerably amjilified ; the subor- 
dinate waves distinct but not strongly developed. 

Transit of anterior trough, Nov. 3. 

Transit of apex „ 12. 

Transit of posterior trough „ 21. 
Amplitude in time, eighteen days. 

1836. The great wave extremely difficult to recognize ; two well-marked 
depressions on the 4th and 17th mark the terminations of a somewhat sym- 
metrical system of movements. If these movements may be considered as 



128 



REPORT — 1846. 



replacing the great wave, they are characterized by the most remarkable 
absence of the central undulation forming its crest ; two of the subordinate 
undulations, equally posited with regard to the anterior and posterior troughs, 
are strongly and strikingly developed; and where the central undulation 
should have occurred, raising the apex above thirty inches, a great depression 
is seen. 

1837. The great M'ave well-developed ; the last posterior subordinate wave 
strongly developed. 

Transit of anterior trough, Nov. 1 or Oct. 31. 

Transit of apex „ 6. 

Transit of posterior trough „ 14. 
Amplitude in time, fourteen days. 

1838. The similarity between the curve of this year during the transit of 
the gseat wave and that of ISiS during the same period is very striking ; the 
antenbr slopes in each case are almost representatives of each other ; the two 
subordinate waves on the anterior slopes are so nearly identical as to leave no 
doubt of the movements of 1842 being a most decided return of those of 1 838 ; 
the similarity of the subordinate waves on the posterior slopes is not so distinct, 
the two are however well-marked. One striking difference between the 
curves must nevertheless be noticed ; in 1838 the anterior trough was lowest, 
in 1842 the posterior was lowest. 

Transit of anterior trough, Nov. 7. 

Transit of apex „ 12. 

Transit of posterior trough „ 21. 
Amplitude in time, fourteen days. 

1839. The great wave in this year is very difficult to recognize. A maxi- 
mum was passed on the 23rd, subordinate waves were developed on each 
side this maximum. There appears to be some similarity in the movements 
of this year to those of 1830, the subordinate waves are however not so 
distinct. Transit of apex, Nov. 23. 

1 840. In this year also the great wave is difficult to detect unless the broad 
maximum of the 26th forms its crest, in which case the posterior slope runs into 
December ; this is borne out by the Greenwich observations, they however ex- 
hibit a large development of one of the subordinate waves on the posterior slope. 

In the following table all the above features are collected. The amplitudes 
(in all cases except two being of the same extent, namely fourteen days) 
strongly confirm the views advanced. These views receive still greater con- 
firmation from the epochs of the transits of the crests, which are arranged 
according to the days of the month on which they occurred in Table III. 









Tabli 


II. 




Year. 


Anterior 
trough. 


Crest. 


Posterior 
trough. 


AmpUtude 
in time. 


Remarks. 


1829 
1830 
1831 
1832 
1833 
1834 
1835 
1836 
1837 

1838 

1839 
1840 


Nov. 9 
15 
6 
9 
7 
7 
3 


Nov. 16 
23 
12 
16 
13 
14 
12 


Nov. 23 


Days. 
14 


Very distinct. 

Symmetry not so apparent. 

Symmetry clearly observable. 

Symmetry very distinct. 

Wave very difficult to recognize. 

Very distinct, 

Very distinct. 

Extremely difficult to recognize. 

Well-developed. 
J Similarity between 1838 & 1842 
\ very great. 

Wave very difficult to recognize. 

Difficult to detect. 


15 

23 
21 
21 
21 

14 

21 


9 
14 
14 
14 

18 


Oct. 31 

Nov. 7 


6 
12 
23 


14 
14 

















ON ATMOSPHERIC WAVES. 

Table III. 



129 



Distinct, well-marked transits. 


Doubtful transits. 


Year. 


Epoch of crest. 


Value. 


Year. 


Epoch of crest. 


Value. 


1837 


Nov. 6 


30-20 


1833 


Nov. 13 


3003 


1831 


12 


30-32 


1830 


23 


30-12 


1835 


12 


30-40 


1839 


23 


30-13 


1838 


12 


30-36 


1840 


26 


30-23 


1845 


13 


29-87 








1834 


14 


30-48 








1843 


14 


30-27 








1829 


16 


30-47 








1832 


16 


30-21 








1842 


17 


30-35 








1841 


25 


' 29-79 









From the above table it appears that with two exceptions in eleven years 
ofdistinct and well-marked transits of the great wave, the epochs of the maxima 
were confined to live days near the middle of the month, namely from the 
12th to the 17th. The greater proportion, twelve years out of seventeen, in- 
cluding 1844, in which the wave has distinctly returned, greatly confirms the 
results noticed in sec. XI., namely that we have obtained the type of the 
barometric movements during fourteen days in November, more or less equally 
disposed about the middle of the month. 

I cannot here avoid noticing another feature of a most interesting character 
which is very strikingly developed in these curves of November; it is apparently 
unconnected with the great wave. I allude to a general tendency to depres- 
sion in the mercurial column about the last four or five days in the month : 
the following are the years in which this depression occurred : — 

Table IV. 



Year. 


Epoch of 
minimum. 


Value. 


1829 


Baromete 


r falling. 


1830 


Nov. 27 


29-40 


1832 


28 


29-19 


1833 


28 


28-31 


1834 


28 


28-97 


1835 


30 


28-86 


1836 


27 


28-81 


1837 


28 


29-26 


1838 


28 


27-77 


1839 


29 


29-08 


1841 


29 


28-47 


1842 


24* 


28-38 


1843 


26 


29-15 


1844 


28 


29-74 


1845 


29 


29-08 



* A minimum occurred on Nov. 27, value 28-56. 

This depression has occurred so regularly, only two exceptions having been 
observed in seventeen years, that it appears highly probable that its return 
may be expected with as much if not more regularity than that of the great 
wave itself, on or near the 28th of the month. 

1846. K 



130 REPORT — 1846. 

Section II. 

Comparison of contemporaneous observations of the return of the Great 

Wave, Nov. \S^5. 

Of the observations that have come to hand, the following have been pro- 
jected in curves in order to exhibit the characters of the great wave at various 
and distant stations ; the epoch of the curves are Nov. 6 to 22, the duration 
of the great wave. 

Scilly. London. Birmingham. 

Helstone. Yarmouth. Stokesley. 

St. Catherine's Puint. Haisboro. Belfast. 

Portsmouth. Heligoland. Galway. 

These stations being considerably less than half the number from which 
observations have been received, it would be premature to draw any conclu- 
sions from a comparison of the curves, as well as appearing to give a preference 
to certain observations to the exclusion of others which have been executed 
with gi'eat care and fidelity, and from which in connection with the whole 
the most valuable results are likely to be arrived at. Every exertion would have 
been made to have completed the rough projection in curves of all the ob- 
servations made during the transit of the great wave, in order to have sub- 
mitted to the present meeting a first approximation to its general characters 
as exhibited at a diversity of stations, had not the publication of Mr. Brown's 
paper directed my attention to the arrangement of the aerial currents over the 
area of the British Isles during the transit of the great wave of Nov. 1842, 
the value of which I have alluded to in my introductory remarks. 

It may however be important on this head to report the progress made, and 
to notice a few particulars merely as indicating the course pursued and the 
highly important results likely to be obtained from a complete discussion of 
the observations, not only for the period during the transit of the great wave, 
but also during the two months over which the observations extend. The 
curves are susceptible of a variety of arrangements, according as it may be 
deemed desirable to exhibit certain characteristic features of the normal or 
secondary waves. In submitting the projected curves to your notice on this 
occasion, I have selected that arrangement best calculated to exhibit, — first, 
the symmetrical character of the wave, and secondly, the direction in which 
this symmetrical character is most departed from. 

The first two curves (Scilly and Helstone) are characterized by ^Aree periods 
of barometric readings of nearly the same value (slightly above twenty-nine 
inches), occurring on the 7th, 11th and 19th ; between the 11th and 19th we 
find the central undulation forming the crest. The Helstone curve gives the 
greatest symmetrical arrangement. From these curves we may conclude that 
Scilly, and especially jHe/stowe, were situated near the line of greatest symmetry. 
It is desirable particularly to notice, that at these stations the depressions of 
the 7th and 11th are about equal ; there appears to have been no fall from the 
commencement of the wave to the depression of the 11th. Two distinct and 
well-marked waves (the two at the commencement of the great wave) are 
very discernible. 

The next two curves, St. Catherine's Point (Isle of Wight) and Portsmouth, 
nearly agree with the two preceding, especially in the depressions of the 1 1th 
and 19th being of equal value. There is however a marked difference be- 
tween these curves and those of Scilly and Helstone, in the two anterior waves 
being less developed, and the barometer e-s.h\h\iing a fall from the commence- 
ment of the great wave to the depression of the 1 1 th ; and this fall is not only 
traced towards the E.N.E. through the stations London, Yarmouth and Har- 



ON ATMOSPHERIC WAVES. 131 

wich, Haisboro and Heligoland, but it increases in value as we approach 
the N.E. Now this state of things would result from a large wave passing 
from W.S.W., the posterior slope from Heligoland to Scilly. The readings 
for contemporaneous epochs at each E.N.E. station would be higher, and the 
fall greater. Taking the Helstone curve as the type of greatest symmetry — 
consisting in the equality of the three depressions above named, and the whole 
of the readings being above these depressions, — we have St. Catherine's 
Point and Portsmouth slightly departing from this symmetry, in the move- 
ments from the 7th to the 1 1th being thrown higher than those at Scilly and 
Helstone. 

The curves in which we have traced the increase of the fall from the 7th 
to the 11th, exhibit a much greater departure from symmetry, in the depres- 
sion of the 19th being lower than that of the 11th; and this difference in- 
creases in the order in which the curves are arranged, viz. London, Yarmouth, 
Haisboro, and Heligoland ; and so great is this difference in the last three 
curves, that when combined with the fall from the 7th to the 11th, the baro- 
metric movements (abstracting the secondary waves) are of a downward 
character, that is, from the 7th to the 19th at these stations the tendency in 
the mercurial column is to fall very slowly and gently. At Scilly, Helstone, 
Portsmouth and the Isle of Wight, this tendency to fall did not exist. 

Birmingham offers a striking difference from the last-named curves ; the 
departure from symmetry is more apparent, but the downward movement is 
confined to the period between the depressions of the 11th and 19th. On 
this hand the Birmingham curve is connected with the south-eastern group, 
and clearly shows that the symmetry is greatly departed from to the N.E. of 
Scilly and Helstone. On the other hand, it is connected with the Scilly and 
Helstone curves by the movements of the 7th to the 11th, with a slighter de- 
velopment of the two anterior waves ; if there is any difference, there is a 
slight rise from the 7th to the 11th. 

Stokesley in Yorkshire presents features nearly approaching Birmingham, 
with a greater departure from symmetry, more especially in the depression of 
the 19th, which is deeper. 

Belfast in Ireland exhibits the same departure from symmetry, in the de- 
pression of the 19th being thrown considerably below that of the 11th ; but 
there is in this curve a certain return to a symmetrical arrangement of a 
somewhat different character to that exhibited by the curves of Scilly and 
Helstone ; this consists in a most decided rise from the depression of the 7th 
to that of the 11th: the depressions of the 7th and 19th are thus brought 
nearer to an equality. In these respects (especially the latter) the curves of 
Belfast and Galway strikingly agree, and offer a decided contrast to the south- 
eastern group, which exhibits a fall to the depression of the 11th. 

We thus have the area included by the angular points, Scilly, St. Catherine's 
Point, Heligoland, Belfast and Galway, parcelled out into three barometric 
areas. Near the extreme southern station the greatest symmetrical move- 
ments occurred ; the south-western portion of our island may therefore be re- 
garded as the area of greatest symmetry. A line passing from Scilly to 
Stokesley will divide the area into two portions, each characterized by dif- 
ferent and opposite barometric movements, as far as the observations from 
the 7th to the 11th are concerned. On the N.W. of this line the barometer 
was rising, while on the S.E. of it, it was falling. 

We noticed that the fall might be occasioned by a wave passing off toward 
the E.N.E. ; now as a rise is occasioned by an anterior slope, a wave coming 
up from the N.W. would occasion the phaenomena observed. In that por- 
tion of the area covered by the advancing wave the barometer would rise ; 

k2 



152 REPORT — 1846. 

in that covered by the receding wave it would fall, while in that in which 
the two waves interfered so as to counteract each other, a quiescent state of 
the atmosphere would result. This appeared to be the case in the area of 
greatest symmetry, in which the larger waves so interfered as to exhibit the 
smaller secondary waves uninfluenced by them. This leads us to the real 
character of the symmetrical wave ; not that there is such a reality in nature, 
as will be shown in the next part of this report, but that it results from the 
combination of large normal waves moving in different directions so as to 
interfere*. 

I do not place any stress upon these deductions, as I have alluded to them 
merely to show the progress I have made, and that a complete discussion of 
the observations is likely to be attended with highly important results. The 
results of the examination of Mr. Brown's observations, as detailed in the next 
part of the report, are I apprehend calculated to throw nmch light on the 
inquiry, and when these observations are discussed with reference to the 
views there set forth, our knowledge of these interesting movements will I 
have no doubt be greatly increased. 

Part II. 

Examination of Mr. Brown's paper on the Oscillations of the Barometer. 

In the Philosophical Magazine for April last, Mr. William Brown has 
published a paper on the oscillations of the barometer, with particular reference 
to the meteorological phaenoniena of November 1842. The object of this 
paper is to show that the barometric oscillations are produced by the meeting 
of opposite or nearly opposite aerial currents ; that one current thus meeting 
or impinging on another, deflects it, and under some circumstances produces 
a rise of the mercurial column, but under others occasions a fall in many 
cases of considerable magnitude. In order to elucidate his views, Mr. Brown 
has collected barometric observations from eleven stations, which are scattered 
over an area included by the following angular points : — The Orkneys, 
Christiania in Norway, Paris, Plymouth and Cork. These observations are 
in most cases given as read off from the scale. In addition to these the 
paper is accompanied by six plates, in which the direction of the wind at 
numerous stations is indicated for every day during twenty-six days in the 
month by arrows. The anemonal observations published in the body of the 
paper not being in all cases for consecutive days, a comparison of them with 
the plates is rendered difficult ; nevertheless the plates form a very valuable 
portion of the communication, and if they have been laid down from accurate 
observations, they furnish us with an important addition to our knowledge of 
the arrangement of the aerial currents, especially with respect to the distribu- 
tion of pressure. It is a matter of regret that Mr. Brown did not so arrange 
his observations and plates, that the accuracy of the latter could have been 
seen by inspection. 

I have alluded to this paper as peculiarly interesting at the present time, 
when the attention of meteorologists is directed to the important and interest- 

* I have accompanied these curves with one on a smaller scale, representing ohservations 
at the Orkneys by the Rev. C. Clouston. The very striking departure from symmetry is 
extremely apparent in this curve by the depression of the 19th sinking considerably below 
the readings at any other station. This curve is more in accordance with those from the Irish 
stations in the rise from the 7th to the llth, and it appeai-sto be connected with Heligoland 
by the depression of the 1 1th being but slightly developed ; in tliis respect it also agrees with 
the Stokesley curve. The depression of the llth is very apparent in the S.W. curves, and it 
gradually decreases as we approach the N.E., where it is much less. The Orkneys appear 
to have been under the anterior slope of the wave coming from the N.W. 



ON ATMOSPHERIC WAVES. 133 

ing problem of the barometric oscillations, — one class of philosophers re- 
garding them as only the effects of currents of air of unequal temperature 
and moisture ; and another as the effects of undulations progressing in the 
manner of loaves of sound, and propagating themselves with great velocity over 
large portions of the earth's surface (Report, 1845, page 30). 

It is not my intention to enter into an examination of the conclusions and 
results which Mr. Brown has arrived at; as the question is open, I apprehend 
I shall not be doing an injustice to that gentleman by employing a rather 
different process to that which he has used, and further discussing the ob- 
servations he has given. I beg to acknowledge the obligations I am under 
to him for these observations, and especially for the plates, of which I have 
before spoken : they are extremely interesting in the present inquiry. 

In accordance with these remarks, I shall select the following stations from 
Mr. Brown's list : — the Orkneys, l3elfast, Shields, Cork, Bristol, Plymouth, 
London, Paris, and Christiania. The reason I have omitted Glasgow and 
Armagh will be apparent from Mr. Brown's notes. As I intend to discuss 
these observations with especial reference to the wave hypothesis, I shall 
most cautiously avoid in my future remarks any thing that may at all bear 
on Mr. Brown's views. The plan I intend to proceed on is as follows. I 
shall select the middle observation of each day ; at those stations where only 
two are given morning and evening ; I shall take a mean of them. These 
observations I shall so arrange that they may exhibit the distribution of 
pressure over the area for each day — the line or lines of the greatest diminu- 
tion of pressure — and the relation of such distribution and of such lines to 
the aerial currents or winds. As a convenient method of readily expressing 
these various relations and giving to the discussion that completeness which 
otherwise it would want, I shall adopt the wave hypothesis, and to every line 
of barometric maxima apply the term crest and to every line of minima the 
term trough. In a word, I shall regard the progress of the barometric and 
anemonal phsenomena as the progress of waves. The observations will re- 
main the same both in Mr. Brown's and my own discussions, the results only 
will be different ; and it will remain for other philosophers, by more closely 
investigating the subject, and submitting the observations to a more rigorous 
and searching discussion, to advance this interesting inquiry and to become 
more intimately acquainted with the causes of these interesting phaenomena. 

Having announced my intention of discussing these observations on the 
wave hypothesis, it will be important before commencing such discussion to 
supply a deficiency in my two former reports, and endeavour to give a com- 
pleteness to them which at present they are destitute of. The nature of the 
inquiry occasioned them to be drawn up and presented to the Association in 
a fragmentary manner, the first detailing the steps I intended to adopt in the 
examination of the great wave of Nov. 1 842, and the second the further in- 
formation I had obtained relative to this and other atmospheric undulatory 
movements ; and to a certain extent the same remark will apply to the present 
report, embodying as it does the progress made since the last meeting of the 
Association. The deficiency to which I allude is the notion we form of an 
atmospheric wave ; T shall therefore, previous to placing the discussion of Mr. 
Brown's observations before you, as clearly as I can, state the idea I entertain 
of such a wave, and in introducing it to your attention I shall avail myself of 
Mr. Scott Kussell's designation of the elements of a wave as in figure 1, and 
then proceed with the definition of an atmospheric wave. 



134 REPORT— 1846, 

Section I. 
Definition and Phcenomena of an Atmospheric Wave. 

When a number of barometric observations are projected on paper accord- 
ing to a suitable scale, and continued for months and years, the eye on 
contemplating them will recognize a variety of curved forms, some of large 
and some of small amplitude ; some rising to a considerable altitude, others 
sinking far below the level, representing the mean barometric pressure at the 
station of observation. At first there appears but little regularity in these 
curvilinear records of the ever-shifting state of our atmosphere, but here and 
there the attentive observer will notice some similarity existing between two 
or more individual curves, and he may notice some which possess a certain 
symmetrical arrangement of the ascents and descents. In consequence of 
this similarity and symmetrical arrangement, he examines more carefully the 
records of barometric pressure, and not only discusses the observations at 
one station, but compares those observations with others made at various 
stations ; and here again he finds apparent irregularity and confusion. The 
curves to a certain extent agree, but in many minor points they differ often 
very considerably, in some cases rising at one station while falling at another ; 
this induces a still more minute and careful investigation : the distribution of 
pressure over the largest area he can command is carefully examined ; and 
whether his stations are few or many at any given time, he finds on this area 
a point of maximum pressure and a point of minimum pressure ; between 
these points he finds various pressures, generally increasing from the point of 
least pressure to the point of greatest pressure. On some occasions he finds 
a line of high pressure, stretching quite across the area, and on others a line of 
low pressure. By continuing his inquiries for successive epochs, he finds 
these lines of high and low pressure move across the area, or in other words, 
the high pressure or low pressure is gradually transferred from one point to 
another. He also finds at still more remote epochs other lines of high and 
low pressure, some having the same direction with the lines originally noticed, 
and others crossing the direction of the original lines at various angles. 

The questions which now suggest themselves are the following. What 
are these movements? How can they be represented? In what manner 
can they be explained ? A simple consideration of the curves suggests the 
idea of waves as explanatory of the phaenomena, and the term atmospheric 
wave has been used to designate that ideal individuality which the mind 
attributes to the process which it observes of the successive change of place 
which the barometric maxima and minima undergo, and by which they re- 
gularly succeed each other over the area under examination ; this ideal in- 
dividuality has been employed as a mean of examining the movements just 
alluded to. The line of high pressure stretching across the area (the figure 
being supposed to cut this line transversely) has been termed the crest, W ; 
the line of low pressure in advance of the crest, the anterior trough, a (the 
origin of Mr. Scott Russell's water wave) ; the line of low pressure succeeding 

'^'^ — >«' Fig. 1. 



h 

W, The crest. «• a, The amplitude. a. The origin *. 

W o, The front. W A, The height. w, The end. 

W w. The back. 

* Mr. Scott Russell designates the point a the origin ; a better term I apprehend would be 
commencement. 



ON ATMOSPHERIC WAVES. 135 

the crest, the posterior trough, w (the end of Mr. Scott Russell's water wave) ; 
the line w...h, as measured by the mercurial column, the altitude of the wave ; 
the slope W a, the anterior slope or front of the wave ; the slope W w, the 
posterior slope or back of the wave ; w a constitutes the amplitude of the wave, 

and X X in the same direction, the axis of translation. 

The existence of atmospheric currents, especially the equatorial and polar, 
has been well-established ; and there is a class of philosophers who attribute 
the barometric oscillations entirely/ to the effects of these currents as con- 
tra-distinguished to the effects of waves such as we have just mentioned. In 
contemplating the transference of the barometric maxima and minima, we 
regard only the wave-motion — but very different must be the air-motion. 
Prof. Dove, in his letter to Col. Sabine relative to the magnetical and mete- 
orological observations, has announced his opinion that the equipoise of the 
atmosphere is maintained in the temperate zone by currents on the same level 
flowing in opposite directions (Report, 1845, page 61) ; thus we have a bed or 
stratum of air moving from the S.W., and on each side of this are strata of 
N.E. winds. We may here inquire, how are these alternate aerial currents 
related to the waves before alluded to ? It is one of the objects of the following 

Xi, Fig. 2. 



sw -^ 



V-^ — «c ■«= — ^ -«f — « < — «s < «s^ -«^ *e T>' 



ws 



< — ^ 



K£ 



discussion to exhibit this relation, which may be thus briefly expressed, at least 
in so far as the examination of the observations has yet extended*. Let the 
strata a a a' a', b' b' b b, fig. 2, represent two parallel aerial currents, a a a' a' 
being from S.W. and b' b' b b from N.E., and conceive them both to advance 
from the N. W. in the direction of the large arrow, that is the strata themselves 
will advance with a lateral motion. Now conceive the barometer to com- 
mence rising just as the edge b b passes any line of country, and to continue 
rising until the edge b' b' arrives at that line, when the maximum is attained. 
The wind now changes and the barometer immediately begins to fall, and 
continues to fall until the edge a a coincides with the line of country on 
which b b first impinged. During this process we have all the phaenomena 
exhibited by an atmospheric wave ; when the edge b b, fig. 2, passes the line 
of country, the point a, fig. 1, of the wave (the anterior trough) transits that 
line of country and the barometer begins to rise with a N.E. wind. During 
the period the stratum b' V b b, fig. 2, transits the line the anterior slope W a, 
fig. 1, passes ; when the conterminous edges of the strata a' a' V b', fig. 2, 
pass, the crest W, fig. 1, extends in the direction of the preceding trough: 
the barometer now begins to fall, and when the edge a a, fig. 2, occupies the 
place of b b, also fig. 2, the descent of the mercurial column is completed ; the 

* For this knowledge I am indebted to Mr. Brown's plates. 



136 REPORT — 1846. 

posterior slope W tv, fig. 1, has passed, and the posterior trough w, fig. 1, now 
occupies the line in which the anterior trough extended. 

From these considerations, we readily see that the wave is a convenient 
method of representing the barometric fluctuations ; we have already noticed 
the wave motion, the lateral transference of the parallel beds of aerial currents. 
We have seen that the rise is due to the anterior slope and the fall to tlie 
posterior ; and we now further learn that the direction of the aerial current on 
the anterior slope is at right angles to the axis of translation directed towards 
the left-hand, while on the posterior slope it is the reverse ; still at right angles 
to the axis of translation, but directed towards the right-hand. 

Having thus noticed the wave-motion with its accompanying air-motion, 
these interesting questions suggest themselves. How are the forces of this 
air-motion arranged ? Do all the particles move with the same velocity ? 
Are there different velocities in different parts of the wave? Our anemo- 
meters will answer these questions. In the troughs, the edges b b a a, the 
forces are strongest; as the barometer rises, the force gradually subsides; when 
the crest passes, it is zero ; and as the barometer falls, it increases until the 
trough passes, when it is again strongest. 

The examination of the transit of a single wave by means of barometric 
and anemonal observations, would be comparatively easy, but it seldom 
happens, from the operation of natural causes, that an isolated or solitary 
wave is produced. In almost every instance (except in those in which the 
generating power is very much greater than any which occasions the pro- 
duction of smaller waves) the wave is contemporaneous with others of equal, 
if not of greater magnitude, so that different systems are in motion at the 
same time, each individual pursuing its own course, and although perfectly 
independent of every other, yet greatly modifying the resulting phcenomena 
as exhibited by the barometer and anemometer. When therefore we pro- 
ceed with the examination of certain barometric and anemonal phsenomena 
in the manner above alluded to, we are speedily perplexed with the baro- 
metric and anemonal effects of cross tvaves ; the flowing of one set of waves 
in a certain direction is apparently interrupted and interfered with by an- 
other in a different direction, and before the first set can be exhibited with 
its proper proportions, and the true altitudes, amplitudes, velocities, and direc- 
tions of its individual waves assigned, all the phsenomena of the other set 
must be carefully disentangled and separated from the aggregate phaenomena 
presented by the contemporaneous systems. The barometric curve, including 
a complete rise and fall at any one station, is not the curve resulting from 
the transit of any one wave ; it does not represent the form of any reality 
in nature ; but it does represent, and is an exponent of the effects resulting 
from the contemporaneous transits of waves, or systems of waves, such as have 
been described. 

The contemporaneous existence of these cross waves, with their appro- 
priate aerial currents, as manifested by the barometer and anemometer, ap- 
pears likely to form an experimentum c'rucis between the conflicting hypotheses, 
the oscillations of the barometer as dependent on waves, in contradistinction 
to that of the same oscillations as dependent only on the aerial currents. 
When a current meets with another at any angle, both are altered in direc- 
tion ; and if the forces are different, the united current proceeds with an 
increased or diminished strength, according to the situation of the station 
relative to the separate currents before confluence. This union would of 
course influence the barometer ; if the station is in a current of slow pro- 
gress, and the air possesses considerable density, the barometer would fall 
upon the new current being established ; while at another station, where the 
force of the wind is great and the pressure low, it would rise when the con- 



ON ATMOSPHERIC WAVES. 137 

fluence took place. These pheenomena, however, could only occur upon the 
impinging of currents ; upon M. Dove's theory of parallel currents in oppo- 
site directions, it does not appear likely that they can exist. M. Dove has 
suggested that these parallel currents may be shifting ones, and we have 
supposed that the parallel currents of N.E. and S.W. winds may advance 
from the N.W. with a lateral motion. The same cause that produces the 
opposite and superposed equatorial and polar currents, will also give rise to 
the same opposite but parallel currents in the temperate zone, namely, the 
ascending column of heated and consequently rarefied air. Now it is well 
known that in stormy weather, when the wind is blowing with great force, 
the barometer being nearly at its minimum, upon the xoind changing the 
barometer commences rising ; the wind however continues to blow with about 
the same force as it did with the previous falling barometer. Upon M. 
Dove's view of parallel and opposite currents, somewhere in or near the line 
forming the boundary between the currents, towards or in the torrid zone, 
we ought to find the point of rarefaction, and to this point the N.E. current 
would rush with the greatest force to supply the ascending column of heated 
air*. This N.E. current would be compensated by a S.W. current of nearly 
or quite the same force, situated just to the S.E. pj__ g^ 

of it, as in fig. 3, in which let a point of rare- 
faction, a for instance, exist in any locality, ro ,^^ 
80 that a N.E. current may be established to y I ^ l 
supply the ascending column ; suppose the . 4. "k T 
greatest force to exist along the line of crossed y y 
arrows b b, the air would be drawn from the , 
end of this line to fill up the vacuum at a, IT 
and a compensating S.W. current, a c, esta- ' 
blished. This S.W. current would be established I j" 
partly by the descent of the overflowing current 1 ^ 
at a, and partly by the rush to supply the air •! I + 'l^ 4 
constantly drawn off to feed the ascending co- 1 ' Y ' 

lumu. When however it is once established, V « ^ >t ^^ ,^ 

the velocity of the line of S.W. current nearest \\\ .,^-- ^^^ — ■ < 
to the N.E. would probably be equal, or nearly x^'' "^^ — "^ — < ' • 

so, to that of the N.E. current itself. •«• 

In this way it is easy to conceive tHat a complete barometric wave may be 
produced ; the lines of greatest velocity of the parallel currents will indicate 
the trough ; the rapidity with which the currents pass in opposite directions 
greatly diminishes the pressure, and according to this view somewhere near 
the direction of the trough and to the S.W. of it, we ought to find the point 
of greatest rarefaction : the velocity decreases on each side this trough, 
and with this decrease of velocity the pressure increases, so that we have a 

* In attributing the greatest force to the N.E. current, I do not by any means wish to put 
forward or support any hypothesis that would at all interfere with the well-known fact, that 
the greatest force is usually manifested by S.W. winds. The point to which I wish more par- 
ticularly to solicit the attention of the Association is this, the cause which induces the south- 
westerly current itself. This must reside in or near the torrid zone. Here we have a suffi- 
cient cause ; we are presented with phsenomena fully adequate to explain an influx of cool air 
from the N.E. This is the current that must first be established, and in the first instance its 
force will be greatest. We have however only to turn to Prof. Dove's letter to Col. Sabine 
(Report, 1845, p. CI), and we shall at once find the reason why S.W. winds manifest by means 
of our instruments the greatest force. The N.E. currents are narrower, and the force soon 
abates as they pass over towards the S.E. ; while on the other hand the same station is not 
only oftener, but longer in the S.W. currents, and as the line of greatest force approaches, the 
force increases, on some occasions very rapidly, until the wind changes. The line of greatest 
force soon passes the station, so that upon a mean of numerous observations the south-westerly 
svind exhibits the greatest force. 



138 REPORT — 1846. 

distribution of pressure of a wave form gradually rising on each side the 
trough, the pressure being dependent on the velocity of the parallel currents. 

The constant ascent of air at the point of rarefaction would continually 
draw off a quantity of air from the S.E. side of the line of greatest velocity 
b — b, fig. 3, and this would be attended with two results ; first, there would 
be a real hollow or trough formed in the line of junction of the parallel cur- 
rents; and secondly, this line would gradually advance towards the S.E.; for 
as more air would be drawn oflP from that side, the whole body of air would 
advance in that direction to supply the deficiency ; and should the rarefying 
process cease, we can readily conceive that not only will the ivave-form be 
continued, but also wave-motion. The establishment of the parallel currents 
will give the air-motion ; the diminution of pressure towards the lines of 
greatest velocity will give the wave-form ; and the drawing-ofl* of air from 
the S.E. will induce the tcave-motion. The wave thus generated is negative ; 
it consists of a hollow produced by the ascending current of heated air 
carrying off a considerable portion of air set in motion by this ascending 
column, and its direction of motion is determined by more air being drawn 
off from the S.E. slope than the N.W. 

It might be expected that as the trough passed, a motion of the air or 
wind from the N.W. (the body of air moving from that quarter to supply 
the constant drain in feeding the ascending current) would be observed ; but 
so strong must the parallel currents be which give rise to the wave, that 
such motion would doubtless be concealed by them. 

The barometric and anemonal phsenomena would present very regular 
phases, provided there was only one system of waves, one set of parallel and 
opposite currents constantly passing from N.W. to S.E. I have however in 
former reports shown that different systems have contemporaneously traversed 
the area over which the observations have extended, and the discussion of 
Mr. Brown's observations has clearly brought to light a set of parallel and 
opposite currents at right angles to those we have just been contemplating, 
namely, from N.W. and S.E, with a wave-motion towards the N.E., pro- 
ducing the cross waves which occasion the complexity before alluded to. 
The late Professor Daniell has remarked that the curves increase in range 
towards the N.W., and in general the neighbourhood of water presents 
curves remarkable for the boldness of their contour and the large extent of 
their range. In venturing a speculation on these cross waves from the S.W. 
with parallel and opposite currents from N.W. and S.E., I should be inclined 
to attribute them to the effect of the solar influence on the terrestrial sur- 
face, extending from Cape Verd in Africa to the extreme north of Lapland 
in Europe. This surface extends from S.W. to N.E., or somewhat in that 
direction. It may be remarked, that to the north-east of Cape Verd is situ- 
ated the Sahara or Great Desert of Africa, and here we have a great rarefy- 
ing surface. To the north-west or west-north-west of this extensive rarefy- 
ing surface, the broadest part of the Atlantic ocean is situated. The relative 
positions of the Great Desert and the broadest extent of the Atlantic will 
produce a great indraught of cool air from the ocean ; the direction of this 
wind will be W.N.W. or N.W. To the north-east of this current, probably 
in the neighbourhood of Morocco, Fez, Algiers, Spain and Portugal, and 
the north-west portions of the Mediterranean sea, we ought to find the 
counter current from the S.E. or E.S.E., the two portions in juxtaposition 
moving with the greatest velocity. Somewhere in the Atlantic the turning- 
point of these oppositely directed currents should exist. The line of junc- 
tion of these parallel currents will determine the trough of the wave, and as 
before shown, in consequence of the air being drawn off from the north-east 
to supply the ascending current, the wave will progress towards that quarter; 



ON ATMOSPHERIC WAVES. 139 

the barometer first descending with the S.E. wind as the trough approaches 
stations to the N.E., and rising with the N.W. as the current produced by 
the rarefaction approaches, until the crest passes, when the new counter cur- 
rent or slope of the next wave would set in *. 

Pursuing this idea further, there can be no question that Ireland and 
Scotland become points, or unitedly constitute a great point of rarefaction, 
forming as they do the nearest land to the northern part of the Atlantic, the 
land becoming hotter than the neighbouring water, and in consequence a 
N.W. current with its compensating current from the S.E. is induced. Not 
only will the rapidity of the currents reduce the pressure, but the ascending 
column from the land will transfer some of the air into the general current 
of the atmosphere, and there will be a real diiFerence in the distribution of 
air as well as pressure ; a section transverse to the line of greatest velocity 
will exhibit a hollow or trough, and the same phsenomena will result from 
this arrangement of the aerial currents as we noticed arising from the N.E. 
and S.W. currents, the only difference being in direction. 

The following marine stations are admirably suited for testing the views 
just advanced, and tracing a wave of this system from the most western 
point of Africa to the north of Europe. 

Cape Verd. Lisbon. Glasgow. 

Cape Verd Islands. Oporto. Inverness. 

The Canaries and Madeiras. Corunna. The Western Isles. 

The Azores as an outlying Brest. The Orkneys. 

station. The Scilly Islands. The Shetland Isles. 

A station in Morocco near Cape Clear. Christiania. 

Cape Cantin. Limerick. Coast of Norway near 

Tangier. Galway. the Arctic Circle. 

Gibraltar. Markree. Hammerfest or Alten. 

Cadiz. 

A station in Iceland as an outlier would be very valuable. 

The following inland stations are calculated to exhibit the influence of the 
land in modifying the waves in their progress towards the N.E. 

St. Petersburg. -■ Prague. Venice. Naples. 

Warsaw. Vienna. Rome. Tunis. 

In thus considering these rectangularly posited systems of parallel and 
opposite currents, many complex anemonal and barometric phaenomena re- 
ceive an easy explanation, particularly the revolution of the vane in one 
uniform direction, and the barometric wind-rose. When the conterminous 
edges of any two currents pass a station, the barometer is either at a maxi- 
mum or minimum with respect to that particular system of currents ; the 
wind also changes at this time. If the barometer has previously been rising 
with a north-easterly wind, it now begins to fall with a south-westerly : the 
cross currents are however passing at this time with a lateral motion towards 
the N.E. ; in this set of cross currents the barometer will rise with a north- 
westerly wind and fall with a south-easterly. Suppose while the posterior slope 
of a N.W. wave transits, wind S.W., and before its trough passes, the trough of 
the cross wave from the S.W. also transits, and is immediately succeeded by the 
following anterior slope with its N.W. current, the wind will paiss from S.W. 
to W. Now while this slope continues, upon the trough of the N.W. system 
passing, the wind changes to N.E., and the resultant of the two currents is N. 
It is easy to pursue this reasoning, and thus trace the changes of the wind 
arising from these two cross systems completely round the compass. 

* In the above suggestion I have considered the northern portion of the African continent 
as inducing the N.W. current, but of course, the entire surface, as far as the extreme north of 
Europe, including Great Britain and Ireland, will act as a rarefying surface. 



140 REPORT — 1846. 

The two systems of cross currents naturally divide themselves into four 
beds of opposite currents, namely, N.E, S.W., N.W. S.E. ; with the first of 
each system, N.E. N.W., the barometer rises, and with the last of each, S.W. 
S.E., it falls, so that in tlie barometric wind-rose the maximum is found 
about the N.E., the prevailing system, and the minimum near the S.W., the 
opposite current of this system. 

The extent of arc which the wind-vane frequently describes, especially in 
stormy weather, also receives an explanation from these systems of ci'oss 
currents. A contemporaneous S.W. with a N.W. wind will occasion large 
arcs to be described between these points ; the south-westerly gusts prevail- 
ing, directing the vane to that quarter; and the north-westerly immediately 
following, instantly occasions a change carrying the vane towards the N.W. 
These sudden and extensive changes are rendered more distinctly perceptible 
by means of a small kite flown with about 250 or 300 feet of string, or even 
more ; the distinctness and independence of the direction of the two currents 
are readily seen, as well as the difference in their strength. 

Col. Sabine has shown in his Report on the Meteorology of Toronto, that 
the intensity of the wind increases as the temperature increases. The con- 
sideration of these cross currents opens up to us another mode of contem- 
plating the force of the wind. It appears probable that the force diminishes 
on each side the line of greatest velocity. Now in order to obtain the true 
expression of this force, its numerical value, it will be important to correct 
the results, either anemometric or those obtained by estimation for the daily 
period ; this will give the value of the force of the currents then passing, and 
will in a great measure test the hypothesis. 

Section II. 
Discussion of Mr. Brown's Observations. 

In the following discussion I have first arranged such of the observations 
collected by Mr. Brown, or deductions from them, as indicate the barometric 
pressure about the middle of each day at the stations before-named, as near 
as the data furnished by that gentleman will allow. These observations or 
deductions will be found in Table V. The arrangement is such that the eye 
may readily ascertain the barometric state of the atmosphere at any station 
on any day embraced by the area and period included in the table. The 
changes at any one station are also readily seen, the altitudes above 30 
inches being distinguished from those between 29 and 30, and those below 
29 also being distinguished from the rest. This table forms the basis of the 
following deductions which have been thus arrived at. Tiie values corre- 
sponding to each day have been arranged with especial reference to the 
maximum and minimum of that day in space, that is, the station exhibiting 
the greatest pressure on any particular day has generally been placed first 
on the list for that day ; and that exhibiting the least, last. At the head of 
each list are placed the directions of the crests as indicated by the observa- 
tions. Crests passing from N.W. to S.E. are distinguished by the odd 
numbers, and those passing from S.W. to N.E. by the even. When the 
observations give two slopes from a crest or trough passing between such 
slopes, the observations have been arranged to exhibit this. After the 
arrangement of the observations, the lines of the greatest diminution of 
pressure corresponding in a majority of cases to tranverse sections of the 
waves, and exhibiting either their anterior or posterior slopes, are inserted. 
These are succeeded by the direction of the wind on each side of the crests 
as given in Mr. Brown's plates, and the discussion of each day's observation 
is concluded by a few explanatory notes. 



ON ATMOSPHERIC WAVES. 

Table V. — Barometric Observations, November 1842. 



141 



Station. 


In. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


10. 


11. 


12. 


13. 


Orkneys . . 
Belfast . . . 
Shields . . . 
Cork. . . . 
Bristol . . . 
Plymouth . . 
London . . . 
Paris . . . 
Christiania. . 


30 
30 
30 
30 
30 
30 
30 
30 
29 


•16 
■33 
•17 
•15 
•18 
•21 
•17 
•04 
•78 


•23 
•18 
•19 
■92 
•05 
■04 
•10 
•86 
•11 


•24 
■18 
•10 
•83 
•96 
•91 
•96 
•73 
•31 


■49 
•45 
•34 
•16 
•14 
•15 
•13 
■80 
•37 


•52 
•55 
•33 
•32 
•20 
•22 
•12 
•75 
•27 


•46 
•51 
•35 
•30 

•24 
•16 
■83 
•21 


•15 
•43 
•27 
•33 
•18 
•24 
•13 
•89 
■02 


•63 
■04 
•97 
•01 
•07 
•13 
•08 
•90 
•67 


•80 
•41 
•28 
•42 
•60 
•72 
•70 
•76 
•37 


•39 
■57 
■58 
•20 
•46 
■48 
•64 
■63 
■24 


•24 
•02 
•99 
•91 
•03 
•12 
•00 
•25 
•48 


•10 
•21 
•07 
•31 
■31 
•46 
•33 
•43 
•20 


•35 
•27 
•24 
•40 

•46 
■26 
•53 
•94 


Station. 


In. 


14. 


15. 


16. 


17. 


18. 


19. 


20. 


21. 


22. 


23. 


24. 


25, 


26. 


Orkneys . . 
Belfast . . , 
Shields . . . 
Cork. . . . 
Bristol . . • 
Plymouth . . 
London . . . 
Paris . . . 
Christiania. . 


29 
29 
29 
29 
29 
29 
29 
29 
29 


•76 
•91 
•82 
•60 
•65 
•68 
•80 
•67 
■35 


•01 
•82 
•83 
•37 
•61 
•64 
•62 
•65 
•70 


•22 
•06 
•03 

•70 
•78 
•70 
79 
•50 
•86 


•35 
•51 
•45 
•31 
•36 
•36 
•36 
•99 
•94 


•18 
•37 
•42 
•18 
•42 
•47 
•53 
•38 
•11 


•91 
•86 

■11 
•92 

•98 

•14 

•06 

•17 

•91 


•96 
•91 
•85 
•80 

•73 

11 
•55 

•60 


■86 
•95 
■89 
•83 
•79 
•79 
•82 

'H 
•54 


•43 
•42 
•30 
•58 
•39 
•53 
•28 
•13 
•55 


•33 
•32 
•27 
•10 

•m 

•30 
•59 
•41 
•63 


•10 
•79 
•78 
•54 
•79 
•91 
•92 
■02 
•66 


•07 

•82 
•82 
•80 
•84 
•93 
•88 
•01 
•62 


•10 
•04 
•99 
•04 
•14 
•20 
•17 
•17 
•57 



The numbers in the columns immediately succeeding the names of the stations indicate the 
initial inch of the barometric readings of the Island 14th of November, the succeeding num- 
bers are decimals of an inch. Observations above 30 inches are not underlined. Those be- 
tween 29 and 30 inches have a single line — , and those below 29 inches a double line =, 

November 1, 1842. 

Crest No. 1. 

N.W. — S.E. 

Crest No. 2. 



S.W. 
Anterior slope, Crest No. I 



Max. 



Belfast 
Shields . . 
Orkneys . . 
Christiania 



30-33 
30-17 
30-16 

29-78 



^— N.E. 

Posterior slope, Crest No. 1. 

Max. Belfast 30-33 

Bristol .... 30-18 
Plymouth.. 30-21 
Cork 30-15 



Anterior slope, Crest No. 2. 

London 30-17 

Paris 30-04 



142 REPORT — 1846. 

Slopes. — Lines of greatest diminution of pressure. 
Anterior slope, Crest No. 1, Belfast to Christiania. ... '55 

» » » 2, „ Paris '29 

Currents. — Wind on N.E. side of Crest No. 1, N.W. 

„ S.W. „ „ „ changing to S.E. 

A decided crest or line of maximum pressure passes across Ireland and 
England with a general direction N.W. — S.E. The stations in the first 
column are N.E. of this crest, the pressure gradually decreasing. The line 
of greatest diminution is Belfast to Christiania. This indicates the anterior 
slope of the wave; altitude from Christiania -55. The wind along the slope 
is N.W. The stations in the second column are S.W. of the crest ; at these 
stations the pressure" but slightly differs from the maximum ; the wind ap- 
pears to be changing to S.E. in the S.W. part of our island ; this wind is 
that due to the posterior slope. The diminution of pressure from Belfast to 
Paris = '29. This indicates the anterior slope of a wave at or nearly at 
right angles to the former. At two stations the wind is N.E., that of the 
anterior slope of this system. 

November 2, 1842. 

Crest No. 1. 
N.W. S.E. 

Crest No. 2. 

S.W. N.E. 

Anterior slope, Crest No. 1. Anterior slope, Crest No. 2. 

Max. Orkneys 30-23 Max. Orkneys . . 30-23 

Christiania . . 30- 11 Shields 30-19 

Belfast 30-18 

London ., 30-10 

Bristol 30-05 

Plymouth.. 30-04 

Cork 29-92 

Paris 29-86 

Slopes. — Lines of greatest diminution of pressure. 

Anterior slope, Crest No. 2, Orkneys to Paris -37 

Posterior slope. Crest No. 1, „ Cork -31 

Currents. — Wind on S.W. side of Crest No. 1, mostly S.E. 
„ S.E. „ „ 2, a few N.E. 

The progression of the crest, which was so distinctly developed on the 1st 
towards the N.E. and the succeeding S.E. current, is most decided. The 
altitudes at Belfast and Christiania are nearly equal, indicating that the crest 
is between them. The wind, with but few exceptions, is S.E. over nearly 
the whole of Great Britain and Ireland, while at Christiania on the anterior 
slope it is N.N.W. The posterior slope from the Orkneys to Cork is well- 
exhibited. Altitude from Cork to Orkneys = -31. 

The line of greatest diminution of pressure this day, Orkneys to Paris, 
crosses that of yesterday nearly at right angles ; this arises from the advance 
of the anterior slope of the wave (Crest No. 2) ; at a few stations the wind 
is N.E. that of the advancing slope, and these in the neighbourhood of a 
line where the wind appears to have been variable. Altitude from Paris to 
Orkneys =: -37. 



ON ATMOSPHERIC WAVES. 143 

November 3, 1842. 

Crest No. 1. 

N.W. S.E. 

Crest No. 2. 

S.W N.E. 

Posterior slope, Crest No. 1. 

Max. Christiania . . 30-31 Bristol 29'96 

Orkneys 30-24 Plymouth . . 29-91 

Belfast 30-18 , Cork 29-83 

Shields 30*10 Paris 29-73 

London 29-96 

Slope — Line of greatest diminution of pressure. 

Posterior slope, Crest No. 1, Christiania to Paris -58 

Currents. — Wind on S.W. side of Crest No. 1, S.E. 

„ S.E. „ „ 2, N.E. towards trough. 

„ N.E. „ „ 1, N.W. Christiania. 

The crest No. 1 is now approaching Christiania. The observations of 
this day oflFera decided contrast to those of the 1st ; the posterior slope of 
crest No. 1 is well-developed, the point of greatest pressure being to the 
west of Christiania : the point of least pressure is still Paris, where the 
barometer has been falling since the 1st: this station appears to be near the 
intersection of the troughs of both waves. The progress of the maximum 
point is extremely interesting. On the 1st we find it at Belfast, on the 2nd 
at the Orkneys, and on the 3rd at Christiania ; the direction of the progres- 
sion is consequently undoubted. The general direction of the wind over 
England, Scotland and Ireland, is S.E. ; that due to the posterior slope, at 
Paris and in the South-east of England, the wind is E. and N.E., the anterior 
slope of crest No. 2. 

November 4, 1842. 

Crest No. 1. 
N.W. S.E. 

Crest No. 2. 
S.W. N.E. 

Anterior slope, Crest No. 2. 

Max. Orkneys 30-49 Plymouth .. 30-15 

Belfast 30-45 Bristol .... 30-14 

Christiania . . 30-37 London 30-13 

Shields 30-34 Paris 29*80 

Cork 30-16 

Slope. — ^Line of greatest diminution of pressure. 

Anterior slope. Crest No. 2, Orkneys to Paris -69 

Currents. — Wind on S.W. side of Crest No. 1, S.E. at a few stations. 

S.E. „ „ 2, N.E. 

The anterior slope of crest No. 2 is well-developed, and the evidence of 
its extending over the whole of the British islands extremely strong ; also the 
establishment of its proper wind N.E. ; a few stations exhibit the S.E. wind 
as the posterior slope of crest No. 1 is passing off. The line of the greatest 
diminution of pressure is identical with that of the 2nd, namely, Orkneys to 
Paris, but it is nearly doubled in value, being now equal to -69, showing 
that the greatest curvature is approaching. 

The crest No. 1 appears now to be over Christiania, or a little to the east of it. 



144 



REPORT — 1846. 



November 5, 1842. 
Crest No. 1. 



N.W.- 

S.W.- 



Crest No. 2. 



-S.E. 
-N.E. 
^ Probable direction of Crest No. 2. 



Max. Belfast 30-55 

Orkneys 3052 

Cork 30-32 

Shields 30-33" 

Christiania 30-27 

Plymouth 30-22 

Bristol 30-20 

London 30-12 

Paris 29-75 J 

Slope. — Line of greatest diminution of pressure. 

Anterior slope, Crest No. 2, Belfast to Paris. 

Currents. — Wind on S.E. side of Crest No. 2, N.E. 



► Anterior slope of Crest No. 2. 



•80. 



The anterior slope of crest No. 2, extending from Cork, Belfast and the 
Orkneys to Paris, is well-developed. Belfast is the highest point, Paris the 
lowest. Altitude from Paris -80. The posterior slope of crest No. 1 is now 
scarcely perceptible. The wind is that due to the anterior slope of crest 
No. 2. The following table will show the gradual approach of the anterior 
slope of this wave. Paris the lowest point: — 

Belfast to Paris. 

November 1 -29 

„ 2 -32 

3 -45 

„ 4 -65 

5 -80 



November 6, 1842. 
Crest No. 2. ■ 



S.W.- 



-N.E. 



Anterior slope, Crest No. 2. 



Max. Belfast 30-51 

Orkneys.... 30-46 

Shields .... 30-35 

Cork 30-30 



Plymouth . , 30-24 

Christiania. . 30-21 

London 30-16 

Paris 29-83 



Slope.- 



-Line of greatest diminution of pressure. 
Anterior slope. Crest No. 2, Belfast to Paris. . . . -68. 
Currents. — Wind, with but few exceptions, N.E., anterior slope of Crest 
No. 2. 

Nearly the same state of the barometer is maintained over the area as on 
the 5th, with nearly similar winds. The anterior slope of crest No. 2 is still 
strikingly developed. The greatest curvature has passed with a very slight 
fall at Belfast and a very slight rise at Paris. 



ON ATMOSPHERIC WAVES. -145 

November 7, ISiS. 

Crest No. 2. 

S.W. N.E. 

Crest No. 3. 

N.W. S.E. 

Crest No. 2. Posterior slope, Crest No. 2. 

Max. Belfast 30-43 Max. Belfast 3043 

Cork 30-33 Orkneys.. 30-15 

Shields .. 30-27 
Plymouth 30-24~] 

Bristol 30-18 | 

London . . 30-13 )> Anterior slope, Crest No. 2. 
Christian ia 30-02 | 

Paris 29-89J 

Slope. — Line of greatest diminution of pressure. 

Anterior slope, Crest No. 2, Belfast to Paris •S'i. 

Currents — Wind on S.E. side of Crest No. 2, N.E. 

;; N.'k ;; »Jf_^V /advancing anterior 

{_ slope 01 new wave. 

The crest No. 2 has now passed the Orkneys, which exhibits a falling 
barometer and the S.W. wind. The trough between crests 1 and 3 is now 
to the N.E. of Belfast and Paris ; the higher readings in the south-west part 
of the area, with the lower in the north-east, clearly indicate the advancing 
anterior slope of the new wave. Diminution of pressure from Belfast to 
Christiania, -^l. 

November 8, 1842. 

Crest No. 2. 

S.W. N.E. 

Crest No. 3. 

N.W. -S.E. 

Anterior slope. Crest No. 2. Posterior slope, Crest No. 2. 

Max. Plymouth.. 30*13 Max. Plymouth.. 30-13 

London . . 30-08 Cork 30-01 

Bristol 30-07 

Paris 29-90 

Anterior slope of Crest No. 3. 

London 30-08 Christiania 29-67 

Belfast 30-04 Orkneys 29-63 

Shields 29-97 

Line of greatest diminution of pressure. Plymouth to Orkneys . . '50 
Currents — Wind on S.E. side of Crest No. 2, N.E. 
N.W. „ 2, S.W. 

N.E. „ 3, N.W. 

The crest No. 2 appears on this day to pass from Plymouth towards Bris- 
tol and London. The direction of the line of greatest diminution of pres- 
sure varies considerably from that of the three preceding days ; this partly 
arises from the great fall which commenced on this day at the northern sta- 
tions, Orkneys, Belfast and Christiania; and from the anterior slope of the 
wave (crest No. 3). The direction of the wind is closely in accordance with 
crest No. 2, passing in the direction from Plymouth towards London, being 
S.W. on the north-west side of the crest. 

1846. L 



146 



REPORT 1846. 



Anterior slope, Crest No. 2. 
The wave (crest No. 2), with its front towards the south-east, has been 
very distinctly developed during the preceding days. The altitude of the 
crest appears to have subsided as the wave progressed ; the highest reading 
at Belfast was 30*55 on the 5th, at London 30-16 on the 6th, and at Paris 
29-90 on the 8th. The following tables exhibit the features of the anterior 
slope. Table VI. shows the barometric rise and fall at stations arranged 
more or less with regard to a line cutting the crest of the wave transversely. 
The depressing influence of the wave, crest No. 1, is clearly seen at London 
and Paris on the 5th. Tables VIL, VIIL and IX. exhibit the depression of 
the south-easterly stations below those to the north-west of them while the 
anterior slope passed. 

Table VI. — Barometric differences arising from Anterior and Posterior 
Slopes of Crest No. 2. 



Epoch. 


Belfast. 


Bristol. 


London. 


Paris. 


Nov. 2 
„ 3 
» 4 
» 5 
» 6 

„ ^ 

„ 8 


-15 
•00 

+•27 
+•10 
-04 
-•08 
-•39 


-13 
-•09 
+•18 
+•06 


-•07 
-14 
+•17 
-01 
+•04 
-03 
-•05 


-18 
-13 

+•07 
-05 
+•08 
+•06 
+•01 


-01? 
-11 



Table VII. — Barometric differences arising from Anterior Slope of Crest No.2. 



Epoch. 


Belfast. 


London. 


London. 

± 


Nov. 1 


30-33 


3017 


-16 


„ 2 


•18 


•10 


-•08 


„ 3 


•18 


2996 


-•22 


., 4 


•45 


3013 


-•32 


,. 5 


•55 


•12 


-•43 


,, 6 


•51 


•16 


-•35 


„ 7 


•43 


•13 


-•30 


„ 8 


3004 


30-08 


+•04 



Table VIII. 



Epoch. 


London. 


Paris. 


Paris. 

± 


Nov. 1 
,, 2 
» 3 
„ 4 
., 5 
» 6 
„ 7 
„ 8 


30^17 
•10 

2996 

3013 
•12 
•16 
•13 

30^08 


3004 
29-86 
•73 
•80 
-75 
-83 
•89 
29^90 


-13 
-•24 
-•23 
-•33 
—37 
-•33 
-•24 
-•18 



Table IX. 



Epoch. 


Belfast. 


Paris. 


Paris. 

± 


Nov. I 


30-33 


30-04 


-•29 


„ 2 


•18 


29-86 


-•32 


,. 3 


•18 


•73 


-•45 


,> 4 


■45 


•80 


-•65 


;; 5 


•55 


•75 


-•80 


„ 6 


•51 


•83 


-•68 


,, 7 


•43 


•89 


-•54 


„ 8 


3004 


29-90 


-14 



ON ATMOSPHERIC WAVES. 



147 



S.W.- 



November 9, 184.2. 
Crest No. 2. 



-N.E. 



Posterior slope, Crest No. 2. 



Max. Paris 29-76 

Plymouth . . 29-72 

London 29-70 

Bristol 29-60 

Cork 29-42 



Belfast 29-41 

Christiania. . 29-37 

Shields 29-28 

Orkneys.... 28*80 



Slope. — Line of the greatest diminution of pressure. Paris to Orkneys, -96 

Current. — Wind on N.W. side of crest, S.W., fully established. 

The posterior slope of crest No. 2 now comes into full view, stretching 
from Paris to the north-west coasts of Ireland and Scotland, with its proper 
wind S.W. The altitude of this slope from the Orkneys to Paris is '96. 
The greatest altitude of the anterior slope, from Paris to Belfast, was -80. 
It will be seen that the greatest oscillation has been in the north-west, Paris 
exhibiting but a very slight oscillation, -17, while that at the Orkneys has 
amounted to 1-72. 

The following table exhibits the fall of the barometer on the 8th and 9th : 

Table X. — Fall of Barometer, November 8 and 9, 1842. 



station. 


November 8. 


November g. 


Orkneys 

Cliristiania 

Belfast 

Shields 

Cork 

Bristol 


•52 
•35 
•39 
•30 
•32 
•11 
•11 
•05 
-^•Ol 


•83 
•30 
•63 
•69 
•59 
-47 
•41 
•38 
•14 


Plymouth 


Paris 



From these numbers we learn that the greatest barometric fall, as well as 
the greatest oscillation, occurred in the N.W. The fall gradually decreases 
as we approach the S.E. 

It appears to me that the difference of oscillation at two stations, as the 
Orkneys and Paris, may be thus explained. The curves in the north-west 
of Ireland, as determined by the discussion of Sir John Herschel's hourly- 
observations, are remarkable for boldness and freedom of contour and great 
range of fluctuation. The late Professor Daniell found, from an examina- 
tion of the Manheim observations, that the range increased towards the north- 
west, and that the greatest oscillation occurred in the neighbourhood of water. 
Now a wave generated in any way and approaching the continent of Europe 
from the north-west, would most probably impinge on it with a high and in 



, but as it passed -onward 




some cases acuminated crest 



the crest would gradually subside -^ "^-~-., _ , so that at sta- 
tions considerably to the south-east the fluctuations would be very much 
less than at or near its point of genesis. Again, a negative wave, with a 

deep trough also approaching from the north-west ^^ y/ , would 

present large fluctuations as it impinged on the laud ; but after passing ou- 

l2 



148 REPORT — 1846. 

wards, the opposite to subsidence would take place ; the depth of trough 
would decrease ~~"--^___.,-^ — , and the oscillations to the south-east 

would also decrease. Such pheenomena appear to be presented by the ob- 
servations from the 5th to the 10th of November 1842. 

November 10, 1842. 
Crest No. 2. 

S.W.— N.E. 

Crest No. 3. 

N.W. — _S.E, 

Max. London 29*64^ 

Paris 29-63 I .. ^ ,. „ 

Shields 29-58 f N^^'" «^^«t No. 3. 

Belfast 29-57 J 

Plymouth.. .. 29-481 

Bristol 29*46 > Under posterior slope, No. 3. 

Cork 29-20 J 

Orkneys 29-39 \ jj , . • i xt o 

Christiania . . 29-21 / ^nder anterior slope. No. 3. 

Slope. — Line of greatest diminution of pressure on posterior slope of 
Crest No. 2. London to Cork '44. 

Altitude of anterior slope. Crest No. 3. Christiania to London . . -40. 

Altitude of posterior slope. Crest No. 3. Cork to Belfast '37. 

Currents. — Wind on N.E. side of Crest No. 3, N.W. 
S.W. „ „ 3, S.E. 
„ posterior slope of Crest No. 2, S.W. 
,, anterior slope of Crest No. 4, N.E. 
Trough succeeding Crest No. 2 now transits Christiania. 

The direction of the crest No. 3 is nearly identical with that of No. 1, 
which passed Great Britain and Ireland on the 1st. This appears to suggest 
that they were either successive crests of the same system of waves, or were 
succeeding waves produced by the same disturbing causes. The altitude of 
crest No. 3 is about half an inch less than that of crest No. 1 ; but between 
the transits of the two crests a wave from the N.W. with a deep posterior 
trough has passed the area, which has probably depressed crest No. 3. The 
interval between the crests Nos. 1 and 3 is equal to nine days. 

It was noticed in the remarks on the 9th, that the great difference in the 
oscillation at the Orkneys and Paris most probably resulted from the sub- 
sidence of the crest as it progressed. The crest No. 3 came from the S.W., 
so that a line from Plymouth to Christiania would cut it more or less trans- 
versely ; the ranges however are nearly the same at both stations. The crest 
which traversed England on the 1st arrived at Christiania on the 4th ; at this 
time the barometer had commenced rising at Plymouth from the anterior 
slope of crest No. 2, and it continued rising until the 7th, when the crest 
passed. At Christiania the barometer had fallen from the posterior slope of 
crest No. 2. It appears from a careful comparison and consideration of the 
barometric movements at Plymouth and Christiania, that crest No. 2 passed 
Christiania about a day earlier than it did Plymouth, that is, the longitudinal 
direction of the crest was such as to cause it to pass over Christiania while 
Plymouth was still under the anterior slope of tjie wave, the sections passing 
over Christiania and Plymouth being separate and distinct. The character 
of the passing wave is well-determined at both stations, the posterior slope 
exhibiting a rapid and deep fall, which took place alike at Christiania and 
Plymouth. 



ON ATMOSPHERIC WAVES. 149 

The crest No. 2 passed Cork, Belfast and the Orkneys on the 5th, Ply- 
mouth on the 7th, and Paris on the 8th, with a diminution of oscillation. 
We find however no diminution of oscillation at Christiania as compared with 
Plymouth. It is highly probable, the subsidence of the crest, as it proceeded 
towards Paris, resulted from the influence of the land in England, while both 
at Plymouth and Christiania the crest was but slightly interfered with by 
the influence of land, the difference of level resulting from the anterior slope 
of crest No. 3. 

These considerations exhibit a large wave of considerable breadth and slow 
motion, extending in a longitudinal direction from the extreme south-west of 
England towards the Swedish capital. 

Nov. 10. — Mr. Brown's diagram for this day very distinctly and beautifully 
exhibits the change of currents resulting from the transit of crest No. 3, as 
well as from the progress of the posterior trough of crest No. 2. The trough 
of the latter wave is now between the Orkneys and Paris (the deep trough 
before mentioned), the wind in the south-eastern portion of the diagram is 
S.W., the strength increasing 'towards the trough. At Thurso and North 
Shields the wind is N.W., the anterior slope of crest No. 3 ; in the south- 
west of Ireland, the wind is S.E. (posterior slope). The N.E. wind on the 
anterior slope of the wave succeeding crest No. 2 is seen on the north-west 
side of the trough. 

November 11, 1842. 
Crest No. 2. 

S.W. —N.E. 

Crest No. 3. 
N.W.— S.E. 

Max. Christiania 29-48 Crest No. 3. 

London . . . . . . . . 29*00 } ^nder posterior slope of Crest No. 2. 

Plymouth 29-12 1 t *u • i,u u i x- 

Bristol 29-03 ^^" , *^^ neighbourhood of posterior 

Shields 28-99*J ^'^^S^' ^''^'^ ^°- ^' 

Cork 28-91t Posterior trough, Crest No. 3. 



"Under anterior slope of wave succeed- 
-N ing Crest No. 2 and posterior slope 
of Crest No. 3. 



Belfast 29-02 

Orkneys 29-24? 

Max. Christiania 29-48 

Orkneys 29-24 

Belfast 2902 ^Under posterior slope, No. 3 

Shields 28-99 | 

Cork 28-91 J 

London 2900 

Paris 29-25 ") 

BilsToT!^. . ; ; ; ; ; ; 29-03 \^''^^^ posterior slope, No. 2. 

Cork 28-91 J 

Shpes. — Lines of greatest diminution of pressure. 

On posterior slope of Crest No. 3. Christiania to Cork. . -57 

On posterior slope of Crest No 2. Paris to Cork -34 

Currents Wind on N.W. side of Crest No. 2, S.W. 

S.W. „ „ 3, S.E. 

* Depressed by posterior trough of Crest No. 2. 
t Depressed by posterior trough of Crest No. 3. 



150 REPORT — 1846. 

The progressive motion of the two posterior slopes is very discernible. 
Crest No. 3 has passed the Orkneys and arrived at Christiania seven days 
after crest No. 1 passed that station. We found an interval between the 
crests as they passed the central parts of England of nine days. Most pro- 
bably a discussion of observations at shorter intervals and more numerous 
stations, especially to the north-east of Christiania, would explain the dis- 
crepancy. The difference in level is very much greater than that which cha- 
racterized the transits of the crests over the centre of England, but in the 
interval between the crests passing England and Sweden, the deep trough of 
crest No. 2 has advanced, which must very considerably have depressed the 
crest at Christiania, as compared with the English stations ; indeed so great 
was the depressing influence of the wave, crest No. 2, that no rise is recorded 
as crest No. 3 passed Plymouth. 

Crest No. 2 is situated considerably to the south-east of Paris, so that its 
progress is not perceptible on the area ; but that of its posterior slope is very 
clear. On the 9th, the deep posterior trough of this wave passed the Orkneys 
with a depression of 28*80; (bearing in mind that its direction was S.W. — N.E.) 
another section passed Christiania on the 10th, 29"24' ; and a third passed 
Plymouth on the 11th, 29-12. 

Symmetrical Wave. — On this day the great symmetrical wave commenced 
at London ; the position of this station was nearly similar under both slopes. 

November 12, 1842. 

Crest No. 3. 

N.W. S.E. 

Crest No. 5. 

N.W. S.E. 

Christiania 29-20"| 

Orkneys 29*10 VUnder posterior slope, No. 3. 

Min. Shields 29-07J 

Trough. 

Belfast 29-2n 

Cork 29-31 

Londt ■::;•.:::: 29-33 r nder anterior slope, No. 5. 

Paris 29-43 

Max. Plymouth 29-46J 

Slopes. — Lines of greatest diminution of pressure. 

On posterior slope of Crest No. 3. Christiania to Shields. . '13 

On anterior slope of Crest No. 5. Plymouth to Shields -39 

Currents. — Wind on S.W. side of trough, N.W. 
A few stations exhibit a S.W. M'ind on the posterior slope of Crest No. 2. 
The receding posterior slope of crest No 3, the intervening trough, and 
the approaching anterior slope of crest No. 5, are brought fully into view 
this day ; the wind also is in close accordance with the transit of these waves. 
The wave (crest No. 3) appears to be much smaller than that of No. 1 ; the 
interval between the crests, as passing the centre of England, was found to 
be nine days ; the epoch of the passage of the intervening trough occurred 
on November 7 ; interval between the troughs five days, taking the last in- 
terval as the amplitude in time of the wave ; it is cleiirly much smaller than 
the preceding. 

Symmetrical Wave. — London is situated under and rising from the ante- 
rior slope of wave No. 5. 



ON ATMOSPHERIC WAVES. 1-51 

November 13, 184-2. 
Crest No. 2. 

S.W.- ^ N.E. 

Crest No. 5. 

N.W. S.E. 

Anterior slope, Crest No. 5. 

Max. Paris 29-53 London 29-26 

Plymouth . . 29-46 Shields 29-24 

Cork 29-40 Orkneys .... 29-35 

Belfast .... 29-27 Christiania . . 28-94. 

Slope. — Line of greatest diminution of pressure. 
Anterior slope, Crest No. 5. Paris to Christiania. . -59. 
Trough succeeding Crest No. 2. Between Paris and Orkneys. 
Trough between Crests Nos. 3 and 5 now transits Christiania. 
Currents.— WmA on S.E. side of trough No. 2, S.W. 
„ N.W. „ „ » » N.E. 
The anterior slope of crest No. 5, the third wave of the S.W. system, is 
well-developed ; but the prevailing winds are those due to the b.E. and r^.W. 
sides of the trough succeeding crest No. 2. _ ^ , . , i a a 

It appears from a consideration of the continuation of the tables appended 
to Nov. 8, that one or two small waves rode in the trough succeeding crest 
No 2, wliich entered on the area on the 9th, and most probably traversed it 
during the next four days, presenting the same slowness of motion as the 

crest itself. ' , . ^ c i. 

Symmetrical TFave.— London is situated under the anterior slope ot crest 

No 5 and the posterior slope of crest No. 2, and falling either from the latter 

slope or the posterior slope of one of the small waves riding m the deep 

trough. 

November 14, 1842. 

Crest No. 5. 
N.W. S.E. 

Crest No. 4. 
S.W. ^ N.E. 

Max. Belfast ^^'^H xt . ^ . xt ^ 

Sljields 29-82 VNear the Crest No. 5. 

London'."..'.".... 29-80 J 

Orkneys J^'^^ 1 Under anterior slope, No. 5. 

Christiania 29-3o J 

Plymouth 29-681 ^ 

Paris 29-67 v Under posterior slope. No. 5. 

Bristol 29-65 f ^ ^ 

Cork 29-60J 

Slope. Line of greatest diminution of pressure. 

Anterior slope, Crest No. 5. Belfast to Christiania. . '56. 
Currmts.—WmA on advancing slope of Crest No. 4, N.E, 
N.E. side „ „ 5, N.W. 

S.W. „ „ » 5, S.E. 

The crest No. 5 now passes over Great Britain and Ireland much in the 
same direction as the crests Nos. 1 and 3 (compare Nov. 1 and 10). Its 
altitude is about -25 higher than that of the last S.W. crest; the advancing 
slope of the second N.W. wave has approached, raising the present crest. 



153 REPORT — 1846. 

Symmetrical Wave. — The crest No. 5 forms the second subordinate maxi- 
mum on the anterior slope of the great symmetrical wave (see plate 2 ap- 
pended to Sir J. Herschel's Report, 1843, and plate 3, illustrating the volume 
of Reports, IS^S). The first strongly developed rise and fall on the anterior 
slope of the symmetrical wave appears to be a small wave riding in the 
trough (A3). 

It appears highly probable that the large wave from the N.W. possessed 
both a broad crest and broad trough, with a very slow progressive motion. 
The anterior slope of crest No. 4 appears to have commenced riding over 
Christiania on the 10th. 

November 15, 184'2. 

Crest No. 5. 
N.W. S.E. 

Crest No. 4. 

S.W. N.E. 

Anterior slope, Crest No. 5. Posterior slope, Crest No. 5. 

Max. Orkneys 30-01 Max. Orkneys 30-01 

Christiania.. 29-70 Shields 29-83 

Belfast .... 29-82 
Plymouth . . 29-64 
London .... 29-62 

Bristol 29-61 

Paris 29-55 

Cork 29-37 

Slope. — Line of the greatest diminution of pressure. 
Posterior slope. Crest No. 5. Orkneys to Cork. . -64 
Currents. — Wind on anterior slope of Crest No. 4, N.E. 
„ posterior „ „ 5, S.E. 

The southern coasts of England exhibit a S.W. wind. 
The progression of the crest No. 5 in the same direction as crest No. 1 
(see Nov. 2), is very discernible ; there can be no doubt of its being a wave 
of the same system. 

Symmetrical Wave. — The barometer at London has fallen from the pos- 
terior slope of No. 5, but the anterior slope of No. 4 is gently raising it. 

November 16, 1842. 

Crest No. 5. 
N.W.—- — S.E. 

Crest No. 4. 
S.W. N.E. 



Anterior slope, Crest No. 5 ? 


Anterior slope, Crest No. 4 


Max. Orkneys.... 30-22 


Max. Orkneys.... 30-22 


Christiania.. 29-86 


Belfast .... 30-06 




Shields 30-03 




London 29-79 




Bristol 29-78 




Plymouth . . 29-70 



Cork 29-70 

Paris 29-50 

Slope. — Line of greatest diminution of pressure. 
Anterior slope, Crest No. 4. Orkneys to Paris. . -72 
Currents. — The winds this day appear to be those due to the anterior slope 
of crest No. 4, or resultants of that and the posterior slope of orest No. 5. 



ON ATMOSPHERIC WAVES. 



153 



The posterior slope of crest No. 5 is well and strikingly developed, and the 
advancing slope of crest No. 4, the succeeding wave to that of No. 2, is in- 
dicated by the line of greatest pressure, Orkneys to Paris, '12 (see Nov. ?, 
when the first wave from the N.W. was coming up). 

Symmetrical Wave. — London is situated under the posterior slope of crest 
No. 5 and anterior slope of crest No. 4, and slightly rising from the latter. 

November lY, 1842. 

Crest No. 7. 

N.W. S.E. 



S.W. 



Crest No. 4. 



-N.E. 



Anterior slope. Crest No. 4. 
Max. 



Belfast ... 


. 30-51 


Shields . . . 


. 30-45 


Bristol . . . 


. 30-36 


Plymouth . 


. 30-36 


London . . . 


. 30-36 


Paris 


. 29-99 



Anterior slope. Crest No. 7. 

Max. Belfast 30-51 

Orkneys " 30-35 

Christiania. , 29-94 

Posterior slope, Crest No. 7. 

Max. Belfast 30-51 

Cork 30-31 



Slopes. — Lines of greatest diminution of pressure. 

Belfast to Paris "52 

Belfast to Christiania . . "57 
Currents. — Wind on anterior slope of Crest No. 4, N.E. 
„ posterior „ „ „ 7, S.E. 

In the south-west of Ireland and England the wind is easterly, being the re- 
sultant of these forces. 

The anterior slope of crest No. 4, extending from Belfast to Paris, is well- 
developed with its proper wind N.E. This anterior slope may be advantage- 
ously compared with the anterior slope of crest No. 2, which occupied the 
same area on the 5th (interval twelve days). The altitude on that occasion 
from Paris to Belfast was equal to -80, on the present it is only equal to '52. 
This may to a certain extent be explained by the presence of crest No. 7, 
which this day passes over Belfast, so that this crest elevates the anterior 
slope of No. 4. The anterior slope of crest No. 7 is well-developed towards 
the Orkneys and Christiania, and the diminution of pressure resulting from 
the posterior slope is conspicuous at Cork ; we have consequently two crests 
traversing the area and crossing each other at Belfast. 

Crest No. 4 passes the Orkneys, Belfast, Shields and Cork this day. 
Symmetrical Wave, — London is situated nearly under the crest of No. 7 
and under the anterior slope of crest No. 4, and rising from the latter. 



N.W.- 



November 18, 1842. 
Crest No. 7. 



-S.E. 



S.W.- 



Paris . . 30-38 



Crest No. 4. 



^^K Transit of the Crest of the Great Symmetrical Wave. 

^^KAnterior slope. Crest No. 4. Crest No. 4. 

^HMax. London .... 30-53 Max. London .... 30-53 

I 



-N.E. 



Plymouth .. 30-47 

Bristol 30-42 

Shields .... 30-42 

Christiania.. 30-11 



154 REPORT — 1846. 

Posterior slope, Crest No. 4. Anterior slope, Crest No. 7. 

Max. London 30-53 Max. London 30'53 

Christiania . . 30*1 1 



London. . . 


. 30-53 


Shields . . . 


. 30-42 


Belfast ... 


. 30-37 


Cork . . . 


. 30-18 


Orkneys . 


. 30-18 



Slopes. — Lines of greatest diminution of pressure. 

Posterior slope, Crest No. 4. London to Cork '35 

„ „ „ „ 4. „ Orkneys -35 

Anterior slope, Crest No. 7. „ Christiania. . -42 

Currents. — Wind on posterior slope of Crest No. 4, S.W. 
A few S.E. directions indicating the posterior slope of Crest No. 7- 
The crest No. 4 has advanced with considerable rapidity as compared with 
No. 2. It now passes London. The depressions to Cork and the Orkneys 
are equal ; these lines are on the posterior slope; the crest No. 7 rises between 
the stations. This crest appears to have a very slow motion ; its anterior slope 
is well-seen in the diminution of pressure from London to Christiania, =-42. 
Crests Nos. 4 and 7 cross at London. 

Symmetrical Wave. — The crest passes over London ; it is identical with 
the crest No. 4. 

November 19, 1842. 

Crest No. 7- 
N.W. S.E. 

Crest No. 9. 

N.W.^=^^^^^ ■ ^^^^:^S.E. 

Crest No. 4. 
S.W. N.E. 

^''' ??^outh- :: 30-1I} U°der anterior slope. Crest No. 9. 

London .... 30-061 rjnder anterior slope, Crest No. 9, at a 
Bristol .... 29-98 Y j j j ^ 

Cork 29-92 J ^"wer levei. 

Min. Ss : : : : 29-?? } N^^'- *^^ *^«"S^ ^^^^^^^ 9 and 7. 
Si; : : 29^9! } ^^^er posterior slope, Crest No. 7. 

Slopes. — Line of greatest diminution of pressure. 

Paris to Shields .... -40 
Currents. — Wind on posterior slope, No. 4, S.W. 

5J » 5J 5> 7, S.E. 

Symmetrical Wave. — London is situated under the posterior slope of crest 
No. 4 and anterior slope of crest No. 9, not far removed from the preceding 
trough. 

The crest No. 4 has now passed considerably to the S.E. of Paris, which 
exhibits the greatest pressure ; the posterior slope extends in the direction to- 
wards Belfast, although Shields is the minimum point. The trough between 
crests 7 and 9 passes somewhat near Belfast and Shields. 



ON ATMOSPHERIC WAVES. 156 

November 20, 184-2. 

Crest No. 9. 

N.W. S.E. 

Crest No. 6. 

S.W. N.E. 

Anterior slope, Crest No. 6. Anterior slope, Crest No. 9. 

Max. Orkneys.... 29-96 Max. Orkneys 29-96 

Belfast 29-91 Christiania. . 29*60 

Cork 29-80 

Shields 29-85 

London 29-77 

Plymouth . . 29-73 

Paris 29-55 

Slope. — Line of greatest diminution of pressure. 
Anterior slope, Crest No. 6. Orkneys to Paris . . -41 
Currents. — Wind on anterior slope of Crest No. 6, N.E. 

„ posterior „ „ „ 9, S.E. and E. 

The observations of this day, both barometric and anemonal, indicate the 
presence of an anterior slope of a wave succeeding crest No. 4 of the N.W. 
system. Crest No. 9 now passes over the Orkneys and between Christiania 
and Paris. 

Symmetrical Wave, two days after transit. — London is situated under the 
anterior slope of crest No. 6 and posterior slope of crest No. 9. 

On the 16th, two days before transit, London was similarly situated with 
respect to waves Nos. 4 and 5, with nearly the same barometric pressure. 

On the 13th a permanent rise took place at London, which continued until 
crest No. 4 passed the station. It appears that this should be carefully distin- 
guished from the rise and fall of the 16th to 19th, the latter being due to 
a separate and distinct wave. 

The curve from noon of the 16th to midnight of the 19th appears to re- 
present the form of the N.W. wave riding on the top or superposed on the 
normal or great symmetrical wave. 

November 21, 1842. 

Crest No. 11. 

N.W. -S.E. 

Crest No. 6. 

S.W. N.E. 

Crest No. 11. 
Max. Belfast .... 29-95 1 ^^^^ ^j,^ ^^^^^^ 

Shields .... 29-89 J 
Max. Belfast .... 29-951 

Orkneys. . . . 29*86 > Under anterior slope. 
Min. Christiania. . 29-54 J 
Max. Belfast .... 29*95 1 tt ^ * • i 

Cork 29-83 / '^""^^ posterior slope. 

Crest No. 6. 

Max. Belfast .... 29*95") 

Cork 29*83 I 

Shields 29*89 a,, j .i, * • i i. 

Tj . . . 90-70 L ^"- ""d^r the anterior slope at 

Plymouth".: 29-79' different levels. 

London 29-82 

Min. Paris 29-57. 



156 



REPORT — 1846. 



Slopes. — Lines of greatest diminution of pressure- 
Anterior slope of Crest No. 11. Belfast to Christiania . . •4'1* 

Anterior slope of Crest No. 6. Belfast to Paris "38 

Currents. — Wind on anterior slope of Crest No. 6, N.E. 

„ posterior ., „ ,,11, E.S.E. 

Another S.W. wave now transits the area, and the crest of No. 6 is rapidly 
approaching. 

Symmetrical Wave, three days after transit. — London is situated under the 
anterior slope of No. 6 and posterior slope of No. 11, and slightly rising from 
the former. 

On the 15th, three days before transit, London was similarly situated with 
respect to waves 5 and 4. 

November 22, 1842. 

Crest No. 11. 

N.W. S.E. 



Crest No. 13. 



N.W. 
S.W.- 



Crest No. 6. 



-S.E. 

-N.E. 



Max. Cork 29*58 



Belfast . . . 


29-42 


Orkneys . . . 


. 29-43 


Plymouth . 


. 29-53 


Bristol . . . 


. 29-39 


Shields ... 


. 29-30 


Chi-istiania . 


. 29-55 


London , . . 


. 29-28 


Paris 


. 29-13 



All under the anterior slope of 
crest No. 6. The trough between 
No. 11 and 13 is seen at Belfast 
and Shields, the stations being ar- 
ranged to exhibit this. 



Slopes. — Lines of greatest diminution of pressure. 

Anterior slope, Crest No. 6. Cork to Paris -45 

Posterior slope, Crest No. 9. Christiania to Paris. . -42 

Currents. — Wind on S.W. side of trough between crests 11 and 13, N.W. 

A few S.W. directions as Crest No. 6 passes. 

The anterior slope of crest No. 6 is still strikingly developed from Cork to 
Paris ; the trough between crests 1 1 and 13 extends in the direction of Belfast 
and Shields. 

Symmetrical Wave, four days after transit. — London is situated under the 
anterior slopes of crests Nos. 6 and 13. The barometer has fallen from the 
posterior slope of crest No. 11. 



N.W. 



November 23, 1842. 
Crest No. 13. 

Crest No. 6. 



Max. 



S.W. 

Christiania . 
London . . . 
Paris 



S.E. 



-N.E. 



29-62 

29-5 

29 



-591 
-41 J 



Under anterior slope. Crest No. 6. 



* From a cnmparison of the St. Petersburg observations, it appears that wave, crest No. 1 1 , 
was a small wave, and that the diminution of pressure, Belfast to Christiania, was compounded 
of the anterior slopes of crests 9 and 1 1, that of 11 b^ipg very small. See Postscript. 



\*%9 I Under posterior slope, Crest No. 6, 



ON ATMOSPHERIC WAVES. 157 

London SQ-SQ^j 

B.SoT'' ■■ 29-26 render posterior slope, Crest No. 6. 

Shields .... 29-27 J 

Cork 29 

Belfast 29^^ r . i i i 

f\ \ on oo I at a lower level. 

Orkneys .... 29-3^ j 

Slopes. — Lines of greatest diminution of pressure. 

Posterior slope of Crest No. 13. Christiania to Cork. ... '52 

Posterior slope of Crest, No. 6. London to Cork "49 

A trough between crests No. 13 and 15 extends in the direction of Cork 
and Bristol*. 

Currents. — Wind on S.W. side of this trough, N.V/. 
N.W. „ Crest No. 6, S.W. 

Crest No. 6, the third of the N.W. system, now passes London five days 
after the transit of crest No. 4, the second of this system. Christiania at this 
time exhibits the greatest pressure; most probably the crest No. 13 is ap- 
proaching this station, and this, combined with crest No. 6, produces the in- 
creased pressure. At the transit of crest No. 4 Christiania exhibited the 
least pressure ; the difference may probably be explained by the transits of the 
cross waves. 

Symmetrical Wave. — London is situated under the crest of No. 6, not far 
removed from the trough, between crests 13 and 15. 

November 24, 1842. 

Crest No. 13. 

N.W S.E. 

Crest No. 6. 
S.W. __N.E. 

Max. Christiania.. 29-66"^ 

Paris 29-02 

London .... 28-92 | 

Plymouth .. 28-91 | ^jj ^^ Christiania under posterior 

Bristol .... 28-/9 > siooe of Crest No 6 

Shields .... 28-78 ''''P^ *"* ^'^^* ^*'- ^• 

Cork 28-54 

Belfast 28-79 

Orkneys 29-loJ 

Max. Christiania.. 29-661 

Orkneys 29-10 j 

Belfast .... 28*79 }► Under posterior slope, No. 13. 

Shields 28-78 | 

Cork 28-54 J 

London 28-92 

* The barometric fall between the 21st and 23rd, that occurred at all tlie stations except 
one, appears to have given rise to an apparent regression of the trough observed in the neigh- 
bourhood of Belfast and Shields on the 22nd. It was shown that on the 22nd the anterior 
slope of crest No. 6 was coming up from the N.W., so that the fall of the 21st resulted from 
the posterior slope of crest No. 9. On the 23rd the crest No. 6 passes London, and we have 
at some stations a much more sudden fall than resulted from the passage of the posterior slope 
of crest No. 9. We may therefore regard the trough in the direction of Cork and Bristol as 
an indication of the approaching trough of crest No. 6, rather than a new trough between 13 
and 15. Should this view be correct, the line of greatest diminution of pressure, Christiania 
to Cork, will be on the posterior slope of crest No. 9. 



158 



REPORT — 1846. 



Paris '29-02 "1 

BrlsToT^ . . . 28-79 '' ^'^^^^ posterior slope, No. 6. 
Cork ..*.".. 28-54. J 
Slopes. — Lines of greatest dinainution of pressure. 

Christiania to Cork. . . . 1'12. Slope, No. 13. 

Paris „ „ 'i8. „ „ 6. 

Currents. — Wind on posterior slope of Crest No. 6, S.W. 

„ „ „ „ ,, IJ, O.Ei. 

East in the north of Scotland and the Orkneys. 

The progressive motions of the two slopes are well-seen from the observa- 
tions of this day, which may be very advantageously compared with those of 
the 11th, when the movements were similar; crest No. 13 has advanced to 
Christiania with the succeeding trough ; crest No. 6 has advanced to or be- 
yond Paris. 

Symmetrical Wave. — London is situated under the posterior slope of crest 
No. 6, not far removed from and to the S.W. of the posterior trough of 
crest No. 13. 

November 25, 184<2. 

Crest No. 13. 
S.E. 



Max. 



Max. 



N.W.- 



s.w. 

Christiania. 

Paris . . 

London 

Plymouth 

Bristol 

Shields 

Cork . . 

Belfast 

Orkneys 

Christiania 

Orkneys 

Belfast 

Shields 

Cork . . 

London . 

Paris ... 

Plymouth 

Bristol . 

Cork . . . 



Crest No. 6. 



-N.E. 



29-62^ 
29-01 

28-88 
28-93 



Most probably all under the posterior 



Qo.QA V. xuusL pruutiuiy an i 

28-82 I ''^^P^ ^^ Crest No. 6. 

28-80 I 

28-82 

29-07J 

29-62"^ 

29-07 I 

28-82 ^Under posterior slope, No. 13. 

28-82 I 

28-80J 

28-88 

29-on 

28-93 
28-84 
28-80 



Under posterior slope. No. 6. 



Slope. — Lines of greatest diminution of pressure. 

Posterior slope, No. 13. Christiania to Cork. , "82 

Posterior slope. No. 6. Paris to Cork "21 

Currents. — Wind on posterior slope of Crest No. 6, S.W. 
„ „ „ „ „ 13, S.E.S. 

„ anterior „ „ „ 15, N.W. 

The barometric state of the atmosphere much the same as yesterday ; the 
greatest ditference occurs at Cork, most probably from the advancing slope 
of the wave, crest No. i5. The wind is closely in accordance with the wave 
slopes. 



ON ATMOSPHERIC WAVES. 



159 



November 26, 1842. 

Crest No. 1 3. 
N.W S.E. 



Crest No. 6. 



S.W.. 



-N.E. 



Crest No. 15. 



N.W.- 



-S.E. 



<2q.-,ij )> Under anterior slope, No. 15. 



Max. Christiania . . 29*57 1 tt j i. • i xr i q 

Orkneys. . . . 29-10 / ^^^^'^ posterior slope, No. 13. 

Min. Shields 28*99 Trough. 

Belfast .-. .. 2904n 

Cork 29-04 | 

Bristol 
London 

Paris 29-17 | 

Plymouth .. 29-2oJ 

Paris 29-17^ 

London 29*17 | 

Bristol .... 29*14 VUnder posterior slope, No. 6. 

Belfast 29*04 | 

Cork 29*04j 

Slopes. — Lines of greatest diminution of pressure. 

Posterior slope. No. 13. Christiania to Shields .. *58 

„ ,, „ 6. Paris to Cork -13 

Anterior „ „ 15- Plymouth to Belfast ... . '16 
Currents. — Wind on posterior slope of Crest No. 6, S.W. 
„ „ „ „ „ 15, N.W. 

The advance of the anterior slope of crest No. 15 is well-seen from the 
observations of this day. The wind proper to it, N.W., has increased in the 
S.W. portion of the area. It appears that the motion of the waves — crests 
Nos. 13 and 15 with the included trough — is slower than that of the waves, 
crests Nos. 3 and 5 (see Nov. 1 1 and 12). The same arrangement of stations 
as to the distribution of pressure which required only one day to establish 
in the case of waves 3 and 5, has occupied ttvo days in the case of waves 13 
and 15. The distribution of pressure was similar on the 11th and 24th ; it 
was also similar on the 12th and 26th. 



Section IIL 
Results of the foregoing Discussion. 

In collecting the results of this discussion, I have arranged in Tables XI. and 
XII. the principal lines of diminution of pressure ; the succession of waves 
as well as the distinct systems become very apparent from these tables. The 
succeeding Tables XIII. and XlV.exhibitthe principal features of the respective 
waves of each system. The most prominent result appears to be the con- 
firmation of Prof. Dove's suggestion of parallel and oppositely directed cur- 
rents. The diagrams of the wind in connection with the barometric obser- 
vations clearly exhibit such currents, and we see by a glance at Tables XIII. 
and XIV. that the beds of these currents varied considerably in breadth. At 
the opening of the observations they were very much broader than at the 
close, and the N.W. system ('waves No. 2, 4, 6) were altogether larger than 
the S.W. We have in fact two systems of waves or currents crossing each 



160 



REPORT — 1 846. 



other at right angles, the individuals in both gradually decreasing in size. In 
the speculation which has been ventured relative to the S.W. system, the 
mass of terrestrial surface forming the N.W. boundary of the great eastern 
continent has been assumed as the rarefying surface, producing the set of 
parallel and oppositely directed S.E. and N.W. winds, the currents gradually 
shifting towards the N.E. The gradual contraction of the beds of each 
system as the observations proceed is a highly interesting feature, which re- 
quires a more extensive discussion for its elucidation. 

Table XL — Exhibiting the principal lines of the greatest diminution of 
pressure of the N.W. system of waves, Nos. 2, 4, and 6. 



Epochs. 


Directions. 


Values. 


Slopes. 


Nov. 1 

2 

4 

5 

6 

7 

9 

10 

11 

16 

17 

18 

18 

20 

21 

22 

24 

25 

26 




•29 

•37 
•69 
•80 
•68 
•54 
•96 
•44 
•34 
•72 
•52 
•35 
•35 
•41 
•38 
•45 
•48 
•21 
■13 


Anterior, No. 2 
2 
2 
2 
2 
2 

Posterior, No. 2 
2 
2 

Anterior, No. 4 
4 

Posterior, No. 4 
4 

Anterior, No. 6 
6 
6 

Posterior, No. 6 
6 
6 






Belfast to Paris 








London to Cork 




Orkne vs to Paris 


Belfast to Paris 




London to Orkneys 


Belfast to Paris 


Cork to Paris 




Paris to Cork 







Table XII. — Exhibiting the principal lines of the greatest diminution of 
pressure of the S.W. system of waves, Nos. 1, 3, 5, 7, 9, 11, 13, 15. 



Epochs. 


Directions. 


Values. 


Slopes. 


Nov. 1 


Belfast to Christiania ... 


•55 


Anterior, No. 1 


3 


Christiania to Paris 


•58 


Posterior, No. 1 


11 


Christiania to Cork 


•57 


3 


12 


Christiania to Shields... 


•13 


3 


12 


Plymouth to Shields ... 


•39 


Anterior, No. 5 


13 


Paris to Christiania 


•59 


5 


14 


Belfast to Christiania ... 


•56 


5 


15 


Orkneys to Cork 


•64 


Posterior, No. 5 


17 


Belfast to Christiania ... 


•57 


Anterior, No. 7 


18 


London to Christiania... 


•42 


7 


21 


Belfast to Christiania ... 


•41 


Anterior, No. 11 


23 


Christiania to Cork 


•52 


Posterior, No. 13 


24 


Christiania to Cork 


112 


13 



ON ATMOSPHERIC WAVES. 



161 



Table XIII. — Exhibiting the principal features of the waves of the N.W. 
system Nos. 2, 4, and 6. 

Wave No. 2. 



Epochs. 


Phases. 


Directions and Localities. 


Altitudes. 


Winds. 


Anterior 
Slope. 


Posterior 
Slope. 


Nov. 1 
2 
3 
4 
5 
6 
7 
8 
9 
9 
10 

11 
11 


Crest. 
Crest. 
Crest. 

Crest. 

Crest. 

Crest. 

Crest. 

Crest. 

Crest. 
Post. Trough. 
Post. Trough. 

Crest. 
Post. Trough. 


N.W. of the United Kingdom 

N.W. of the United Kingdom 

N.W. of the United Kingdom 

N.W. of the United Kingdom 

From Cork to the Orkneys 


30-49+ 

30-55 

30-51 + 

30-43+ 

30-13 

29-90+ 

28-80 

29-24 

29-12 


N.E. 
N.E. 
N.E. 
N.E. 
N.E. 
N.E. 


S.W. 

s.w. 

S.W. 

s.w. 
s.w. 


S.E. of Belfast 


S.E. of the Orkneys 




S.E. of Paris 




Near the eastern coast of Ireland 




Passes Plymouth 




Wave No. 4. 


Nov. 17 
18 
19 


Crest. 
Crest. 
Crest. 


Passes Belfast 


30-51 
30-53 


N.E. E. 
N.E. 


s.w. 




Considerably S.E. of Paris 




Wave No. 6. 


Nov. 21 
22 
23 
24 


Crest. 
Crest. 
Crest. 
Crest. 


N.W. of Belfast and Cork 


29'58 
29-59 


N.E. 


s.w. 
s.w. 


Mear Cork, Belfast and Orkneys 


Near to or S.E. of Paris 





Table XIV. — Exhibiting the principal features of the waves of the S.W. 
system Nos. 1, 3, 5, 7, 9, II, and 13. 

Wave No. 1. 



Epochs. 


Phases. 


Directions and Localities. 


Altitudes. 


Winds. 1 


Anterior 
Slope. 


Posterior 
Slope. 


Nov. 1 
2 
3 

7 


Crest. 

Crest. 

Crest. 

Post. Trough. 




30-33 
30-23+ 
30-31 + 


N.W. 

N.N.W. 
N.W. 


S.E. 
S.E. 
S.E. 


Between Belfast and Clmstiania ... 
W. or s.w. of Christiania 


N.E. of Belfast and Paris 




Wave No. 3. 


Nov. 7 

8 

10 

11 

12 


Ant. Trough. 

Crest. 

Crest. 

Crest. 
Post. Trough. 


N.E. of Belfast and Paris 


29'64 
29-48 


N.W. 
N.W. 


S.E. 

S.E. 


s.w. of Belfast and London 















1846. 



162 



REPORT — 1846, 
Table XIV. (continued). 







Wave No, 5. 








Epochs. 


Phases. 


Directions and Loealitiea. 


Altitudes. 


Winds. 


Anterior 
Slope. 


Posterior 
Slope. 


Nov. 12 
13 
14 
15 
16 


Ant. Trough. 
Crest. 
Crest. 
Crest. 
Crest. 




29-91 
30-01 


N.W. 


S.E. 
S.E. 
E." 


S.W. of Cork, Plymouth and Paris... 




Between Orkneys and Christiania... 


Wa\'e No. 7. 


Nov, 17 Crest, 

18 Crest. 

19 1 Post. Trough. 


Passes Belfast 


30-51 




S.E.E.i- 
S.E. 


Between Cork and the Orkneys 

Near Belfast and Shields 






Wave No. 9. 


Nov. 19 
20 


Ant. Trough, 


Near Belfast and Shields 


29-96 




S.E. 












Wave No. 11. 


Nov. 21 Crest. 

22 Post. Trough. 




29-95 


N.W. 


S.E.E.b 






Wave No. 13. 


Nov. 23 Crest. 

Pnct TrniiO'Vi. 


S.W. of Christiania 


29-62 




Cork to Bristol 




1 






1 



^ Resultants of N.E. and S.E. currents. 

Part III, — Desiderata. 

In addition to briefly reviewing the progress made in this inquiry, it may 
be well to glance at the desiderata that now present themselves to our notice. 
The object of this report, in connection with the preceding ones, has been to 
show that we have observed on some occasions the successive returns of ex- 
tensive barometric undulations, that these undulations have exhibited a certain 
physiognomy, and that we have been able to recognize and characterize them ; 
that when these undulations have been observed at various and distant 
stations, and the observations carefully reduced and compared, they have been 
resolved into separate and distinct waves of pressure, each having an advan- 
cing front, a crest extending in a certain direction, a receding posterior slope, 
and bounded by an anterior and a posterior trough. It is the object of the 
second part of this report particularly to show that these characteristic 
features of a wave are intimately connected with a certain arrangement of 
the aerial currents first suggested by Prof, Dove, consisting of horizontal 
and parallel beds of oppositely directed winds. In his letter to Col, Sabine, 
the Professor speaks of these currents as northerly and southerfy, the mean 
direction being converted into south-westerly in the northern hemisphere by 
the rotation of the earth. The examination of Mr. Brown's data has clearly 
developed a set of parallel and opposite currents at right angles to these, 



ON ATMOSPHERIC WAVES. 163 

namely, from N,W. aud S.E. ; and it has been suggested that these two sets 
of oppositely directed currents, N.E. — S.W. andN.W. — S.E., continually cross- 
ing each other, occasion the complexity of the barometric and anemometric 
phaenomena, and that in future discussions these should be particularly taken 
into account. Should the views thus advanced be substantiated, we are 
beginning to unravel some of the complicated problems of meteorological 
research : still much remains to be done ; we are as yet only on the thresh- 
hold of the vast meteorological arena now opening upon us. The subject, over- 
whelming with interest, naturally divides itself into two branches. First, the 
determination of thephases of the larger undulations, — the barometric curves 
which include complete elevations and depressions of the barometer, and 
which represent, and are exponents of, the effects resulting fromcontemporane- 
ous transits of waves or systems of waves such as have been previously 
noticed. These, with the smaller secondary waves superposed on their slopes, 
form the types of the various seasons of the year. Second, the absolute extent 
of each normal wave of each system in space, as it exists with the smaller 
superposed waves riding on its slopes. The direction of its crest, its am- 
plitude in miles, the altitude of its crest above, and the depression of its 
troughs below the surface of general repose of the atmosphere, the place of 
its formation, the manner in which it is propagated, the precise direction and 
extent of its motion, the force with which it is translated from place to place, 
and the locality of its final extinction, are questions which the present state 
of our knowledge is inadequate to resolve. 

These desiderata regarding the waves as resulting from parallel and op- 
positely directed currents, may be thus expressed. The absolute extent, both 
as regards length and breadth of each current with that of its counter and 
oppositely directed current, together forming the two slopes of the wave 
with its included trough ; the points or lines of intersection of the two systems 
of parallel and oppositely directed currents ; the precise direction of the 
conterminous edges of the currents, the lines in which the velocity of the 
wind is greatest answering to the included trough ; the amount of the dimi- 
nution of pressure resulting from this velocity below the mean pressure of 
the atmosphere ; the locality of the formation of these currents ; the direction 
in which they advance with a latei'al motion ; the force with which they are 
translated by means of such lateral motion from place to place, and the 
locality of their final extinction or disappearance. 

With respect to the first branch of inquiry, the phases of the larger un- 
dulations, the seasonal barometric types, but little has yet been done towards 
its accomplishment. There is some hope, as mentioned in the foregoing re- 
marks, that we have obtained the type o^ fourteen days in November for one 
locality only ; and we have also a glimpse of the character of the movements 
during a portion of October. This is however very small compared with 
the extent of the problem. At the utmost it will only amount to the twelfth 
partof the annual type; even the S^th cannot yet be said to be fully established. 
Again, the station of observation is to be taken into account ; were the 
entire year's observations for one station projected, and year after year such 
observations compared, we should onlj"^ have the annual type at that station. 
The examination of the symmetrical wave of 1842 has already shown that 
there is a line of greatest symmetry as far as that wave is concerned, namely 
Dublin, Birmingham, Brussels and Munich ; and the discussion of the equi- 
noctial and solstitial observations, 1835 to 1838, has clearly established 
Brussels as a nodal point, and we find it situated on this line of greatest 
symmetry. At very short distances N.E. and S.W. of the line of greatest 
symmetry of 1842, the symmetry is departed from. On the return of the 
great wave in the autumn of IS^S, the line of greatest symmetry appears to 

m2 



164 REPORT — 1846. 

have been confined to the southern shores of England : Brussels is not far 
removed from this line ; so that while the symmetry on the last return is 
considerably departed from at Dublin, it is highly probable that at Brussels 
the movements are more in accordance with its nodal character. It is there- 
fore important for the complete determination of the problem, not only to 
obtain the annual type at one station ; we also require it at numerous stations ; 
and we ought to be furnished with local types similar to those, but more ex- 
tensive, which Sir John Herschel has established for different stations from 
the observations of 1835 to 1838. 

It has already been observed that the barometric curves at any one station 
do not give sections of waves passing the station, that is, the curve as pro- 
jected is not a section of the wave then transiting, but exhibits the effects 
of two or more systems of waves passing at the same time. Now as like 
causes produce like effects, it is highly probable that there may be a general 
flowing of the larger normal waves in the same direction, about the same 
season of the year ; and as we have seen in the case of the symmetrical wave 
that the secondary waves are erratic, sometimes falling on one point and 
sometimes on another of the normal waves, these normal waves may be 
crossed at these seasons by similar systems of secondary waves slightly re- 
moved from a normal epoch year after year, giving rise to a similarity, within 
certain limits, between the combined barometric curves as observed at the 
stations. These combined curves furnish us with the local and annual types. 

While this labour is accomplishing, and we are in progress of obtaining 
annual and local types, we may be accumulating information that will bear 
considerably on the second branch of our inquiry. At present we are un- 
able to answer these questions fully. We have obtained some glimpses of 
the vast extent of these waves, and in our contemplation of them we must, 
as Sir John Herschel beautifully observes, enlarge our conception till in the 
extent of their sweep and the majestic regularity of their progress they 
approach in some degree to the tide waves of the ocean ; still our knowledge 
of them is very small. The volume of any one atmospheric wave, the extent 
of surface it covers, indeed any particular feature we may name and which 
we may wish to be exhibited to us in all its details, we must still reckon among 
our desiderata. 

In closing this report, I beg to acknowledge the valuable assistance I have 
received from the following public bodies and gentlemen. 

To THE British Government I am indebted, through the hands of the 
Astronomer Royal and Lieut-Col. Sabine, R.A., for the volumes of Greenwich 
Magnetical and Meteorological Observations for the years 184-0 to 1843, and 
the volume of similar observations made at the Colonial Observatory, Toronto, 
in the years 1840, 1841 and 1842. I am also indebted to both the above- 
named gentlemen for the readiness Avith which they have furnished me with 
extracts from the records of their respective observatories, and for many 
valuable suggestions which I have received, especially from Col. Sabine. 

To the Lords Commissioners of the Admiralty I am indebted for 
several valuable sets of observations made on board our surveying vessels, 
and received through Rear-Admiral Beaufort, our excellent hydrographer. 
To this gentleman I am peculiarly indebted for the lively interest he has 
taken in forwarding the inquirj-, and also to the officers under whose directions 
the observations have been made, for the care and fidelity with which they 
have been executed. The names of the respective officers of Her Majesty's 
surveying vessels will be found in Table I. 

To the Honourable the Corporation of the Trinity House, I am 
indebted for the ready access which has been afforded me to the records of 
meteorological observations kept at certain lighthouses ; and I take this op- 



ON ATMOSPHERIC WAVES. 165 

portunity of testifying to the care and fidelity with which the observations 
are made daily by the lighthouse keepers. The situation of the lighthouses 
at which observations have been made especially to assist in this inquiry, will 
be found in Table I. ; and I am greatly indebted to the Corporation for certain 
modifications in the observations at these lighthouses, which have been made at 
my suggestion in order that the subject should receive the fullest investigation. 

To THE Royal Society I am indebted for several sets of observations 
extracted from records preserved in its archives. In connection with the 
Society, I may mention the kind assistance I received from the late Pro- 
fessor Daniell ; and I take this opportunity of recording the kindness 
and urbanity which he ever manifested, when applied to in reference to this 
or any other scientific inquiry. 

To Sir John F. W. Herschel, Bart., I am under peculiar and 
especial obligation : the kindness I experienced from that gentleman while 
engaged in discussing the quarterly observations, called for and collected by 
himself, demands the most lively gratitude ; and I take this opportunity of ac- 
knowledging this kindness, and particularly the publication, in the report 
drawn up by Sir John, of the remarks which had been suggested in the course 
of my labours, and which I had communicated at intervals. I need scarcely 
mention that this report forms the foundation of all my subsequent labours, 
and that we must ever regard Sir John as the first individual who has given 
an impetus to this inquiry, and who has first trodden the field to which Prof. 
Forbes some years since directed the attention of meteorologists. It has been 
well-said by Col. Sabine, " that Sir John Herschel is the father of all our 
modern researches in meteorology ; to him we owe all our hourly observations, 
and to him we are indebted for those systematic arrangements by which 
meteorology will take its due place among the sciences." The observation 
of the great symmetrical wave in November 18¥2, was an immediate con- 
sequence of the discussion of Sir John's hourly observations. It resulted in 
fact from a continuance (at such intervals as I could command) of the 
observations until a complete rise and fall of the barometer had been observed, 
and projected in a curve on a similar but reduced scale to that used in the 
projections of the quarterly observations. My former reports carry on the 
history ; in them I have mentioned the further assistance I have received from 
Sir John, which I have now great pleasure in acknowledging. 

To Capt. I.arcom, R.E., I am indebted for a valuable series of observa- 
tions, accompanied with curves during October, November, and December 
1845. I have already alluded to these in the body of the report. 

To Professor Phillips and Dr. Stevelly I am indebted for some 
valuable series from the north of England and Ireland. I am also indebted 
to Dr. Lloyd for several extracts from the records of the Dublin observatory : 
also to Sir Thomas Brisbane, Professor Nichol and Silvanus Thomp- 
son, for observations made at their respective observatories. 

To Professor Quetelet I am indebted for a valuable series from the 
observatory at Brussels, and for several series of the quarterly observations 
collected by himself, which may be most advantageously used in such Inquiries 
as the present. 

To E. W. Brayley, Jun., Esq., of the London Institution, I am greatly 
indebted for the valuable assistance which he has on several occasions in con- 
nection with this inquiry most readily afforded me ; especially the great in- 
terest which he manifested at the commencement of the discussion of Sir 
John Herschel's quarterly observations which materially contributed to the 
reductions being entrusted to my hands ; and I take this opportunity of acknow- 
ledging, not only such assistance, but also the direction which that gentleman 
has given to my earlier studies, and the advice he has offered me in prosecuting 



166 REPORT — 1846. 

my inquiries. The interesting conversations on this and kindred subjects that 
I have had with him during the last ten years, have greatly assisted me in my 
labours. I am also indebted to George Gwilt, Esq. and to E.Johnston, 
Esq., for the assistance afibrded by those gentlemen, in my earliest endeavours 
to observe and trace a complete wave. 

To the gentlemen named in the third column of Table I., I am indebted 
for observations at the stations recorded in the first column of that table, 
which have been made with great care, and mostly at the hours named in the 
instructions. 

I cannot close this report without remarking, that many of the observations 
which have thus been collected and partially discussed, owe their existence en- 
tirely to the auspices of this Association ; and should the further discussion 
of them be entrusted to my hands, the same care shall be manifested which 
I have endeavoured to exhibit in my previous labours ; and by examining 
them in every point of view and under every possible aspect, I trust the re- 
sult will be such as fully to accord with the great object of the Association ; 
and should no new facts be elicited, yet it is to be hoped that these observa- 
tions, called for as they have been by the Association, will confirm the sug- 
gestions, and throw considerable light on the labours of several eminent 
meteorologists, so that in these respects subjects at which we have only ob- 
tained a glance, may be brought fully into view, and thus by means of these 
observations the science in some degree advanced. 

Of the grant of £7 placed at my disposal I have expended £3 3s. 3c?. As 
nearly the whole of the observations on the return of the great wave in the 
autumn of 184'5, as well as those during the previous October are at present 
unreduced, I respectfully request a continuance of the grant. 

W. R. BiHT. 
Postscript, April 10, 1847. 

During the period between the sitting of the Association and this report 
passing through the press, I have been furnished, by the liberality of the 
Royal Society, with the magnetical and meteorological observations made 
during the year 1842 at various stations in the Russian empire. These 
stations embrace an area extending over 195 degrees of longitude. The ob- 
servations at St. Petersbui-gh, the nearest station to those given by Mr. Brown, 
in a great majority of cases fully confirm the results arrived at in the pre- 
ceding discussion, and in others the views obtained by means of Mr. Brown's 
observations are corrected, and considerable light thrown on the real character 
of the smaller waves traversing Great Britain and Ireland. In addition to 
these advantages, the Russian observations, in connexion with others, exhibit 
to us the vast area over which the slopes of these waves extend, so vast that 
they actually approach in the extent of their sweep and the regularity of their 
progress to the tide-waves of the ocean. But this is not all ; the records of the 
Russian observatories contain ample materials for carrying out the suggestion 
of Sir John Herschel, expressed in the close of his Report on Meteorological 
Reductions (Report, 184-3, page 98), " that when dealing with undulations of 
such extent, it is by no means a visionary speculation to consider the possi- 
bility of tracing them over the whole of our globe." The area embraced by 
Mr. Brown's and the Russian observations extends over 235 degi-ees of lon- 
gitude ; and it is apparent from the observations themselves, that the greater 
fluctuations are readily traceable. Our Colonial Observatory at Toronto will 
carry on the observations from Sitka, and the stations on the eastern shores 
of America will enable us to trace the waves from the eastern to the western 
shores of the Atlantic, over the vast continents of Europe and Asia. 

The following table contains the altitude of the barometer at St. Peters- 
burgh during the twenty-six days included by Mr. Brown's observations. 



ON ATMOSPHERIC WAVES. 



167 



Whenever the results obtained by means of Mr. Brown's observations are 
either confirmed or illustrated by them, a reference is made to the day on 
which the particular wave, a;s indicated by the observations given in page 141, 
is either identified with one as developed by my previous investigations, or 
more clearly exhibited and its true character more distinctly brought to light. 

Table XV. — Barometric readings at St. Petersburgh, 1842. Nov. 1 to 26, 
at noon, illustrating Table V. 



Date. 


Eng. In. 


Date. 


Eng. In. 


Date. 


Eng. In. 


Nov.l 


29-449 


Nov. 10 


29-890 


Nov. 19 


30155 


2 


29-795 


11 


•659 


20 


29-866 


3 


30-213 


12 


•870 


21 


■776 


4 


•222 


13 


•617 


22 


•419 


5 


•251 


14 


•530 


23 


29-585 


6 


30-164 


15 


•139 


24 


30-032 


7 


29-961 


16 


•450 


25 


•080 


8 


-963 


17 


•682 


26 


■110 


9 


29-916 


18 


•848 







November 5. Crest No. 1. — The observations of Nov. 1 indicated a crest 
which passed across England and Ireland with a general direction N.W. — S.E. 
This crest is now vertically over St. Petersburgh. We have traced it from 
Belfast, past the Orkneys to Christiania, and we now find it at St. Petersburgh. 
The observations of this day, as given by Mr. Brown, clearly indicate the 
direction of crest No. 2, so that the point of intersection of the two crests, 
Nos. 1 and 2, must have been situated towards the north-west of Norway. 
This at once explains the greater amount of pressure in the north-west of 
Europe in the early part of November. 

November 8. Posterior slope, crest No. 2. — This slope was characterized 
by a deep barometric fall, which was very considerable, especially at the north- 
western stations. A very careful comparison of Mr. Brown's observations 
with those made at St. Petersburgh and those given in my last report, Sec- 
tion III. (Report, 184^5, pp. 124 to 128), identifies crest No. 2 with wave A° 
of the last report. The direction of the crest, from a comparison of the num- 
bers over the larger area, including St. Petersburgh, appeared to extend from 
the south-west of England past Norfolk to the east of Christiania, and be- 
tween this station and St. Petersburgh. 

It has been observed in the remarks under the head anterior slope, crest 
No. 2 (page 146), that the altitude of this crest (No. 2) appears to have sub- 
sided as the wave progressed. This subsidence was also observed at Chris- 
tiania and St. Petersburgh. At Christiania the crest passed with about the 
same altitude as it passed Plymouth, 30*27, and at St. Petersburgh it was 
slightly under 30 inches. 

November 9. Posterior slope, crest No. 2. — On this day this posterior 
slope comes into full view. We have already identified crest No. 2 with wave 
A° of my former investigations. The observations of this day give us the 
direction of the posterior slope, which more or less accords with the sections 
of atmospheric pressure at 3 p.m. of this day, as exhibited in plates 45 and 46, 
Report 1844. 

November 10. Crest No. 3. — The direction of this crest is nearly identi- 
cal with that of No. 1, which passed over Great Britain and Ireland on the 
1st. A comparison of Mr. Brown's observations with those at St. Peters- 
burgh and those given in my last report, identifies this wave, crest No. 3, 
with B° (see Report, 1845, page 125). 

Crest No. 2. — This crest is situated to the east of St. Petersburgh on this 
day, at the same time that its posterior trough is situated to the east of the 



168 



REPORT— 1846. 



Orkneys. This will give some idea of the extent of country covered by this 
wave and the vast amount of its amplitude. The direction of the trough is 
indicated on plate 42, Report, ISl'i. 

November 11. Posterior troughs, crests Nos. 2 and 3. — On Nov. 5 we 
distinctly traced the directions of the crests Nos. 1 and 2, and the observa- 
tions at St. Petersburgh assisted us in indicating the locality of their inter- 
section. The observations of this day, Nov. 11, indicate the contemporane- 
ous existence of the posterior troughs of crests Nos. 2 and 3. 

It appears probable, from a consideration of the observations of November 
12, that the depression which occurred at Plymouth (see page 150) was oc- 
casioned by the crossing of the posterior troughs of crests Nos. 2 and 3. If 
so, we are enabled to form a correct estimate of the direction of these contem- 
poraneous troughs. The posterior trough of crest No. 2 now passes St. 
Petersburgh. Table IX. (Report, 184^5, page 125) clearly indicates that the 
depression of this day, in the south of England, was due to the posterior 
trough of crest No. 3. We find Paris under the posterior slope of crest No. 2, 
so that the intersections of the troughs must have been situated in the neigh- 
bourhood of Plymouth, or between Cork and Plymouth : the direction of the 
trough of crest No. 3 appears to have extended from Paris towards Cork 
while the crest was passing Christiania. It appears the velocity of crest No. 
3 was greater than that of crest No. 1, which may, to a certain extent, explain 
the discrepancy noticed in page 150. 

November 12. Crest No. 3. — This crest now passes St. Petersburgh, while 
its posterior trough passes Belfast and Shields. 

November 15. Anterior trough, crest No. 5. — This trough now transits 
St. Petersburgh. The fact of the succeeding crest (No. 5) passing the Ork- 
neys at the same time, clearly indicates the wave to be much smaller than the 
preceding two. 

November 17. Crest No. 7. — It appears from a comparison of the Chris- 
tiania and St. Petersburgh observations that the wave, crest No. 7, was very 
small. 

November 18. Crest No. 4. — This wave, which forms the crown of the 
great symmetrical wave, has been very distinctly developed by the discussion 
of Mr. Brown's observations. The altitude it attained, especially in the 
south-east of England, has contributed to bring it prominently into view. 
The observations at St. Petersburgh make us acquainted with the great ex- 
tent of its longitudinal direction. Its crest passed Dublin on the 17tb, Lon- 
don on the 18th, and Munich on the 19th (see plate 2, appended to Sir John 
Herschel's Report on Meteorological Reductions, Report, 1843). In the fol- 
lowing table the transit of its crest at the three northern stations, the Orkneys, 
Christiania and St. Petersburgh, is readily seen. The direction in which tlie 
wave progressed being N.W. to S.E., the section which passed over St. Peters- 
burgh was more northerly than the others. The maximum occurred at St. 
Petersburgh at least twelve hours later than at Munich, and about two days 
later than at Dublin. 

Table XVI. 



Epoch. 


Orkneys. 


Christiania. 


St. Petersburgh. 


Nov. 17 
18 
19 


3035' 

30-18 

29-91 


29-94 
30-11= 
29-91 


29-68 
29-85 
30-16'^ 



Crest. 



[For Addenda to this Report see end of the Reports."] 



[ 169 1 



Report on the Archetype and Homologies of the Vertebrate Skeleton. 
By Prof. Owen, F.R.S. 

Part I. — Special Homology. 
Introduction. 
When the structure of organized beings began to be investigated, the parts, 
as they were observed, were described under names or phrases suggested 
by their forms, proportions, relative position, or liiieness to some familiar ob- 
ject. Much of the nomenclature of human anatomy has thus arisen, espe- 
cially that of the osseous system, which, with the rest of man's frame, was 
studied originally from an insulated point of view, and irrespective of any 
other animal structure or any common type. 

So when the exigences of the veterinary surgeon, or the desire of the 
naturalist to penetrate beneath the superficial characters of his favourite 
class, led them to anatomise the lower animals, they, in like manner, seldom 
glanced beyond their immediate subject, and often gave arbitrary names 
to the parts which they detected. Thus the dissector of the horse, whose 
attention was more especially called to the leg as the most common seat 
of disease in that animal, specified its 'cannon-bone,' its 'great' and 'small' 
.pastern-bones, its ' cofiin-bone,' and its « nut-bone' or 'coronet': some 
cranial bones were also named agreeably with their shape, as the ' os qua- 
dratum,' for example. The ornithotomist described, in the same irrelative 
manner, the ' ossa homoidea,' ' ossa communicantia' or ' interarticularia,' 
the ' columella ' and ' os furcatorium.' Petit * had his ' os grele ' and ' os 
en massue;' Herissantf his 'os carre'; which, however, is by no means the 
same bone with the 'os carre' or 'os quadratum' of the hippotomist. The 
investigator of reptilian osteology described ' hatchet-bones ' and chevron- 
bones, an ' OS annulare' or ' os en ceinture,' and an 'os transversum': he 
likewise defined a 'columella'; but this was a bone quite distinct from that 
so called in the bird. The ichthyotomist had also an ' os transversum,' which 
again Avas distinct from that in reptiles, and he demonstrated his ' os discoi- 
deum,' ' OS ccenosteon,' ' os mystaceum,' ' ossa symplectica prima,' ' secunda,' 
'tertia,' 'supreraa,' 'postrema,' &c. Similar examples of arbitrary names might 
easily be multiplied ; many distinct ones signifying the same part in different 
animals, whilst essentially distinct parts often received the same name from 
different anatomical authors, occupied exclusively by particular species. 
Each, at the beginning, viewed his subject independently ; and finding, there- 
fore, new organs, created a new nomenclature for them; just as the anthro- 
potomist had done, of necessity, when, with a view to the cure or relief of 
disease and injury, he entered upon the vast domain of anatomical science by 
the structure of Man, or of the mammals most resembling man. 

* Observations Anatomiques sur les mouvemens du bee des Oiseaux, Memoires de I'Acad. 
des Sciences, 1748, p. 345. 

t Mem. de I'Aead. des Sciences, 1774, p. 497. 
18i6. N 



lyO REPORT — 1846. 

It may well be conceived with what a formidable load of names the me- 
mory must have been burthened, if any could have been found equal to it, 
had the anatomy of animals continued and made progress under its primitive 
condition of an assemblage of arbitrarily described and uncompared facts. 

Happily the natural tendency of the human mind to sort and generalize its 
ideas could not long permit such a state of the science, if science it could be 
called, to remain. A large and valuable portion of the labours of the com- 
parative anatomists who have honoured the present century, has been devoted 
to the determination of those bones in the lower animals which correspond 
with bones in the human skeleton ; the results being usually expressed by 
applying to the parts so determined the same names, as far as the nomen- 
clature of anthropotomy allowed. Few, however, of the parts of the human 
body have received single substantive names ; they are for the most part in- 
dicated by shorter or longer descriptive phrases, like the species and parts of 
plants before Linnaeus reformed botanical nomenclature. 

The temptation to devise a systematic Nomenclature of Anatomy, generally 
applicable to all animals, increases with the advance of the science, and from 
the analogy of what has taken place in other sciences it may one day be 
yielded to and exercise the ingenuity of some ardent reformer. But the same 
analogy, especially that afforded by chemical science since the time of Lavoi- 
sier, would rather lead the true friend of anatomy to deprecate the attempt 
to impose an entirely new nomenclature of parts, however closely expressive 
of the nature and results of tlie science at the period when it might be devised. 
For there is no stability in such descriptive or enunciative nomenclature ; it 
changes, and must change with the progress of the science, and thus becomes 
a heavy tax upon such progress. 

If the arbitrary term ' calomel,' which, like ' house' and 'dog,' signifies the 
thing in its totality, without forcing any particular quality of its subject 
prominently upon the mind, be preferable, on that account as well as its 
brevity, to the descriptive phrases 'submuriate of mercury,' 'chloride of 
mercury,' or ' proto-chloride of mercury,' in enunciating propositions respect- 
ing the substance to which it is applied ; and if it possesses the additional ad- 
vantage of fixity, of a steady meaning not liable to be affected, like a descrip- 
tive name or phrase, by every additional knowledge of the properties of the 
substance; the anatomist, zealous for the best interests of his science, will feel 
strongly the desirableness of retaining and securing for the subjects of his 
propositions similar single, arbitrary terms, especially if they are also capable 
of being inflected and used as noun adjectives. 

The practice of anatomists of the soundest judgement has usually been, 
to transfer the anthropotomical term or phrase to the answerable part when 
detected in other animals. The objection that the original descriptive or 
otherwise allusive meaning of the term seldom applies to the part with equal 
force in other animals, and sometimes not at all, is one of really little moment; 
for the term borrowed from anthropotomy is soon understood in an arbitrary 
sense, and without regard to its applicability to the modified form which 
the namesake of the human bone commonly assumes to suit the ends required 
in the lower species. No anatomist, for example, troubles himself with the 
question of the amount of resemblance to a crow's or other bird's beak in the 
'coracoid' bone of a reptile, or with the want of likeness of the kangaroo's 
'coccyx' to the beak of a cuckoo; or of the whale's 'vomer' to a plough- 
share ; or ever associates the idea of the original mystic allusion in the ana- 
tomical term ' sacrum ' with his description of that bone in the megatherium 
or other monster. Common sense gratefully accepts such names when they 
become as arbitrary as cat or calomel, and when such concretes or adjectives 
as 'coccygeal,' 'vomerine' and 'sacral' can be employed to teach the pro- 
perties or accidents of their subjects. 



ox THE VERTEORATE SKELETON. I7l 

To substitute names for phrases is not only allowable, but I believe it to be 
indispensable to the right progress of anatomy ; but such names must be arbi- 
trary, or, at least, should have no other signification than the homological one, 
if anatomy, as the science of the structure of all animals, is to enjoy the inesti- 
mable benefit of a steady and universal nomenclature. I am far from being in- 
sensible to the advantages which other sciences have derived from revolutions 
in their technical language ; but experience has also demonstrated attendant 
evils ; and these, it is to be feared, would preponderate in the case of anatomy, 
on account of the peculiar character of its origin, and the fact of its cultivators 
being for the most part introduced to the science through the portal of anthro- 
potomy. So long, likewise, as due deference continues to be paid to the deep 
and vital importance of the practical applications of the parent science in 
medicine and surgery, it will be in vain for any man to expect that his sole 
authority would suffice for the general reception of an entirely new nomen- 
clature, however philosophically devised or clearly enunciative of the highest 
and most comprehensive truths of the science at the time of its formation. 

After maturely considering this subject in its various relations, I have ar^ 
rived at the conviction that the best interests of anatomical science will be 
consulted by basing the nomenclature applicable to the vertebrate subking- 
dom upon the terms and phrases in which the great anthropotomists of the 
16th, 17th and 18th centuries have communicated to us the fruits of their 
immortal labours. For it is only on this firm foundation that we may hope 
to avoid that ceaseless change of terms which follows the device of a syste- 
matic nomenclature significant of a given progress and result of scientific 
research. But the names of the parts of the vertebrate animals so based on 
or deduced from the language of anthropotomy must divest themselves of 
their original descriptive signification, and must stand simply and arbitra- 
rily as the signs of such parts, or at least with the sole additional meaning 
of indicating the relation of the part in the lower animal to its namesake or 
homologue in Man. It is an old maxim accepted by the best logicians, that 
no name is so good as that which signifies the total idea or whole subject, 
without calling prominently to mind any one particular quality, which is 
thereby apt to be deemed, undeservedly, more essential than the rest. 

The chief improvement which the language of anatomy, based upon that 
of anthropotomy, must receive in order to do its requisite duty, is the substi- 
tution of ' names ' for ' phrases ' and ' definitions ' ; and this is less a change 
of nomenclature than the giving to anatomy what it did not before possess, 
but which is absolutely requisite to express briefly and clearly, and without 
periphrasis, propositions respecting the parts of animal bodies. Such names 
should be derived from a universal or dead language, and when anglicized, 
or translated into other modern equivalents, ought to be capable of being 
inflected adjectively. 

A few examples will suffice to show how greatly the advantage of such 
names preponderates over the trouble of substituting them in the memory 
for the definitions which previously signified the ideas. 

In the classical Anthropotomy of Soemmerring, a well-defined part of the 
skull, which is a distinct bone in the human embryo, and permanently so in 
all cold-blooded Vertebrata, is called " pars occipitalis stricte sic dicta partis 
occipitalis ossis spheno-occipitalis*." Monro, in his justly-esteemed treatise 
* On the Human Bonesf,' defines the same bone as " all the part of the (oc- 
cipital) bone above the great foramen." In the ' Elements of Anatomy,' by 
Dr. Quaini, a work of repute for its clearness and minuteness of detail, the 

* De Corporis Humani Fabrica, 1794, t. i. p. 162. f Kirby's edition, 8vo, 1820, p. 76. 
X Elements of Descriptive and Practical Anatomy, 8vo, 1828, p. 50. 

n2 



172 REPORT — 1846. 

part in question is neither named nor described. The term supra-occipitale, 
luat. (siq)ra-occipital,Eng.,stir-occipital, Fr.), is obviously a gain to anatomical 
science in all propositions respecting this part in the vertebrate series. 

Certain parts of a vertebra, distinct bones at an early period in man, and 
throughout life in most reptiles, are defined by Soemmerring as 'radices ar- 
cus posterioris vertebrae,' or ' arcus posterior vertebrae ' collectively *. Monro 
describes the same parts separately, as " a broad oblique bony plate extended 
backwards," and together, as " a bony arch produced backwards" : he names, 
defines and minutely describes the processes, &c. of these bony plates, which 
in the series of Vertebrata are soon found to be non-essential characters ; but 
for the plates themselves, which are the most constant and essential consti- 
tuents of a vertebra, he has no name. Dr. Quain defines the same parts as " two 
plates of bone, the lamellae or arches, which complete the central foramen f." 
They are sometimes more briefly but vaguely spoken of in English works 
of Comparative Anatomy as " the vertebral lamellae " or " vertebral laminae," 
or " perivertebral elements." The term ' neurapophysis,' Lat. and Eng. (' neur- 
apophyse', Fr.), applicable to each element individually, under which all its 
properties may be predicated of by the adjective ' neurapophysial,' without 
periphrasis, seems by its adoption in the classical works of MM. Agassiz 
and Stannius, to be as acceptable as the term 'sur-occipital' substituted by 
Cuvier for the definitions in anthropotomy above cited. 

Similar instances of the absence of determinate names, capable of in- 
flection, for parts of the human frame, will be seen in the last column of 
Table I., and others will occur to the anatomist, even in regard to most 
important parts, as the primary natural divisions of the neural axis, for 
example, to the great hindrance of brief, clear and intelligible descriptions. 
So long as the phrases 'marrow of the spine,' 'chord of the spine,' continue 
to usurp the place of a proper name, all propositions concerning their sub- 
ject must continue to be periphrastic, and often also dubious. Thus if the 
pathologist, speaking of diseases of the spinal marrow, desires to abbreviate 
his proposition by speaking of ' spinal disease,' he is liable to be misunder- 
stood as referring to disease of the spinal or vertebral column. The vague, 
but often-used phrase 'chorda dorsalis' for the embryonic fibro-gelatinous 
basis of the spine, adds another source of confusion likely to arise from the 
use of the term ' spinal chord,' as applied to that most important part of the 
neural axis which I have proposed to call 'Myelou:];,' a term which, if adopted, 
would be attended by this advantage, that no ambiguity could arise in speak- 
ing of ' myelonal functions,' ' myelonal affections,' or other properties of this 
part of the central axis of the nervous system. 

Anthropotomy, in respect to its nomenclature, or rather the want of one, 
is, as I have already remarked, not unlike v\ hat botany was before the time of 
Linnaeus, and we may anticipate the happiest efi'ects from a judiciously re- 
formed technical language in the advances ent of the true and philosophic 
knowledge of the human structure, from the rapid progress of botany when 
the opposition raised by sloth or envy to the Linnaean reforms was overcome. 
For a good general anatomical nomenclature, based and regulated upon the 
principles above defined, must reflect its benefits upon anthropotomy. 1 dare 
not flatter myself that the names adopted or proposed for the Osseous System 
of the Vertebrata in my'HunterianLectures'and in the first column of Tablel. 
will meet at once with acceptance, but the attempt to establish such a nomen- 
clature will be felt to have been an indispensable step in undertaking a general 
survey of the homological relations of the vertebrate skeleton. 

* De Corporis Humaiii Fabrica, 1794, t. i. pp. 235, 236. 

t Elements of Descriptive and Practical Anatomy, 8vo, 1828, p. 121. 

X Hunterian Lectures, vol. ii. ' Vertebrata,' part i.p. 172. 



ox THE VERTEBRATE SKELETON. 173 

In proposing a definite name for each distinct bone, declaratory of its 
special homology throughout the vertebrate kingdom, 1 have sought earnestly 
to reduce the amount of reform to the minimum allowed by the exigences 
of the case. Agreeably with Aphorism III. of the ' Philosophy of the In- 
ductive Sciences' (p. Ixvii.), the nomenclature of anthropotomy forms the 
basis, and all the names given to parts by one or other of the great French 
anatomists have been accepted, with the modifications of a Latin or an En. 
glish termination, wherever such names had not been applied, as is the case 
with some proposed by Geoifroy St. Hilaire, to two different parts. In sub- 
stituting names for phrases, I have endeavoured, conformably with another 
of Dr. Whewell's canons (Aph. XVII. op. cit. p. cxvii.), to approximate the 
sound of the name as nearly as possible to those of the leading terms of the 
definition or phrase, as e. g. alisphenoid for ' ala media, &c. sphenoidalis ' and 
for ' grande aile du sphenoide ' ; orbitosphenoid for ' ala superior seu orbi- 
talis, &c. sphenoidalis,' and for 'aile orbitaire du sphenoide*.' 

The corresponding parts in different animals being thus made namesakes, 
are called technically ' homologues.' The term is used by logicians as syno- 
nymous with ' homonyms,' and by geometricians as signifying ' the sides of 
similar figures which are opposite to equal and corresponding angles,' or to 
parts having the same proportions t : it appears to have been first applied in 
anatomy by the philosophical cultivators of that science in Germany. Geof- 
froy St. Hilaire says, " Les organes des sens sont homologues, comme s'ex- 
primerait la philosophic AUemande ; c'est-a-dire qu'ils sont analogues dans 
leur mode de developpement, s'il existe veritablement en eux un meme prin- 
cipe de formation, une tendance uniforme a se r6peter, a se reproduire de la 
meme fa9ont." The French anatomist, however, seems not rightly to 
define the sense in which the German philosophers have used the term : 
there is a looseness in the expression ' analogous in their mode of develop- 
ment,' which may mean either identical or similar, and also different kinds of 
similarity. Parts are homologous in the sense in which the term is used in 
this Report, which are not always similarly developed: thus the 'pars occi- 
pitalis stricte sic dicta,' &c. of Soemmerring is the special homologue of the 
supraoccipital bone of the cod, although it is developed out of pre-existing 
cartilage in the fish and out of aponeurotic membrane in the human subject. 
I also regard the supraoccipital as the serial homologue of the parietal and 
the midfrontal, although these are developed out of the epicranial membrane 
in the fish, and not out of pre-existing cartilage, like the supraoccipital. 
The femur of the cow is not the less homologous with the femur of the cro- 
codile, because in the one it is developed from four separate ossific centres, and 
the other from only one such centre. In like manner the compound mandi- 
bular ramus of the fish is the homologue of the simple mandibular ramus of 
the mammal, as the compound tympanic pedicle of the fish is homologous 
with the simple tympanic pedicle of the bird, the differences expressed by 
the terms ' simple ' and ' compound ' depending entirely on a difference of 
development. 

Without knowing the precise sense in which Geoffroy St. Hilaire under- 

* The happy facility of combination which the German language enjoys has long enabled 
the very eminent anatomists of that intellectual part of Europe to condense the definitions of 
anthropotomy into single words ; but these cannot become cosmopolitan ; such terms as 
' Hinterhauptbeinkorper,' ' Schlafbeinschiippen.'and 'Zwischenkiemendeckelstiick/are likely 
to be restricted to the anatomists of the country where the vocal powers have been trained 
from infancy to their utterance. 

t This is' the sense in which the term is defined in the French Dictionary and in our 
Johnson's Dictionary. 

X Annales des Sciences Naturelles, torn. vi. 1825, p. 341. 



174 REPORT— 1846. 

stood ' analogous development,' one cannot determine how much or how little 
it is applicable to the determination of homologies or to the definition of 
homologous parts. Dr. Reichert seems to have been unduly influenced by the 
idea of ' analogy or similarity of development in the determination of homo- 
logous parts ' when he rejected the parietal and frontal bones from the system 
of the endo-skeleton, because they were not developed from a pre-existing 
cartilaginous basis*, or, because they could be easily detached from subja- 
cent persistent cartilage in certain fishes ; the essential distinction between 
these and the supra- occipital in regard to development being, that whereas 
the cartilaginous stage intervened in the latter between the membranous and 
the osseous stages, in the other, usually more expanded, cranial spines, the 
osseous change appears to be immediately superinduced upon the primitive 
aponeurotic histological 'condition. 

M. Agassiz seems, in like manner, to give undue importance to similarity 
of development in the determination of homologies, where he repudiates the 
general homology of the basi-sphenoid with the vertebral centrum, and con- 
sequently its serial homology with the basi-occipital, because the pointed end 
of the chorda dorsalis has not been traced further forwards along the basis 
of the cranium in the embryo osseous fish than the basi-occipital f. But the 
development of the centrum of every vertebra begins, not in the gelatinous 
chord, but in its aponeurotic capsule, and it is in the expanded aponeurosis 
directly continued from the 'chorda' along the 'basis cranii' that the thin 
stratum of cartilage cells is formed from which the ossification of the basi- 
sphenoid, presphenoid and vomer proceeds. 

There exists doubtless a close general resemblance in the mode of deve- 
lopment of homologous parts ; but this is subject to modification, like the 
forms, proportions, functions and very substance of such parts, without their 
essential homological relationships being thereby obliterated. These rela- 
tionships are mainly, if not wholly, determined by the relative position and 
connection of the parts, and may exist independently of form, proportion, 
substance, function and similarity of development. But the connections 
must be sought for at every period of development, and the changes of rela- 
tive position, if any, during growth, must be compared with the connections 
which the part presents in the classes where vegetative repetition is greatest 
and adaptive modification least. 

Relations of homology are often not only confounded with those of analogy, 
but in some recent and highly estimable works on comparative anatomy the 
terms ' analogy ' and ' analogue ' continue to be used to express the ideas of 
homology and homologue, or are so used as to leave in doubt the meaning of 
tile author. Thus when we read in the latest edition of the ' Le9ons d' Ana- 
tomic Comparee ' of Cuvier, " Les branchies sont les poumons des animaux 
absolument aquatiques," t. vii. p. 164; and with regard to the cartilaginous 
or osseous supports of the gills, " elles sont, a notre avis, aux branchies des 
poissons, ce que les cerceaux cartilagineux ou osseux des voies aeriennes sont 
aux poumons des trois classes superieures," Ibid. p. 177, we are left in doubt 
whether it is meant that the gills and their mechanical supports merely perform 
the same function in fishes which the lungs and windpipe do in mammals, or 
whether they are not also actually the same parts differently piodified in re- 
lation to the diflFerent respiratory media in the two classes of animals. The 
deeper-thinking GeoflProy leaves no doubt as to his meaning where he argues 

* Vergleichende Entwickelungsgeschichte des Kopfes der nackten Reptilien, 4to, 1838, 
pp.212, 218. 

t Recherches siir les Poissons Fossiles, 4to, 1843, i. p. 127. 



ON THE VERTEBRATE SKELETON. l75 

in the ' Philosophie Anatomique' (8vo, 1818, 4ieme memoire, p. 205), that 
the branchial arches of fishes are the modified tracheal rings of the air- 
breathing vertebi-ates : we perceive at once that he is enunciating a relation 
of homology. 

. I have elsewhere* discussed the relations, both homological and analogical, 
of the respiratory organs of the air-breathing and water-breathing vertebrate 
animals, and have here adverted to them merely to illustrate the essential 
distinction of those relations. In the ' Glossary ' appended to the first volume 
of my ' Hunterian Lectures,' the terms in question are defined as follows : — 

" Analogue." — A part or organ in one animal which has the same func- 
tion as another part or organ in a different animal. 

" HoMOLOGUE." — The same organ in different animals under every variety 
of form and function-f-." 

The little ' Draco volans ' offers a good illustration of both relations. Its 
fore-limbs being composed of essentially the same parts as the wings of a bird 
are homologous with them ; but the parachute being composed of different 
parts, yet performing the same function as the wings of a bird, is analogous 
to them. Homologous parts are always, indeed, analogous parts in one sense, 
inasmuch as, being repetitions of the same parts of the body, they bear in 
that respect the same relation to different animals. But homologous parts 
may be, and often are, also analogous parts in a fuller sense, viz. as perform- 
ing the same functions : thus the fin or pectoral limb of a Porpoise is homo- 
logous with that of a Fish, inasmuch as it is composed of the same or answerable 
parts; and they are the analogues of each other, inasmuch as they have the 
same relation of subserviency to swimming. So, likewise, the pectoral fin of 
the flying fish is analogous to the wing of the Bird, but, unlike the wing of 
the Dragon, it is also homologous with it. 

Relations of homology are of three kinds : the first is that above defined, 
viz. the correspondency of a part or organ, determined by its relative position 
and connections, with a part or organ in a different animal ; the determination 
of which homology indicates that such animals are constructed on a common 
<ype: when, for example, the correspondence of the basilar process of the 
human occipital bone with the distinct bone called ' basi-occipital ' in a fish 
or crocodile is shown, the special homology of that process is determined. 

A higher relation of homology is that in which a part or series of parts 
stands to the fundamental or general type, and its enunciation involves 
and implies a knowledge of the type on which a natural group of animals, 
the vertebrate for example, is constructed. Thus when the basilar process of 
the human occipital bone is determined to be the • centrum ' or ' body of the 
last cranial vertebra,' its general homology is enunciated. 

If it be admitted that the general type of the vertebrate endo-skeleton is 
rightly represented by the idea of a series of essentially similar segments 
succeeding each other longitudinally from one end of the body to the other, 
such segments being for the most part composed of pieces similar in number 
and arrangement, and though sometimes extremely modified for special func- 
tions, yet never so as to wholly mask their typical character, — then any 
given part of one segment may be repeated in the rest of the series, just as 
one bone may be reproduced in the skeletons of different species, and this 
kind of repetition or representative relation in the segments of the same 
skeleton I call ' serial homology.' As, however, the parts can be namesakes 
only in a general sense, as centrums, neurapophyses, ribs, &c.; and since 

* Lectures on Vertebrata, 1846, p. 279. 

t Lectures on Invertebrate Animals, 8vo, 1843. Glossary, pp. 374, 379. My ingenious 
and learned friend Mr. Hugh Strickland has made a strong and able appeal to the good 
sense of comparative anatomists in favour of the restriction of these terms to the senses in 
which they are here defined. — Phil. Mag. 1846, pp. 358, 362. 



I7fi REPORT — 1846. 

they must be distinguished by different special names according to their par- 
ticular modifications in the same skeleton, as e. g. mandible, coracoid, ilium, 
&c., I call such serially related or repeated parts ' homotypes.' The basi- 
occipital is the homotype of the basi-sphenoid ; or in other words, when the 
basi-occipital is said to repeat in its vertebra or natural segment of the ske- 
leton the basi-sphenoid or body of the parietal vertebra, or the bodies of the 
atlas and succeeding vertebrae, its serial homology is indicated. The study 
of this kind of homologies was commenced by Vicq d'Azyr, in his ingenious 
memoir ' On the Parallelism of the Fore and Hind Limbs.' If we except 
the complex and extremely diversified and modified parts of the radiated 
appendages of the vertebral segments, to which Vicq d'Azyr restricted his 
comparisons, the serial homologies of the skeleton are necessarily demon- 
^trated when the general and special homologies have been determined. 

In the present section of this Report I propose to consider some of those 
examples of special homology which are least satisfactorily determined and 
respecting which different opinions still sway different anatomists. Such 
instances are fortunately few, thanks to the persevering and successful labours 
of the great comparative anatomists of the last half-century : pre-eminent 
amongst whom will ever stand the name of Cuvier, in whose classical works, 
' Ossemens Fossiles,' 'Histoire des Poissons,' ' Le(;:ons d' Anatomic Comparee ' 
(posthumous edition), and ' Regno Animal,' 1828, will be found the richest 
illustrations of the special homological relations of the bones in the four classes 
of vertebrate animals. 

Second only to Cuvier must be named Geoffroy St. Hilaire, whose 
memoir on theBonesof the Skull in Birds as compared with those in Mammals, 
in the ' Annales du Museum,t. x. (1807)> forms an early and brilliant example 
of the quest of special homologies, which could not fail, with other and similar 
investigations of the same ingenious author, to impart a stimulus to that 
philosophical department of anatomical inquiry*. In regard to the osteology 
of the crocodile, we find Cuvier and Geoffroy engaged in a long parallel series 
of rival researches, the results of which have had the happiest effects in de- 
termining some of the most difficult questions of special homology. 

Nor was the co-operation of zealous cultivators of comparative anatomy 
wanting in the eminent schools and universities of Germany. Goethe, in- 
deed, had taken the lead in inquiries of this nature in his determination, in 1787, 
of the special homology of that anterior part of the human upper maxillary 
bone which is separated by a more or less extensive suture from the rest of 
the bone in the f^oetus ; and the philosophical principles propounded in the 
great poets famous anatomical essays called forth the valuable labours of the 
kindred spirits, Oken, Bojanus, Meckel, Carus, and other eminent culti- 
vators of anatomical philosophy in Germany. 

It is not requisite for the purpose I have in view, to trace step bj' step the 
progress of the special homological department of anatomy. Its present 
state, as regards the skull of the Vertebrata, will be best exposed by the sub- 
joined tabular view of the fruits of the latest inquiries. 

Table I. (See end of the Report.) 
This table gives at one view the general results of the researches into 
the conformity of structure of the skull throughout the vertebrate series, 

* Oken's famous "Programm, Uber die Bedeutung der Schadelknochen" was published 
in the same year (1807) as Geoffrey's Memoir on the Bird's skull; but it is devoted less to 
the determination of ' special ' than of ' general homologies ' : it has, in fact, a much higher 
aim than the contemporary publication of the French anatomist, in which we seek in vain 
for any glimpse of those higher relations of the l)ones of the skull, the discovery of which 
has conferred immortality on the name of Okejt. 



ON THE VERTEBRATE SKELETON. 177 

by the two great French anatomists who have most advanced this part of 
osteological science ; by the authors of two classical German works on 
Comparative Anatomy ; and by their countryman Dr. Kallmann, who has 
detailed in an elaborate treatise his especial investigations of some of the most 
difficult parts of this difficult inquiry. I have added the synonyms of the 
bones of the bead of fishes from the great work of the celebrated Swiss na- 
turalist, who has, so happily for ichthyology, devoted himself to the advance- 
ment of that interesting branch of Natural History ; and also, the antiiropo- 
tomical terms for the corresponding parts in the human skeleton. These, 
after much comparison and deliberation, 1 have chosen from the justly-cele- 
brated work of SoEMMERRiNG, the high reputation of which has been sanc- 
tioned by the new edition to which some of the most eminent of the German 
professors of anthropotomy and physiology have recently devoted their com- 
bined labours. The English teacher of these sciences will find some of the 
descriptive designations of the parts by Soemmerring not agreeing with 
those which he may be in the habit of using, and which are current in the 
later Manuals of Anthropotomy published in this country : the ' ossa la- 
teralia lingualia' are more commonly called, with us, the ' cornua majora 
Oisis hyoidei' ; the 'os spheno-occipitale' is generally described as two di- 
stinct bones, the ' os occipitis' and ' os sphenoide'; the 'pars occipitalis 
stricte sic dicta,' &c. is sometimes called ' squama occipitalis,' or occipital 
plate ; and other synonyms might easily be multiplied from the osteolo- 
gical treatises of Monro and later authors of repute. The fact of such a 
conflicting and unsettled synonymy still pervading the monographs relating 
to the human structure, should stimulate the well-wisher to the right progress 
of anatomy to lend an earnest aid to the establishment of a fixed and deter- 
minate nomenclature. A little present labour and the example of adoption, 
where the reasonableness and necessity of the reform are plain and undeni- 
able, will much accelerate the future progress of anatomical science ; and I 
would respectfully appeal to the Professors and Demonstrators of Human 
Anatomy for an unbiassed consideration of the advantages of the terms pro- 
posed in the first column in Table I. It is designed to express the results of 
a long series of investigations into the special homologies of the bones of the 
head, in simple and definite terms, capable of every requisite inflection to 
express the properties of the parts, and applicable to the same bones from 
the highest to the lowest of the vertebrate series. 

The degree and extent of the diversity of my determinations from those 
of other anatomists are shown in the succeeding columns, headed by their 
names ; and I proceed now to give the reasons which have compelled me, in 
such instances, to dissent from the high authority of Cuvier, Geoff'roy, Meckel, 
Hallmann and Agassiz : these reasons will exonerate me, I trust, from the 
reproach of underrating their justly-esteemed opinions, which have been 
abandoned only where nature seemed clearly to refuse her sanction to them. 
The instances of such dissent are much fewer than they appear to be at first 
sight. In most cases, where the names diflPer, the determinations are the 
same. For ' basilaire,' which Cuvier exclusively applies to the ' pars basilaris' 
of the occiput, and which Geoffrey as exclusively applies (in birds) to the 
' pars basilaris' of the sphenoid, I have substituted the term ' basioccipital' 
(hasi-occipitale, Lat.) ; a term which, as it is more descriptive of the bone in 
question (i figs. 1 to 25), will, perhaps, be the more acceptable to those 
who prefer a determinate to a variable nomenclature, since Cuvier himself 
has almost as frequently applied to that bone the term 'occipital inferieur' 
as the term ' basilaire.' For the descriptive phrase ' occipital lateral,' the 
term 'exoccipital' (exoccipifale, Lat.), proposed by Geotfroy, is preferable for 



178 



REPORT — 1846. 




the bones 2,2, figs. 1 to 25; especially since the paroccipital is the ntiost ' lateral' 
of the elements of the occipital bone, in the definite sense in which the term 
'lateral' is used in the precise and excellent anatomical nomenclature of 
Dr. Barclay. For the numerous syno- 
nyms borne by the element :» of the oc- 
cipital segment of the skull, the term 
' supraoccipital ' (sitpra-occipifale, Lat.) 
seemed to best agree with the truest de- 
scriptive phrase of the part, viz. ' occipital 
superieur.' The interparietal is no con- 
stant cranial element, nor is it a dismem- 
berment of one and the same bone of the 
skull. It is at best only the largest and 
most common of the accidentally interca- 
lated ' ossa wormiana.' Sometimes, for 
example, in the Cehus monkey, it is a 
dismemberment of the backwardly-pro- 
duced frontal bone : more frequently it is 
the detached upper angle of the supra- 
occipital. But by this term ' supraoccipi- 
tal,' 1 signify the totality of the bone 3 (in 
figs. 1, 5, 18, 22, 23, 24, 25), confining 
the term interparietal to its superior and i''?'""''<'"^at<5<i<^pencepiiiJ'eorneur-occipitaiarch, 

. • 1 1 1 u I i iL viewed from behind : Cod (Morrhua mtlgaria). 

anterior apex when detached, or to the * 

superior and posterior apex of the frontal, when it is in like manner detached 
and wedged between the parietal bones. The inapplicability of the term ' in- 
terparietal' to the whole of the supraoccipital is strongly manifested in those 
fishes, e.g. the carp and tench, in which the supraoccipital is withdrawn from 
between the parietals to the back part of the skull, leaving those bones to come 
into contact and unite by the normal sagittal suture on the mesial line of 
the vertex. GeofFroy's error is of the same kind, and scarcely greater than 
Cuvier's, where he applies the term ' interparietal' to the whole of the parietal 
bones in Birds*. The supraoccipital thus defined can never be mistaken for 
the 'sur-occipital' of Geoffroy, who. by this term signifies the elements called 
' occipitaux externes' by Cuvier. At the same time the term ' sur- occipital' is 
too near in sound to ' supraoccipital,' and too significant of the highest part of 
the occipital segment to be retained for elements, which, like the ' paroccipi- 
tals'(fig. 1,4,4), are usually inferior in position to the supraoccipital. Geoffroy; 
moreover, is not consistent in his application of the term ' sur-occipital.' In 
his memoir on the skull of the crocodile in the ' Annales des Sciences' for 
1824, he applies that term to a part of the bonef, the whole of which he calls 
'exoccipital ' in his later memoir, on the skull of the crocodile, of 1833 J ; 
whilst in the memoir illustrated by the skull of the Sea-perch {Serranus 
ffigas) in the ' Annales des Sciences' for 1825, the term ' suroccipital' is ap- 
plied to the whole of the bones described as ' occipitaux externes' by Cuvier. 
I trust, therefore, to have shown the necessity for the definite name of 
* paroccipital' (paroccipitale, Lat.) which is here proposed for the elements, 4, 
of the occipital segment of the cranium (figs. 1 and 5). The name has re- 
ference to the general homology of the bones in question, as ' parapophyses' 
or transverse processes of the occipital vertebra. And if the purists who are 
distressed by such harmless hybrids as 'mineralogy,' 'terminology' and 'mam- 

* Annales du Museum, x. p. 363, pi. 27. 

t PI. 16. fig. 5z+R. " Plur-occipital forme du sur-occipital et de I'ex-occipital." 

X Meraoires de I'Acad. Royale des Sciences, t. xii. Atlas, p. 43. 



ON THE VERTEBRATE SKELETON. 



179 




Disarticulated mesencephalic or neuro-parietal arch, viewed 
from behind : Cod-fish. 



malogy,' should protest against the combination of the Greek prefix to the 
Latin noun, I can only plead that servility to a particular source of the fluc- 
tuating sounds of vocal language is a matter of taste ; and that it seems no 
unreasonable privilege to use such elements as the servants of thought ; 
and, in the interests of science, to combine them, even though they come from 
different countries, where the required duty is best and most expeditiously 
performed by such association. 

For the same motive that suggested the term basi-occipital, viz. because 
the anthropotomist has been 

long accustomed to hear 'S* -" 

that and the corresponding 
element of the s])henoid 
bone described as ' basilar 
processes,' I propose to sub- 
stitute the term ' basisphe- 
noid' (basisphenoideum,\j&t.) 
for the three different de- 
scriptive phrases applied to 
the part (5, figs. 2, 5, 19,&c.) 
by Cuvier, for the two ad- 
ditional synonyms of Geof- 
froy, and for the ' sphenoi- 
deumbasilare' of Kallmann. 
' Alisphenoid ' (alisphenoi- 
deum, Lat., e, 6, figs. 2.5, 19, 
&c.) seemed to retain most of 
the old anthropotoraical term 
of ' alse majores,' or wings ' par excellence' of the os sphenoideum ; as ' orbito- 
sphenoid' (orbito- sphenoideum, 10, 10, figs. 3 and 20) best recalls or expresses 
the idea conveyed by the descriptive phrase ' alae orbitales,' or 'ailesorbi- 
taires,' often applied to the homologous bones, regarded as processes of the 
sphenoid in human anatomy. Here, however, in reference to the alisphenoid, 
we find the first marked discrepancy in the conclusions of the anatomists 
who have particularly studied its special homologies. The bone which ap- 
pears as the 'grande aile du sphenoide' to Cuvier and Agassiz in fishes, is 
the ' petrosura' to Hallmann and Wagner ; it is also ' rocher' (petrosal) to 
Cuvier himself in reptiles, and is again ' grande aile du sphenoide' in birds 
and mammals. The reasons which have led me to the conclusion that the 
bones so denominated, as well as the ' ptereal' and ' prerupeal' of Geoffroy, 
are homologously one and the same, are so intimately linked with the con- 
sideration of the true petrosal and of other elements of the anthropotomist's 
' temporal bone,' that I reserve the discussion of these questions until I have 
completed the apology for the names proposed in the first column of Table I. 
The 'parietal' (j9amto/e,Lat.,7,7, figs.2,5, 19, &c.)and ' mastoid' (masfoi- 
deum, Lat., s, s, figs. 2, 5, 19, &c.) are amongst the few bones that have had 
the good fortune to receive, originally, definite names, applicable to then? 
throughout the vertebrate series ; although the mastoid, being like the par- 
occipital, essentially a parapophysis, loses its individuality sooner than do 
other bones of its segment, and becomes, therefore, a ' processus mastoideus 
ossis temporis,' in the language of anthropotomy. The homology of the 
* parietal' has fortunately been, with a single exception, universally recog- 
nised throughout the vertebrate subkingdom ; the exception being furnished 
by the eccentric homologist Geoffroy, who is, as usual, inconsistent. with 
himself, even on this plainest and least mistakeabie point. 



180 



REPORT — 1846. 



Theterm 'presphenoid'(prespke?ioid€um,'La.t.o,^gs.3,5,^0, 2i,25,&c.) ispro- 
posed forthe'sphenoide anterieur,'on the principleof substituting, as thebetter 
instrument of thought, a definite name for a descriptive phrase. For the same 
reason ' postfrontal' (postfron- 
tale, Lat., 12, 12, figs. 3,5,20,&c.) 
is substituted for Cuvier's ' fron- 
tal posterieur' and its sj'nonyms. 
The 'frontal' {frontale, Lat. u, 
figs. 3, 5, 20, &c.) and ' vomer' 
(vomer, l^ai., i3,figs.4, 5, 20, 25), 
are among the few bones which 
have had their special homolo- 
gies recognised unanimously 
throughout the vertebrate sub- 
kingdom ; in the one case even 
without departure from the 
original anthropotomical name, 
and in the other, with but a 
single deviation from the esta- 
blished nomenclature. But when 
GeoflProy was induced to reject 
the term ' vomer' as being ap- 
plicable only to the peculiar 
form of the bone in a small 
proportion of the vertebrata, he 
appears not to have considered 
that the old term, in its wider 
application, would be used with- 
out reference to its primary 
allusion to the ploughshare, and 
that becoming, as it has, a purely arbitrary term, it is superior and prefer- 
able to any partially descriptive one. ' Rhinosphenal,' it is true, recalls the 
idea of the vomer forming the continuation in the nasal segment of the skull 
of the basi- and pre-sphenoidal series of bones in other segments; but 'vomer,' 
used arbitrarily, summons equally every idea derived to form the complex 
whole from the general study of the bone throughout the vertebrate series. 
•Prefrontal' {prefrontale, Lat., 14, 14, 
figs. 4, 5, 21, &c.) claims the same pre- 
ference over anterior frontal, and its 
foreign equivalents, as does postfrontal 
over its synonymous phrases. There is 
also another reason for proposing the 
term ; viz. because it is applied to bones 
in the vertebrate series generally, accord- 
ing to conclusions as to their homologi- 
cal relations, which differ from those to 
which Cuvier and GeofFroy had arrived. 
The discussion of the discordant deno- 
minations at present applied to this im- 
portant element of the skull will be fully 
carried out in the sequel. 'Nasal' 
(nasale, 15, figs. 4, 5, 21, &c.) is another 
of the few instances in which it is possible to retain and generally apply an 
old and received anthropotomical term. No one, it is presumed, will con- 




Disarticulated prosencephalic or neuro-frontal arch, viewed 
from behind: Cod-lish. 



Fig. 4. 




Disarticulated rhinencephalic, or neuro-nasal arch, 
viewed from behind : Cod-fish. 



ON THE VERTEBRATE SKELETON. 1^1 

tend for the perpetual expression or insertion of the understood generic word 
'bone' or 'os' in this case any more than in the parietal, frontal, &c., which, 
from being originally specific adjectives, have been properly and conveni- 
ently converted into definite nouns. 

In conformity with this mode of acquiring an improved as well as briet 
and precise expression of anatomical facts, I have substituted for 'pars petrosa 
or ' OS petrosum' the substantive term 'petrosal' (Lat.petrosum,;Rgs.5,^5,lb). 
The necessity for some such designation for an essentially and often physically 
distinct bone in the vertebrate skull has been felt by both Cuvier and 
Geofl'roy, when they respectively proposed the names ' rocher and ' rupeal 
for the element in question. ' Petrosal' has appeared to me to be the best 
EnMish equivalent of Cuvier's ' rocher' ; as containing the most character- 
istic vocable of the old anthropotomical descriptive phrase ' pars petrosa 
ossis temporis,' &c. ' Rupeal' unfortunately has no determinate meaning : it 
is applied by its author with certain prefixes to several distinct bones, which 
already had their proper names. ' Sclerotal' (sclerofale, Lat., figs. 5, 22, 23, 17) 
for ' ossicula sen laminse osseae membranse scleroticse,' is proposed on the same 
grounds as exoccipital, postfrontal, &c., viz. the substitution of a name for a 
phrase. The sclerotals have not been usually included amongst the bones ot 
the head, though they have precisely the same claims to that rank as the pe- 
trosals, or other bony capsules of the organs of special sense. Retaining the 
old anthropotomical term ' ethmoid,' I restrict its application to the very irre- 
gular and inconstant developments of bone in the cartilage or membrane 
which is applied to the anterior outlet of the cranium proper, for the support 
or defence of the cranial part of the organ of smell. The ' ossa turbinata supe- 
riora,' and the ' cellulae sethmoideae' are parts of the capsule of that sense, ex- 
tensively developed in the mammalia, to which the term ethmoid may properly 
apply ; but they must always be distinguished from the modified though con- 
stant neurapophyses of the nasal vertebra, called ' prefrontals,' with which the 
above developments of the olfactory capsule usually coalesce in birds and mam- 
mals. 'Turbinal' (<Mr6ma/e,Lat.,figs.5, 25,19), like petrosal, is a substitute for 
the phrase 'os turbinatum inferius,' and its synonym 'os spongiosum inferius. 
' Palatine' {palatinum, Lat., ib. 20) is another of the few fortunate instances 
of the general recognition of the homologous bone throughout the vertebrate 
kingdom, with the further advantage of a steady retention of a good old name. 
' Maxillary' (maxilla, Lat., ib. 21) is a similar instance ; but Geofl'roy, as 
usual, makes himself singular by adding an uncalled-for synonym. If 
Soemmerring's term 'mandibula' for the lower jaw were universally adopted 
and constantly understood to signify the totality of that part of the tympano- 
mandibular arch throughout the vertebrate series, it would be unnecessary 
to encumber ' maxilla' 'with the distinctive epithet 'superior,' which, indeed, 
expresses a character peculiar only to Man and a few mammalia : in the ver- 
tebrate series the ' maxilla' is more commonly anterior than superior to the 
' mandibula.' 

I have adopted the term ' premaxillary ' (premaxillare, Lat. ib. 22), as used 
by M. de Blainville and some other distinguished continental osteologists, in 
preference to ' intermaxillary ;' because that term has already been applied 
(by Schneider) to another bone of the skull (the tympanic in birds), of which 
it is more accurately descriptive, than it is of a bone which is^ more com- 
monly before than between the maxillary bones. ' Entopterygoid' {entoptery- 
goideum, Lat.) claims preference to the phrases 'pterygoide interne' of Cuvier 
and Agassiz, on the same logical grounds as have already been urged in favour 
of ' exoccipital,' ' prefrontal,' &c. But I have also another reason for pro- 
posing a definite term for the bone 23, fig. 5, which I regard as a peculiarly 



182 REPORT— 1846. 

ichthyic development. Cuvier has applied the term * pterygo'ide interne* 
to another part of the diverging appendage of the palato-maxillary arch, 
which part, 1 concur with Dr. Kostlin in regarding as homologically distinct 
from the ' entopterygoid' of fishes. For the part in question, viz. the ' os 
transverse' of Cuvier in the skull of fishes (24, fig. 5), and its homologue in 
reptiles, which he calls ' pterygoidien interne' (24, fig. 22), I retain the terra 
'pterygoid' { pterygoideum, Lat.), meaning pterygoid proper: and to the 
bone which Cuvier calls 'transverse' in reptiles (24', fig. 22), I apply the 
term 'ectopterygoid' {ectopterygoideum, Lat.); but this, as the table demon- 
strates, does not signify Cuvier's 'os transverse' in the skull of fishes. En- 
topterygoid, pterygoid and ectopterygoid, have, therefore, both the advantages 
of substantive terms, and of being applied steadily each to a distinct bony 
element. The ' herisseal' of Geofi'roy, like the ' pterygoTde interne' of Cuvier, 
means one thing in a fish and another in a crocodile ; GeofFroy has also en- 
cumbered the latter bone with a third synonym. ' Malar' {malare or os mala, 
Lat.) is preferable to 'jugal,' because Cuvier applies that name to one bone 
in a fish, to another in a mammal, and to two essentially distinct though 
coalesced bones in a bird. Malar is also the name most commonly applied 
by English anthropotomists to the bone, to the true homologue of which I 
would restrict its application throughout the vertebrate series. 

With regard to the 'squamosal' (s5'?<a??io5M?M, Lat. pars squamosa, &c., figs. 
22—25,2?), it may be asked M'hy the term ' temporal' might not have been re- 
tained for this bone. I reply, because that term has long been, and is now uni- 
versally, understood in human anatomy to signify a peculiarly anthropotomical 
coalesced congeries of bones which includes the 'squamosal' together with the 
' petrosal,' the 'tympanic,' the ' mastoid,' and the ' stylohyal.' It seems prefer- 
able, therefore, to restrict the signification of the term ' temporal' to the whole 
(in Man) of which the ' squamosal' is a part. To this part Cuvier has unfor- 
tunately applied the term 'temporal' in one class and 'jugal' in another : and 
he has also transferred the term ' temporal' to a third equally distinct bone in 
fishes; whilst to increase the confusion, M. Agassiz has shifted the name to a 
fourth different bone in the skull of fishes. Whatever, therefore, may be the 
value assigned to the arguments which will be presently set forth, as to the spe- 
cial homologies of the ' pars squamosa ossis temporis,' 1 have felt compelled to 
express the conclusion by a definite term, and, in the present instance, have 
selected that which recalls best the accepted anthropotomical designation of the 
part, although 'squamosal' must be understood and applied in an arbitrary 
sense, and not as descriptive of a scale-like form, which, in reference to the bone 
so called, is rather its exceptional than normal figure in the vertebrate series. 

The term ' tympanic' (tympanicum, Lat.) appears to have received the most 
general acceptance as applied to that bone which the early ornithotomists have 
called 'OS quadratum' and ' os intermaxillare,' (fig. 23, 2s) and which as a pro- 
cess of the human temporal, sometimes called 'external auditory,' supports the 
tympanic membrane (fig. 25, 2s). 'Caisse' is the French and 'pauke' the Ger- 
man equivalent ; but Cuvier more commonly uses the phrase ' os tympanique.' 
The chief point, in reference to that term, as applied by Cuvier, from which 
I find myself compelled to dissent from the great and ever-to-be-revered 
anatomist, relates to the view which he has taken of the large and long pe- 
dicle which supports the mandible in fishes, and which, in that class, is sub- 
divided into sometimes two, sometimes three, and commonly into four pieces. 
I regard this subdivisiori of the elongated supporting pedicle as explicable 
chiefly, if not solely, by reference to a final purpose, viz. to combine strength 
with a certain elastic yielding and power of recovery, in the constant and 
powerful movements to which it is subject in the transmission of the respi- 



ON THE VERTEBRATE SKELETON. 183 

ratory currents, and in the prehension and deglutition of the food. Cuvier 
himself regards in the same light the analogous subdivision of the mandibular 
or lower half of the arch, and both Conybeare* and Bucklandf have well 
illustrated the final purpose which the subdivision of the lower jaw of the 
Crocodile into overlapping pieces, subserves. Cuvier has given distinct and 
convenient names to these several pieces of the mandible, but he views them 
collectively as answering to the simple mandible of the mammal and the bird. 
I, in like manner, regard the subdivided pedicle supporting the mandible in 
fishes as answering to the undivided pedicle supporting the mandible in ophi- 
dians, lizards and birds. There is the same necessity or convenience for a 
distinct name to each distinct part of the tympanic pedicle, or upper part of the 
tympano-mandibular arch, as for the divisions of the mandible or lower part of 
that arch. But Cuvier unfortunately persuaded himself that the subdivisions 
of the tympanic pedicle in fishes represented other bones in higher vertebrates 
besides the tympanic, and applied to them the names of such bones. I have 
been compelled, therefore, in dissenting from this view to propose new names 
for the peculiar ichthyic subdivisions of the tympanic, and in doing so I have 
been careful to retain the dominant term, and to distinguish the parts by 
prefixes indicative of their relative position. Time and the judgement of 
succeeding homologists will determine the accuracy or otherwise of thi.s 
view; and, should it be ultimately adopted, I feel great confidence that the 
terms 'epitympanic' (epitympanicum, Lat., fig. 5, 28a), mesotympanic {ineso- 
tympanicum, 286), pretympanic (^pretympanicum, asc) and hypotympauic 
{hypotympaniciim, 2sd), will be preferred to the names proposed by Geoffroy 
St. Hilaire for the same parts. With regard to the subdivisions of the man- 
dible in cold-blooded vertebrates, I adopt most of those proposed by Cuvier. 
As, however, ' operculaire ' had been applied by the great anatomist to a 
distinct bone in fishes, it was necessary, in order to avoid its use in a double 
sense, to substitute a distinct name for the part of the jaw in question, and as 
it is always applied, like a surgeon's splint or plaster to the inner side of most 
of the other pieces, that of ' splenial' (splenium, Lat., figs. 22,23, 31) suggested 
itself to me as the most appropriate name. For an obvious reason I have 
restored the term ' coronoid' (coronoideicm, 31') in place of ' complementary,' 
for the piece into which the crotaphite muscle is always more or less inserted 
in the mandible of reptiles. There is no ground for disturbing the appropriate 
names given by Cuvier to the parts of the diverging appendage of the tym- 
pano-mandibular arch in fishes; and the same principle which he has adopted 
in distinguishing the different opercular bones (fig. 5, 34—37), has guided me 
in naming the different parts of the bony pedicle which supports them. 

I have gladly adopted as many of the well-devised terms which Geoff"roy 
proposed for the elements of the hyoid arch, as his unsteadiness in their ap- 
plication would permit to be retained. They are obviously preferable to the 
descriptive phrases by which Cuvier designates the homologous parts. 

The substantive terms applied to the corresponding divisions of the bran- 
chial arches have been modelled on those of the hyoid system ; but I have 
deviated in one instance from the rule which has governed throughout my 
nomenclature of the bones, in proposing a second name for a modified homo- 
logue in the air-bi-eathing animals, of a part of the branchial apparatus in 
fishes, viz. that part which is retained even in the human hyoid, and which 
is known in anthropotomy as the ' os laterale linguale,' or ' cornu majus ossis 
hyoidei ;' for this part I have proposed the name 'thyrohyal,' for the reasons 
assigned in the note (2) to Table I. 

The names assigned to the bones of the scapular arch (figs. 5, 22, 23, 24-, 25, 
* Geol. Trans., vol. v. p. 565. f Bridgewater Treatise, vol. i. p. 176. 



184 REPORT — 1846. 

23, 50-52) and its appendages (ib. as-ss) agree so closely with those which 
they have always borne as to require no explanation here. The chief 
surprise of the anthropotomist will be occasioned by their being included 
amongst the bones of the head. That the upper or pectoral extremity 
and its supporting arch form actually integrant parts of the occipital seg- 
ment of the skull, will be proved in the memoir on the general homologies 
of the bones of the head. I may, here, however, in reference to the terms 
' ulna' and ' radius,' request the anatomist to compare the skeletons of the 
perch or cod with that of the porpoise. The pectoral extremity is in the 
form of a fin, and in both fish and marine mammal it is applied, in a state of 
rest, prone to the side of the trunk ; in this position it will be seen in the 
Delphinus, that the radius is downward, and the ulna with its projecting 
olecranon upwards. I take this as the guide to the homology of the two bones 
that support the carpal series of the pectoral fin in fishes. Cuvier, however, 
gives the name of 'cubital,' perhaps on account of its angular olecranoid 
prolongation, to the lower bone, and 'radial' to the upper bone: and in 
these determinations he is followed by M. Agassiz. Both bones coalesce 
with the supporting arch in the lophius and some other fishes ; and since, in 
the lophius, two of the carpal bones are unusually elongated, Geoffroy mistook 
these for homologues of the radius and ulna. The condition of the pelvic 
member or ventral fin is, in fact, here repeated in the pectoral ; there being 
no homologous segment of thigh or leg interposed in any ventrals between 
the supporting (pelvic) arch and the fin-rays representing the tarso-me- 
tatarse and phalanges. The earlier stages in the development of all loco- 
motive extremities are permanently retained or represented in the paired fins 
of fishes. First the essential part of the member, the hand or foot, appears : 
then the fore-arm or leg ; both much shortened, flattened and expanded, as 
in all fins and all embryonic rudiments of limbs : finally comes the humeral 
and femoral segments ; but this stage I have not found attained in any fish. 
It is with considerable doubt that I place, qualified by a note of interroga- 
tion, Cuvier's " troisieme os qui porte la nagoire pectorale" as the homologue 
or rudimental representative of a ' humerus.' Normally, I believe this proxi- 
mal member of the radiated appendage of the scapular arch not to be di- 
stinctly eliminated from that arch in the class of fishes. The Siluroids are 
examples of a similar confluence of the first segment (preoperculum) of the 
diverging appendage of the tympanic arch with that arch. With regard to 
the lower, distal or apical element of the scapulo-coracoid arch, always the 
largest bone of the arch in fishes, Cuvier's idea that it is the ' humerus,' far 
less accords with the law of the development, the connections, and the essen- 
tial nature of that bone, than the more prevalent view, that it represents the 
clavicle: a view entertained by Spix, Meckel, and Agassiz, by Wagner, 
who calls it ' vordere Schliisselbein,' and by Geoffroy, who calls it ' furculaire.' 
I have, however, been induced to regard the lower element of the scapular 
arch, in fishes (fig. 5, 52), as homologous with that bone, the ' coracoid,' which 
progressively acquires a more constant and larger development in descending 
fiom mammals to fishes, and which is manifestly a more essential part of the 
arch than the clavicle, since it is more constant in its existence, and always 
more completely developed in birds and reptiles ; and especially since it con- 
tributes more or less of the surface of attachment for the radiated appendage, 
which the clavicle never does. With reference, also, to the Cuvierian deter- 
mination of the haemapophysial portion of the occipital inverted arch in fishes, 
this is unquestionably as essential an element of the arch as is the ' coracoide' 
in other vertebrates ; and it is the most important part in the piscine class, in 
no member of which does it present the slightest approach to the character of 




1846. 



ON THE VERTEBRATE SKELETON. 187 

a diverging appendage, such as the humerus essentially is, whenever it lias an 
independent existence. I3y some ichthyotomists, the Ijone which I call cora- 
coid (52) has received the special name of ' ccenosteon.' 

Cuvier's usual judgement and acumen seem to have been in abeyance, 
when, having determined the rays of the pectoral fin to represent the bones 
of the hand, and the two bones which support them in fishes to be those of 
the fore-arm, he concluded that, therefore, the great bone which completed 
the scapidar arch "repondra done necessairement a I'humerus." — Hist, des 
Poissons, ^to. i. p. 274. The great anatomist assigns no other reason : but 
the arch supporting the ventral fin does not necessarily answer to the tibia 
or the femur, because neither of these segments are interposed between the 
arch and its appendage — the modified foot*. The scapula of many reptiles, 
especially of the batrachia, is manifestly, he proceeds to state, composed of 
two bones. But in those reptiles the arch is completed below by a third 
bone, which neither Cuvier nor any other anatomist has called ' humerus.' 
Now Cuvier's 'humerale' in fishes precisely ans\\ers to that third bone in 
reptiles which he rightly calls the 'coracoid' in that class. 

The coracoid of fishes being thus determined, it necessarily follows that 
that inconstant bone, or pair of bones (ss) posterior to it on each side, cannot 
be, as Cuvier, Geoftroy, Meckel and Agassiz have supposed, the representa- 
tive of the ' OS coracoidien' of the reptile and bird. It holds, indeed, as they 
have said, the same relative position to the bone 52, here called coracoid, 
which the coracoid in the lizard and bird holds to the clavicle in those ani- 
mals. But is no account to be taken of the remarkably though normally ad- 
vanced position of the scapulo-coracoid arch in fishes? Granting, as I shall 
give evidence to prove in treating of the general homologies of the bones, 
that the bone (ss) called by Cuvier 'coracoidien' in fishes appertains to a 
vertebral segment posterior to the occipital one, yet in the extraordinary back- 
ward displacement which the true scapulo-coracoid arch undergoes in the 
air-breathing vertebrates, may not its relative position to that arch become 
reversed, and the part which is behind in fishes become before in birds ? I 
entertain no unmeet confidence in the correctness of my view of the special 
homology of Cuvier's ' os coracoidien' in fishes with the furculum or ' clavicle' 
of air-breathing vertebrates: the argument against such a view, from its pos- 
terior position in fishes, has not, however, the same weight with me as it ap- 
pears to have had with Cuvier and his followers : and, leaving this as one of 
the undecided points in special homology, with the proposition of the pro- 
visional name of ' epicoracoid ' (epicoracoidettm, Lat.) for the bone in ques- 
tion, I proceed to consider other mooted points of special homology, of which 
there are better and surer grounds for the determination. 

The first discrepancy, demanding special consideration, which meets the 
eye in the Table I. is that which relates to the determination of no. e. The 
German authorities regard what I believe to be the homologue of the human 
' ala major sphenoidalis' in the cold-blooded Vertebrata, to be the homologue 
of the ' pars petrosa ossis temporis.' Cuvier recognises the 'grande aile du 
sphenoide' in mammals, birds and fishes, but regards my 'alisphenoid' in 
reptiles as the ' rocher ' or ' pars petrosa.' GeofTroy concurs with Cuvier and 
the German anatomists so far as to view my 'alisphenoid' in the Crocodile 
as a dismemberment of the petrosal, calling it ' pr^rupeal ;' but he recognises, 
like Agassiz and Cuvier, the true alisphenoid in fishes, and with them differs 
in that respect from the German homologists. It does not appear that the 
alisphenoid has been mistaken for any other bone than the petrosal, and the 

* The great Liniiseus indicates his appreciation of the homology of the ventral fins of 
fishes hy stvling the fishes without those fins ' Apodal.' 

o 2 



188 



REPORT — 1846. 



question to he determined, therefore, is, what are the essential characters re- 
spectively of the 'alisphenoid' and the 'petrosal' in the vertebrate series? 

Those of the alispherioid appear to me to be the following : — 1st, its con- 
nection below with the basisphenoid and behind with the petrosal, where it 
forms the forepart of the ' otocrane' or cavity for the reception of that osseous 
or cartilaginous immediate capsule of the labyrinth or internal organ of hear- 
ing: the alisphenoid is also commonly, but not constantly, joined before 
with the orbitosphenoid, and above with the parietal : it has other less con- 
stant connections with the squamosal, the exoccipital, the supraoccipital and 
the basioccipital : 2ndly, with regard to its essential functions, the alisphenoid 
protects more or less of the side of the mesencephalon, or (in mammals) of 
the middle lobe of the hemisphere : it gives exit, by notches or foramina, to 
the third, and usually, also, to the second divisions of the trigeminal or fifth 
pair of nerves. 

The essential character of the petrosal is to envelope immediately the 
whole of the vascular and nervous tunics of the labyrinth or internal organs 
of hearing, either in a membranous, a cartilaginous or an osseous state ; 
its histological condition being much less constant than that of the alisphe- 
noid. 

On viewing the alisphenoid on the interior surface of the human skull 
(fig. 6, e), it seems to be ihe least significant and important part of the lateral 

Fig. 6. 




Vciticul longitudinal section of the human cranium. 

walls of the cranial cavity : it forms their smallest portion : it is much sur- 
passed in extent by the squamosal (ib. 27) and the supra-occipital (ib. 3), 
and still more so by the enormously expanded parietal (7) and frontal (u). 
Nevertheless we find it connected, anchylosed indeed, below to the basisphe- 
noid (5), bounding anteriorly the space into which the petrosal (le) is 
wedged ; connected in front with the orbito-sphenoid (10), and usually 
articulating by its superior apex with the parietal : I purposely omit the 
mention of other connections of the alisphenoid in Man which are less 
constant in the vertebrate series. But it is important to observe, notwith- 
standing the displacement which the alisphenoid has undergone through the 
intercalation of the extraordinarily developed squamosal into the lateral walls 



ON THE VERTEBRATE SKELETON. 189 

of the cranium, that it is still perforated by the third (ib. tr) and second 
divisions of the fifth or trigeminal nerve. 

In tracing the alisphenoid downwards through themammalian series, wecan- 
not but be impressed with the conviction of its true character and importance 
as an essential part of the cranium, from its constancy in the formation of its 
walls, and by observing that, whilst the share which the squamosal takes in them 
progressively decreases, — until in the sheep, for example, it is quite excluded 

Fig. 7. 




Veitical longitudinal sectioa of the cranium of a sheep (Obi's Aries). 

from the cranial cavity, — that of the alisphenoid (fig. 7, e) increases as the 
cavity itself diminishes in size ; and, further, that this increase is not accom- 
panied with any material change in the relative size of the alisphenoid to the 
basisphenoid. The share which the alisphenoid takes in forming the ante- 
rior boundary of the otocrane increases; as does also the extent of its supe- 
rior connections, especially of that with the parietal (7). It is important, 
in tracing these modifications, to note, also, the change in the relative position 
of the foramen ovale in the mammalian series. In Man the foramen ovale 
(fig. 6, tr) is close to the hinder border of the alisphenoid ; and in some 
quadrumanes the third division of the fifth escapes through a notch in the 
same border. This position of the foramen ovale relates to the alisphenoid 
being pushed forward by the intrusion not only of a large ossified petrosal 
(le), but of a still larger squamosal (27). In the sheep, however, the fora- 
men ovale is no longer at the posterior margin ; but, the alisphenoid, having 
retrograded by the recession of the squamosal towards its more normal ex- 
terior position in the vertebrate series, the third division of the trigeminal 
now perforates its middle part (fig. 7, tr). It may be observed that, con- 
comitantly with this retrogradation of the alisphenoid, the orbito-sphenoid 
{ib. lo) acquires larger propoi'tional dimensions than in Man (fig. 6, 10). 

In the bird the alisphenoid (fig. 8, e) is recognizable by the repetition of 
the connections which it presented in the sheep; the squamosal being quite 
excluded from the cranial parietes, and, indeed, never again presenting itself 
in the capacity of a cranial bone in any of the oviparous vertebrates. The 
alisphenoid (fig. 23, 0) is in contact posteriorly with the petrosal (ih. le), 
which soon becomes anchylosed with it, as well as with the exoccipital (2), 
mastoid (s), and other bones forming the cavity for its reception, in all birds. 
The alisphenoid further manifests its true homology in the bird by its other 
constant character of transmitting the third and also the second or maxillary 
division of the trigeminal nerve ; which divisions, in the young ostrich, I 



190 



REPORT — 1846. 

Fig. 8. 




Partly disarticulated cranium of a young ostrich {Stnithio camelus), natural size. 

found distinctly perforating the middle of its lower border (fig. 8, 6, tr). The 
alisphenoid is deeply impressed by the chief ganglions of the mesencephalon, 
viz. the optic lobes. The prosencephalon or hemispheres are still defended 
principally by expanded parietals (ib.r) and frontals (ib. ii)*. 

In the crocodile these spinal elements of the cranium are much restricted 
in their development, and a larger proportion of the hemispheres is defended 
by the orbitosphenoid (fig. 9, lo), which here surpasses the alisphenoid (ib. e) 
in size. This, however, still performs its essential and characteristic func- 

Fig. 9. 




Vertical longitudinal section of the cranium of a crocodile (Crocodilus acutus). 

tions of protecting the sides of the mesencephalon, and giving issue to the 
chief part of the trigeminal nerve. Owing to the diminution in size of the 

* The right frontal has been removed to show better the extent and connections of the 
orbitosphenoid (lo) and the prefrontal (u). 



ON THE VERTEBRATE SKELETON. 191 

petrosal (le), and the retention by a great proportion of this capsule of the 
acoustic labyrinth of its primitive cartilaginous state, it occupies a smaller 
interval between the alisphenoid (e) and exoccipital (2). It no longer pro- 
trudes as a large bony wedge (as in figs. 6 and 7, lo) into the cranial cavity, 
but permits the alisphenoid to come into connection with the exoccipital. 
The result of this further retrogradation of the alisphenoid, in regard to the 
relative position of the outlet of the third division of the fifth, is analogous 
to that which occurs in the sheep. We saw in that mammal, through the 
recession of the squamosal, the foramen ovale advanced from the posterior to 
the middle part of the alisphenoid ; in the crocodile, through the further re- 
moval from the cranial cavity of the interposed petrosal, the foramen ovale is 
advanced to the anterior border of the alisphenoid ; which border, in fact, it 
notches, the nerve escaping by a common foramen or ' trou du conjugaison' 
between the alisphenoid and the orbitosphenoid, the hole, however, being 
principally formed by the alisphenoid (fig. 9, tr). This position of the ' fora- 
men ovale' loses all its value as an argument in favour of the petrosal cha- 
racter of no. 6, by analogy with the position of the foramen ovale in man 
or the ape, when we take into consideration the necessary consequences of 
the successive withdrawal of the squamosal and true petrosal from the inner 
surface of the cranium in descending to the reptiles. The orbitosphenoid 
(fi-^. 9, 10), notwithstanding its great relative size, retains all its essential cha- 
racters: it is perforated or notched for the exit of the optic nerves (op) and 
first division of the fifth pair (5); it rests upon the presphenoid (9) Uelow, 
and likewise, through its backward development, partly upon the basisphe- 
noid, and it articulates with the frontal (n) above, and also through the 
same backward extension with the parietal (7) ; it constitutes the anterior 
border of the lateral bony parietes of the cranium, which are interrupted 
by the orbits, and separated by their interposition in saurians and fishes 
from the rhinencephalic part of the cranial cavity (at i4, fig. 9). The cha- 
racters, in fact, of the orbitosphenoid are so clearly manifested 111 the cro- 
codile, that Cuvier, having been led by the increased share, as compared 
with mammals, which the crocodile's alisphenoid (fig. 9, e) takes in the form- 
ation of the otocrane, to regard it as the petrosal, and yet perceiving the 
essential characters of the orbitosphenoid in the bone (ib. 10) anterior to it, 
was driven to the conclusion that that bone represented both orbitosphe- 
noid ('aile orbitaire du spheno'ide') and alisphenoid (aile temporaie du sphe- 
noide). The cold-blooded crocodile, however, is not exactly the animal in 
which we should expect to find so unusual an instance of obliteration of 
sutures, as that between the alisphenoid and orbitosphenoid *. The actual 
and most characteristic modification of the orbitosphenoid in the crocodile's 
skull, is its retrogradation together with the alisphenoid, or rather the main- 
tenance of its normal connection therewith by increased antero-posterior 
development, whereby it comes into communication above with the parietal 
(7) and below with the basisphenoid (5); whilst the alisphenoid, in like 
manner, gains a connection with the supra-occipital (3) above and the basi- 
occipital (1) below ; although it still retains its more normal relations with the 
parietal, and rests in great part on the basisphenoid (5), as the orbitosphe- 
noid rests in great part upon the pre-sphenoid (9.) The superior connec- 
* No one better appreciated the characteristic persistence of the sutures in the crocodile 
than Cuvier, when his attention was not diverted from it by a favourite hypothesis. " Le 
crocodile a cela d'avantageux a I'etude de son osteologie, que ses sutures ne s'effacent point, 
du moins n'en a-t-il disparu aucune dans nos plus vieilles tetes," is the remark with which 
he commences his article on the determination of the bones of the head of that reptile 
(Ossemens Fossiles, 4to. v. pt. ii. p. 69) : but at p. 76, a suture is assumed to be effaced, 
which is present in most mammals and all cold-blooded vertebrates, where a wider space 
does not intervene between the alisphenoid and orbitosphenoid. 



192 REPORT — 184G. 

tions of the orbitosphenoid and alisphenoid are always less constant than 
tiieir inferior ones. By these latter characters, and still better by their nerve- 
outlets and their relations to the primary divisions of the eucephalon, are 
they rightly and truly determinable. The German authors who have fol- 
lowed Cuvier in his views of the special homology of the alisphenoid in rep- 
tiles, are more consistent than the great French anatomist in regard to the 
alisphenoid of fishes. Dr. Kallmann, accepting Cuvior's characters of the pe- 
trosal, taken from its internal position and lodgement of the whole or part 
of the labyrinth*, naturally applies them to the alisphenoid in fishes, and 
adds to the grounds for regarding that bone as the ' petrosal,' that it is in 
some fishes perforated by the opercular branch of the great trigeminal nervef. 
But, admitting the homology of the opercular nerve with the facial nerve of 
mainmals, yet its wider homology and essential character as a motor division 
of the great ti-igeminal nerve must not be lost sight of: its origin in close 
contiguity with tlie great sensory portions of the trigeminal in fishes accords 
better with the character of that nerve as the great spinal nerve of the brain, 
than it usually presents in higher classes; and it is surely no important de- 
jjarture of the alisphenoid from its normal character, that it should give exit 
to both motory and sensory divisions of the great nerve with which it is so 
intimately associated from man down to the fish. Indeed, the progressive 
withdrawal of the bony petrosal from the interior of the skull and the con- 
comitant backward extension, or retrogradation of the alisphenoid, ought to 
prepare us to expect that nerves which traverse the petrosal in mammals 
should perforate the alisphenoid in reptiles and fishes. And so we find 
in the carp that the glosso pharyngeal even perforates the posterior border 
of the alisphenoid ; but its origin close to the acoustic and facial nerves 
in fishes diminishes the force of the argument which might be drawn from 
tills exceptional perforation, in favour of the petrosal character of the ali- 
sphenoid I concur entirely with Cuvier and M. Agassiz in their determi- 
nation of the alisphenoid in fishes ; but, if the great share which that bone 
in reptiles (figs. 9 and 10, e) contributes to the formation of the otocrane, 
if the anterior position of the foramen ovale, and the superior connection of 
the bone with the supra-occipital, are proofs (as Cuvier believed) of its ho- 
mology with the petrosal in the class Beptilia, they ought also, as Kallmann 
and Wagner contend, to establish the same special homology of the bone (o) 
in the class Pisces. But none of these are essential characters of the petrosal. 
The petrosal is a conttntum and not SLparics, or any part of the parietes of the 
cranial chamber or otocrane lodging the organ of hearing : it is the outermost 
tunic, membranous, gristly, or bony, of the labyrinth or essential part of the 
acoustic organ. Kad the above-cited anatomists clearly appreciated the 
general homology of the petrosal, they could scarcely have failed to detect 
its special homologies in the vertebrate series. Cuvier was evidently guided 
to the true determination of the alisphenoid in fishes, less by its own essen- 
tial characters, than by observing in certain fishes, the perch and cod for ex- 
ample, a ])artial ossification of the acoustic capsule, to which, therefore, he 
assigned the name 'rocher.' And, having thus satisfied himself of the ex- 
istence of the homologue of the ' pars petrosa,' &c., he could not but assign 
to the bone which rested below upon the basisphenoid, which protected late- 
rally the optic lobes and gave exit to the third division of the trigeminal nerve, 
the name of ' grande aile du sphenoide.' But all these characters equally 
coexist in the bone which Cuvier calls ' rocher' (petrosal) in the crocodile and 
other reptilia. Ke was not aware, however, that in both gavials and cro- 
coililes a distinct ossicle, the veritable homologue of the intra-cranial p) ra- 

* Ossemens Fossiles, Ito, t. v. pt. i. p 81. 

t Dcr vergk'idiende Ostcologic tics Schlafeiibeiiis, p. 61. 



ON THE VERTEBRATE SKELETON. 193 

midal-shaped petrosal oF mammals and birds, makes its appearance between 
the alisphenoid, exoccipital and basioccipital, as at id, fig. 9. Here, however, 
it is necessary to offer a few observations on the sense in which I use the 
term 'petrosal' as applied to that ossicle. 

The petrosal, properly so called, considered in its totality, as the immediately 
investing capsule of the labyrinth or internal organ of hearing, is wholly carti- 
laginous in many fishes and saurians, and in all batrachians, ophidians and 
chelonians, and is contained in a cavity or orbit (otocrane) which most, or all 
of the elements of the occipital and parietal vertebrae concur in forming. A 
part of the ear-capsule remains cartilaginous in the crocodile; but several 
portions become ossified around the semicircular canals and rudimental 
cochlea, which ossifications contract slender adhesions to the smooth oto- 
cranial surfaces of the supraoccipital, exoccipital and alisphenoid ; and to 
one of these portions (on the principle on which Cuvier applies the term 
'rocher' in fishes) the name petrosal might more particularly be given, as it 
is more distinct and moveable than the other partial ossifications of the cap- 
sule, and contributes to form the ' meatus internus' towards the cranial cavity, 
surrounds nearly the whole of the ' fenestra rotunda', and one-half of the ' fe- 
nestra ovalis' towards the tympanic cavity. Looking upon the inner surface 
of the lateral walls of the cranium (as at fig. 9), one sees at the bottom of 
the T-shaped suture* uniting the otocranial laminae of the exoccipital, ali- 
sphenoid, and supraoccipital bones, a fourth osseous element (le), presenting 
a convex extremity towards the cranial cavity, and completing, with the exocci- 
pital, the lower half of the foramen for the nervus vagus. If this little bone 
be pressed upon with a needle or probe, it yields and moves, being divided 
by smooth harmoniae from both the exoccipital (2) and alisphenoid (e). 

The protuberance in question, which thus projects into the cranial cavity, 
is the rounded angle of the border of the inferior plate of the petrosal, which 
joins the exoccipital. This lower horizontal plate of the petrosal forms the 
upper wall of the ' fissura lacera posterior,* and the lower wall of the ' fenestra 
cochleae': the fore-part of the horizontal plate bends upwards, twisting 
and expanding into a vertical oval plate, articulated by its anterior surface to 
a corresponding sutural surface of the alisphenoid. The lower margin of 
this plate forms the upper boundary of the ' fenestra cochleae,' and is con- 
tinued into a thin plate of bone which divides the ' fenestra cochleae' from the 
' fenestra vestibuli' above. This thin plate of the petrosal joins and is usually 
anchyiosed to the exoccipital : it is the only part of the true petrosal noticed 
by Cuvier, who describes it as a slender filament of bone which separates 
the two fenestraef. Seen edgewise, looking into the tympanic cavity, the 
plate appears like a filament : and this plate forms the sole connection, wlien 
any exists, between the petrosal and the exoccipital. I have always found 
the sutures persistent between the petrosal and the alisphenoid. The upper 
border of the 'fenestra vestibuli' is formed by a petrosal, or rather otocra- 
nial, process of the alisphenoid. 

The part (fig. 9, le) entering into the formation of the lateral walls of the 
brain-case, and which is here specially indicated by the name of ' petrosal,' 
seems to have been overlooked : it is, however, relatively to the alisphenoid 
or exoccipital, as large as is the petrosal (Cuvier's rocher) in the perch: it 
has a true osseous texture, and is quite distinct from the lenticular mass of 
calcareous matter in the adjacent cochlear chamber which Cuvier compares 
to starch (' amidon durci '). 

* Suture a trois branches, Cuvier, I. e. p. 165. 

t Du cote de la caisse la parol est percee de deux fenetres trausversalement oblongues et 
scpavces par uu filet mince." /. c. p. 82. 



194 REPORT— 1846. 

Neither the figure of the interior surface of the cranium of the crocodile, 
which Spix gives as that of the Nilotic species in his great 'Cephalogenesis,' 
tab. ii. fig. 6 ; nor the figure given by Geoffroy of the skull of his Crocodihts 
SHchus in the ' Annales des Sciences,' torn. iii. pi. 16, fig. 2; nor that of the 
Crocodilus biporcatus, Avhich illustrates the later memoir by the same author 
in the 'Memoires de I'Academie Royale des Sciences,' t. xii. (1833), pi. 1, 
fig. 2.; nor that (if it be an original figure) published by Dr. Kallmann in 
his 'Comparative Anatomy of the Temporal Bone' (taf. iii. fig. 49), give any 
indication of this, in the determination of the homology of the alisphenoid 
and petrosal, most significant and important ossicle. The proof of its normal 
character will be afforded by comparisons of the description and figure of 
the part here given with a section of the cranium of any true CrocodUvs, 
Alligator or Gavial. In the latter, the otocranial plates of the alisphenoid, 
exoccipital and supra-occipital, project considerably into the cranial cavity. 
Any one of these plates might be called ' petrosal,' for such reasons as have 
induced Cuvier to apply that name to the alisphenoid in the crocodile and 
other reptiles*. We find, indeed, that Geoffroy has applied the equivalent 
term, by turns, to each. But the true idea of the petrosal should include all 
those gristly and bony parts of the immediately investing capsule of the la- 
byrinth which occupy the otocranial excavations of the exoccipital, supraoc- 
cipital and alisphenoid ; and as the ossified portions of the true petrosal, in the 
crocodile, usually contract a bony union with the parietes of the otocrane, 
all these bony portions of the immediate capsule of the labyrinth might be 
called 'petrosal processes' of the bones to which they respectively adhere. 
That portion which unites to the exoccipital is attached by two lamellae ; it 
forms a great part of the cochlear cavity, the lower half of the posterior semi- 
circular canal and the hinder half of the external or upper semicircular canals: 
that plate which belongs to the supra-occipital is attached to its otocranial 
surface by three points, and forms the upper third part of the anterior semi- 
circular canal and the crus of the posterior canal which communicates there- 
with : that part which adheres to the alisphenoid forms the anterior crus of the 
anterior (in Man superior) semicircular canal and the anterior beginningof the 
external canal. The proper and usually distinct bony portion of the petrosal 
(fig. 9, le), which articulates with both alisphenoid and exoccipital, forms 
part of the ' meatus internus,' nearly the whole of the ' fenestra cochleae,' and 
half of the ' fenestra vestibuli ' : it can only be regarded a ' petrosal process' 
of the exoccipital by virtue of the very limited anchylosis occasionally con- 
tracted by the thin plate dividing the two ' fenestrse,' along with the true 
petrosal process of the exoccipital above described. 

If we compare with 
the inner wall of the cro- rig. 10. 

codile's cranium that of 
an ophidian, the python 
for example (fig. 1 0), we 
shall find the walls of the , 
' otocrane ' or chamber 
of the labyrinth to be 
contributed by the ex- 
occipital, (2) supra-oc- 
cipital(3)and alisphenoid 
(e) in nearly equal pro- 
portions ; the basioccipi- 'ir ac tr^ 

tal (1^) also, being ac- Cranium of a python partially bisected. Natural size. 

* Ossemens Fossiles, 4to. 1824, v. ii. pp. 81, 180, 258. 




ON THK VERTEBRATE SKELETON. 195 

cessory to the formation of the floor of the ear-chamber : the three principal 
bones are united, as in the crocodile, by a triradiate suture. The petrosal, 
which, like the squamosal, was gradually more and more withdrawn and 
shut out from the cranial cavity, as we decended from mammals, now entirely 
disappears from view : and it retains its primitive cartilaginous state in ser- 
pents as it does in chelonians, lizards and batrachians. The essential cha- 
racters of the exoccipital (2) are manifested by its relative position and con- 
nections; by its affording exit for the vagal (v) and hypoglossal (/ig) nerves, 
and by its protecting the sides of the epencephalon. The alisphenoid (e) is 
not less clearly indicated by its constant and essential characters ; it rests below 
upon the basisphenoid (5), it articulates above with the parietal (7), and 
behind with the cartilaginous petrosal ; but the otocranial plate being, as in 
the crocodile, unusually extended backwards, unites with the basioccipital 
(1), exoccipital (2) and supraoccipital (3), in almost equal proportions, and 
becomes directly perforated by the acoustic nerve (ac). Its chief foramen 
(<r), however, is, as usual, that which answers to the foramen ovale in the 
human alisphenoid, and which gives passage, as in fishes, to the great third 
division of the fifth, and to the branch which is homologous with the 
contribution by the fifth to the 'nervus lateralis' in many fishes, and at 
the same time with the nerve called ' chorda tympani ' in anthropotomy. 

In the frog I have given an external view of the alisphenoid (e) and the 
cartilaginous petrosal (le) in their undistui'bed connections, in fig. 13, with 
the surrounding bones. The alisphenoid is here perforated, as in Man, by 
both a foramen ovale and foramen rotundum (tr.) : it forms posteriorly the 
fore-part of the chamber for the cartilaginous petrosal, and usually coalesces 
with the mastoid (s), which overarches the petrosal : the back wall of the 
otocrane is contributed, as usual, by the exoccipital (2) ; the floor by the 
homologue of the coalesced basisphenoid and basioccipital. Had the outer 
part of the petrosal (le) been the seat of a partial ossification, a bone would 
have resulted corresponding precisely with Cuvier's ' rocher ' in the cod and 
perch : but the immediate capsule of the labyrinth retains the same histolo- 
gical condition in the batrachia as it does in the carp and pike, and as in the 
salamandroid polypterus and lepidosteus : in the latter fish, at most, the only 
ossified part of the petrosal forms a small bony cup covering the posterior 
extremity of the outer semicircular canal*. 

The attention of the justly celebrated ichthyotomist of Neuchatel appear* 
to have been too exclusively occupied with the persistent embryonic condi- 
tion of the ' petrosal ' in these highly organized fishes, to gain that true and 
clear idea of the essential nature of the petrosal of which its partial ossifica- 
tion in the perch and cod is indicative. Adopting the opinion of Cuvier, in 
preference to that of Meckel and Kallmann, touching the special homology 
of the alisphenoid, M. Agassiz originally diverged into the opposite extreme 
of repudiating altogether the existence of a petrosal in the class of fishes. 
Thus, he says, " II devrait suffire ce me semble de voir I'organe de I'ouie 
presenter des modifications graduees dans toute la serie des vertebres, pour 
se convaincre que le rocher n'existe pas du tout chez les poissons, par plus 
que les osselets de la cavite du tympan. S'il y avait un rocher chez les 
poissons, ce devrait etre un os qui entourerait le labyrinthe et les canaux 
seniicirculaires; mais nous avons vu que ces parties de I'oreille interne se 
trouvent dans la cavite du crane sans enveloppe osseuse particuliere, et pro- 
tegees seulement par les parois des os qui eutourent le rocher, la ou il existef ." 

* This condition answers to that in the human emhiyo of ahout the fourth month, in which 
a hght porous bony crust begins to be formed upon the cochlea and semicircular canals 
commencing with the outer and upper ones, the rest of the petrosal being cartilaginous. 

t Recherches sur les Poissons Fossiles, tom. v. p. 66. 



196 REPORT — 1846. 

M. Agassiz is perfectly accurate in his character of the petrosal, according 
to its relative position, as completelj' investing the entire labyrinth (of which, 
by the way, the semicircular canals are an integrant part in all vertebrates 
and form almost the whole in fishes) ; but he takes a narrow view of its 
histological characters. The sclerotic is not less essentially a sclerotic in the 
shark, where it is cartilaginous, tlian it is in the cod, whore it is osseous; neither 
is it less the eye-capsule and homotype of the petrosal in the mammal because 
it retains the earliest histological condition of tiie skeleton, viz. that of a fibrous 
membrane. And, in point of fact, in those fishes where the essential parts of 
the internal organ of hearing appear to be protected solely by the parietes of 
the bones, which, in the animals where the petrosal is ossified, or, as M. Agassiz 
expresses the fact, ' exists,' surround such petrosal, the vascular and nervous 
parts of the labyrinth are actually in such fishes more immediately enveloped 
by the petrosal in its membranous or cartilaginous states. What is peculiar 
to the petrosal in fishes is, that it is never entirely ossified ; and, furthermore, 
that whenever it is partially ossified, the bony part is external and appears on 
the outside of the skull instead of the inside, as in the crocodile and birds. 

In the chelonia, a larger proportion of the petrosal intervenes between the 
alisphenoid and exoccipital upon the inner wall of the cranial cavity than in 
the crocodile ; but it is wholly cartilaginous. In the bird, on the contrary, the 
whole petrosal capsule of the organ of hearing soon ossifies and becomes 
firmly anchylosed to the parts of the exoccipital, mastoid, alisphenoid and 
basisphenoid that form its primitive chamber or otocrane ; owing, however, 
to the larger relative size of the ossified part of the proper capsule (petrosal 
proper) which penetrates the cranial cavity, none of the surrounding bones 
which contribute accessory protection, have received the name of ' rocher,' 
or pars petrosa. It is chiefly from not recognizing or appreciating the general 
nature or homology of the ' petrosal ' that Cuvier failed to perceive its special 
homology in reptiles. Speaking of the skull of the crocodile, he says that 
the petrosal, or ' rocher,' is not less recognizable than the ' tympanic ' and 
other so-called dismemberments of the temporal by its internal position, 
by its lodging a great part of the labyrinth, and by its contributing essen- 
tially to the formation of one of the fenestrae (/. c. p. 81). But the part in 
the crocodile which I regard as homologous with Cuvier's ' rocher ' in the 
perch, is more completely internal in position than is Cuvier's so-called 
' rocher ' in the crocodile : it contributes a greater share to the formatiou 
of the ' fenestra vestibuli,' and it forms almost the whole of the ' fenestra 
cochleae.' It is not true of the alisphenoid (Cuvier's '■rocker'^ in the cro- 
codile, that it lodges a great proportion of the labyrinth*: the otocranial 
or petrosal process of the alisphenoid lodges a part only of the anterior 
semicircular canal, and no part at all of the other semicircular canals. The 
exoccipital is that tributary of the otocrane which lodges the major part 
of the labyrinth ; it contains, for example, parts of two semicircular canals, 
and the rudimental cochlea: and, when the middle, usually distinct part 
of the petrosal is joined to it, the exoccipital may be said to form the 
whole ' fenestra cochleae ' and a greater part of the ' fenestra vestibuli.' We 
see, then, that the characters by which Cuvier deems his 'rocher' to be so 
easily recognizable, are more prominent in the exoccipital than in the ali- 
sphenoid : and the choice of the latter by Cuvier as the representative of 
the ' rocher,' seems chiefly to have been influenced by the more obvious and 
unmistakeable essential (neurapophysial) characters of the ' occipital lateral ' 
(fig. 9, 2), whilst the accessory character which this bone derives from its 
lodging and becoming confluent with part of the true petrosal, was not allowed 

* " 11 loge en grande partie le labyrinthe," I. c. p. 81. 



ON THE VERTEBRATE SKELETON. 



197 



to prevail, as in the case of the alisphenoid, in the determination of its special 
homologj'. 

The supraoccipital, by virtue of its internal position and lodgment of part 
of the labyrinth, has equal claims to the name of ' rocher,' according to the 
Cuvierian characters of that bone, and Geoffroy St. Hilaire did not make a 
less arbitrary choice in singling out this element as 'le seul rupeal*,' than 
Cuvier did in choosing the alisphenoid, or, as any other anatomist would do 
in preferring any other element of a cranial vertebra in the crocodile to 
represent the ossified ear-capsule of the fish or mammal, because portions of 
that ossified capsule are protected by, or have coalesced with, such vertebral 
elements. Had Cuvier looked beyond the special homology of the bones of 
the head of the crocodile, and permitted himself to appreciate their higher and 
more general relations, he could scarcely have failed to perceive the corre- 
spondence of his so-called 'rocher' in batrachians, ophidians, chelonians and 
saurians, to the bone which he so well recognizes as ' the great wing of the 
sphenoid' in the perch and cod-fish. 

The Mastoid. — In the human embryo of the fifth month a centre of ossi- 
fication is established on the outer surface of the mass of cartilage occu- 
pying the interspace between the basioccipital (fig. 11, i) and exoccipital 

(2) below, the tympanic (2s) and squamosal (27) in front, the supraoccipital 

(3) behind, and the parietal (7) above: this mass of cartilage incloses the 
membranous labyrinth, about which a light osseous crust has begun to be 
formed ; and, from the centre (s) established near the outer border of the 
posterior semicircular canal, ossification radiates to complete that part of the 
cranial parietes, which, in the adult skull, is impressed on its inner surface by 
the great venous channel called ' fossa sigmoidea,' and developes from its 



Fig. 11. 



outer surface the ' processus mastoi- 
deus.' The primitive independence 
of the base of this process, which 
Kerkringius so clearly and accurately 
delineates in his tab. xxxv.^ff. iii. as 
the posterior of his ' tria petrosi ossis 
distincta ossiculaf,' is a fact of much 
more significance than its brief and 
transitory manifestation would lead 
the anthropotomist to divine. The 
coalescence of the primitively distinct 
mastoid with the ossifying capsule of 
the labyrinth is very speedy, being 
usually complete before the foetus has 
passed its fifth month, and a com- 
posite ' petro-mastoid' bone is thus 
formed, which, retaining its indivi- 
duality in monotremes, marsupials, 
ruminants and many rodents, pro- 
ceeds to coalesce with the additional 
elements of the ' temporal ' bone in man, and with other surrounding cranial 
bones in birds. In the cold-blooded vertebrata, the mastoid retains, with a few 
exceptions, its primary embryonic distinctness, as an independent element of 
the skull. In tracing the modifications of this element downwards from man, 
we find the external process from which its anthropotomical name originated, 




Skull of the human embryo ; fifth month. 
Natiiral size. 



* Annales des Sciences Naturelles, torn. iii. 1824, p. 271, pi. 16. 
t Spicilegium Anatomicum, 4to. 1670, Osteogenia Foetuum, p. 269. 



198 



REPORT — 1846. 



inconstant, its functions being transferred in many mammals to another pro- 
cess, sometimes udder-shaped, sometimes of great length (fig. 24', 4), but 
M'hich is developed from the exoccipital, and is represented in the human skull 
by the 'eminentiaaspera,' &c. of Soemmerring (Table I. 4-), and bythe "sca- 
brous ridge extended from the middle of the condyle towards the root of the 
mastoid process" of Munro {pp. cit. p. 72) ; but sometimes also here deve- 
loped, as a rare anomaly, on one or both sides, into a process like a second 
but smaller posterior mastoid*. The more constant and essential (tharacters 
of the mastoid are its contribution to the walls of the acoustic chamber, 
carried to anchylosis with the petrosal in birds and mammals, and its sutural 
connection in the latter with tiie exoccipital, parietal, and squamosal (the 
squamo-mastoid suture becoming obliterated in many species, e. g. the hog, 
fig. 2-t, 8, 27) : it is also grooved, notched or perforated by a greater or less 
proportion of the lateral venous sinus, whether this is continued to the ' fora- 
men jugulare,' as in man, or sends a large division to escape by the 'meatus 
temporalis ' Mhich forms the large orifice between the mastoid and squamosal 
above the meatus auditorius in the horse and ruminants, and which directly 
perforates the mastoid in the echidna (fig. 12, tri). 

Fi?. 12. 




Partially disarticulated cranium of the Echidna setosa. Natural size. 

It is important to keep these essential characters steadily in view,and to avoid 
giving undue importance to the apophysial character of the mastoid, which has 
led to so common a transference of its name, in the great osteological works of 
Cuvier and De Blainville, to a quite distinct element (paroccipital) of the 
ci-anial wallsf. It is necessary, also, to be prepared for that change of the 

* The continuators of Cuvier make mention of an example of this kind and propose tlie name 
of ' paramastoid ' for the process (Legons d'Anat. Comp. ii. (1837) p. 312). I have observed 
it in the skull of a New Zealander and in that of an Irishman, preserved in the iluseum of 
Anatomy in Richmond Street, Dubhn. Believing it to be the homologiie of the ' paroccipital ' 
(4), which is developed independently in chelonia and most fishes, I retain that name for it : 
it must not be confounded w ith that ar.gle of the occipital which projects into the ' foramen 
jugulare' in the human skull, and which has received the name of ' processus jugularis,' in 
some systems of anthropotomy. 

+ How essential a correct view of special homology becomes to the appreciation of the 



ON THE VERTEBRATE SKELETON. 199 

connections of the mastoid, which results from the gradual withdrawal, in the 
mammalian class, of the squamosal from the proper cranial walls. With much 
inconstancy of relative size in the mastoid, of which the dugong and the walrus 
offer two extremes, we discern upon the whole a progressive increase in de- 
scending through the mammalian class: in the walrus, for example, the mastoid, 
or petromastoid, forms as large a proportion of the outer lateral walls of the 
cranium as does the squamosal ; and, in the sheep, the removal of the squamosal 
exposes the connection of the petromastoid with the alisphenoid, — a return to a 
relation common in the oviparous vertebrata : it is shown from the inner side 
of the cranium in the sheep, in fig. 7, le and e. The mastoid of the echidna 
(fig. 12, s) presents a most interesting and instructive combination of both the 
modification of expansion and of that of direct union with the alisphenoid (o), 
which is here effected by the mastoid plate independently of the petrosal (le). 
In fig. 12 these characters are well exposed by the removal of the squamosal 
2r, and tympanic as, which retain their primitive independence throughout 
life in the echidna. If now we compare the bone s and le with the carti- 
laginous and osseous mass s and lo in the skull of the human embryo (fig. 11), 
and allow for the change produced in the position of the alisphenoid (e) by 
the gradual withdrawal of the squamosal (27), traceable in the intervening 
forms of mammalia, the special homology of the petromastoids at the two ex- 
tremes of the mammalian class will be obvious and unmistakeable. The bone 
8 and 16 in the echidna, fig. 12, is connected below and behind with the basi- 
occipital and exoccipital (2), behind and above with the supraoccipital (3) and 
parietal (r), in front with the tympanic, the squamosal, and also, as a conse- 
quence of the modified position of the latter and of its own increased deve- 
lopment, with the alisphenoid (e). • All the connections, save that with the 
alisphenoid, are identical with those of 8 and 16 in the human embryo ; and 
the supervening alisphenoidal connection in the echidna affords an additional 
light to the determination of the bone in the lower vertebrata, since it is a 
.consequence of the progressive advance to a lower (oviparous) type, in the 
descent through the mammalian scale. In regard to the essential functions 
of the petromastoid, we find the petrosal portion inclosing the membranous 
labyrinth, and the mastoidal portion giving exit to the blood from the great 
lateral venous sinus and supporting the tympanic*. It will be unnecessary 
to dwell further on the broad and obvious characters by which the homology 
of the bone s and le in the echidna is established with the equally independent 
petromastoid in the sheep and walrus, and with the petromastoid portion of 
the human 'temporal bone.' 

The continuators of the ' Le9ons d'Anatomie Comparee,' influenced by the 
large proportional size of the petromastoid in the echidna and the share 
which it consequently takes in the formation of the cranial parietes, supposed 
it to be the squamosal: — "le veritable temporal, qui n'aurait pour toute 
apophyse zygomatique qu'un tres petit tubercule pres de la facette glenoide," 

higher law of general homology may be learnt from the application by Ciivier of his idea of 
the mammalian mastoid to the refutation of the vertebral theory of the skull. " On a aussi 
trouve quelque rapport entre I'apophyse mastoide qui, dans la plupart des animaux, appar- 
tient a I'occipital, et I'apophyse transverse de I'atlas et des autres vertebres ; sur quoi il faut 
reraarquer que ces rapports sent moindres dans I'homme a certains egards que dans les qua- 
drupedes, puisque I'atlas n'y a ordinairement qu'une echancrure pour le passage de I'artere, 
et que I'apophyse mastoide y appai-tient entiere au roclier." — Resume sur le question — ' Le 
crane est-il une vertebre ou uu compose de trois ou quatre vertebres ?' Lecons d'Anatomie 
Comparee, t. ii. (1837) p. 711. 

* In the article ' Monotremata,' Cyclopaedia of Anatomy and Physiology, 1841; influenced, 
then, by the absence of the external character of the process, I described the petromastoid as 
the petrous bone. 



200 REPORT— 1846. 

op. cil, t. ii. (18S7) p. 377. Tliis tubercle is the rudiment of tlie mastoid 
process, which is so largely developed in birds, and which, in the echidna, 
overhangs the tympanic cavity. There is no glenoid articular surface upon 
the bone s and lo. We find, on the other hand, the squamosal under its proper 
mammalian form and connections, with a long and slender zygomatic process, 
and performing the function, peculiar to the class Mammalia, of supporting 
the mandible by the true glenoid articular surface in the echidna (fig. 12, 27). 

Dr. Kostlin, whose painstaking and minutely accurate description of the 
osteology of the vertebrate skull renders his conclusions as to their homo- 
logies worthy of respectful consideration, concui's with me in regard to the 
squamosal (27) of the monotremes, but regards the bone b-ig in the 
echidna as a dismemberment of the alisphenoid. In no mammal, however, 
do we find the alisphenoid concerned in immediately protecting the semicir- 
cular canals — this is the function of the petrosal: in neither mammal nor 
bird does the alisphenoid extend its connections so far back as to the basi- 
ex- and supra-occipitals. In the echidna, as in every other mammal and bird, 
the alisphenoid (e) exists, exclusively exercising its essential function of trans- 
mitting the third division of the fiith pair by the large vacuity (tr) and with 
its normal connections modified onlj', as in the sheep and some other inferior 
mammalia, through the recession of the squamosal, by joining the mastoid, 
in addition to those which it unites with in man. I confess that I can perceive 
no other gain to anatomy by Dr. Kostlin's new determination of s and le in 
the echidna as 'hintere Abtheilung des Schlafenfliigels ' or 'hintern Schla- 
fenfliigel*' (posterior alisphenoid), than an additional phrase to the synonyms 
of the mastoid. 

The discussion of the homologies of this bone under its modifications in 
the mammalia, and especially in the monotremata, will not be deemed super- 
fluous or too detailed, when it is remembered how valuable a key the cranial 
organization of the implacental monotremes with their bird-like heads becomes 
to the comprehension of the modifications of the cranial structure in birds 
themselves. If we pass from the comparison of the echidna's skull, as re- 
presented in fig. 12, to that of the ostrich (fig. 8), we shall find there a bone 
(s) articulated in front to the alisphenoid (b), behind to the exoccipital (2), 
below to the basi-occipital and basi-sphenoid, above to the parietal 7, and 
coalescing by its inner surface with the petrosal. The sole modification of 
note in regard to connective characters, as compared with the mammalian 
petromastoid, is the loss of the connection with the squamosal, for which we 
have been progressively prepared by the conditions of that bone in rodents, ru- 
minants and monotremes. In the bird this least constant element of the cranial 
walls (fig. 21,27) has undergone a further degradation, is now dismissed en- 
tirely from any share in the formation of even the outer surface of the cranial 
parietes, and is reduced to its mere zygomatic form and function, serving 
exclusively to connect the jugal (fig. 21, 25) with the tympanic (28); which 
function it performs in the echidna and in man, besides other superadded 
offices arising out of its peculiarly mammalian expansion into a scale-like 
lamina, or as compensatory of the reduction of the tympanic bone. Dr. 
Hallmann, however, in his elaborate monograph on the temporal bone, con- 
siders the bone s (fig. 8) to be the squamous or zygomatic element, and cites 
the following characters of the bone, in the young cassowary f, as establishing 
its homology with the squamosal: — "its junction above with the parietal, in 
front with the alisphenoid and post-frontal and behind with the occipital ; also 
its formation of the upper border of the meatus auditorius externus, and its 

* Op. cit. pp. 29, 126. 

t Die vergleichende Osteologie des Schliifensbeins, p. 8. pi. 1. fig. 5. 



ON THE VERTEBRATE SKELETON. 201 

contribution of the articular surface for the tympanic bone," which surface 
he regards as homologous with the glenoid cavity of the squamosal for the 
lower jaw in mammals. 

Cuvier, whose homology of no. s he thus adopts, describes it in the bird 
as being on the outer side of the parietal, advancing also to beneath the 
frontals, occupying the region of the temporal fossa and giving origin to the 
temporal muscle, and as forming the superior border of the tympanic cavity. 
" The temporal fossa," adds Cuvier, " is in great part excavated in the tem- 
poral bone, and is bounded behind by a special process which might be re- 
garded as the analogue of the zygomatic did it not remain far removed from 
the malar bone*." The annotators add, " that there are some species of bird 
in which, nevertheless, such zygomatic process does approach very close to 
the jugal-f-." 

First, then, with regard to the character which appears to have most 
weighed with Cuvier, from his twice citing it in the above brief definition 
of no. 8, — the marks of the origin of the temporal muscle. To conclude that 
the bone impressed by the so-called ' temporal fossa' in the skull of the bird, 
is therefore the temporal bone, because such fossa impresses a bone called 
' temporal ' in the mammal, is an example of that fallacy which logicians call 
arguing in a circle. The two propositions by no means reciprocally prove 
each other. Suppose, for example, that the bone no. s in the bird had been 
determined, bj' way of ascensive comparison from the fish (fig. 5) and cro- 
codile (fig. 16), to be the homologue of the bone no. s in those animals, which 
we will assume to have been rightly called ' mastoid ' by Cuvier, and that he 
had arrived at the determination of no. s in the bird by this surer method, 
than by the descent from placental mammals ; and supposing that, having thus 
recognized no. s as the mastoid, the fossa and muscle with which it is im- 
pressed in the bird had been called ' mastoidal ' instead of ' temporal '; then, 
ascending to the mammalian cranium, Cuvier might with equal reason have 
said that the bone 27, figs. 1 1 and 22, was the ' mastoid,' because it occupied the 
region of the mastoidal fossa and gave origin to the mastoidal muscle. The 
origins of muscles are not, however, sufficiently constant to be included amongst 
the characters of connection or function determinative of special homologies. 
The transference of the * stern o-raastoideus ' from the true mastoid process 
(Man, Carnivoraand Rodentia) to the angle of the mandible (horse), and to 
both this part and the second cervical vertebra (Ruminants), shows that the 
attachments of a muscle must be determined after the recognition of the bone, 
and not the homology of the bone by muscular attachments. With the very 
case in question the uncertainty of the character is illustrated : in the skull 
of the ostrich, for example (fig. 8), the temporal fossa is chiefly formed by the 
conjoined portions of the parietal (7) and alisphenoid(6), which intervene be- 
tween the mastoid (s) and the post-frontal, the mastoid forming not more of 
the posterior part of the fossa than the post-frontal does of the anterior part. 
Dr. Kallmann probably appreciated the unsoundness of the argument from 
the muscular impression, since he does not cite it ; he repeats, however, the 
character adduced by Cuvier, from the relation of no. s to the tympanic 
cavity, or as Hallmann expresses it, the meatus auditorius (aussern Gehor 
offnung), the value of which therefore I next proceed to consider. 

In the skull of the ostrich, with the tympanic bone and ear-drum in place, 
the upper border of the meatus, as defined by the periphery of the membrana 
. tympani, is formed, not by no. s, but by the tympanic anteriorly, and by the 
paroccipital process (4) posteriorly. When the tympanic bone and mem- 
brane are removed, then the descending process of no. s overarches the 

* Lefons d'Anat. Comp. u. (1837), p. 580. f ^*- P- 581. 

1846. P 



202 REPORT— 1846. 

upper and forepart of the tympanic cavity so exposed. So much for the 
facts of the argument*. 

We may next ask, Is the formation of the upper boundary of the meatus 
externus an essential character of the squamosal in mammals; or is it not 
rather a secondary consequence of the expansion and application of that bone 
to the side of the cranium in this particular class? If we were desirous of 
obtaining a homological character by comparison of the contour of the 
meatus externus or the tympanic cavity in mammals and birds, ought we 
not rather to select tli€ lowest and most ornithoid of mammals, as best cal- 
culated to throw light upon the real nature of the modifications of this part 
of the skull in the respective classes? In the echidna, then, we find that 
the squamosal does not form the whole of the superior border of the shallow 
tympanic cavity, but that the mastoid forms the posterior half of that border, 
and sends a short obtuse process downwards (at le, fig. 12), which overhangs 
the cavity and gives attachment to the tympanic (2s). Behind the mastoid 
is the exoccipital. Now in birds the antero-posterior extent of the cranium 
between the exoccipital and post- frontal bones is much shortened as compared 
with mammals, and this modification I interpret as the result, in a great de- 
gree, of the entire removal of the squamosal from the cranial parietes. Of 
the homology of no. 4 as a part of the exoccipital there has been no question, 
although its development, and the share it takes in the lateral parietes of the 
head, is increased, as compared with most mammals, rather than diminished. 
The exoccipital constantly unites anteriorly with the mastoid in mammals, 
from man down to the echidna; but the extension of the squamosal back- 
wards to articulate with the exoccipital is far from being a constant character 
in mammals. We ought on that ground therefore to conclude that the bone 8, 
which articulates with the fore-part of the exoccipital in the bird, is the 
' mastoid,' rather than that it is the ' squamosal.' It overhangs the tympanic 
cavity by a longer or shorter process ; but being more advanced in position, 
partly by the development of tlie exoccipital behind, and the non-interposition 
of a squamosal between it and the alisphenoid in front, it overarches the 
middle of the upper instead of the posterior part of the upper border of the 
tympanic cavity in the bird ; but it is still in great part posterior to the tym- 
panic pedicle, a relative position which is foreign to the squamosal. The 
process of no. s resembles the mastoid process in mammalia, inasmuch as 
it terminates freely in most birds ; and in those, the parrot for example, in 
which it joins another process to form a zygoma or bridge over the temporal 
fossa, that process answers to the post-frontal, the very bone which the mas- 
toid similarly joins in the crocodile, and does not answer to the malar bone, 
which the squamosal joins in both mammals and crocodiles. 

The mastoid always coalesces with the petrosal, rarely with the squa- 
mosal, in the mammalia ; such coalescence is therefore a more constant cha- 
racter of the mastoid than of the squamosal, and the argument becomes 
cumulative in favour of the mastoid or petromastoid character of no. s in the 
bird. When we remove the squamosal in the sheep we bring away the man- 
dible which articulates with it, but we leave the distinct and independent tym- 
panic closely articulated to the petromastoid. Precisely the same thing 
happens in the rodentia, in the marsupialia, and especially in the echidna, 
in which the tympanic has the slightest connection with the squamosal. The 
articulation of the tympanic therefore with the petromastoid is a more con- 
stant character than its articulation with the squamosal ; therefore the arti- 
culation of the unquestioned tympanic bone in birds with the bone no. s is a 

* The same formation of the upper boundary of the meatus externus is shown by Geoflfroy 
in the young fowl. — Annales du Museum, x. pi. 27. fig. 2. V. Q. 



ON THE VERTEBRATE SKELETON. 203 

stronger proof of no. 8 being the petromastoid than of its being the squamosal : 
and for the same reasons that the articulation of no. s with the exoccipital, and 
its coalescence with the petrosal, are more essential characters of the petro- 
mastoid than they are of the squamosal, so I regard the articular surface 
furnished by no. s to the tympanic bone to be homologous with the articular 
surface of the petromastoid for the tympanic in the ruminants, rodents 
and other mammals, and am compelled to dissent from Dr. Kallmann's idea 
of its answering to the articular surface furnished by the squamosal to the 
mandible in mammals. In the ostrich a part of the articular cavity for the 
tympanic is excavated in the exoccipital, and would afford as good an argu- 
ment to prove that bone to be the squamosal as the one which Dr. Kallmann 
has deduced from the same character in favour of the petromastoid in the 
bird being the squamosal. Dr. Kallmann cites the junction of no. s (his t, 
taf. i. fig. 5, op. cit.) with the post-frontal in a young cassowary as evidence 
of its squamous character. I have not met with this union in the young 
ostrich nor in the young emeu, in which latter bird there is a distinct post- 
frontal : the anterior inferior angle of the parietal descends and meets the 
alisphenoid in both these StruthionidcB, at the part where the post-frontal is 
marked (a.tf") in Dr. Kallmann's figure above cited. The extremity of the 
mastoid process does, however, arch over the temporal fossa to join the post- 
frontal process in certain birds, as above mentioned ; but this junction, when 
we ascend in our pursuit of the homologies of the elements of the composite 
temporal bone of mammals, as it is safest to do, from fishes to reptiles, 
and from these to birds, forms a repetition of a very characteristic feature 
of the mastoid in the cold-blooded classes, and one that is quite intelligible 
when we rise to the appreciation of the higher relations of both mastoid and 
post-frontal as parapophyses of their respective vertebrae. 

In every mammal the squamosal is applied to the cranial parietes, and at- 
tached by a peculiar suture called squamous ; the outer surface of the bone 
exceeding the inner surface. In no bird is the mastoid so united to the sur- 
rounding bones, but joins them by harmoniae vertical to the surface, as the 
other true cranial bones are joined before they coalesce; and the outer very 
little, if at all, surpasses the inner surface, to which the petrosal is confluent. 
The petromastoid of the mammal resembles that of the bird in this respect. 

There is no difficulty in the ascensive survey in appreciating the special 
homology of no. s in the bird (fig. 23) with no. s in the crocodile (fig. 22) 
and in the fish (fig. 5) ; and Dr. Kallmann, retaining a firmer and more 
consistent view of their common characters than Cuvier, enunciates clearly 
this homology : but having persuaded himself that the ' mastoid ' of the bird 
was its ' squamosal,' he concludes that the bone which Cuvier had called mas- 
toid in the crocodile and fish must also be their squamosal. I believe Cuvier 
to have rightly determined the bone (no. s) in the cold-blooded classes to be 
the mastoid ; but he is not consi^tent with himself when he adopts a different 
conclusion with regard to no. s in the bird. The greater development of 
the bird's brain, as compared with the crocodile's, requires a greater expan- 
sion of the cranial part of the mastoid, just as the still greater development 
of the brain in mammals calls forth a peculiar expansion and application of 
the cranial end of the squamosal, involving a transference of the mandibular 
joint to that expanded end. 

Cuvier, in descending from mammals to the consideration of the homolo- 
gies of no. 8 in the bird, passed too abruptly to the comparison, lacking the 
instructive link furnished by the monotremes. It might have sufficed for 
the present report to have demonstrated the homology of no. s in the bird, 
ascensively, with Cuvier's well-determined mastoids in fishes and reptiles; 

p2 



204 REPORT — 1846. 

but since both Cuvier and Dr. Hallmann have elucidated their views of its 
homology by characters drawn from the mammalian class, I have endeavoured, 
and I trust satisfactorily, to meet their objections and to determine the true 
homology of the bone by other arguments drawn from modifications of the 
petiomastoid in the same class. 

Pursuing therefore the comparison descensively, I proceed in the next place 
to consider the characters of the mastoid in the crocodile (figs. 19 and 22, s). 
Cuvier premises his determination of the bone in that reptile by citing the 
following as its characters in the mammalia : — " La partie niasto'idienne qui 
recouvre le rocher en arriere de I'ecailleuse et de la caisse, mais qui se sonde 
de si bonne heure a ce rocher que Ion paroient a peine a la reconnaitre 
comme distincte dans les plus jeunes fetus ou elle est quelquefois double*." 
The squamosal he defines as a bone " qui devient de plus en plus etrangere 
au crane a mesure qu'on descend dans I'echelle des quadrupedes, en sorte 
que dans les ruminans elle est plutot collee dessus qu'elle n'entre dans la 
composition de ses paroisf." If we pause to apply these characters to the de- 
termination of nos. 8 and it respectively in the bird, before proceeding to 
the crocodile, we shall see how far they sustain the conclusions I have ar- 
rived at, in opposition to the views of Cuvier and his followers, in reference 
to the true homologue of the mammalian squamosal in birds. With regard 
to the mastoid in the crocodile, Cuvier says, " Le mastoidien des crocodiles 
proprement dits et des gavials a cela de particulier, qu'il s'avance laterale- 
raent jusqu'a s'unir au frontal posterieur, et a entourer avec lui et le pari- 
etal le trou de la face superieure du crane qui communiq!ie avec la fosse 
temporale ; dans quelques caimans il s'unit meme a ces trois os pour couvrir 
entierement cette fosse en dessus, et dans les tortues de mer, non-seulement 
lis font la meme chose, le temporale et le jugal venant aussi a s'unir au mas- 
toidien et au frontal posterieure, ils eouvrent la fosse temporale, meme par 
dehors." J 

Doubtless the German anatomists who dissent from Cuvier's determination 
of the bone s in the crocodile (fig. 22) have been influenced in some degree 
by the little conformity between the character above assigned to the mastoid 
in that reptile and the character Cuvier had previously assigned to the mas- 
toid in mammalia. The confluence of the mastoid with the petrosal, for 
example, is a modification peculiar to the warm-blooded vertebrates, whilst 
the relative position of the mastoid, above and external to the petrosal and 
behind the squamosal and tympanic, is a constant character in all vertebrates; 
to which nmst be added, that in most mammals and all other vertebrates the 
mastoid affords an articular surface for the tympanic bone, and developes an 
outstanding (mastoid) process for the attachment of strong muscles moving 
the head upon the trunk. 

With regard to the relative position of the mastoid process to the cranial 
walls, its origin ascends as the expansion of the parietal diminishes with the 
decreasing size of the cerebrum : in mammals, the process, when present, 
extends from the lower border of the postero-lateral wall of the cranium : 
in birds it projects from near the middle of that wall, and nearer the upper 
surface in the flat-headed Dinornis: in the crocodile it has ascended to a 
level with the upper surface of the cranium, and forms the posterior angle of 
that surface. The paroccipital presents a similar progressive ascent, but later 
in the series traced descensively ; it does not gain the level of the mastoid 
until we arrive at the class of fishes. 

The mastoid, thus determined in the crocodile, is recognized with ease 
and certainty in chelonia, lacertia and ophidia. It is a distinct bone in all 
* Op. cit. t. V. pt. ii. p. 81. t lb. p. 81. % lb. p. 84. 



ON THE VERTEBRATE SKELETON. 205 

these reptiles, and preserves with singular constancy its normal relative po- 
sition anterior to the exoccipital, superior to and supporting the tympanic, 
and anterior to the squamosal when this is present. In lizards the mastoid 
is much reduced in size : in serpents it attains a considerable length. In the 
python and most serpents it forms no part of the proper wall of the cranium, 
but overlaps the contiguous parts of the parietal, alisphenoid, supra-occipital, 
and exoccipital, projecting backwards beyond the latter. It is large in the 
serpentiform batrachia, but presents in CcBcilia (Cuvier, Regno Animal, 1817, 
pi. 6. figs. 1 & 2, ^) its normal connections with the occipital (/), parietal 
(e), tympanic (Ji), and also with the post-frontal, which has coalesced or is 
connate with the frontal (at d, 1. c). Cuvier does not admit of this conflu- 
ence in the caecilia ; and although he assigns the character ' point des fron- 
taux posterieures' to the typical batrachia*, gives the name ' posterior frontal ' 
with a note of doubt, indeed, to g, and assigns to the bone h, which suspends 
the mandible, the name of "mastoidiens et caisses reunisf." There is no 
actual necessity for assuming so rare a confluence to characterize the caecilia. 
The mastoid exists with all its normal connections, and beautifully manifests 
by its independence and large size the affinity of the caecilia to the true 
ophidia. In the typical batrachia, where the cranium is remarkably cha- 
racterized by instances of confluence which seem borrowed from the warm- 
blooded classes, the mastoid sometimes loses its independence, and appears 
as an exogenous process from the external and posterior part of the parietal, 
retaining however its normal office of suspending the tympanic : but in a skull 
of the Rana hoans now before me, the suture between the mastoid (fig. 13, s) 
and parietal (7) is not obliterated, and it further articulates with the exocci- 
pital (3) behind and the alisphenoid (e) in front. Cuvier, in his description of 
the tympanic of the Rana esculentaX, says, that its upper branch articulates 
with the ' rocher.' In Rana loans that branch articulates exclusively with 
the truncated extremity of the broad outstanding mastoid, which mastoid 
overhangs, as in all fishes, the petrosal, which is chiefly cartilaginous in the 
Rana boons (ih. id). In Rana esculenta the mastoid (Duges, Recherches 
sur les Batrachiens, fig. 1, 12) appears to have coalesced with the alisphenoid 
(ib. figs. 2, 6 & 7, 12) ; and the compound bone has received the name of 
'rocher' from Cuvier and that of ' rupeo-ptereal ' from Duges. The fora- 
men ovale however marks the alisphenoidal part (a distinct bone in my Rana 
boans), and the suspension of the tympanic marks the mastoid, which, with 
its other connections, overhangs also in Rana viridis that mass of cartilage § 
which immediately invests the membranous labyrinth and forms the ' fenestra 
ovalis' against which the plate of the columelliform stapes is applied. 

Prof. J. MuUer has well recognized the homologue of this sense capsule in 
the CcBcilia hypocyanea, in which he describes it as " petrosum cum operculo 
fenestras ovalis Ij." It is situated further back than in Rana, and appears poste- 
rior to the tympanic («) and the large suspending mastoid (Ji), to which Muller 
gives the name of ' temporale.' In the singularly modified cranium of the 
Tythlops the mastoid articulates above with the parietal and supraoccipital, 
behind with the exoccipital, coalesces in front with the alisphenoid, as in 
some batrachia, and aff'ords the usual articulation below to the tympanic. 
How necessary it is to retain a clear and consistent appreciation of these evi- 

* Ossem. Fossiles, v. pt. i. p. 386. f Regne Animal, ed. 1817, t. iv. p. 102. 

X Ossem. Fossiles v. pt. ii. p. 390. 

§ The precocious development of this capsule in the larva of the frog is well shown by 
Reichert, ' Entwickelungsgeschichte des Kopfes,' 4to, pi. i. figs. 13 — 15, ^r : it resembles 
that in the myxinoids and lampreys. 

II Beitrage zur Anatomie der Amphibien ; Tiedemann's Zeitschrift fur Phvsiologie, 
Bd. iv. 1831, p. 218, pi. 18. tig. v. *. 



206 RKPORT — 1846. 

dences of the homology of the mastoid is shown by the second synonym, ' os 
petrosum,' which it has received from the justly-celebrated author of this 
instructive memoir (pi. 20. figg. 10, 12, 13, H,/>). The actual capsule of 
the membranous labyrinth is covered by the mastoid and exoccipital, and 
remains wholly cartilaginous, as in other ophidia ; and as it likewise does in 
Rhinophis, where its name ' petrosum ' is in like manner transferred by Prof. 
MuUer to the coalesced mastoid and alisphenoid. In Cheirotes the course 
of confluence proceeds to obliterate not only the suture between the mastoid 
and alisphenoid, but that between the mastoid and parietal ; as also of those 
between the frontal, parietal and supra-occipital; the whole cranium pre- 
senting almost the extent of coalescence which characterizes the hot-blooded 
bird. Only the immediate covering of the membranous labyrinth remains 
cartilaginous. 

The sides of the superior surface of the cranium of bony fishes usually 
extend outwards as a strong irregular ridge, from which three processes more 
particularly project, which are supported by three distinct bones, suturally 
united, and each impicssed with an articular glenoid cavity. And here I 
cannot avoid remarking how beautifully the principle of vegetative repe- 
tition* is exemplified in the lowest class of the Ve cebrata, where conse- 
quently the relations of serial homology of the parapophyses in question are 
unmistakeable. The posterior process or bone which sustains (in part) the 
scapular arch is the paroccipital (fig. 5, 4) ; the anterior one, which sustains 
in part the tympano-mandibular arch, is the post-frontal (i6. 12) ; and the 
intermediate and usually most prominent bone (ib.s), which sustains in part 
the epitympanic (-isa), and through that the hyoid arch, is the homologue of 
the bone whose essential characters have been discussed under the name of 
' mastoid.' The paroccipital having now risen to a level with the mastoid, 
this forms the second strong transverse process at each side of the cranium. 
The process is developed from the outer margin of the mastoid ; the inner 
side of the bone is expanded, and enters slightly into the formation of the 
walls of the cranial or rather the otocranial cavity, its inner, usually cartila- 
ginous surface lodging the fibro-cartilaginous continuation of the petrosal 
which immediately covers the external semicircular canal. It is wedged into 
the interspace of the ex- and par-occipitals, the petrosal, the alisphenoid, the 
parietal and post-frontal bones. The projecting process lodges above the 
chief mucous canal of the head, and below affords attachment to the epi- 
tympanic or upper piece of the bony pedicle from which the mandibular, 
hyoid, and opercular bones are suspended : its extremity gives attachment to 
the strong tendon of the dorso-lateral muscles of the trunk. 

It might have been supposed that this contribution to the walls of the 
cranial cavity, this articulation to the occipital and tympanic bones, all of 
which are constant characters of the mastoid in mammals, and but occasional 
ones in the squamosal — not to speak of the apophysial form and functions of 
the bone in question in the skull of fishes — would have made the balance in- 
cline to the choice of the ' mastoid ' rather than of the ' squamosal ' elements 
of the human temporal in the judgement of every unbiassed investigator of 
its homologies. The German anatomists, however, in falling with Cuvier 
into the mistake respecting the homology of the 'mastoid' (no. s) in birds, 
with the squamosal in mammals, adhere more consistently to their error and 
continue to apply the name 'squamosal' or its equivalents to the homologous 
bone in reptiles (fig. 22, s) and fishes (fig. 5, s). 

* This principle or law is explained in the first volume of ray Hunterian Lectures ' On the 
Itivertebrata,' in wliich classes of animals it is necessarily most strikingly and fully exem- 
plified. 



ON THE VERTEBRATE SKELETON. 207 

The high repute which M. Agassiz has so justly earned in ichthyotomy 
renders the accession of his name in support of Drs. Hallmann, Reichert, 
and Kostlin's determination of the bone in question, one to which those able 
homologists and their followers will naturally attach great weight, and which 
indeed has caused me to pause and retrace more than once, and with the 
utmost pains and care, every step in the series of comparisons which have 
finally brought conviction of the accuracy of the Cuvierian determination of 
no. s in fishes. 

I am not aware that any anatomist has replied to the objections to the 
Cuvierian view propounded by M. Agassiz. Drs. Hallmann and Kostlin, 
who have published the most elaborate monographs on the temporal and 
other bones of the skull since the time of Cuvier, concur entirely with the 
learned Swiss naturalist. Dr. Reichert, in giving the name of ' squama tem- 
poralis' to no. 8, and that of 'processus temporalis posterior' to its process, 
transfers the name ' processus mastoideus' to the paroccipital (no. 4, fig. 5)*. 
It becomes then necessary to consider the arguments of M. Agassiz in favour 
of the homology of no. s. in fishes with the squamosal no. 27 in mammals. 
In the valuable monograph on the osteology of the pike (Esox) in the 15th 
'Livraison' of the ' Recherches sur les Poissons Fossiles,' the author says 
(p. 66), " Un OS de la tete plac6 entre le frontal posterieur, le frontal prin- 
cipal, le parietal, la grand aile sphenoidale et Foccipital lateral, ne saurait 
jamais etre envisage comme correspondant a I'apophyse mastoidienne du 
temporal. D'apres ses liaisons, je crois done qu'il faut envisager le mastdidien 
de Cuvier comme I'analogue de 1' ecaille du temporal ou comme le temporal 
proprement dit. C'etait deja I'opinion de Spix, qui est tombe juste sur ce 
point." To this I reply that, in regard to the connections of the mastoid, those 
with the parietal, alisphenoid and exoccipital, are more constant than that 
with the frontal, which is interrupted in mammalia by the interposition of 
the expanded squamosal, peculiar to that class ; but the mastoid retains its 
piscine connection with the postfrontal in many reptiles and some birds. On 
the other hand, the union of the squamosal with the frontal is by no means 
a constant character in mammalia : it is rarely found in the orang, still more 
rarely in man, never in the cetacea and monotremes, nor in certain ruminants, 
nor in the myrmecophaga, &c. The connection of the mastoid with the 
frontal is more common than is the connection of the squamosal with the 
exoccipital. It is a bold leap to take from the mammal to the fish in the de- 
termination of a variable bone like the squamosal : nevertheless, 1 would re- 
quest the unbiassed reader to glance at fig. 12, whilst he reads M. Agassiz's 
precis of the character of the squamosal above cited, and see how far no. s de- 
viates from it, save in regard to the frontal connection. Spix, who appears 
not to have traced the beautiful gradation of the mastoid in the mammalia, 
and who was unacquainted with the decisive step to its normal condition in 
the oviparous vertebrates made by the monotremes, — and who was influenced, 
therefore, by seeing that bone in higher mammals pushed back from any con- 
nection with the alisphenoid and postfrontal by the interposed squamosal, 
which usurps these connections and combines them with others, as with the 
parietal and tympanic, which the mastoid (no. s) presents in fishes, — not un- 
reasonably concluded that no. 3 represented the squamosal in that class ; and 
it is probable that M. Agassiz, who received his anatomical rudiments at 
Munich, and was early engaged in describing the fishes collected in Brazil by 
the author of the ' Cephalogenesis,' might have derived a bias in favour of this 
view which prevented his assigning their due value to the connection of no. s 
in fishes with the paroccipital, and its contribution to the otocranial cavity. 
* Op. cit. tab. ill. figs. 9 and 13,/), q. 



208 REPORT — 1846. 

lu urging a reconsideration of the value and significancy of these charac- 
ters, I may repeat that in mammals the mastoid constantly presents them, 
whilst the squamosal very rarely has the first, and not often the second cha- 
racter. It must also be remembered that the squamosal loses its connection 
with the frontal and progressively decreases in the manmialian class to less than 
the dimensions of the mastoid itself, as e. g. in echidna (fig. 12), whilst in this 
monotreme the mastoid, s, besides its connections with the parietal and exocci- 
pital, extends forwards to articulate with the alisphenoid, 6. If ossification 
were restricted in mammals to no. s, fig. 11, in reference to i6, which re- 
mained cartilaginous, then no. s would have the same relation to the otocrane, 
or in other words, would contribute the same protection to the acoustic laby- 
rinth, which no. s, fig. 5, performs in fishes; the external semicircular 
canal at least would be protected in the mastoid by both : only in mammalia 
the mastoid would also extend over the posterior canal. The petrosal loses 
no part of its essential character as the capsule or outer tunic of the laby- 
rinth by becoming ossified, nor is it less recognisable in fishes within the 
mastoid, by remaining membranous or cartilaginous, than is the sclerotic 
capsule of the eye in its chamber or orbit ; which capsule, in like manner, 
presents all the corresponding histological modifications in one or other part 
of the vertebrate series. The mask which has concealed the true features of 
resemblance in the human mastoid to that of fishes, is simply the petrosal 
ossified and cemented to it. But the squamosal presents no such relations to 
the bony capsule of the semicircular canals in any mammal. Even the 
connection of the squamosal with the tympanic bone is, as we have seen, far 
less constant and intimate in mammals than the connection of the mastoid 
with the tympanic*. 

In the anatomical description of the existing ganoid fishes which M. 
Agassiz has unfortunately called ' Sauroidf,' the bone no. s is described as 

* From the remark in p. 53, t. ii. pt. ii. ' Rechercbes sur les Poiss. Foss.,' it vrould seem 
that the circumstance of the extension of the tympanic air-cells into the mastoid, in certain 
mammalia, had weighed with M. Agassiz in determining its homological characters. 

t All the characters by which these highly organized fishes approximate the Reptilia are 
found, not in the highest, but in the lowest order of that class, viz. in the batrachia, and herein 
more especially in the salamanders. The air-bladder of Lcpidosteus resembles the lung of 
the serpent in its singleness, and those of the salamander in the degree of its cellularity ; 
some parts of the structure being peculiarly piscine. The bifid air-bladder of Polypterns 
resembles the lungs of the salamandroid menopome and proteus, in the want of cellular 
walls. The characteristic large bulbus arteriosus and its numerous rows of valves, which 
distinguish the ganoids from most other osseous fishes, are retained in the menopome, but 
are not present iu any siiurian. The anterior ball and posterior cup of the vertebrae of Le- 
pidosteus are repeated in the salamander and pipa, but in no existing saurian. The laby- 
rinthodont character of the teeth of Lcpidosteus was developed to its maximiim in the great 
extinct reptiles {Salamandroides, Jager), which, by their double occipital condyle, denti- 
gcrous double vomer, and biconcave vertebrae, were essentially Batrachia, not Sauria ; and 
which combined characters now found in the lower salamandroid Batrachia, with the dental 
ones borrowed from fishes, and but feebly manifested by the most fish -like of saurians 
(Ichthyosaurus). All the so-called sauroid fishes retain the characteristic piscine articular 
concavity on the basioccipital for the. atlas : it is, however, very shallow in the polyptenis ; 
and is also extended transversely, with the lateral borders or angles so prominent, that, as 
M. Agassiz well remarks, " it needs very little to change this transverse articulation with its 
two lateral ridges into two distinct articular condyles," /. e. p. 71. But this would convert, 
pro tanto, the polypterus into a batrachian, not into a saurian. So far as the character of a 
single convex occipital condyle is valuable as a mark of affinity to the Sauria, it is present 
in a fish of a different order from the ganoids, and with much fewer approximations in other 
respects to the reptilian class, viz. in the Fistularia tabaccaria. There remains, therefore, 
only the character of the enamelled scales which the polypterus and lcpidosteus present in 
coramon with all the lower organized ganoids, and which to a certain extent resemble the 
bony scutes of the crocodilia. If the deposition of calcareous matter in and upon the skin 
were not essentially a retention of a very low type of skeleton ; if it were not presented by 



ON THE VERTEBRATE SKELETON. 209 

taking part, by its large size, in the formation of both the internal and ex- 
ternal surfaces of the cranial* box, which size depends essentially on the 
degree of development of the frontals, parietals and occipitals : it is further 
urged that the suborbitals ('apophyse jugale') are likewise attached to it; that 
the preopercular(' apophyse styloide') diverges, and is directed or abuts against 
it ; that, finally, the bone in question (no. 8, fig. 5) is, with the exception of the 
petrosal, the sole part of the temporal bone which takes a direct part in 
the formation of the cranial box. " D'apres ces considerations," M. Agassiz 
proceeds, "il est impossible de prendre I'os No. 12 [no. s, in fig. 5], que 
Cuvier a uomme mastdidien, pour autre chose que pour la veritable 6caille du 
temporal. II prend part a la formation de la boite cerebrale, il donne inser- 
tion a I'arcade zygomatique, enfin, il prete une articulation au preopercule, 
que nous regardons maintenant comme le veritable representant de I'apo- 
physe styloide du temporal," I. c. p. 63. Admitting, for the sake of the argu- 
ment, that the preopercular is the homologue of the stylohyal, and that it arti- 
culates with the so-called ' ecaille du temporal,' which is not the case in the 
majority of fishes, yet this would prove more for the 'mastoid' than for the 
' squamosal' character of no. s, fig. 5. The stylohyal unquestionably articu- 
lates in many mammals with the mastoid or petromastoid, between which 
and the tympanic it is anchylosed in man, and it rests with M. Agassiz to 
demonstrate the species in which it articulates with the true squamous part 
of the temporalf. 

With regard to the connection with the suborbital chain of ossicles, which 
M. Agassiz regards, with Geoffroy, as the jugal or zygomatic arch, even 
admitting such connection to be the rule and not the exception, all its 
force as an argument in favour of the squamosal character of no. s will 
depend on the ultimate decision of comparative anatomists as to the respect- 
ive claims of the upper and lower zygomata in the macaw's skull, for 
example, to a special homology with the zygomatic arch in man and other 
mammals. The orbit in the bird cited, as in other Psittacidce, is circum- 
scribed below by a bony frame continued from the lacrymal to the post- 
frontal, and thence to the bone (no. s) which I regard as the mastoid. 
Below this frame, the slender bone, considered by Cuvier as the jugal, and 
by me as the coalesced jugal (26) and squamosal (27, fig. 23), extends from 
the maxillary (21) backwards to the tympanic (23), and forms a second arch 
or zygoma. According to the Cuvierian and generally-received view of the 
homology of no. s in the bird, the bridge which it sends forward over the 
temporal fossa to join the above-described inferior boundary of the orbit, 
in the macaw, would be the zygomatic process ; and that boundary would be 
what M. Agassiz calls its homologue in fishes, viz. the jugal or ' arcade zygo- 
matique.' But what then is the parallel zygomatic arch below, connecting 

many fishes of different grades of organization, and by some, as the sturgeons and siluroids, 
e. g. under a scattered arrangement, more like that in the crocodiles than is seen in the scale 
armour of the typical ganoids, it might have some weight in proving the affinity of such 
ganoids to the highest order of reptiha ; but, viewing this character under all its relations, 
I am not disposed to regard it as establishing that affinity more directly, than it would the 
affinity of the crocodile to the mammalian genus Dasypus. It is for the reasons above assigned 
that I have been accustomed to treat, in my Lectures, of the anatomical characters of the 
group represented by the Polij]iterus and Lepidosteus, as those of a Salamandroid, rather than 
of a Sauroid family of fishes ; the characters being carried out in the direction of the batra- 
chian order by the remarkable genera Protopterus and Lepidosiren. 

* More properly ' otocranial,' in lepidosteus at least. 

t In my notes on the osteology of Mammalia, I find that the stylohyal sometimes artieu- 
lates with the petrosal, sometimes with the mastoid, exclusively, as in most mammals, 
sometimes with the tympanic, sometimes with the paroccipital process : but no instance is 
recorded of its articulation witii the squamous portion of the temporal. 



210 REPORT — 1816. 

the maxillarj' with the tympanic, and marked «™' in fig. 7, taf. i. of Dr. Kall- 
mann's monograph ? If Cuvier had been correct in regarding no. 8 as the 
squamosal, the name 'jugal' ought to have been transferred from the lower 
zygoma to the upper one connected with such squamosal in the macaw : and 
with a like consistency the name '.jugal' ought to have been retained for the 
suborbital chain of dermal bones in fishes, to which it had been applied by 
Geoffroy St. Hilaire, and to which it has been restored by M. Agassiz. But, 
in truth, there may be clearly discerned in the beautiful modification which 
has been adduced from the Psittacidce, a proof of Cuvier's erroneous homo- 
logy of the bone no. 8 in the class of birds, and at the same time of his 
accurate homology of the same bone in that of fishes. 

Is there no significance in the fact of the bone anterior to the orbit, which 
we call lacrymal in man down to the lowest reptile, being constantly per- 
forated by a mucous duct ? Can we not recognize in this function and 
glandular relation, as in the commonly thin scale-like character of that bone, 
and its connections in front of the orbit, the repetition of the characters of 
the largest, most anterior, and most constant of the suborbitals in fishes ? If 
the rest of that chain be sometimes wanting, but more commonly present in 
that class ; if it should present the condition occasionally of a strong conti- 
nuous bony inverted arch, spanning the orbit below from prefrontal to post- 
frontal, as in the right orbit of the Hippoglossus and the left orbit of Rhombus; 
ought we to lose our grasp of the guiding thread of ' connections' by being 
confronted with a repetition of that condition in the skulls of certain birds, 
caused by a continuous ossification from the lacrymal to the post- frontal, 
seeing that a diverging bony appendage of the maxillary arch, unknown in the 
class of fishes, has there established a second and true ' zygoma' below the 
suborbital one ? The extension of the ossification from the post-frontal crus 
of the suborbital arch to the mastoid is, in truth, a beautiful repetition of an 
ichthyic cranial character, not unknown however in the reptilia ; and whilst 
it adds a proof of the mastoidal character of no. 8 in the bird, it reflects 
reciprocal confirmation of the accuracy of Cuvier's determination of that 
bone in fishes. 

The true signification and homologies of the bones in that interesting 
class could never have been elicited from an exclusive study of it, however 
extensive, detailed or profound ; nor will the feeble rays reflected from an- 
thropotomical reminiscences lend sufficient light in their determination : they 
can be clearly discerned only by the full illumination of the beams concen- 
trated from all the grades of organic structure. M. Agassiz, descending to 
the determination of the squamosal in fishes from its characters in man, con- 
cludes that it must be the bone no. s, fig. 5, because that bone takes part in 
the formation of the inner as well as the outer walls of the cranial cavity. But 
this protective function is an exceptional one in the squamosal (fig. 6, 27); 
it is peculiar to that bone only in one class, and, as we have seen, is not con- 
stant even there ; whilst, on the other hand, the mastoid is recognizable 
from the inner surface of the cranial walls of the highest mammal (in the 
human cranium where it is impressed with the fossa sigmoidea, fig. 6, s), and 
in a still greater degree in that of the lowest mammal (^Echidna, fig. 12, s) ; 
whilst in almost every mammal, by its coalescence with the outer surface of 
the petrosal, it closely repeats the protective character in relation to the ex- 
ternal semicircular canal, which it presents in fishes, — a function which is 
altogether foreign to the squamosal in every mammal. I have dwelt thus 
long, perhaps tediously, and it may be thought unnecessarily, on the true 
characters and homologies of the petrosal and mastoid. But their determina- 
tion is essential to, and, indeed, involves that of the squamosal and other 



ON THE VERTEBRATE SKELETON. 211 

dismemberments of the human temporal bone ; and we cannot climb to the 
higher generalizations of anatomical science, except by the firm steps of true 
and assured special homologies. There are more important subjects than 
homologies, no doubt ; but nothing is more important than truth, in whatever 
path we may be in pursuit of her. 

Orhitosphenoid. — As evidence will be given in the section on ' General 
Homology' that both squamosal and tympanic belong to a quite distinct 
category of bones from the parts of the 'temporal' which have just been 
discussed. I shall proceed next to the neurapophyses that precede the 
alisphenoid. 

As the determination of this bone (e in all the figures) involves that of 
the orhitosphenoid (lo), which has rarely been mistaken* for any other bone 
than 6, there remains little to be added in proof of its homology after 
what has been advanced respecting the alisphenoid. The most constant 
character of the orbitosplienoid is its relation to the optic nerve, which either 
perforates or notches it, whenever the ossification of the primitive cartilage 
or membrane holding the place of the bone is sufficiently advanced, which 
is not always the case in fishes, especially those with broad and depressed 
heads, and still more rarely in lacertine saurians. The recognition of the 
orbitosphenoid is also often obscured by another cause, viz. the tendency in 
the class Reptilia, and especially in ophidians and chelonians, to an extension 
of ossification downwards into the primitive membranous or cartilaginous 
neurapophysial walls of the brain-case, directly from the parietal and frontal 
bones. 

In the fishes with ordinary-shaped, or with high and compressed heads, 
the orbitosphenoids are usually well-developed : they are, however, repre- 
sented by descending plates of the frontal in the garpike ; and they are, like the 
alisphenoids, mere processes of the basisphenoid in the polypterus, which thus 
oflTers so unexpected a repetition of the human character of the correspond- 
ing parts f. In the cod (fig. 5, lo) they are semielliptic, raised above the pre- 
sphenoid (9), suspended, as it were, between the alisphenoid (e) and the 
frontal (n), and bounding the sides of the interorbital outlet of the cranium : 
the optic nerves pierce the unossified cartilage closing that aperture, imme- 
diately beneath the bone itself. In the malacopterous fishes with higher 
and more compressed heads, the orbitosphenoids are more developed ; they are 
directly pierced or deeply grooved by the optic nerves, and are pierced also 
by the ' nervi pathetici' in the carp. The crura of the olfactory ganglions 
(rhinencephala) pass out of the interorbital aperture of the cranium by the 
upper interspace of the orbitosphenoid, into the continuation of the cranial 
cavity which grooves the under surface of the frontal, in their course between 
the orbits to the prefrontals. The orbitosphenoids protect, more or less, the 
sides of the prosencephalon ; and this function, their transmission of the optic 
nerves, their anterior position to the alisphenoids, and their articulation 

I above with the frontals, establish their special homology from the-fish up to 
man. 
In certain fishes a distinct centre of ossification is set up in the median 
line of the fibrous membrane or cartilage, closing the interorbital aperture 
of the cranium, below the orbitosphenoids, and extending forwards as the in- 
terorbital septum. The bone (represented in outline in fig. 5, at 9') extends 
downwards to rest upon the presphenoid {ib. 9), and bifurcates, as it ascends, 
iioi 
tur 



* Geoffroy in his memoir on the skull of birds (Ann. du Mus. x.), indicates the orbitosphe- 
'noid at P, fig. 2, pi. 27, as the 'rocher': and Cuvier describes it as part of his 'os en cein- 
ture ' in anourous batrachia. 
t Agassiz, Recherches sur les Poissons Fossiles, ii. p. 38. 



212 REPORT — 1846. 

to join and prop up the elevated orbitosphenoids in the perch and carp (not 
in the cod). The relations of this ossicle are precisely those of the part 
forming the conjoined bases of the orbitosphenoids in mannnals, and usually 
called the ' body of the anterior sphenoid," in them ; though this is deve- 
loped from two distinct centres. In the young whale I found it supported 
by a direct extension of the basisphenoid forwards, which joins the back- 
wardly prolonged vomer, as in fishes. The common base of the orbitosphe- 
noids is peculiar, as a distinct bone, so far as I know, to fishes. It has been 
called by Bojanus* the ' basis alarum minorum sphenoidei seu rostrum sphe- 
noidei'; by GeofFroy ' entosphenal' ; and by Cuvier 'le sphenoide anterieure.' 
M. Agassiz opposes these determinations by the following remarks, founded 
on the embryological researches of the ingenious Dr. Vogt : — "In fishes 
with a short and thick muzzle, the cartilaginous embryonal plate (' plaque 
faciale' of Vogt), which serves as the base of support to the prosencepha- 
lon and the nasal fossae, is transformed into an independent bone, " se trans- 
forme integralement en os." It is then, he says, " represented by the cranial 
ethmoid (le sphenoide anterieure of Cuvier), an azygous bone, ' os impair,' 
short, of an almost square form, in which are pierced the canals for the 
transmission of the olfactory nerves. But in the fishes with elongated 
muzzles, and of which the eyes in place of preserving their primitive lateral 
position at the sides of the mesencephalon are carried forwards in advance 
of the cranium between that and the nasal fossae, the relations of the 
'plaque faciale' are necessarily altered: part of the plate remaining in its 
primitive situation is transformed into the ' cranial ethmoid,' the other part 
is carried forwards, but is never transformed into a distinct bone : it re- 
mains cartilaginous as the nucleus of the muzzle; or if, indeed, the ossifi- 
cation of the muzzle is completed, it disappears by virtue of the progressive 
encroachment of the exterior ossification. This is tiie reason why fishes 
have never a true ' nasal ethmoid' (the bones called ethmoid by Cuvier are 
the nasals), but only a cranial ethmoid f." Influenced by the deservedly 
high authority of M. Agassiz, I adopted his homology of the bone 9' in the 
' Hunterian Lectures on Vertebrata,' delivered in ISli. But since the notes of 
those lectures were printed, having been charged with the formation of a new 
Osteological Catalogue of the Hunterian Museum, I have carefully reconsi- 
dered this question. Passing over, for the present, the assertion that the homo- 
logue of the ' nasal ethmo'ide' does not exist in fishes, I would first observe, 
that if the orbital aperture (or what appears to those who deem the rhinen- 
cephalic crura to be olfactory nerves, the anterior aperture) of the cranium 
were homologous with the aperture closed by the cribriform plate in man, then 
any bony bar or place tending to close that aperture might be held to be homo- 
logous with the cribriform plate or crista galli of the ethmoid : but the inter- 
orbital aperture of the cranium is always bounded laterally, in fishes, by the 
orbitosphenoid ; and the rhinencephala and their crura extend forwards, to a 
considerable distance in most fishes, before the olfactory nerves sent oflT from 
the rhinencepliala escape by those perforations in the prefrontals, which are the 
true homologues of the single foramina of the olfactory nerves in the so-called 
ethmoid of birds, and of the cribriform foramina in n>ammals. The inter- 
orbital groove or canal in the skull of fishes, which is continued from the 
presphenoidal or interorbital aperture to the prefrontal foramina, is as essen- 
tially a part of the cranial cavity as is that contracted anterior olfactory 
chamber of the cranium of mammals, which, in the thylacine, for example, 
extends forwards, from where the orbitosphenoids sustain the frontals, ex- 

* Oken's Isis, 1818, p. 508. 

t Recberches sur les Poissons Fossiles, t. i. p. 120. 



ON THE VfiRTEBRATE SKELETON. 213 

panding, to where the frontals and the modified prefrontals (ethmoid) form 
the actual anterior boundary wall of the cranial cavity ; the chief distinc- 
tion between the condition of this boundary in the mammal and the fish, 
being, that whereas it is perforated by numerous apertures in the mammal, 
the olfactory nerves in the fish escape each by a single foramen or groove 
in the homologous bones. As beautiful as true was that clear perception 
by Bojanus of the homology of the simply perforated prefrontal of the fish, 
with its sieve like homologue in the class in which the olfactorj"^ sense reaches 
its maximum of development and activity, and modifies all around it. The 
coalesced bases of the orbitosphenoids, forming the anterior boundary of the 
bed of the optic chiasma, answer to the separate ossification called ' eth- 
mo'ide cranien' by Agassiz, in fishes : it has the same relation with that con- 
tracted area of the cranium answering to the interorbital aperture of the cra- 
nium in fishes, which the so-called «ranial ethmoid (entosphenoid) presents 
in fishes ; and this same entosphenoid (fig. 5, 9') has as little relation to the 
formation of the canals pierced by the olfactory nerves in fishes, as the 
orbitosphenoid has in mammals. The olfactory, rhinencephalic or anterior 
division of the cranial cavity in most fishes has its lateral bony walls incom- 
plete, and it opens freely, in the dry skull, into the large orbital chambers 
below, which are then said to have no septum : we see a similar want of de- 
finition of the cranial cavity in relation to the great acoustic chambers in most 
fishes. But in mammals the orbits are always excluded from the rhinence- 
phalic, or olfactory compartment of the cranium*; and a like exclusion 
obtains in some of the highly organized ganoid fishes and in the plagiostomes. 
As the prosencephalic parts of the brain progressively predominate, and the 
rhinencephalic parts diminish, in the higher mammals, the compartment of 
the cranium appropriated to the latter loses its individuality, and becomes 
more and more blended with the general cavity. In the elaborate ' Icono- 
graphy of Human Anatomy' by Jules Cloquet, for examplef , the small pe- 
culiarities of the 'trou borgne' and the 'apophyse crista galli' are both in- 
dicated, and very properly; but tlie rhinencephalic or olfactory division of 
the cranial cavity, though defined by the suture between the orbitosphe- 
noids and prefrontals and lodging the olfactory ganglia or rhinencephala, — 
so important an evidence of the unity of organization manifested in man's 
frame and traceable in characters, strengthening as we descend to the lowest 
osseous fishes — is wholly unnoticed. Thus, very minute scrutiny, con- 
ducted with great acuteness of perception of individual features, qualities 
highly characteristic of the anthropotomists of the school of Cloquet, being 
directed from an insulated point oif view, prove inadequate to the apprecia- 
tion of sometimes the most constant and important features of their exclusive 
subject. 

But to return to the homology pj„ 10 

of the orbitosphenoids. Intheme- 
Dopome these neurapophyses are 
elongated parallelograms, perfo- 
rated by the optic nerves, and are 
distinct bones. In the great bull- 
frog (^Rana hoans) they present a 
similar form (fig. 13, 10), but are 

confluent with the prefrontals (14) : side view of cranium {Runa boans), nat. size. 

in both batrachians an unossified sPace intervenes between them and the ali- 

* This is not to be confounded with the olfactory chamber itself, lodging the organ of 
smell, 
t Manuel d'Anatomie Descriptive, 4to, Atlas, pi. 8, fig. 2. 




214 REPORT — 1846. 

sphenoid (e). In most lizards the wider roof of the cranium, supported by the 
long mastoids, squamosals, postfrontals and malars, like a bony scaffolding 
on each side, is independent of its proper (neurai)ophysial) walls for support, 
and these retain, through the oeconomy of nature, tlieir primitive semi-mem- 
branous, semi-cartilaginous state. A dismemberment of the alisphenoid 
(which may be discerned as a process of that bone in the piscine genera 
Xiphias, Sphyraena) props up the parietal upon the pterygoid, so like a post 
or pillar, that the name 'columella' may well be retained for it. At the 
sides of the membrane forming the orbital aperture, rudiments of the orbi- 
tosphenoids may be seen in most lacertia: I find them, e. g. in the form of 
a slender osseous filament on each side, slightly bent inwards and bifurcate 
above, in a large Australian lizard {Cyclodus gigas). In the crocodile (figs. 
9, 20, and 22, lo) the orbitosphenoids attain their maximum of development, 
but retain all their typical characters: tney bound the orbital aperture of the 
cranium ; are notched below, as in many fishes, by the optic nerves (op) ; 
are perforated by the pathetic and other orbital nerves at the ' foramen spheno- 
orbitale' («) ; they protect the sides of the prosencephalon ; support above the 
frontals (and by their backward development also the parietals) ; and they 
rest below upon a peculiar development of the presphenoid (9), which seems 
to answer to the entosphenoid in fishes. 

Some salient points of resemblance between the cranial organization of fishes 
and birds have elicited remarks from more than one comparative anatomist. 
Not to dwell upon the more obvious correspondence arising out of the mo- 
bility of the upper jaw, chiefly through its connection with the pedicle of the 
lower jaw, I may indicate the overhanging position of the orbitosphenoid 
(figs. 8, 23, 10), raised high above the presphenoid (9), at the back part of the 
interorbital septum : we see exactly the same position of the orbitosphenoid 
in many fishes. Cuvier accurately represents it in the skull of the perch*. 
This beautiful trait of unity of organization is completely put out of sight by 
the false homology of the orbitosphenoid in fisiies with the alisphenoid in 
birds and mammals. The progressive recession of the orbitosphenoid and 
alisphenoid, as we descend from mammals to fishes, transfers indeed their 
characteristic nerve-notches or foramina from their posterior to their ante- 
rior margins. But the notch {op, fig. 8; at the posterior margin of the orbito- 
sphenoid in the bird for the escape of the optic nerve by a foramen common 
to it and the nerves of the orbit, is not less significant of its true homology 
than is the anterior notch in the crocodile or fish ; the osseous connections 
w'ith the sphenoid below, with the frontal above, and with the alisphenoid 
behind, being the same. 

Prefrontals If the cranium of a cod-fish be bisected horizontally and 

longitudinally, its most contracted part will be found at the upper part of 
the interorbital aperture, bounded by the orbitosphenoids, which mark the 
division between the prosencephalic and rhinencephalic compartments of the 
cavity : the latter extends as a triangular channel or groove on the under 
part of the frontal, opening below into the orbits, gradually expanding as it 
advances forwards, and dividing into two canals, which diverge to the inter- 
spaces left on each side of the nasal, between it and the bones (fig. 4, 14), that, 
meeting behind the anterior expanded end of the nasal, bound the anterior 
extremity of the true and entire cranium. The diverging canals of the rhi- 
nencephalic compartment are formed by the two bones in question: the rhinen- 
cephalaor olfactory ganglions are sometimes lodged at the extremities of these 
canals, and they send out the olfactory nerves Ijy the apertures formed be- 
tween the bones 14 and 15, which then ramify upon the vascular olfactory sacs, 
* Histoire des Poissons, pi. ii. figg. J. vii. 14. 



ON THE VERTEBRATE SKELETON. 215 

supported by the bones lo, fig. 5. For the arguments by which the olfactory 
ganglions in the cod are shown to be homologous with the olfactory ganglions 
that rest upon the cribrii'orm plate in man, and by which the medullary cords 
or crura connecting them to the rest of the brain are shown to be homologous 
with the so-called ' olfactory nerves' in the human cranium, and for the ge- 
neral homology of both as primary divisions and peduncles of the encephalon, 
the reader is referred to Dr. Desmoulins, 'i\natomie des Systemes nerveux 
des Animaux a Vertebres,' 1825, 8vo. t. i. p. 169 ; to Mr. Solly's excellent 
treatise 'On the Human Brain,' 1836, p. 78; and to my 'Lectures on the 
Vertebrata,' 1836, p. 184. I there adopt the expressive name applied by 
MM. Vogt and Agassiz to this most anterior of the four primary divisions 
of the brain of fishes, and apply to the peduncles of the ' rhinencephala,' 
which are frequently of great length in fishes, the name of ' rhinencephalic 
crura,' since they are serially homologous with the prosencephalic or cerebral 
crura; and I call that division of the cranial cavity which specially lodges 
these crura and their lobes the 'rhinencephalic' chamber or compartment. 
The right appreciation of the above essential characters of the most anterior 
division of the brain and the brain-case is indispensable to the accurate pur- 
suit of the homologies of the bones is, 14 and 15, whose development, espe- 
cially of the pair no. u, is governed by that of the rhinencephalon. In man 
the all-predominating cerebrum, overarching the mesencephalon and epen- 
cephalon behind, and the rhinencephalon in front, so modifies the surround- 
ing cranial bones as to obliterate every part of the rhinencephalic division, 
save the terminal fossa that immediately supports the so-called ' olfactory 
ganglia,' which fossa seems, as it were, to be unnaturally drawn in and 
blended with the great prosencephalic chamber, by reason of the enormous 
outswelling development of the proper spines or roof-bones of that chamber, 
the frontals. Still, even here, through the absence of any commissural band 
connecting together the rhinencephala, a fibro-membranous process of the 
endoskeleton extends between them, and into this septum ossification extends 
from below, called the 'crista galli.' In the cod-fish the homologous parti- 
tion between the rhinencephala is cartilaginous, and it extends some way back 
between their crura, not being opposed by a coextended overhanging cere- 
brum with great transverse commissures. In many fishes (e. g. Xiphias) the 
outlet of the olfactory nerves, which notches the inner side of no. 14 in 
the cod, is converted into a foramen by the extension of ossification around 
the mesial surface of the nerves. Where the olfactory nerves are sent off 
from the ganglions in great numbers (e. g. Raia), they perforate a mem--, 
brane before reaching and ramifying upon the vascular pituitary sac. In 
man, the homologous membrane, or basis of the olfactory capsules, is ossi- 
fied, and called from its numerous apertures the cribriform plate. The holes 
which these cribriform plates fill up are homologous with the foramina, or 
grooves forming the outlets of the olfactory nerves in the bones no. 14 in fishes 
(figs. 4 and 5). 

The grounds for this homology are so plain that we cannot be surprised 
that they should have been early appreciated, as e. g. by the painstaking and 
philosophic Bojanus in 1818*. I never could comprehend the precise mean- 
ing of the statement with which Cuvier opposed his view : — " M Bojanus, par- 
tant sans doute du trou qu'il a dans plusieurs poissons pour le nerf olfactif, en 
fait une lame cribleuse de I'ethmoide ; mais cette ojinion, qui n'a pas ce soutien 
dans toutes les especes, est refutee d'ailleurs par les autres rapports de cet os 
avec les os voisinsf ." Cuvier seems to have thought the ground of Bojanus's 
opinion to be cut away by the fact that in the cod and some other fishes the 
• Isis, heft iii. p. 503. f Ilistoire des Poissons, i. p. 235. 



216 REPORT — 1846. 

olfactory nerves groove instead of perforate the bones no. 14. But the trige- 
minal still determines tiie alisphenoid, whether it perforates or notches that 
neurapophysis in its escape : the relation of the alisplienoid to the division 
of the 5th, including the gustatory nerve, and that of the orbitosphenoid to 
the nerve of sight, are not more constant than is the relation of no. 14 to the 
nerve of smell. The differences of connection of no. 14 — ' les autres rap- 
ports' — -are not specified by Cuvier, and I know none that affect its essential 
character. 

No. 14 is however the. most anterior of the neurapophysial or lateral 
bones of the true cranium, and is in relation with the anterior terminal divi- 
sion of the encephalon and with the first or anterior terminal pair of nerves. 
Like all extreme or peripheral parts, it is subject, as we should be prepared 
to find it, to a greater extent and variety of modifications than the more 
central neurapophyses. The difference between its connections in the fish 
and that of the cribriform plates and their sustaining basis in man may 
therefore be expected to reach the extremes of possible homology. It will 
be interesting to inquire whether there are intermediate modifications by 
which the nature of that difference may be appreciated, and how many of 
such links are permanently retained in the intervening species. 

We might anticipate the smallest amount of departure from the fun- 
damental vertebrate type, as respects form, size and connections of the bones 
in question, in that class where the principle of vegetative repetition most 
prevails and the archetypal plan is least obscured by teleological adaptations. 
Adopting the name modified from the phrase applied to these bones by Cu- 
vier in those vertebrata in which they present their most typical characters, 
we find the ' prefrontals' in all bony fishes resting below upon the vomer (figs. 
4 and 5, u) and on part of the presphenoid (9), sustaining by their mesial and 
upper surfaces the nasal (15) and fore-part of the frontal (11), aflTording the 
whole or part of the surface of articulation for the palatine (-20) or the palato- 
maxillary arch, and giving attachment exteriorly to the large suborbital or 
laci-ymal bone (figs. 22 and 25, 73), when this exists. Besides their protec- 
tive functions, in relation to the olfactory ganglions and nerves, they close the 
cranial cavity and bound the orbits anteriorly. The most constant and cha- 
racteristic connections appear to be with the vomer, nasal, palatine and frontal. 
In the muraenoid fishes, where confluence begins to prevail in the cranial bones, 
we find that the prefrontals coalesce with the vomer and nasal, not with the 
true frontal. This fact, though not of a class materially affecting relations 
of homology, is not devoid of significancy in regard to the real character of 
the bone usually described as one of the ' deux demembremens du frontal*.' 
A clew not to be neglected in tracing the homologies of the prefrontals is 
their histological progress, although the value of such embryonic characters 
has been overrated and their application sometimes abused. The substramen 
of their ossification, like that of the exoccipitals, mastoids and post-frontals, 
is a cartilaginous mass, a part of that which M. Duges has called ' cartilage 
cranio-faciale,' and M. Vogt ' plaques protectrices laterales.' The frontals 
and parietals, being ossified in supra-cranial fibrous membrane with so rapid 
and transitory a cartilaginous change as to have escaped general recognition, 
have been, on that account, rejected from the vertebral or endo-skeletal system 
of bones by Dr. Reichert, and with as little real ground as the rejection of the 
vomer and sphenoid from the same system, because they are ossified in mem- 
brane extended from the under and fore-part of the sheath of an evanescent 
subcranial ' chorda dorsalis,' like the homologous basal ossification beneath 
the coalesced anterior abdominal vertebra of the siluroids. 
* Agassiz, op. cit. i. p. 123. 



ON THE VERTEBRATE SKELETON. 21? 

M. Duges, who has accurately figured the 'cranio-facial' cartilage of a 
gadoid fish in pi. ii. of his valuable Monograph*, gives as accurate a figure 
of the same cartilage in the Rana viridis (pi. i. figs. 6, 7, of the same work), 
out of which has been ossified a bone which transmits the olfactory nerve to 
its sense-capsule : this bone (is in the figures cited) rests below upon the di- 
vided vomer and on the end of the presphenoid, sustains above the nasal and 
forepart of the frontal, affords an articular surface ou its outer part for the 
palatine, and only fails to repeat every characteristic connection of the pre- 
frontals in fishes, because (as likewise happens in certain of that class) there 
is no lachrymal bone developed in the Batrachia. The sole modification 
of any consequence tending to mask the homology is this ; that whereas we 
find in many fishes ossification extending into the persistent part of the cra- 
niofacial cartilage connecting, whilst it separates, the prefrontals, so as to 
circumscribe the canals for the transmission of the olfactory nerves, such ossi- 
fication proceeds in the anourous batrachia to anchylose the prefrontals with 
each other, and convert them into a single bone. This difference however 
sufficed with Cuvier to make of it a new and peculiar bone — an « os en cein- 
turef .' It would have been as reasonable to have given a new name to the 
supraoccipital in the Lepidosteus, because it is divided in the middle line in- 
stead of being single, or to the frontal in the species where it is single instead 
of being divided, or to the vomer in the fro^ because it is double instead of 
single, or to the exoccipitals in the same reptile, which manifest the same 
mesial and annular confluence as the prefrontals. But, adds Cuvier, in refer- 
ence to the single bone (fig. 13, u) resulting from this modification, " Je ne 
I'ai pas trouve divise, meme dans des individus tres-jeunes qui avoient encore 
un grand espace membraneux entre les os du dessus du crane." Nor did the 
great anatomist ever find the rudiments of the radius and ulna distinct at any 
period of development of the single bone of the Batrachia, which he never- 
theless rightly describes as representing both bones of the fore-arm : nor 
did he ever find a division of the single parietal in the embryo crocodile, 
which he equally well recognized, nevertheless, as the homologue of the two 
parietals, which in most fishes have been subject to greater modifications in 
their connections and relative position than the single prefrontal presents in 
the anourous batrachia. These are not the only instances where relations of 
homology are by no means obscured, nor ought to be, by reason of the con- 
fluence or even connation;}: of essentially distinct elements. The capsule of 
the olfactory organ, partly protected by the anterior infundibular expansions 
of the connate prefrontals, undergoes no partial ossification homologous with 
the 'turbinal' (19, fig. 5) of fishes, but remains cartilaginous, like the scle- 
rotal and petrosal. 

The prefrontals, however, are not only connate with each other in the 
frog, but coalesce with the contiguous neurapophyses — the orbitosphenoids 
(10, fig. 13). And this modification has led Cuvier, notwithstanding the 
connection of the bone 10 with the presphenoid below, with the frontal 
above, and with the prosencephalon, optic nerve (op) and orbit, to charac- 
terise the batrachian skull as having " un seul sphenoide sans ailes tempo- 
rales ni orbitaires ;" the true and distinct ' alisphenoid ' (e, fig. 13), with its 
typical connections and nerve-perforations (tr), being described as the pe- 

* Recherches sur I'Dsteologie, &c. des Batraciens, 4to, 1835. 

t Ossemens Fossiles, 4to, t. v. pt. ii. p. 387. He had before applied the name of ' ceinture 
osseuse ' to the scapular arch in fishes. — Lemons d' Anat. Comp. i. (1800) p. 332. 

X I use these terms in the same definite sense as the botanists ; those essentially distinct 
parts are connate which are not physically distinct at any stage of development, those united 
parts are confluent which were originallv distinct. 

1846. ' Q 



218 REPORT— 1846, 

trosal, 'rocher*.' But the real difficulties which beset the quest of general 
truths in comparative osteology ai'e such that we maj' well dispense with any 
over-statements of the amount of deviation from the cranial archetype which 
much -modified skulls like those of the anourous batrachia may present. 
Fortunately the light which the development of such skulls throws upon 
their mature characters, is aided by the persistent larval stages manifested 
by the perennibranchiate species. 

In the menopome, for example, the prefrontals remain distinct, both from 
each other and from the orbitosphenoidsf , their characteristic connections 
and functions being the same as those of their coalesced homologues in the 
frog, except that they are notched, instead of being perforated by the olfac- 
tory nerve, which grooves their inner border, as in the cod and some other 
fishes. Cuvier just hints at the possibility of his ' os en ceinture ' in the frog 
representing " a la fois le frontal principal et I'ethmoiideJ," or as having an 
equal pretence to one or the other name. 

The suture, however, which marks the limits between the frontal u and 
parietal 7 is persistent in the menopome, and indeed in all batrachians but 
the anourans ; and even in the very young larvae of these, Cuvier admits 
(and the observations of M. Duges warrant the admission) " que Ton separe 
une partie posterieure de forme ronde de I'anterieure qui est allongee" (Ibid. 
p. 387). The permanently distinct frontals present a similarly elongated form 
in the urodeles, and are therefore recognized by Cuvier in the salamander, 
e. g. at c, pi. xxv. fig. ] , op. cit. ; in the newt, pi. xxvi. fig. 6 ; in the menopome, 
fig. 4 ; in the axolotl, pi. xxvii. fig. 24 ; in the siren, ib. fig. 2 ; and in the am- 
phiuma, ib. fig. 6. In all these crania the true frontals are indicated by the 
same letter c ; in none of them do they close the cranial cavity or bound the 
orbits anteriorly, or are perforated by the olfactory nerves, or articulate with 
the vomer below, or perform any of the essential functions, or combine the cha- 
racteristic connections of the prefrontals of fishes, all of which concur in the 
' OS en ceinture.' But the frontals do present the chief connections and occupy 
the relative position of the anterior half of the bone (11 — 7, fig. 13) which 
Cuvier calls the parietal in the frog. The evident tendency to coalescence of 
essentially distinct bones which pervades the skeleton in the adult anourans 
greatly diminishes the difficultj^, through the loss of the suture between the 
parietal and frontal, of recognizing the homology of the latter bone, which, 
with that exception, not only repeats the characters of the frontals in fishes, 
but of those in most tailed batrachians. 

Next, then, with regard to the ethmoid, the second of the two bones to 
which Cuvier restricts the choice of the homologues of the ' os en ceinture,' 
no. 14. No name has been applied more vaguely or with a less definite 
meaning than this same ' ethmoide.' In the sense in which Cuvier would 
permit its application in the present instance, it is a bone which forms the 

* Op. cit. p. 386. 

t The menopome, which represents a gigantic tadpole of the tailless batrachia, manifests 
a lieautiful conformity to the general type, and well illustrates the real nature of the apparent 
deviations which take place in the course of the remarkable metamorphoses of the anourans. 
At first sight the orbitosplienoids seem to be barred out from their normal connection with 
the frontal by the junction of the parietal with the prefrontal in the menopome, as appears, 
for example, in the figure given by Cuvier in the ' Ossemens Fossiles,' v. pt. ii. pi. xxvi. fig. 4, 
where c h divides c from u. Remove, however, the -prefrontal h from the parietal c (which 
may be readily done, the suture, which is not indicated in the figure cited, being persistent), 
and the anterior and mesial half of the orbitosphenoid {ti) is then seen extending inwards 
(mesiad), beneath the parietal and prefrontal, to join a triangular surface formed by a de- 
scending process from the middle of the outer edge of the frontal. 

X Op. cit. p. 388. 



ON THE VERTEBRATE SKELETON. 219 

anterior and antero-lateral walls of the cranium, defends the rhinencephala 
and transmits the olfactory nerves, but is altogether distinct from and pos- 
terior to the capsules of the organs on which those nerves are ramified. 
In the crocodile Cuvier restricts the term ethmoid to the cartilaginous 
laminae, capsules, or supports of the olfactory ramifications after the nerves 
have left the cranium. In mammals the ethmoid is made to include both the 
bones that close the cranium anteriorly, support the rhinencephala, give exit 
to the olfactory nerves, and those which defend and sustain the enormously 
developed and complex superior parts of the organ of smell*. Whilst this 
confusion is permitted to vitiate osteology, it is plain that no intelligible 
homological or other proposition can be predicated of the ' ethmoid.' 

When Cuvier, with. reference to the hypothetical possibility of the homo- 
logue of the frontal forming part of the bone 7 — u in the frog, adverts to 
the second chance of bringing the ' os en ceintnre' into the ordinary cate- 
gory of cranial bones, by viewing it as the 'ethmoide,' he adds, that it would 
then be "un ethmoide ossifie, ce que sera une grande singularite" (ib. 
p. 388). Here it is obvious that the predominating idea of the ethmoid was 
that presented to his mind by the capsules of the olfactory organ in the 
crocodile^and other reptiles.'which he had so called, and which are wholly or 
in great part cartilaginous. But the parts of Cuvier's ethmoid in birds and 
mammals, which are in functional and physical relation with the cranial cavity, 
rhinencephala and olfactory nerves, are ossified : the bone, also, to which he 
gives the name * ethmoid' in fishes (fig. 5, is) is ossified ; and, what is more 
to the purpose, the bones (14) in fishes, ophidians, chelonians and saurians, 
which repeat the essential characters of the batrachian ' os en ceinture,' are 
likewise ossified. 

General homology teaches that the bone or bones in relation to the defence 
of the rhinencephala and the transmission of their nerves belong to one class, 
and that the parts of the skeleton, whether membranous, gristly or bony, 
which form the capsule or sustain the olfactory organ itself, belong to another 
and very different class of parts of the skeleton. But, not to anticipate what 
belongs more properly to a subsequent section of this report, observation 
shows the two parts to be physically distinct in all vertebrates except mam- 
mals, and to be distinct in the foetus of these. Whether we restrict the term 
< ethmoid ' to the neurapophysis or to the sense-capsule (which in mammals 
constitutes the ' conchae superiores' and cells of the ethmoid), the term must 
be applied arbitrarily in its extended or homological signification, since the 
neurapophysis dismisses the nerve by a single foramen or groove in all the 
vertebrates below mammals. The multiplied foramina in the neurapophysial 
or cranial part of the anthropotomical ' ethmoid,' whence that name, as well 
as the special designation of the part called ' lamina cribrosa,' are modifica- 
tions peculiar to the mammalian class, but not constant here, and they form 
no essential homological character of the bone in question. It appears to 
me preferable, since we have two essentially distinct parts of the skeleton 
combined in the mammalian and human ethmoid, to restrict the term to the 

* Objecting to Oken's idea, that the prefrontal in the crocodile was homologous with the 
part of the ethmoid called ' os planum' in anthropotomy, Cuvier says, " Or I'os planum ne 
paroit jamais sur la joue ; il ne se montre plus dans I'orbite a compter des makis si ce n'est 
un petit point dans les galeopitheques et dans quelques chats. Dans tons les autres mam- 
miferes I'ethmoide est entierement enveloppe et cache par le palatin" (note that significant 
connection) " et par le frontal et specialement par cette partie du frontal dont il est main- 
tenant question et qui se detache dans les ovipares. Le veritable ethmoide est enveloppe 
de la meme maniere dans le crocodile, quoique presque toutes ces parties restent cartilagi- 
neuses." — Ossem. Foss., v. pt. i. p. 73. 

q2 



220 KEPORT — 1846. 

part which appertauis to the sense-capsule, i. e. which is directly concerned 
in the support of the membrane and cells of the olfactory organ. 

But leaving for the present the question of names, and returning to things, 
let us pursue our search and comparisons of the bones which continue in the 
higher classes to repeat the essential characters of those called ' prefrontals ' 
in fishes. Were it necessary to add to the reasons above assigned for regarding 
no, 14, fig. 13, as the homologues of i4 in the fish; notwitlistanding they are 
connate in the batrachian, I would cite the structure and relations of those 
bones in the sword-fish. The whole of the anterior part of the, extensive 
interorbital space is occupied by the prefrontals, which join each other at the 
median line by an extensive vertical cellular surface : they form the anterior 
border of the orbit, and the posterior wall of the nasal fossa; they close the 
cranial cavity anteriorly, and transmit the olfactory nerve to the capsule by 
a central foramen. They are almost entirely covered by the frontals above, 
which they support by a broad flat surface ; a very small portion only ap- 
pearing on the upper surface of the skull at the anterior angle of the orbital 
ridge. Were the frontals separated, the prefrontals would then appear, as in 
the frog, at the median line : were the suture between the two prefrontals 
to be obliterated in Xiphias, an ' os en ceinture' would be produced like that 
of the frog. The nasal bone of the sword-fish, which Cuvier calls * ethmoide,' 
presents a cellular structure of its base, designed to break the force of the 
concussion arising from the blow which is delivered by the ' sword.' But the 
prefrontals manifest more extensively this peculiar cellular structure, which 
Cuvier well says, "I'on prendrait presque pour les cellules de I'ethmoide d'un 
quadrupede*." 

Cuvier, not perceiving or not appreciating the grounds of the homology of 
the 'os en ceinture' with the prefrontals, describes the divided nasal (is, fig. 
13), in the batrachia as the ' frontaux anterieures' ; and reciprocally, having 
called the bones in fishes, homologous with the bone i4, (which he thought 
might represent the ethmoid in the frog) 'frontaux anterieures,' he gives the 
name ' ethmoide ' to the bone is, fig. 5, whether single or divided, in 'fishes. 
It is not necessary to add anything to the arguments by which M. Agassiz 
has sustained the conclusion of Spix, that Cuvier's ' ethmoid ' in fishes is the 
' nasal.' And it needs, I think, only to compare the connections of the 
bones is, fig. 13, wath either the single or the divided nasals in fishes, and to 
glance at the obvious homology of the bones h in Cuvier's pi. xxiv. fig. 1 — 6, 
with the bones g g'm figs. 4 & 6 of pi. xxvi. (' Ossemens Fossiles,' t. v. pt. 2), 
to ensure the acceptance of the conclusion, that his ' frontaux anterieures ' 
in the frog and the other anourans are the true nasal bones. 

In the python Cuvier transfers the name ' frontaux anterieures ' to the 
lacrymal bones. The bones in this serpent, which are in neurapophysial 
relation with the olfactory nerves, and which present other essential charac- 
ters of the prefrontals (14) in fishes, are also two in number, in the form of 
thin osseous plates, intervening on each side, anterior to the frontal, between 
the vomerine and nasal bones, bent outwards, in the form of a semicylinder 
about the olfactory nerves, which they support and guide to the cartilaginous 
capsule of the organ of smell, and having the palatine bones articulated to 
their under and outer sides. The bones, which thus present every essential 
character of the prefrontals, are those (5 s in pi. ix. figs. 1 , 2, 3, ' Regne 
Animal,' t. iii. 1830) which Cuvier there calls 'cornets inferieures.' But 
the true ' cornets ' (turbinals) are cartilaginous in serpents as in every other 
reptile, and give attachment to the palatines in no animal. The bones bb \n 

* Hist, des Poissons. t. viii. p. 194. 



ON THE VERTEBRATE SKELETON. 221 

the same figures, to which the name of 'anterior frontal' is given, have no 
relation whatever to the protection of the rhinencephala or the exit of the 
olfactory nerves, but they have a large perforation for the passage of the 
muco-lacrymal duct from the eye. They repeat indeed the single and 
\east essential character of the prefrontals, in standing anterior to the fron- 
tals and the orbits ; but these are characters common to the great anterior 
mucous scale-bone in fishes, whose essential function — the transmission of a 
mucous duct — they superadd to the repetition of its connections, viz. with 
the prefrontal, nasal and superior maxillary bones*. 

The bones, which more resemble the anchylosed prefrontals in the frog, are 
the frontals of the python ; but the resemblance is confined to one character 
only, and that an exaggeration of a character common to the frontal bones of 
many birds, and of the ornithorhynchus among mammals, viz. a develop- 
ment of a median bony partition from the line of the frontal suture into the 
median interspace of the encephalon. In the python each frontal sends 
down at the fore-part of this suture such a partition, which is therefore double, 
as the falx essentially is in man and the mammalia, in which it retains its 
primitive histological condition of a fibrous membrane. The ossified laminae 
of the falx in the python bend outwards and coalesce below with the external 
or orbitosphenoidal plates of the frontal, and thus surround the lateral divi- 
sions of the fore-part of the brain ; in fact, the olfactory nerves, drawn back 
in the progress of the concentrative movement of the cerebral centres, so as 
also to occupy the prosencephalic segment of the cranium, the prosencepha- 
lon being, in like manner, protected chiefly by the mesencephalic bony arch. 
The change is precisely analogous to that which takes place at the opposite 
extremity of the neural axis in higher animals. In the python every segment 
of the spinal chord retains its primitive relation to the segment of the endo- 
skeleton, through which it transmits its pair of nerves. In the mammal the 
concentrative movements of the spinal chord draw its segments in advance 
of their proper vertebrae, and the primary relation is indicated by the nerves 
which these vertebrae continue to transmit, and by which alone we are guided 
from the segment of the endoskeleton to that of the neural axis which origi- 
nally governed its development. 

So, likewise, at the opposite end of the skeleton, we trace the relation of 
the anterior osseous segment, which transmits the olfactory nerves to their cap- 
sule, to its proper segment of the neural axis, by following those nerves back 
to the retracted ganglions (rhinencephala) from which they take their origin. 
The connections of the annular frontals of the python with the parietals 
and post-frontals behind, with the connate orbitosphenoids, and through 
them with the presphenoid below, prevent their homology being mistaken ; 
for they are far from completely representing or repeating the essential cha- 
racters of the coalesced annular prefrontals of the frog. 

Not to lengthen unnecessarily this exposition of the homologues of the pre- 
frontals (i4, figs.4 and 5) in fishes, I pass at once to the highest of existing rep- 
tiles, the crocodile. Here we find, in the dry skull, the condition of the cranial 

* No one could better appreciate the Talue of the functional character of the lacrymal 
perforation in a homological discussion than Cuvier, when the more obvious features of the 
prefrontals of fishes were so repeated in any higher animal as to have led him to distinguish 
the prefrontals in that animal from the lacrymal bone. Thus with regard to the pre- 
frontals of the crocodile, Cuvier says, " Quant a M. Spix, entraine par un autre systeme et 
negligent le trou lacrymal, qui cependant est bien visible, et qui, specialement dans le cro- 
codile, est perce tout entier dans I'os auquel je donne ou plutot auquel je maintiens le meme 
nom, c'est mon frontal anterieur qu'il appelle lacrymal." (Ossemens Fossiles.) Change 
python for crocodile and Cuvier for Spix, and the criticism equally applies in the present 
instance to its original author. 



222 REPORT— 1846. 

cavity in the fish beautifully and closely repeated : the prosencephalic part 
opens freely by the aperture bounded by the orbitosphenoids (fig. 9, lo) into 
the common orbital cavity (o?-), and the rhinencephalic division of the cranium 
is prolonged, as a groove upon the under surface of the coalesced frontals 
(ib. ii) above the orbits, expanding as it advances, until it is arrested by a 
boundary formed by two bones (ib. 14), which rest below upon the vomer 
and give attachment there to an ascending process of the palatines (20), which 
sustain by their mesial and upper expanded surfaces the nasal (15) and fore- 
part of the frontal (n) ; and articulate exteriorly with the large lacrymal 
bone (fig. 22, 73) perforated as in the fish and serpent by a mucous duct from 
the orbit. They are each grooved on their inner or mesial surface (indicated 
by the numerals 14, in fig. 9) by the olfactory nerve, where it escapes from 
the cranium to spread upon the membranes sustained by the cartilaginous 
capsules anterior to the bones in question ; below these grooves the bones 
(14) extend inwards and meet at the mesial line; but do not coalesce there 
as in the frog, nor extend their mesial union upwards, so as to convert the 
olfactory grooves into two complete canals. They, therefore, retain or resume 
much more of their primitive piscine character than do their homologues in 
the frog or serpent, and manifest it conspicuously by developing a subtrian- 
gular external plate which appears on the upper surface of the cranium at 
the anterior angle of the orbit between the frontal, the lacrymal and the 
nasal bones. In short, the homology of the bones 14 in the crocodile (figs. 9, 
21, 22) with those so numbered in the fish (figs. 4 and 5), was quite unmis- 
takeable ; and, with the exception of Spix, all anatomists have concurred in 
this respect with Cuvier : only some of them have extended further and 
expressed differently the homologies of the bones in question. 

Now, bearing in mind the small brain of the cold-blooded crocodile, and 
the concomitantly restricted development of the spine or roof-bone in special 
relation with the cerebrum, viz. the frontal (11), which is aided in its se- 
condary function in relation to the orbit by distinct supraorbital bones in all 
crocodiles, and contrasting the condition of the part of the brain which 
chiefly governs the development of the frontal bone with that of the same 
division of the brain of mammalia, — let us proceed to make the comparison 
which Cuvier recommends*, in order to trace the homologues of the croco- 
dile's prefrontals in the mammalian class. 

We place the skull of a ruminant (the red deer, e. g.) by the side of that 
of a crocodile, and delineate a suture which would detach a portion from the 
frontal, having the same superficial connections as the upper peripheral plate 
of the prefrontal has in the crocodile. It appears to be far from presenting 
the same figure ; but most assuredly such artificially detached portion of 
the ruminant's frontal has not the same functions (' emploi') as the pre- 
frontal has in the crocodile. For if we even include with the part so 
detached the anterior portion of the descending orbital plate of the frontal, 
we find it joining below the orbitosphenoid without any connection with the 
vomer, or any attachment to the palatine : it forms no immediate part of the 
supporting plate of the rhinencephalon, nor of the foramina for the exit of 
the olfactory nerves. Such artificially detached portions of the mammalian 
frontal are entirely separated from each other ; whilst one of the important 

* " II suffit en effet de placer une tete de mammifere, de ruminant par example, a cote 
d'une tete de crocodile, pour s'assurer qu'il s'est fait ici (' du frontal anterieur') un demem- 
brement du frontal. On pourroit, sans rien deranger, dessiner sur le frontal du mammifere 
la suture qui existe dans le crocodile, et on detacheroit ainsi dans le premier un frontal 
anterieur qui auroit la meme position, presque la meme figure, et absoluraent le meme emploi 
que dans le crocodile." — Ossem. Fossiles, v. pt. ii. p. 73. 



ON THE VERTEBRATE SKELETON. 223 

points of resemblance between the prefrontals of the crocodile and those of 
the fish are the mesial approximation and junction of their descending (neu- 
ropophysial or rhinencephalic) plates — the most constant and important parts 
of the bones in question. 

If the frontal of the ruminant or other mammal were expanded only at 
the parts corresponding with the detached bones called " frontaux ante- 
rieures" in the crocodile, there might then be a, prima facie probability that 
such expansions were connate parts, dismembered in the crocodile's skull. 
But the vastly increased lateral as well as anteroposterior development, and 
the more or less vertical convex expansion of the frontal in the highest 
vertebrate class, naturally indicate, in the first place, an inquiry into the 
concomitant modification of the nervous centres by which the development 
of that bone is mainly governed ; and if such modification should then be 
found to exist, in the cerebrum, for example, which, from the ascertained 
correlative progress of the frontal in other classes, ought to cause or be 
associated with such a general development of that bone as characterises the 
skull in the mammalian class, it must surely be superfluous and gratuitous 
to explain that development by the hypothesis of a coalescence of another 
essentially distinct element of the cranial parietes : especially if that element 
be proved by a similar tracing of its relations to the progressive development 
of the cerebral centres, to have as essential and exclusive a dependence 
upon the rhinencephalon as the frontal bone has upon the prosencephalon. 

The position of the upper peripheral part of the prefrontal in the situation 
in which it is seen in the crocodile, is, in fact, the least constant and import- 
ant of the characters of that bone. In the bull-frog, for example, the ex- 
posed part of the prefrontal is mesiad of the conjoined parts of the nasals 
and frontals instead of being lateral : in the sword-fish the prefrontals barely 
appear, and in the python they do not appear at all upon the upper surface 
of the skull ; but they retain in each their more typical neurapophysial po- 
sition, M'ith all their more constant and essential characters. The enormously 
developed frontal of the mammal masks these characters, and usurps the 
less constant and least important one, viz. superficial position, on which alone 
Cuvier insists as proving the prefrontal of the crocodile, with its complex 
functions and connections, to be such a dismemberment of the true frontals 
of the ruminant, as may be marked off with the pen on the upper surface of 
the skull. 

The descending [rhinencephalic] plates of the prefrontal in the crocodile 
(fig. 9, 14) are subcompressed in the axis of the skull, and expanded laterally, 
especially at their upper part ; where, in the alligator, I find them forming a 
shallow cup, concave forwards for the lodgment of the cartilaginous olfactory 
capsule, — of that part, namely, which is ossified in mammalia, and there de- 
veloped into the great labyrinth of the superior turbinals and ethmoidal cells. 
The vertical plates, continued forwards from the prefrontals, which extend 
above to the nasal suture and descend into the vomerine groove below, to aid 
in forming the ' septum narium,' are cartilaginous in the crocodile : they are 
more or less ossified, and form the 'lamina perpendicularis ethmoidei' in 
mammals. The median plate, dividing the olfactory nerves at their exit, and 
developed backwards as a partial septum of the rhinencephalic chamber of 
the cranium, and continued into the simple interorbital septum of the croco- 
dile, also remains cartilaginous: when ossified in mammals, it forms the 
' crista galli.' Now not one of these cartilaginous representatives of the parts 
of the compound bone called ' ethmoid' in anthropotomy, is united or con- 
nected with the portions of the frontal in mammals which Cuvier has assumed 
to be the homologues of the prefrontals in the crocodile ; those bones being 



224 REPORT — 1846. 

in that reptile, as the prefrontals are in fishes, chiefly concerned in closing 
the anterior end of the cranial cavity, in giving exit to the olfactory nerves, 
in suspending the palatine arch, in connecting the vomer with the nasal ver- 
tically, and the nasal with the frontal and lacrymal horizontally, repeating in 
the crocodile for the latter purpose the development of the upper or horizontal 
plate which had almost or entirely disappeared in some of the intervening 
forms of reptiles. In most chelonians this portion of the prefrontal coalesces 
or is connate with the short nasal : but I have found the instructive exception 
presented by the existing freshwater tortoise {Hydromedusa) of the persistent 
suture between the nasals and prefrontals, repeated in two fossil chelonians 
{Chelone playiiceps and Chelone pulchriceps)* . 

Proceeding in the ascensive track of the homologies of the prefrontals, 
I have selected from the class of birds the skull of the ostrich (figs. 8 and 23), 
the representative of an aberrant order, in which every deviation from the 
type of the class that has been supposed to tend towards the Mammalia, tends 
equally or more towards the Repti.lia\, and in which, conformably with the 
lower development of the respiratory system, the original sutures of the 
cranium, or in other words, the signs of the vertebrate archetype on which it 
is constructed, are longest retained. Were we to cut off the corresponding an- 
terior angles of the frontals, no. n, to those supposed to represent in mammals 
the bones we are in quest of, we should have even fewer of their characters 
than in the higher class alluded to, because the descending orbital plate is 
less developed, and the frontal, though its general size is much augmented, 
retains more of its oviparous horizontality as an expanded spine or roof-bone 
of the cranium. 

There is a large bone (fig. 23, 73) bounding the anterior border of the orbit, 
and from which, as we have seen in the parrots, ossification sometimes extends 
backwards along the inferior contour of the orbit to the postfrontal. But this 
bone, besides its repetition of the connections of the lacrymal in the fish and 
crocodile, resting as in the latter animal upon the true malar bone, is either 
perforated or grooved by the lachrymal duct, which it defends in its course 
from the eye to the nose, and has none of the essential characteristics of the 
prefrontal. But we see on the exterior of the skull of the ostrich and other 
struthious birds J, a distinct rhomboidal plate of bone interposed between the 
frontals and nasals, precisely in the situation in which the upper surface of 
the coalesced prefrontals appears in the skull of the frog and other anourous 
batrachians. In a nearly full-grown ostrich's skull, I removed the left fron- 
tal, nasal, lacrymal and tympanic bones, and the zygomatic arch, as in fig. 8, 
and found the facet in question to be the upper and posterior expanded 
surface of a large irregularly subquadrated compressed bone (ib. 14), consist- 
ing of two vertical compact plates coalesced at their periphery, and including 
a loose cancellous texture. The upper and posterior expanded surface of the 
bone extends a short way back beneath the frontals, descends and closes the 
anterior aperture of the cranium, and sends out from each side a plate of 
bone which arches over the olfactory nerves and forms the canals by which 
they are conducted along the upper part of the orbits. The anterior and upper 
surface of the bone again expands (at 14', figs. 8 and 23), and there sustains, 
and is covered by, the nasal bones, and again overarches, and is sometimes 

* Report on British Fossil Reptiles, Trans. Brit. Assoc. 1841, pp. 169, 172. 

t The urinary bladder and intromittent organ, e. g. : the modification of the feathers in 
the Struthionidm is a degeneration of a peculiarly ornithic character ; but not, therefore, an 
approximation to the hairy covering of mammals. 

X In the emeu {Droniaius ater) at u, fig. 1. pi. 39. Zool. Trans, t. iii. : and in the casso- 
wary at h, fig. 3, taf. i. in Kallmann's • Vergleichende Osteologie des Schliifcnbeins.' 



ON THE VEETEBRATE SKELETON. 225 

perforated by the olfactory nerves (the course of which along the rhinen- 
cephalic continuation of the cranial cavity, is shown by the arrows, ol. 14, 
figs. 8 and 23) prior to their final expansion on the olfactory organ ; the 
main body of the bone forms the fore-part of the interorbital septum and 
the back part of the nasal septum, a slight outstanding ridge or angle 
dividing the two surfaces : it rests below upon the rostral prolongation of 
the presphenoid, which, however, barely divides it from the semicylindrical 
grooved vomer (13) which sheathes the under part of that process. The 
posterior extremities of the palatines develope broad horizontal plates mesiad 
and upwards (fig. 23, 20), which join the lower border of no. 14, where it rests 
upon the presphenoid. The outer margins of the anterosuperior expansion 
of no. 14 come into contact with the lacrymals : the posterior border of the 
vertical or rhinencephalic plate joins and soon coalesces with the orbitosphe- 
noids (10). Thus we have all the essential characters of the prefrontals in 
the fish, the frog and the crocodile, with a repetition of their first important 
modification in the tail-less batrachians, viz. that of median confluence ; and 
it is not unimportant to observe that this is associated with the obliteration of 
other cranial sutures, by which also those batrachians resemble birds. The 
first step in the progress of this median approximation of the prefrontals, is 
the development of the plates which, in certain fishes, convert the olfactory 
grooves into foramina; these mesial plates next come into contact at the middle 
line, e. g. in Xiphias and Ephippus ; they proceed to coalesce in the frog, and 
the prefrontals are so much further compressed in the bird that the olfactory 
grooves open upon the outer or lateral instead of the inner or mesial surfaces of 
the rhinencephalic plates : they are, however, very deep grooves in the ostrich, 
and in the apteryx are canals protected by a distinct external plate. The 
interruption of the direct vomerine connection by the prolonged presphenoid 
is the chief secondary modification of the prefrontals in the bird. No other 
bone in the bird's skull repeats the more essential characters of the prefrontals 
in fishes and reptiles, save the bone no. 14, figs. 8 and 23. Cuvier calls this bone 
the ' ethmoide '; but blames the clear-sighted and consistent German anato- 
mists who applied that name to the prefrontals in fishes and reptiles ; yet the 
part of Cuvier's ethmoid in the bird answering to the ' lamina cribrosa' of the 
mammal, sometimes gives passage to the olfactory nerve by a single foramen, 
sometimes by merely a groove, a difference which does not prevent him 
adopting the homology here, though he opposes it to the adoption, by 
Bojanus, of the homology of the same part in the fish (ante, p. 215). The 
smooth plate forming, with the orbitosphenoid, the interorbital septum, is 
the ' OS planum,' or papyraceous plate of the bird's ethmoid, with Cuvier : 
the masking of this part in most mammals by the downward development 
of the orbital plates of the frontal, offered no difficulty to the ethmoidal de- 
termination of no. 14 in the bird ; and it forms as little valid objection to 
Oken's mode of expressing the ethmoidal homology of the prefrontals in the 
cold-blooded ovipara. 

For the reasons before assigned, viz. that the terms ' frontal anterieur' 
had been given to the bone in question, no. 14, in those animals in which it 
deviates least from its general type, as the nasal neurapophysis, I retain the 
name prefrontal for it under all its metamorphoses. Cuvier, after balancing 
the characters of the bones nos. 15, m and 73 (fig. 23) in birds, inclines to the 
opinion that 15 is the true nasal, and 22' an essential part (nasal process) of 
the premaxillary : with regard to 73, he says, " les os externes et plus voisins 
de I'orbite seraient presque comrae on le voudrait, ou des frontaux ante- 
rieurs ou des lacrymaux," In which case, no. 14 having been described as 
the ' ethmoid,' one or other of the above-named bones would be wholly absent 



226 REPORT— 1846. 

in birds. " Ce que pourrait faire croire que c'est le frontal anterieur qui 
manque, c'est que dans les oiseaux il n'y a point de frontal posterieur, et que 
la paroi anterieur de I'orbite, a I'endroit ou le frontal anterieure se trouve 
ordinairement, est raanifestenient formee en grande partie par une lame 
transverse de rethnioide*." But the postfrontal is not always absent in 
birds : it is present as a distinct bone, though small, in the emeu's skull, 
figured in the ' Memoir on the Dinornis' above-cited ; and it is still more 
developed in the remarkable extinct (?) genus, the immediate subject of that 
memoir. Besides, to anticipate the subject of a subsequent part of this report, 
a parapophysis always disappears from a typical segment of the skeleton 
sooner than a neurapophysis. The rest of Cuvier's difficulty in the recog- 
nition of the prefrontal in birds was more nominal than real. 

The ethmoid, in the restricted sense in which Cuvier applies the term in the 
crocodile and other animals with divided prefrontals, and in which I would 
apply it in those animals also in which the prefrontals have coalesced, is 
present but remains cartilaginous in the bird. In the mammal it becomes 
bony and contracts anchyloses not only with the still more reduced debris of 
the coalesced prefrontals, but also, in consequence of the change of position 
of the prefrontals through the further progress of concentration, whereby 
they are drawn backwards closer to the prosencephalic part of the cranium, 
and in consequence of the concomitant expansion of the true frontals, — with 
the orbital plates of the frontals ; whereby these plates usurp in most mammals 
the oflHice and the position of the external parts of the prefrontals in the cold- 
blooded vertebrataf . 

The posterior part of the coalesced prefrontals (figs. 24 & 25, u) divides 
the anterior aperture of the cranium into two outlets, upon the inner circum- 
ference of wiiich tile rhinencephala rest ; each ontlet being commonly closed 
by part of the olfactory capsules, which are ossified and perforated to receive 
the divisions of the olfactory nerves. When the prefrontals extend backwards 
and beyond the cribriform plates, they form what is termed the ' crista galli': 
this exists in comparatively few mammalia ; but is as large in the seal tribe 
as in man. In the tapirs the prefrontals expand above and overarch the ol- 
factory capsules, but their upper horizontal plates are overlapped by the 
nasals and true fi'ontals. In the DelphinidcE, where the olfactory capsules 
are absent, the prefrontals expand posteriorly, and diverge from their median 
coalesced portions constituting the septum of the nasal passage, in order to 
form the posterior boundaries of those passages and the anterior wall of the 
cranial cavity. They again expand and form a thick irregular mass anterior 
to the nasal passages in some DelphinidcE, and in Ziphius ossification extends 
along the fibrous continuation of the prefrontals forwards to near the end of 
the premaxillariesj. They are connate with the orbitosphenoids behind, and 
soon coalesce with the vomer below ; they rise anterior to the frontals and 
support the stunted nasals which are wedged between the prefrontals and 
frontals. The cetacea are the only mammalia in M'hich the prefrontals appear 
upon the exterior of the skull, and which in this respect resemble the reptilia. 

* Lefons d'Anat. Comp. 1837, t. ii. p. 580. 

f Cuvier takes this gi-ound in objecting to Oken's ethmoidal homology of the prefrontal 
in the crocodile, and says, "the ethmoid coexists in a cartilaginous state with, and is enve- 
loped hy, the prefrontal, ' comme la partie anterieure du frontal enveloppe rethmoide des 
ruminans.' " — Hist, des Poissons, v. p. 235. The correspondence is exaggerated, but it 
matters not. There are other characters of the mammalian ethmoid, as the closing of the 
cranium anteriorly, the transmitting the olfactory nerves, &c., which are nowise manifested 
by Cuvier's cartilaginous ' ethmoide' in the crocodile, and are very satisfactorily so by the 
prefrontals in that animal. 

X Ossem. Foss. v. pt. i. p. 351. 



ox THE VEKTEBRATE SKELETON. 22? 

Cuvier describes the posterior aud superior expanded and diverging plates 
of the prefrontals as "la lame cribreuse de I'ethinoide :" the coalesced part 
forming the septum, he ascribes to the vomer*. Dr. Kostlinf, also, who 
rightly recognises the ethmoid to be no proper bone of the skull, but only 
an ossified organ of sense, yet describes, after the anthropotomists, tiie coa- 
lesced prefrontals as the cribriform and azygos processes of the ethmoid 
(' Siebplatte' and ' Scheidewand des Siebbeins,' pp. 85. 89) in cetacea which 
have no organ of smell. In a young balaenoptera, in which tlie frontals, the 
vomer and the nasals were ossified, I find the prefrontals as two cartilaginous 
plates, extending from the nasals above to the groove of the vomer below. In 
the manatee the essential parts of the prefrontals which close the cranial 
cavity anteriorly, and give exit to the olfactory nerves, are thick and unu- 
sually expanded. But in no mammal do these parts, with their continuation, 
the ' lamina perpendicularis,' which, as the coalesced neurapophysial plates 
of prefrontals, bring the vomer below in connection with the nasals above, 
ever undergo such modifications as to obliterate their true and essential ho- 
niological characters. 

In proceeding next to consider the special homologies of the bones of the 
arch closed by the premaxillaries (22) and constituting the ' upper jaw,' I 
commence with the palatines (20), because they form, throughout the verte- 
brate series, the most constant medium of suspension of that arch to the 
anterior cranial segment formed by the vomer, prefrontals and nasal. This 
' secret affinity,' as Goethe would have termed it, before the knowledge of 
the general type had revealed its nature, is manifested by the process of the 
palatine in man, which creeps up, as it were, into the orbit to effect its wonted 
union with the prefrontal, to that part of the bone, viz. of which Cuvier had 
recognised the homologue in his 'ethmo'ide' of tlie bird J. It is the very 
constancy, indeed, of these and other connections which has exempted the 
palatine from the diff"erent determinations and denominations attached to 
other bones, and which renders further discussion of its special homology 
unnecessary here. 

Passing over, for the same reason, the maxillary (21) and premaxillary (22), 
and referring to the excellent treatise by Dr. Kostlin § for the grounds of 
the determination of the 'pterygoid' (24), I proceed to notice other bones 
which, diverging from the maxillary arch, serve to give it additional fixation 
and strength in the air-breathing vertebrates. The first of these is the malar 
bone (fig. 11, 26), the homology of which has been traced without difference 
of opinion throughout the mammalian class ; where, however, the inconstancy 
of its proportions, number of connections, and very existence, is sufficient to 
indicate its comparative unimportance as an element of the maxillary arch. 
It is absent in many insectivores (Ce?ifetes, Echinops, Soi-ex): it has not 
been detected as a distinct bone in the zygomatic arch in the monotremes, on 
account perhaps of its early coalescence, as in birds, with the maxillary 
(fig. 12, 21, 26) : m Myr7necophaga gigantea and Manis, it projects back- 
wards, as a styliform appendage, from the maxillary, but does not attain the 
squamosal; whilst in the sloths and their extinct congeners the gigantic 
megatherioids, the malar presents its maximum of development and complex- 
ity y. In the DeJphinidcB, again, the malar is much reduced : its slightly ex- 
panded maxillary end forms part of the orbit and joins the frontal ; the rest 
extending backwards, as a verj-^ slender style, beneath the orbit to the squa- 

* Ossem. Foss. v. pt. i. pi. xxvii. fig. 3, Ti. 

t Der Bau des Knochernen Kopfes, p. 11. 

X See the passage above quoted from the ' Lepons d'Anat. Comp.' ii. p. 580. 

§ Op. cit. p. 328. II Description of the Mylodon robustus, 4to, p. 19. 



228 REPORT— 1846. 

mosal. The malar joins the post-orbital process of the frontal in the Mana- 
tus senegalensis, the hippopotamus, the solipeds, and ruminants, some carni- 
vores and the lemurs ; in the true quadrumanes and man it joins the alisphe- 
noid, and sometimes also the parietal. 

The presence, form and connections of the malar are much more constant 
in the class of birds ; where, however, it must be sought for as an indepen- 
dent bone at an early period. In the young ostrich (fig. 23, le) it is reduced 
to the form of a simple, straight, slender style, and coalesces first with the 
similarly-shaped squamosal (27), and next with the malar process of the 
maxillary (21"). In the crocodile the malar bone (fig. 22, 26 ) becomes more 
developed, and adds the connections with the postfrontal (12) and the ecto- 
pterygoid (24') to the more constant ones with the maxillary (21) and squa- 
mosal (27), which alone sustain it in birds. In most of the chelonians the 
malar presents the same connections as in the crocodile, but is transmuted 
from a ' long' to a ' flat' bone. It retains the expanded shape in the agama ; 
but in most other lizards it resumes the styloid form ; being broadest, how- 
ever, in those genera, e. g. Iguana, Thorictes, Tejus, in which it extends from 
the maxillary to the postfrontal and squamosal ; in the Varani it projects 
freely backwards, like a styliform appendage of the maxillary, as in the 
toothless mammalian JBruta, above-cited. 

There is no malar bone in ophidians and batrachians. The lower portion 
of the tympanic pedicle in the Anoura sends forward a process which joins a 
backward prolongation of the maxillary : in all other batrachia the lower 
portion of the tympanic pedicle is restricted to its normal connections and to 
its function of affording articulation to the lower jaw. With regard, there- 
fore, to the zygomatic modification of this portion of the pedicle in anourous 
Batrachia, some may deem it the homologue of the malar ; and, in marsu- 
pial quadrupeds, the malar actually forms part of the glenoid cavity for the 
lower jaw : or it may be regarded as the squamosal, which constantly sup- 
ports the lower jaw in mammals : or it may be viewed as the coalesced homo- 
logue of both bones : or finally, as a simple modified dismembennent of the 
tympanic pedicle of the higher reptiles and birds ; effecting a union with 
the maxillary bone which makes it analogous to, but not, therefore, homolo- 
gous M'ith, the distinct malar and squamosal in those higher vertebrates. This 
is a question of special homology on which I am unwilling at present to 
express a decided opinion : but viewing the inconstancy of the squamosal in 
reptilia, and its deprivation of the function of exclusively supporting the 
mandible in all ovipara, I am disinclined to adopt the idea of its sudden resti- 
tution to that mammalian function in frogs and fishes ; yet, if either of the 
bones 20 and 27 are to be selected as the homologue of the hypotympanic (28^) 
of batrachians and fishes, I should regard the claims of the squamosal to be 
stronger than those of the malar, which Cuvier has chosen. The further sub- 
division, however, of the tympanic pedicle in fishes, prepares us, in the as- 
censive comparison, for the simple division of the pedicle in batrachia, and 
for recognising in the lower articular portion a vegetative dismemberment of 
23 in the crocodile. 

The characters and chief changes, in respect of connections and functions, 
of the squamosal (27) in the mammalia have already been noticed in the dis- 
cussion of the homologies of other elements of the complex ' temporal bone' 
in that class. In birds the bone (fig. 23, 27) undergoes the same change of 
form which has been noticed in the jugal, viz. from the squamous to the 
styloid. It continues, however, to connect the malar with the tympanic as 
it does in figs. 11 and 12, but it has no connections with other bones. Cu- 
vier having been led to recognise the squamosal in the mastoid (fig. 23, ») of 



ox THE VERTEBRATE SKELETON. 229 

birds, does not distinguish 27 from 26, the true ' jugal :' and Geoffroy v iewing 
the ' portion ecailleuse' of the temporal in that cranial bone of the bird, which 
he figures under the letter R, fig. 17, pi. 27 (Annales du Museum, x.), calls 
the true squamosal, the original separation of which from the malar he had 
noticed in the chick, ' jugal posterieure.' He did not admit that this division 
of the zygomatic style was constant or common in the osteogeny of the skull 
of birds : but I have always found such division in the embryo, and it con- 
tinues longer than usual in those very species, e. g. the duck and ostrich 
(fig. 23, 26, 27), in which Geoffroy denies its existence {I. c, p. 361). Oken 
accurately describes the two constituents of the zygoma in the skull of the 
goose, in his characteristic and original Essay*, where he calls the posterior 
piece (27) the humerus, and the anterior one (26 ) the radius of the head. 
Bojanusf, who also recognised the fact of the essential individuality of the 
bone (27) in birds, but who saw the homologue of the squamosal rather in the 
tympanic (23), calls it ' os zygomaticum posterius.' I could cite other testi- 
monies to the primitive existence of the distinct bone in birds connecting the 
malar with the tympanic ; but the fact which chiefly concerns us here is, that 
if the special homology of no. s with the mastoid, and that of no. 28 with 
the tympanic be proved, we then have a bone presenting the most constant 
connections of the squamosal in no. 27 : if, however, that name be transferred, 
as has been done by Cuvier, BojanusJ and Geoffroy, to other bones, then a 
new bone and a new name must be introduced into vertebrate craniology, 
for which, as I trust I have shown, there is no sufficient ground. 

Both Oken and Bojanus rightly discern in the permanently distinct bone 
which, in the crocodiles (fig. 22, 27) and chelonians, connects the malar (25) 
with the tympanic (as), the homologue of the bone they call ' cranial hume- 
rus,' or ' zygomaticum posterius' in the bird. Cuvier is more accurate in his 
determination of this bone (fig. 23, 27) as the ' squamosal' in reptiles ; but 
again at the expense of his consistency in regard to the characters of his 
squamosal in the bird: for the homology of no. s (Cuvier's 'squamosal') in 
fig. 22 with no. s (Cuvier's ' mastoid') in fig. 23, is as obvious and unmistake- 
able as is that of no. 27 (Cuvier's ' squamosal') in fig. 22 with no. 27 (his dis- 
memberment of the jugal) in fig. 23. The squamosal is relatively stronger in 
crocodiles than in birds, and in many chelonians resumes its flat, scale-like 
form ; although, as Cuvier well observes, it answers, in function, only to the 
zygomatic part of the mammalian squamosal : — " c'est un temporal dont la 
partie craniale a disparu§." In lizards the squamosal again resumes the zy- 
gomatic or styloid shape, connecting the mastoid and tympanic with the 
postfrontal, and usually also with the malar ; the posterior connections being 
here, as in mammals, the more constant ones. 

As the squamosal varies in form with the malar, so it likewise disappears 
with it in ophidians ; unless the anatoftiist, tracing it descensively, prefers to 
see it again in the peculiarly developed hypotympanic of the anourans. Ac- 
cording to this view of the sudden resumption of its mammalian function in 
regard to the lower jaw in batrachia, the name 'squamosal' may be trans- 
ferred to the hypotympanic in fishes ; and, if we must view the pedicle 
(2s a — d, fig. 5) as ' homologically compound,' and not, like the mandibular 
ramus, ' teleologically compound,' i^d seems to me a less arbitrary selection 
from the pieces of that long and subdivided pedicle, for the representative 

* Ueber die Bedeutung der Schadelknochen, 4to, 1807, p. 12. 
+ Anatome Testudinis Europaese, fol. Parergon, 1821, p. 178, fig. 196, i. 
X The tympanic bone 28 is described in the same work as ' squamosum sive quadratum,' 
(fig. 196, g.) : the mastoid is rightly named. 
§ Ossemens Fossiles, 4to. t. v. pt. ii. p. 85. 



230 REPORT — 1846. 

of the squamosal, than the proximal or uppermost piece (na) to which Cu- 
vier has applied that name. If, indeed, Bojanus could have determined to 
his own satisfaction or that of other anatomists, that the pedicle (2s, tig. 23), 
articulated by one end to the mastoid, and by the other to the mandible, in 
birds, was the ' squamosum,' then there would have been some ground for 
regarding the bone (28a, fig. 5) connected in fishes, with the mastoid as the 
' squamosum.' 

But when Cuvier had persuaded himself that the bone no. s, fig. 23, in 
birds, to which the tympanic pedicle is articulated, was the * ecaille du tem- 
poral,' we feel at a loss to know on what principles special homologies can 
be traced, when we find the name transferred to the upper part of the tym- 
panic pedicle in fishes (fig. 5 28 a), which is articulated to the bone (s) un- 
equivocally answering to Cuvier's ' ecaille du temporal' in birds. M. Agassiz 
is more consistent, and abandons with reason the Cuvieriau determination of 
the squamosal in fishes : if, however, the grounds assigned are conclusive as 
to the homology of no. 8, figs. 8 & 23 in birds with the mastoid of mammals 
and reptiles, M. Agassiz cannot be correct in regarding the bone no. 8, fig. 
5 in the fish, as the ' ecaille du temporal.' 

With reference to the idea entertained by Spix, GeoflProy and Agassiz, of 
the homology of the suborbital muciferous scale- bones in fishes with the malar 
bones of higher vertebrates, I may refer to what has already been said in 
regard to the actual repetition of the osseous arch connecting the prefrontal 
with the postfrontal in certain birds, where that arcii coexists with, and in- 
dependently of, the bone recognised as the 'malar' by both Spix and Geof- 
frey. The connection of the malar with the lacrymal and post-frontal is 
less constant and characteristic of the bone than that with the maxillary and 
squamosal. And it may further be remarked, that the functional character 
of circumscribing a mucous duct, manifested by the lacrymal or anterior 
end of the upper zygomatic or suborbital arch in the parrot, is superadded to 
the character of connections in proof that such arch, and not the true zygo- 
matic arch below, is homologous with the suborbital chain of bones in fishes. 
All these discrepancies as to the jugal and squamosal in fishes arise, in my 
opinion, out of the circumstance that those bones are normally absent in 
that class; both 26 and 27, figs. 11, 22, 23, 24, 25, being accessory parts, de- 
veloped only in saurians, chelonians, birds and mammals, for additional fixa- 
tion of the upper jaw, or for additional expansion of the cranium, or for both 
purposes*. 

According to this view, I regard the tympanic (2$) as essentially charac- 
terized in the oviparous vertebrates (fishes, reptiles, birds) by its free articu- 
lation by a convex condyle with the mastoid above, and by a convex condyle 
with the mandible below ; and I regard its subdivisions in the lowest of 
these vertebrates, in the same light as the subdivisions of the mandible itself. 
The formation of the tympanic cavity and support of the tympanic membrane 
are secondary functions. The tympanic pedicle is essentially a single cranial 
element, and actually so in all air-breathing vertebrates above batrachians. 
We see plainly, even in the frog, that the portion which supports the ' mem- 
brana tympani ' is a mere exogenous process of the pedicle : it has still less the 
appearance of a distinct part or process in the saurians, chelonians and birds : 
and when the tympanic is excluded by the squamosal in manmials from its 
normal office of supporting the mandible, it still manifests its character of 

* The inconstant ossicle suspended to the back part of the free extremity of the maxillary 
in the percoid fishes would have the best claim to homology witti the malar, if the further 
subdivision of the maxillary in the herring and lepidosteus did not indicate it to be a vege- 
tative dismemberment of that bone. 



ON THE VERTEBRATE SKELETON. 231 

unity, whether it be expanded into a ' bulla ossea,' extended into a long tube 
or meatus, or both, as in fig. 24', 2S, or whether, as in fig. 25, it be reduced to 
a mere ring or hoop supporting the tympanic membrane, until it coalesces 
with other parts of the temporal, to form the tympanic or ' external auditory 
process' of that bone. In no air-breathing vertebrate have I ever found, or 
seen described, the separation of the part of the tympanic forming the wall 
of the tympanic chamber from the part supporting the tympanic membrane, 
or this distinct, save in batrachia, from the part supporting the lower jaw*. 
The tympanic pedicle is still further subdivided in fishes; but M. Agassiz's 
original idea of the 'epitympanic' as a dismemberment of the pedicle, which 
he proposed to call ' os carre superieur,' is, in my opinion, much more consist- 
ent with nature than his later determination of that bone as the ' mastoid,' 
or than Cuvier's attempts to find the homologues of both the mammalian 
'squamosal' and 'jugal' in the piscine subdivisions of the same pedicle. 
There is as little ground for making the zygomatic process a distinct element 
from the squamous portion, as for severing the annular process from the rest 
of the tympanic. This idea of the zygomatic as an independent piece, which 
Dr. Kostlin has also adopted, seems to rest only on the mal-determination 
by Bojanus and Oken of the true squamosal in birds and reptiles as the 
'zygoniaticum' or 'jugale posterius': and the idea was perhaps further 
strengthened in the mind of M. Agassiz, by what he deems to be the essen- 
tial and characteristic function of the squamosal. But its protective cere- 
bral or cranial scale is a peculiarly mammalian development ; much reduced 
in the ruminants and cetacea, and totally disappearing in the oviparous ver- 
tebrates. The zygomatic functions and connections are, notwithstanding a 
few exceptions, as in the scaly manis and a few lizards, the essential homo- 
logical characters of the ' squamosal.' The necessity for ibrming an opinion 
of the essential nature and general homologies of the parts blended together 
in the human ' os temporis' by the ascensive or synthetic method, is strikingly 
exemplified by the results of the application of M. Agassiz's id^a of its nature 
to his determination of the bones in the head of fishes. 

As the palato-maxillary arch in most air-breathing vertebrates supports, ac- 
cording to my views, certain appendages, e. g. the malar and squamosal, which 
are not present in fishes ; so, I believe, with Cuvier, that the tympano-man- 
dibular arch supports in fishes, certain appendages, which are not developed 
in any other class. It is this fact, chiefly, that has led to so much discrepancy 
in the attempts to determine by reference to bones in higher vertebrates the 
opercular bones of fishes, — the chief battle-field of homological controversy. 
All the four opercular bones forming the diverging appendage of the tym- 
pano-mandibular arch (fig. 5, 34 to ar) were deemed by Cuvier to be peculiar 
ichthyic super-additions to the ordinary vertebrate skeleton ; whilst by Spix, 
GeotFroy, and De Blainville they are held to be modifications of parts which 

* M. Agassiz applies the subjoined analysis of the ' temporal bone' to elucidate the homo- 
logies of the skull of fishes : — " Nous distinguoiis encore dans le temporal complet les parties 
suivantes : Vecaille, servant de complement a la parol laterale du crane dans sa partie poste- 
rieure ; le mastdidien, servant de rempart posterieur a la cavite tympanal ; la caiise, logeant 
les parties principales de la cavite tynipanale; I'anneau tympanique, servant d'appni a la 
membrane du tympan ; Yapophyse jugal, formant I'appui posterieur de I'arcade zygomatique ; 
Yapophyse styldide, ofFrant une insertion a I'os hyo'ide, par laquelle ce dernier se fixe au crane ; 
et enfin I'os carre, formant la surface articulaire sur laquelle la machoire inferieure exerce 
ses raouvemens. La maniere variee dont ces ditferentes pieces se soudent ensemble, se separent 
et se combinent, occasionnent ces innombrables variations auxquelles le temporal est sujet 
dans son ensemble, h'ecaille du temporal est destinee, comme nous venons de le voir, a pro- 
teger les parties cerebrales posterieures de la tete, sur la face laterale du crane." — Recherches 
sur les Poissons Fossiles, t. ii. pt. 2, 1843, p. 62. 



232 REPORT— 1846. 

exist in the ordinary or endo-skeleton of other vertebrata. The learned 
Professor of Comparative Anatomy in King's College, London, who regards 
this as "the more philosophical mode of considering them*," has briefly 
stated the homologies proposed by the supporters of this view, viz. that the 
opercular bones are gigantic representatives of the ossicles of the ear (Spix, 
Geoffroy, Dr. Grantf): or that they are dismemberments of the lower jaw 
(De Blainville, Bqjanus), — a view refuted by the discovery of the compli- 
cated structure of the lower jaw in certain fishes, which likewise possess the 
opercular bones : he then cites a third view, viz. that they are parts of the 
dermal skeleton ; " in short, scales modified in subserviency to the breathing 
function ;" an opinion which Professor Jones correctly states that he derived 
from my Lectures on Comparative Anatomy, delivered at St. Bartholomew's 
Hospital in 1835, and which he adopts, although its accordance with his first 
proposition is not very clear. I have subsequently seen reason to modify that 
view, though it has received the sanction of the greatest ichthyologist of the 
present day, M. Agassiz ; and, as I have since found, had presented itself so 
early as 1826, under a peculiar aspect to the philosophical mind of Professor 
Von Baer. In his admirable paper on the endo- and exo-skeleton, M. Von Baer 
expresses his opinion, that the opercular bones are (dermal) ribs or lateral 
portions of the external cincture of the head J. The idea of the relationship 
of the opercular flaps to locomotive organs is presented by Carus, under the 
fanciful view of their homology with the wing-covers of beetles and the valves 
of a bivalve shell §. In 1836, M. Agassiz propounded his idea of the relation 
of the opercular bones to scales in a very precise and definite manner; 
though, as I have elsewhere shown ||, the chief ground of his opinion is erro- 
neous. He says, "Les pieces operculaires des poissons ne croissent pas, 
comme les os des vertebres en general, par irradiation d'un ou de plusieurs 
points d'ossification ; ce sont, au contraire, des veritables ecailles, formees, 
comme celles qui recouvrent le tronc, de lames deposees successivement 
les unes sous les autres, et dont les bords sont souvent meme denteles 
comme ceux des 6cailles du corps. Tels sont I'opercule, le sub-opercule, et 

* Professor Rymer Jones, General Outline of the Animal Kingdom, 8vo, 1841, p. 509. 

t Lectures, Lancet, Jan. 11, 1834, p. 573 ; Outlines of Comp. Anat. p. 64. 

X " In mancher Beziehung gehoren die Kiemendeckel zu ihr, und ich halte sie um so 
mehr fur(Haut) Rippen, d. h. fiir Seitentheile der iiussern Ringe desKopfes, da ich sieauch 
iu den gewohnlichen Knockenfischen fiir nichts anderes ansehen kann. Hat bei diesen auch 
der oberste Knochen des Kiemendeckels wenig Aehnlichkeit mit Rippen, so geht dagegen 
der unterste so uuverkennbar in die strahlender Kiemenhaut iiber, das der Uebergang gar 
nicht zu verkennen ist." — Meckel's Archiv, 1826, 3 heft, p. 369. 

An analogous idea of the relation of the opercular bones to the inferior or costal arches was 
proposed by Geoffroy St. Hilaire (Annales des Sciences, t. iii. pi. 9), and Cuvier (Hist, des 
Poissons, i. p. 232), and has been adopted by the learned Professor of Comparative Ana- 
tomy in University College, who, speaking of the occipital vertebrae, says, " The two external 
and the two lateral occipitals form the upper arch, and the two opercular and two sub- 
opercular bones constitute the lower arch." (Lectures, Lancet, 1834, p. 543.) He subse- 
quently, however, adopts and illustrates (p. 573) the homology of the opercular bones with 
the ' ossicula auditus' of mammalia; and in the ' OutUnes of Comp. Anat.' cites only the 
Spixian and Blainvillian hypotheses (pp. 64, 65). In my Hunterian Lectures (vol. ii. 1836, 
pp. 113, 130), I have adduced the grounds which have led me to the conclusion that the 
opercular bones are neither ribs of the exo-skeleton, nor inferior arches of the endo-skeleton, 
but persistent radiating appendages of an inferior (haemal) arch ; not, however, of the occipital 
vertebra, but of the frontal ; just as the branchiostegal rays are the appendages of the haemal 
arch of the parietal, and the pectoral fins of that of the occipital vertebrae. That parts of 
both endo- and exo-skeleton may combine to constitute the opercular fin is the more pro- 
bable, inasmuch as we see the same combination of cartilaginous and dermal rays in the 
pectoral fins of the plagiostomes, and in the median fins of most fishes. 

§ Urtheilen des Knochen und Schalengeriistes, fol. p. 122. 

II Lectures on Vertebrata, p. 139. 



ON THE VERTEBRATE SKELETON. 233 

I'inter-opercule. Le supra-scapulaire mfeme peut etre envisage comme la 
premiere 6caille de la ligne laterale, dont le bord est egalement dentele. On 
pourrait dire aussi que le scapulaire n'est qu'une tres grande ecaille de la 
partie anterieure des flancs*." And he adds, "L'opinion que j'ai emise a 
leur egard prouve que je suis loin d'admettre les rapports que Ton a cru 
trouver entre les pieces operculaires et les osselets de Toreille internet." 

I apprehend that the idea of the development of the opercular bones by 
the successive excretion or deposition of layers, one beneath the other, ac- 
cording to the mode in which M. Agassiz supposes scales to be formed, was 
derived merely from the appearance of the concentric lines on the opercular, 
subopercniar, and interopercular bones in many fishes. I have examined 
the development of the opercular bone in young gold-fish and carp, and I 
find that it is effected in precisely the same manner as that of the frontal and 
parietal bones. The cells which regulate the intussusception and deposition 
of the earthy particles make their appearance in the primitive blastema in 
successive concentric layers, according to the same law which presides over 
the concentric arrangements of the radiated cells around the medullary canals 
in the bones of the higher vertebrata : and the term ' successive deposition,' 
in the sense of excretion, is inapplicable to the formation of the opercular 
bones. The argument in favour of their dermal character drawn from the 
phaenomena of the development of the opercular flap, would equally apply to 
prove the bones (ulna, radius, carpus, &c.) supporting the pectoral fin, to be 
'dermal' bones J. 

The interopercular as well as the preopercular bones exist in the Lepi- 
dosiren annectens with all the characters, even to the green colour, of the rest 
of the ossified parts of the endo-skeleton ; the preopercular, as an appendage 
to the tympanic arch, retaining its primitive embryonal subcylindrical form, 
the interopercular being partly attached to the hyoid arch. Of the supra- 
scapular there is no trace in the lepidosiren ; but in the sturgeon it plainly 
exists as part of the cartilaginous endo-skeleton, under the same bifurcate 
form, and double connection with the cartilaginous skull, which it presents 
in most osseous fishes. The large triangular bony dermal scale firmly adheres 
to its broad, triangular, flat, outer surface. The epi- and meso-tympanic 
cartilages in like manner expand posteriorly, and give a similar support to 
the large opercular ganoid scale. Were the supporting cartilages of the 
opercular and suprascapular scales to become ossified in the sturgeon, they 
might become anchylosed to the dermal bony plates, and bones, truly homo- 
logous with the opercular and suprascapular in ordinary osseous fishes, 
would thus be composed of parts of the endo- and exo-skeleton blended 
together. I cannot, therefore, concur with Von Baer in the opinion that the 
opercular bones are ribs of the exoskeleton, nor with Agassiz that both the 
opercular and suprascapular bones are merely modified scales. In explaining 
my views of the opercular bones, I am compelled, believing them to have no 
special homologues in higher animals, to express those views in the terms of 
a higher generalization. The suprascapular bone (fig. 5, 40) is the upper or 
first part of the haemal arch of the occipital segment of the skull, and corre- 
sponds in serial homology with the epi-tympanic portion (28 a) of the mandi- 
bular arch, and with the palatine portion (20) of the maxillary arch. The 
opercular bones are the diverging appendages of the tympauo-mandibular 

* Recherches sur les Poissons Fossiles, livraison 6me, 1836, torn. iv. p. 69. 

t lb. p. 73. 

X " L'embryologie nous prouve, en effet, que la formation de I'appareil operculaire n'est 
qu'un simple produit de la peau, qui peu-a-peu s'etend par dessus les branchjes, d'abord 
entierement degagees dans I'embryon." — lb. p. 64. 

1846. R 



324 REPORT — 1846. 

arch, and correspond, in serial homology, with the brancliiostegal appendages 
of the hyoid and the pectoral appendages of the scapular arches, and have 
the same title to be regarded as cephalic fins, and as parts of the normal 
system of the vertebrate endo-skeleton ; but neither opercular bones nor 
branchiostegal rays are retained in the skeletons of higher vertebrata. All 
diverging appendages of vertebral segments make their first appearance in 
the vertebrate series as ' rays '; and the opercular bones are actually repre- 
sented by cartilaginous rays, retaining their primitive form in the plagio- 
stomes. In the conger the subopercular still presents the form of a long and 
slender fin-ray. 

The opercular and subopercular, in ordinary osseous fishes, may frequently 
coalesce, like the suprascapular, with their representative scales of the dermal 
system ; but they are essentially something more than peculiarly developed 
representatives of those scales. M. Agassiz, indeed, excepts the preoper- 
cular bone from the category of "pieces cutanees," believing it to be the 
homologue of the styloid process of the temporal bone in anthropotomy, or 
the 'stylo-hyal' of vertebrate anatomy, as the piece, viz. which completes the 
hyoid arch above. " C'est en effet," he says, " cet os a la face interne duquel 
I'os hyoide des poissons est suspendu, qui s'articule en haut avec le mastoi- 
dien et tres sou vent meme snr I'ecaiile du temporal." So far as my obser- 
vation has gone, it is a rare exception to find the hyoid arch suspended to 
the preoperculum ; the rule in osseous fishes is to find the upper styliform 
piece of the hyoid arch (fig. 5, as) attached to the epi-tympanic (28 a) close 
to its junction with the meso-tympanic bone (236). It is equally the rule to 
find the preopercular (34) articulated with the epi-, meso-, and hypo-tym- 
panics ; and it is an exception, when it rises so high as to be connected with 
the mastoid (' ecaille du temporal ' of Agassiz). If the styio-hyal be not the 
upper piece of the hyoid arch displaced, and if the upper piece connecting 
that arch with the mastoid is to be sought for in osseous fishes, I should 
rather view it in the posterior half of the epi-tympanic (as «), which is usually 
bifurcate below and very commonly also above, when the posterior upper 
division articulates with the mastoid, and one of the lower divisions with the 
hyoid arch. 

The normal position, form, and connections of the preoperculum clearly 
bespeak it to be the first or proximal segment of the radiated appendage of 
the tympano-mandibular arch : the opercular, subopercular, and interoper- 
cular bones form the distal segment of the same appendage. 

M. Vogt, in supporting M. Agassiz's views of the Ganoid order, reiterates 
his original idea that the preopercular bone is the proximal piece (styloid) 
of an arch distinct from the tympano-mandibular one ; but as the chief ground 
of this opinion rests on a simple question of fact easily determinable, viz. 
whether, as a rule, the hyoid arch is suspended from the preoperculum, and 
this from the mastoid in fishes, neither of which accord with my observation 
of their connections of those parts, the verdict may be left to the experience 
of other observers. From a remark of M. Vogt's*, viz. that " M. Miiller 
attache, a ce qu'il parait, trop peu d'importance a ce fait, que toujours le 
preopercule, et cela aussi chez les Siluroides, sert de point d'attache a Tare 
hyoidien," it would seem that, perhaps, the accomplished physiologist and 
ichthyologist of Berlin had not found the fact; and, therefore, gave not more 
than its due importance to the rare exceptional circumstance of such an at- 
tachment. The preopercular can be removed in most fishes, except where, 
as in the siluroids, it coalesces with the tympanic arch, without dislocating 

* Annales des Sciences, 1845, p. 56. 



ON THE VERTEBRATE SKELETON. 235 

or disturbing the connections of the true stylo-hyal (fig. 5, ss) with the epi- 
tympanic (asa) from which it is normally suspended. 

M. Vogt correctly observes that the ' temporal' ^epitympanic, 2s a), ' sym- 
plectique' (mesotympanic, 2s 6), and ' jugulaire' (hypotympanic, 2S rf), "a 
eux seals forment deja un arc suspensoir complet, a la face posterieure 
duquel le preopercule est seulement accole*." But this only proves that the 
preoperculum is an appendage to such arch, not that it is a suspensory pier 
of a second arch. 

The only essential modification which the siluroids present is the confluence 
of the preoperculum with the true tympanic pedicle, here reduced to a single 
piece. But this does not disprove its character as an appendage of the 
tympano-mandibular arch, any more than does the confluence of the ulna and 
radius with the scapular arch in the sturgeon disprove the character of those 
elements as appendages of that arch. 1 have not been able to trace in the 
siluroids the primitive boundaries of the coalesced preoperculum to such an 
extent as to justify the statement, that it is intercalated between the epitym- 
panic and hypotympanic, replacing the mesotympanic : but, if the preopercular 
should extend in any siluroid fish so far as M. Vogt describes, this excep- 
tional development would rather prove it to belong essentially to the tym- 
panic and not to the hyoidean arch : at least it is only through this abnor- 
mal encroachment that the preopercular can detach the stylohyal from the 
epitympanic. 

As the otosteals, or ' ossicula auditus,' have borne a prominent share in the 
discussions of the special homologies of the tympanic pedicle and its append- 
ages, I may here remark that the extension in the embryo mammal of the 
long and slender process of the malleus in the direction of the mandible, and 
its continuation or connection with the cylindrical cartilage (haemal portion 
of the tympano-mandibular arch) from which the lower jaw is subsequently 
developed, is a circumstance which renders the idea of the malleus, at least, 
being a modified element of the tympano-mandibular arch in batrachiaiis 
and fishes, worthy of consideration. The prolongation from the mesotym- 
panic of the cylindrical cartilage, described by Meckel, and around which 
the mandible is ossified in fishes, and the characteristic cylindrical or styloid 
form of the mesotympanic, have induced M. Vogtf to view that bone, the 
'symplectique' of Cuvier, as the homologue of at least part of the malleus; 
and at the same time of the bone called * tympano-malleal' by Duges (my 
'hypotympanic') in the batrachians. M. Vogt offers no other reasons for 
the determination. I find that the cartilage which in the batrachians forms 
the medium of communication between the semi-ellipsoid ossicle (stapes) 
closing the fenestra ovalis and the tympanic membrane, is repeated or repro- 
duced in the more malleiform cartilage connecting the columelliform stapes 
of the saurian reptiles to the membrana tympani. In birds a portion of the 
cartilage attached to the tympanum becomes ossified and coalesces with the 
columelliform stapes ; and at the angle of union one or two cartilaginous^ 
processes exist, which some anatomists have compared with the incus. But 
all anatomists have concurred in recognising the homology of the peripheral 
bent-down portion of the long columella, which adheres to the membrana 
tympani, with the part of the malleus called ' manubrium,' or handle, in 
mammalia. The superadded modifications characteristic of the otosteals in 
this class, have their seat between the manubrium mallei and the stapes, and 
chiefly result in the development of the new bone called ' incus ' and its epi- 
physis, which has been termed the ' os orbiculare.' Notwithstanding, there- 
fore, the connection of the ' processus gracilis mallei' with the embryonic 
* Annales des Sciences, 1845, p. 53. f Loc. cit. p. 58. 

r2 



236 REPORT — 1846. 

haemal or visceral cartilage of the mandibular arch in mammals, the homo- 
logy of the malleus is so clearly traceable down ta its first independent ma- 
nifestation in coexistence with the tympanic membrane of the batrachia, to 
which it connects the unequivocally acoustic ossicle representing the ' stapes,' 
that the reference of all the additional ossicular mechanism of the ear-drum 
to the same system of the skeleton as the petrosal itself, appears to me to be 
most consonant with the recognised facts in their development and compara- 
tive anatomy. 

M. Agassiz has never countenanced the idea of the reproduction of the 
mammalian tympanic ossicles in a magnified form in either the tympanic 
arch or its opercular appendages. Returning to the consideration of these 
bones in the last volume (p. 68) of his admirable ' Recherches,' he reaffirms 
his opinion, that the opercular, subopercular, and interopercular are ' osse- 
lets particuliers de la peau ;' but calls them ' branchiostegal rays.' If he 
had meant that they were parts essentially distinct, but comparable to the 
true branchiostegals, he would have accurately enunciated their 'serial ho- 
mology.' M. Agassiz, however, expressly repudiates this idea of represen- 
tative relation, and affirms them to be part of one and the same series of 
ravs. " Mais en disant que les pieces operculaires sont des rayons branchio- 
stegues, je n'entends point faire une simple comparaison, mais bien affirmer, 
que je considere ces plaques osseuses simplement comme les rayons bran- 
chiostegues superieurs*." This idea is, in fact, a necessary consequence of 
M. Vogt's conclusion, that the preoperculum is the upper or styloid element 
of the hyoidean arch. The combination of the opercular rays or bones with 
the branchiostegals in the support and movements of the continuous gill- 
cover and gill-membrane, does not prove them to be diverging appendages 
of the same arch, any more than the similar combination of the rays of the 
pectoral and ventral fins in the sucker of the Cycloplerus proves those rays 
to be parts of the same arch. And I may repeat that, admitting the humerus 
to be, as Bakker surmised, confluent in all fishes with the bone 52, fig. 5 ; 
and since in the plagiostomes, sturgeons and lophioids, the second segment of 
the rudimental fore-limb is not liberated from the supporting arch ; so, like- 
wise, the proximal member of the opercular limb may remain, or become in 
some instances confluent with its sustaining arch, without that exceptional 
state invalidating the determination deduced from its more constant and re- 
gular character as the proximal element of the free appendage to that arch. 

The third inverted arch of the skull is suspended in fishes by a slender 
styliform bone, the 'stylohyal' (fig. 5, ss), from the lower end of the epi- 
tympanic (2s a) close to the joint of the styliform 'mesotyrapanic' (i%b); 
and it is connected, through the medium of the posterior division and 
joint of the epitympanic, with the mastoid (s). Now, either that division 
of the epitympanic may be viewed, by virtue of its proper articular condyle 
above, and its connection with a distinct inverted arch below, as the proximal 
piece of that arch, coalesced with the proximal piece of the next arch in 
advance, which articulates with the post-frontal; or, it may be viewed as an 
excessive development of the proximal piece of the tympano-mandibular arch, 
which, extending backwards, has displaced the hyoid from the mastoid, just 
as the squamosal, by a similar backward development, in mammals, displaces 
the mandibular arch from the tympanic. 

According to the first view, the bone no. 38 would be a dismemberment 
of the proximal element of the hyoid arch ; according to the second view, it 
would be the entire element reduced and displaced : in both cases it would 
be homologous with the proximal slender piece of the hyoid arch in all 

* Recherches sur les Poissons Fossiles, v. pt. ii. p. 68. 



ON THE VERTEBRATE SKELETON. 237 

vertebrata, and to which piece the term 'styloid' or 'stiliform' has been 
given from the fish up to man (see Table!.). The homology, indeed, is so 
obvious, that M. Agassiz, in accepting the conclusion of M. Vogt, that the 
bone (fig. 5, 34), peculiar to osseous fishes, which so rarely articulates di- 
rectly with the mastoid or with the hyoid arch, and so constantly sustains 
the distal segment of the operculum, was the homologue of the 'processus 
stiliformis ossis temporis,' nevertheless retains the name 'styloide' for the 
part no. ss in question. 

The true homology of no. 34, already explained, removes the anomaly of 
viewing that peculiarly piscine bone as the homologue of a constant element 
of the hyoid arch in all the vertebrate classes, and the greater anomaly of 
the introduction of a new element — a styloid piece of the os hyoides — in 
addition to the 'styloid process of the temporal' in fishes. The 'stylohyal' 
articulates below to the apex of a triangular piece (39), which is pretty con- 
stant in fishes, and to which I give the name of ' epihyal,' as being the upper 
of the two principal parts of the cornu or arch : the third longer and stronger 
piece is the ' ceratohyal ' (ih. 40). 

The keystone or body of the inverted hyoid arch is formed by two small 
subciibical bones on each side, the 'basihyals' (ib. 41). These complete the 
bony arch in some fishes : in most others there is a median styliform ossicle, 
extended forwards from the basi-hyal symphysis into the substance of the 
tongue, called the ' glossohyal ' (ib. 42), or ' os linguale'; and another symme- 
trical, but usually triangular, flattened bone, which expands vertically as it 
extends backwards, in the middle line, from the basihyals ; this is the ' urohyal' 
{ib. 43). It is connected with the symphysis of the coracoids, which closes below 
the fourth of the cranial inverted arches, and it thus forms the isthmus which 
separates below the two branchial apertures. In the conger the hyoidean 
arch is simplified by the persistent ligamentous state of the stylohyal, and 
by the confluence of the basihyals with the ceratohyals : a long glossohyal 
is articulated to the upper part of the ligamentous symphysis, and a long 
compressed urohyal to the under part of the same junction of the hyoid arch. 
The glossohyal is wanting in the MurcBnophis. 

The appendages of the hyoidean arch in fishes retain the form of simple, 
elongated, slender, slightly curved rays, articulated to depressions in the outer 
and posterior margins of the epi- and cerato-hyals : they are called " bran- 
chiostegals," or gill-cover rays, because they support the membrane which 
closes externally the branchial chamber. The number of these rays varies, 
and their presence is not constant even in the bony fishes : there are but 
three broad and flat rays in the carp ; whilst the clupeoid Elops has more 
than thirty rays in each gill-cover : the most common number is seven, as 
in the cod (fig. 30, 44). They are of enormous length in the angler, and 
serve to support the membrane which is developed to form a great receptacle 
on each side of the head of that singular fish. 

Branchial Arches. — In the class of fishes, certain bony arches, which ap- 
pertain to the system of the visceral skeleton, succeed the hyoidean arch, 
with the keystone of which they are more or less closely connected. Six of 
these arches are primarily developed, and five usually retained ; the first four 
of these support the gills, the fifth is beset with teeth and guards the opening 
of the gullet : this latter is termed the ' pharyngeal arch,' the rest the ' bran- 
chial arches.' 

The lower extremities of these arches adhere to the sides of a median chain 
of ossicles, which is continued from the posterior angle of the basi-hyal, or 
from above the uro-hyal, when this is ossified : the bones which form those 
extremities are the ' hypobranchials'; and they support longer bent pieces, 



238 REPORT— 1846. 

called ' cerato-branchials.' It is with these elements of the branchial arches 
in fishes and perennibranchiate batrachians that we are chiefly concerned 
in tracing the homology of the liyoid apparatus in the air-breathing verte- 
brates. With regard to the branchial and pharyngeal arches, which attain 
their full development only in the class of fishes, I regard them as appertain- 
ing to the system of the splanchno-skeleton, or to that category of bones to 
which the heart-bone of the ruminants and the hard jaw-like pieces support- 
ing the teeth of the stomach of the lobster belong. The branchial arches 
are sometimes cartilaginous when the true endo-skeleton is ossified : they are 
never ossified in the perennibranchiate batrachians, and are the first to dis- 
appear in the larvae of the caducibranchiate species ; and both their place 
and mode of attachment to the skull demonstrate that they have no essential 
homological relation to its endo skeletal segments. 

The hyoid arch or apparatus retains most resemblance to that of fishes in 
the Siren lacertina ; the basihyal is simplified into a single osseous spatu- 
late piece, with the bowl of the spoon anterior, and supporting a broad and 
flat semicircular glossohyal. A strong and thick ceratohyal is articulated 
by means of a small cartilage to the side of the expanded part of the basi- 
hyal, and a cartilaginous epihyal arches backwards from its upper end. A 
cartilaginous urohyal extends from the hind end of the basihyal, and ex- 
pands into a radiated disc, which supports the membranous trachea and the 
simple glottis. One pair of bony ' hypobranchials' is articulated to the 
basi-uro-hyal joint and a second pair to the sides of the urohyal : and to the 
upper and outer ends of these are attached four pairs of cartilaginous ' cerato- 
branchials.' The fimbriated branchiae are attached to the three anterior 
ceratobranchials. 

In the proteus the urohyal is absent, and it is not again developed in any 
batrachian. The long subcylindrical basihyal supports a subcircular carti- 
laginous discoid glossohyal, and at the angle of union the bony ceratohyals 
are sent ofiP. A pair of hypobranchials diverge from the end of the basihyal ; 
to which a second small pair of basibranchials are loosely connected by an 
aponeurosis. These support three ceratobranchials on each side, which are 
bony. 

In the newts there is neither a glossohyal nor urohyal, or but a rudiment 
of the latter, to each side of which are articulated two hypobranchials, whose 
distal ends converge on each side to support a single cartilaginous gill-less 
rudiment of a ceratobranchial. The special homologies of all those parts of 
the complex hyoid, rendered more complex by the retention of part of the 
branchial skeleton, are clearly demonstrated by pursuing the metamorphoses 
of the hyo-branchial skeleton in the larvae of the anourous batrachians. In 
the fuU-gilled tadpole a short and simple basihyal supports laterally two 
thick and strong ceratohyals, and posteriorly two short and broad hypo- 
branchials, to which four ceratobranchials are attached : all the parts are 
cartilaginous. The type of this stage is retained in the siren, with the histo- 
logical progress to bone in the hyoid and hypo-branchial pieces. The second 
well-marked stage in the tadpole shows an extension of the external and 
posterior angles of the hypobranchials, with progressive absorption of the 
cartilaginous ceratobranchials. The growth and divergence of the posterior 
angles of the hypobranchials refer to the development of the larynx, now 
commencing, which part they are destined to support. That period may be 
described as the third stage at which the ceratobranchials have disappeared, 
and the posterior angles of the hypobranchials increase in length and assume 
the character of posterior cornua of the os hyoides. The last and adult 
stage shows the ossification of the elongated angles of the hypobranchials, 



ON THE VfiRTEBBATE SKELETON. 239 

the coalescence of their cartilaginous bases with the basihyal, the expansion 
of the basihyal and extension of its anterior and external angles ; in front of 
which the now long and slender ceratohyals usually coalesce with the basi- 
hyal ; their opposite ends having shifted their attachments and retrograded, 
like other haemal arches of the skull, in the course of the metamorphosis. 
In the case of the hyoid arch of the frog, the change of place is from the 
tympanic pedicle backwards to the persistent cartilaginous petrosal : and 
this is a very suggestive and significant change. All the parts of the hyoid 
remain cartilaginous except the appended and persistent detachments from 
the visceral system of the brancliial arches: these long 'hypobranchials' 
(' comes thyroidiennes' of Cuvier and Duges) diverge and include the larynx 
in their fork. The relative position, connexions and office in subserviency 
to the larynx, to which the retained parts of the splanchno-branchial arches 
are introduced in the lowest of the air-breathing vertebrates, are preserved in 
all the higher classes. The 'hypobranchials' are as constant in their ex- 
istence, therefore, as the upper larynx itself, and attach themselves more 
especially to the thyroid element of that larynx. We recognise them by this 
relation in birds and man (46, figs. 23 and 25), where they always much ex- 
ceed the parts of the true hyoid arch (cerato- and epi-hyals) in length ; and 
in birds, where these elements (40, fig. 23) are sometimes obsolete and always 
rudimental, the hypobranchials have been mistaken by both Cuvier and 
Geoffroy * for the ceratohyals or anterior cornua. 

For the modifications and special homologies of the complex hyoid appa- 
ratus in lizards, I refer to my ' Lectures on the Vertebrata.' The crocodiles 
offer a well-marked ordinal difference from those inferior sauria in this as 
in most other parts of their structure. The basihyal and thyrohyals have 
coalesced to form a broad cartilaginous plate, the anterior border rising like a 
valve to close the fauces, and the posterior angles extending beyond and sus- 
taining the thyroid and other parts of the larynx. A long bony ' ceratohyal' 
(fig. 22, 40), and a commonly cartilaginous 'epihyal' (ib. 39), are suspended 
by a ligamentous 'stylohyal' to the paroccipital process ; the whole arch 
having, like the mandibular one, retrograded from the connection it presents 
in fishes. 

In birds as in chelonians, the ceratohyals are much reduced, and the chief 
'cornua' of the hyoid are represented by the hypo- and epi-branchials (thy- 
rohyals), which here attain their maximum of length and tenuity. The basi- 
hyal (fig. 23, 41), as in Chelys, is long and slender, but is always a simple 
piece ; and, as in lizards, is usually most expanded posteriorly, from which 
expansion the thyrohyals (40) are sent off. Conforming with the long and 
slender tongue in most birds, the basihyal extends forwards, and is articu- 
lated with the rudimental ceratohyals (40), when these exist, at some distance 
from the thyrohyals. A commonly long and slender, sometimes spatulate 
glossohyal (42), is articulated to the fore-part of the basihyal ; and a con- 
stantly long, slender and pointed urohyal (43) is articulated with the posterior 
end of the basihyal, and extends backwards beneath the trachea. The thyro- 
hyals (46) diverge and include the larynx in their fork ; and support at their 
extremities a bony or gristly (cerato-branchial) style (47). This is never 
attached by ligament to the base of the skull, but is suspended freely, as in 
the chelonia, by the glossohyoid and omohyoid muscles ; it, however, curves 
over the back and upper part of the cranium in the woodpeckers, and the 
extremities of both cerato-branchials are inserted, by way of rare exception 
in that bird, into the right nostril. 

* Duges appears to have first pointed out tliis error, but without, however, perceiving the 
true homology of his ' comes thyro'idiens ' with the hypobranchials of fishes. 



240 REPORT — 1846. 

In mammals the normal completion of the hyoidean arch, as it first ap- 
pears in fishes, is again resumed, and that not by a slender cartilage, as in 
the frog, but by a chain of bones, in which we again recognise the cerato- 
(fig. 24, 4o), epi- (39) and stylo- (as) hyals suspending the basihyal (41) and 
the tongue to the base of the skull, often to the petrosal, sometimes to the 
tympanic, or to the mastoid, or to the exoccipital. The ungulates and the 
true carnivora best display this type. 

In man (fig. 25) the ceratohyals are reduced, as in birds, to mere tuber- 
cles of bone (40), and the extent of the arch between them and the stylo- 
hyals, which become anchylosed to the temporal bones, retains its primitive 
ligamentous condition. Occasionally, however, ossification extends along 
the stylohyoid ligament, and marks out, as in the specimen figured by 
GeoflTroy St. Hilaire (Philosophic Anatomique, pi. 4, fig. 87), the more nor- 
mal proportions of the ceratohyal, and also the epihyal. Other examples of 
this ' monstrosity' are recorded in works on anthropotomy. The thyro- 
hyal (40) — the last remnant of the branchial arches — maintains more con- 
stancy in its existence and proportions ; but manifests its true character of 
free suspension below the skull, and an articulation by short ligaments to the 
angles or horns of the thyroid cartilage. 

The remarks already made on the special homologies of the parts of the 
scapular arch and its appendages, preclude the necessity of further extending 
the present part of this Report. 

Part II. — General Homology. 

On taking a retrospect of the results of the researches of anatomists into 
the special homologies of the cranial bones, the student of the science, how 
little soever practised in such inquiries, cannot but be struck with the amount 
of concordance in those results. It must surely appear a most remarkable 
circumstance to one acquainted only with the osteology of the human frame, 
that so many bones should be, by the common consent of comparative ana- 
tomists, determinable in the skull of every animal down to the lowest osseous 
fish. This fact alone, so significant of the unity of plan pervading the ver- 
tebrate structure, has afforded me, at least, a large ground of hope and 
much encouragement to perseverance in the reconsideration of those points 
on which a difference of opinion has prevailed ; and in the re-investigation of 
what is truly constant and essential in characters determinative of special 
homologies. 

In this, as in every other inquiry into nature, the first labours are neces- 
sarily more or less tentative and approximative : but if errors have to be 
eliminated in the course of successive applications of fresh minds to the 
task, truths become confirmed and established. And I regard the body of 
such truths (see Table I.) to be now so great, in respect of the determination 
of the homologous bones in the heads of all vertebrate animals, as to impe- 
ratively press upon the thinking mind the consideration of the more general 
condition upon which the existence of relations of special homology depends. 

Upon this point the anatomical world is at present divided, lacking the 
required demonstration. The majority of existing authors on comparative 
anatomy have tacitly abandoned*, or with Cuvier and M. Agassiz, have 

* Wagner, ' Lehrbuch der Zootomie,' 8vo, 1843, 1844. Siebold and Stannius, ' Lehr- 
buch der Vergleichende Anatomie,' 8vo, 1845, 1846. Milne-Edwards, ' Elemens de 
Zoologie,' 8vo, 1834. Prof. Rymer Jones, ' Outline of the Animal Kingdom and Manual 
of Comparative Anatomy,' 8vo. 1841. The sentiments which this pleasing and instructive 
writer expresses, are probably akin to those which have influenced tlie above-cited authors 



ox THE VERTEBRATE SKELETON. 241 

directly opposed the idea of ' special homology ' being included in a higher 
law of uniformity of type. 

Yet the attempt to explain, by the Cuvierian principles, the facts of special 
homology on the hypothesis of the subserviency of the parts so determined 
to similar ends in different animals, — to say that the same or answerable bones 
occur in them because they have to perform similar functions — involve many 
difficulties, and are opposed by numerous phsenomena. We may admit that 
the multiplied points of ossification in the skull of the human foetus facilitate, 
and were designed to facilitate, childbirth ; yet something more than such a 
final purpose lies beneath the fact, that most of those osseous centres repre- 
sent permanently distinct bones in the cold-blooded vertebrates. The cra- 
nium of the bird, which is composed in the adult of a single bone, is ossified 
from the same number of points as in the human embryo, without the pos- 
sibility of a similar purpose being subserved thereby, in the extrication of 
the chick from the fractured egg-shell. The composite structure is repeated 
in the minute and prematurely-born embryo of the marsupial quadrupeds. 
Moreover, in the bird and marsupial, as in the human subject, the different 
points of ossification have the same relative position and plan of arrange- 
ment as in the skull of the young crocodile, in which, as in most other rep- 
tiles and in most fishes, the bones so commencing maintain throughout life 
their primitive distinctness. These and a hundred such facts force upon the 
equal and knowing anatomist the inadequacy of the teleological hypothesis 
to account for the acknowledged concordances expressed in this report by 
the term ' special homology.' If, therefore, the attempt to explain them as 
the results of a similarity of the functions to be performed by such homo-' 
logous parts entirely fails to satisfy the conditions of the problem ; and if, 
nevertheless, we are, with Cuvier, to reject the idea of their being manifes- 
tations of some higher law of organic conformity on which it has pleased 
the divine Architect to build up certain of his diversified living works, 
there then remains only the alternative that special homologies are matters 
of chance. 

This conclusion, I apprehend, will be entertained by no reasonable mind ; 
and reverting, therefore, to the more probable hypothesis of the dependence 
of the special resemblances upon a more general law of conformity, we 
have next to inquire, what is the vertebrate archetype? The gifted and 
deep-thinking anatomist, Oken, obtained the first clew to this discovery by 

on this subject. " It is not by any means our intention to engage our readers in discussing 
all the conflicting and, sometimes, visionary opinions entertained by different authors re- 
lative to the exact homology of the individual bones forming this part of the skeleton ; and 
■we shall, therefore, content oiu'selves by placing before them, divested as far as possible of 
superfluous argumentation, Cuvier's masterly analysis of the labours of the principal inquiries 
concerning this intricate part of anatomy." — p. 494. A later English author, who has em- 
bodied a most valuable amount of careful and exact osteological observation in the article 
" Zoology" of the ' Encyclopaedia Metropolitana' (4to, 1845), seems scarcely to regard even 
the determination of special homologies as a necessary object of anatomical research. Thus, 
in discussing the differences of opinion respecting the coracoid (fig. 5, 4S). he says, " Bakker's 
view, however, if it be absolutely necessary to hunt up analogies, seems more correct." — 
p. 302. 

This reserve is, however, perhaps less obstructive to the philosophical progress of anatomy 
and to the requisite resumption of original inquiry to that end, than the mere reproduction 
of the transcendental views of others without criticism or attempt to explain or refute the 
objections to such views which have been promulgated by so great authorities as Cuvier and 
Agassiz. Thus Bojanus's 4-vertebral theory of the cranial part of the skull is adopted by 
M. De Blainville (Osteographie, 4to) ; -whilst Dr. Grant (Outlines of Comparative Anatomy, 
8vo, 1835, p. 63) deems the composition of the skull, in fishes, to correspond nearly with 
Geoflfroy's theory of this part of the skeleton being composed of seven vertebrae, each con- 
sisting of a body with four elements above and four elements below. 



242 REPORT — 1846. 

the idea of the arrangement of the cranial bones of the skull into segments, 
like the vertebrae of the trunk. He informs us that walking one day in the 
Hartz forest, he stumbled upon the blanched skull of a deer, picked up the 
partially dislocated bones, and contemplating them for a while, the truth 
flashed across his mind, and he exclaimed " It is a vertebral column !*" Oken 
afterwards tested and matured this happy inspiration by examining the skulls 
of a cetacean, a chelonian, and a cod-fish in Dr. Albers's museum at Bremen ; 
and on his return to Jena in 1807, he published his beautiful generalization in 
a now very scarce Introductory Lecture, or " Programm beim Antritt der Pro- 
fessur," entitled ' On the signification of the bones of the skull't- He illus- 
trates his views by reference to the skull of a ruminant. " Take," he says, 
" a young sheep's skull, separate from it the bones of the orbit, also those 
cranial bones which take no share in the formation of the ' basis cranii,' e.g. 
the frontal, parietal, ethmoid and temporal, and there will remain an osseous 
column which any anatomist, at first glance, would recognise as three bodies 
of a kind of vertebrae with transverse processes and foramina. Replace the 
cranial bones with the exception of the temporals, for, without these, the 
cavity is still closed, and you have a cranial vertebral column, which diff"ers 
from the true one ('von der wahren') only by its more expanded neural 
canal (Ruckenmarkshohle). As the brain is a more voluminously developed 
spinal chord, so is the brain-case a more voluminous spinal column. As 
the cranium includes, then, three vertebral bodies, so must it have as many 
vertebral arches. These are next to be sought out and determined. One 
sees the sphenoid divided into two vertebrae ; through the foremost pass the 
optic nerves, through the hindmost the maxillary nerves {par trigeminurn). 
I call one the 'eye-vertebra' (Augwirbel), the other the 'jaw-vertebra' 
(Kieferwirbel). Upon this latter abuts the basilar process of the occipital 
bone and the petrous bones : both belong to one whole. As the optic nerve 
perforates the ' eye-vertebra,' and the trigeminus the 'jaw-vertebra,' so the 
acoustic nerve takes possession of the hindmost vertebra. I call it, there- 
fore, ' ear-vertebra' (Ohrwirbel) : and I regard this as the first cranial ver- 
tebra ; the jaw-vertebra as the second, and the eye-vertebra as the third." — 
ih. p. 6. 

After entering upon the difficulties which beset him in determining whether 
the petrosal belonged to the first (Ohrwirbel) or the second (Kieferwirbel), 
and enunciating his views on the essential relations of each cranial vertebra 
with a single special sense (excluding, however, smell and taste, as being 
inferior in dignity to the others), Oken proceeds, in his characteristic bold 
metaphorical language : — " Bones are the earthy hardened nervous system : 
Nerves are the spiritual soft osseous system — Continens et contentum." 

" Between the sphenoid and occipital, between the sphenoid and petrosal, 
between the parietal (the temporal being removed) and the occipital, theie 
runs a line which defines the anterior boundary of the first vertebra. In the 
line between the two sphenoids, or that which in man extends anterior to 

* " Im August 1806 machte ich eine Reise uber den Hartz," — " ich rutschte an der Siid- 
seite durch den Wald herunter — und siehe da ; es lag der schonste gebleichte Schiidel einer 
Hirschkuh vor meinen FUssen. Aufgehoben, umgekehrt, angesehen, und es war geschehen. 
E» ist eine Wirhelmule ! fuhr es mir wie ein Blitz durch Mark und Bein — und seit dieser 
Zeit ist der Scbadel eine Wirbelsiiule." — Isis, 1818, p. 511. 

t Uber die Bedeutung der Schadelknochen, 4to, 1807. I am indebted to my friend 
Mr. Tulk, the able translator of ' Wagner's Comparative Anatomy,' for the opportunity of 
perusing this most suggestive and original essay, which does not exist in either the Library 
of the British Museum, that of the College of Surgeons, or that of the Medico-Chirurgical 
Society. Mr. Tulk is at present engaged in the arduous task of translating the " Lehrbuch 
der Natur-philosophie " of Oken for the ' Ray Society.' 



ON THE VERTEBRATE SKELETON. 243 

the pterj'goid processes laterally and upwards through the fissura orbitalis 
superior, anterior to the great ala, and finally between the frontal and the 
parietal bones, we trace another line, which divides the second from the 
third vertebra " {ib. p. 7). 

" Now," says Oken, " take the ear-vertebra from a foetus of any mammal 
or of man, place near it an immature dorsal vertebra, or the third cervical 
of a crocodile, and compare the pieces of which they consist, their form, their 
contents, and the outlets for the nerves. 

" According to Albinus and all anthropotomists, each vertebra of the 
foetus consists of three distinct parts — the body and the two neurapophyses 
(bogentheile). You have the same in the occipital bone, but more clearly 
and more distinctly: the 'pars basilaris' is separated as the body of the ver- 
tebra from the ' partes condyloideae,' which form the lateral parts : these 
are still more distinct from the ' pars occipitalis' which forms the spinous 
process : even this part is often bifid, like the spinous processes in spina 
bifida." 

" Since then the foramen magnum is the hinder or lower opening of a 
vertebral canal, the condyles true oblique vertebral processes, the foramen 
lacerum an intervertebral foramen, and the crista occipitalis a spinous pro- 
cess, proved to be such by both its position and the muscles inserted into it, 

since lastly the whole occipital bone in relation to its form as well as its 

function — inclosing the cerebellum as a production of the spinal chord, is 

a true and in every sense characteristic vertebra, it is unnecessary to dwell 
more diffusely on parts, the bare mention of which suffices to make their 
nature recognizable." — ih. p. 7. 

This will serve as an example of the close observation of facts, the philo- 
sophical appreciation of their relations and analogies, and, in a word, of the 
spirit in which Oken determines the vertebral relations of the cranial bones 
of the skull : and I refer to Table II. for his conclusions as to the parts of 
the second and third cranial vertebrse. 

Reverting to the petrosal, Oken thus beautifully and clearly enunciates 
its essential nature and homology : — " You will say I have forgotten the 
' pars petrosa.' No ! It seems not to belong to a vertebra, as such ; but to 
be a 'sense-organ' (Sinnorgan), in which the vertebral- or ear-nerve loses 
itself; and, therefore, is as distinct an organ from a vertebral element as is 
any other viscus (Eingeweide), or as is the eyeball itself. The (cause of) 
delusion (as to the homology of the petrosal) lies in this, viz. that it must be 
ossified agreeably with its nature (wesen),just as the eye must be crystallized." 

Although Oken does not in this essay formally admit a fourth vertebra 
anterior to the ' eye-vertebra,' he recognises the vertebral strijcture as being 
carried out rudimentally or evanescently, by the vomer, as the prolongation 
of the cranio-vertebral bodies, by the lacrymal bones, as their neurapo- 
physes, and by the nasal bones, as the spinous process. His ideas of a 
vertebra have evidently at this period not extended beyond the ordinary 
anthropotomical one of centrum and neural arch with its transverse, oblique, 
and spinous processes. When he indicates (beautifully and truly) the general 
homology of the palatine bones, as pleurapophyses, under the name of an- 
chylosed or immoveable ribs of the head, it has reference to the transcen- 
dental idea of the repetition in the head of all the parts of the body. Thus 
the squamosal in mammals and the tympanic in birds represent the 'scapula' 
of the head, and at the same time, also, the ilium. The homologue of the 
squamosal (fig. 21, 27) in the bird is the 'humerus capitis': the malar (ae) 
and the maxillary (21) are the ' oberarm' (i-adius and ulna capitis) : the pre- 
maxillary (22) is the ' manus capitis.' The segments of the hind limb are 



244 REPORT — 1846. 

represented by divisions of the compound lower jaw in the crocodile and 
embryo bird (see Table, No. III.). The pterygoids (24), the essential di- 
stinction of which from the sphenoid Oken clearly recognises, are his ' clavi- 
culae capitis.' Oken hints at, without accepting, the (serial) homology of 
the hyoid arch with the pelvis ; but he regards the stylohyal (ss) as the 
' sacrum capitis' (ib. p. 16). 

The year after the publication of Oken's famous ' Introductory Lecture,' 
Prof. Dumeril, apparently unacquainted with its existence, communicated 
to the French Institute a memoir entitled ' Considerations generales sur 
I'analogie qui existe entre tons les os et les muscles du tronc dans les ani- 
maux,' the second paragraph of which is headed *' De la tete consideree 
conime une vertebre, de ses muscles et de ses mouvements." In this para- 
graph, repeating the homological correspondences, demonstrated by Oken, 
between the basioccipital as a vertebral centrum, the condyles as ' obli(|ue 
processes,' and the occipital protuberance as a spinous process, he adds, that 
the mastoid processes are entirely conformable to transverse processes. And 
M. Dumeril has, I believe, here the merit of having first enunciated the 
general homology of the mastoids, although he does not aim at showing to 
which vertebral segment of the skull they properly belong. Nor, indeed, 
with the exception of an observation that " very often the body of the splie- 
noid, like the ' apophyse basilaire' of the occiput, resembles the body of a 
vertebra," does he push the transcendental comparisons further. GeofFroy 
St. Hilaire tells us*, that even the moderate and very obvious illustrations 
of the general homologies of the cranial bones, which M. Dumeril deduced 
from the anatomy of the occiput, excited an unfavourable sensation in the 
bosom of the ' Academic;' and that the phrase ' vertebre pensante,' which a 
facetious member proposed as an equivalent for the word ' skull,' and which 
circulated, not without some risibility, along the benches of the learned 
during the reading of the memoir, reaching the ears of the ingenious author, 
the dread of ridicule checked his further progress in the path to the higher 
generalizations of his science, and even induced him to modify considerably 
many of the (doubtless happy) original expressions and statements in the 
printed report, so as to adapt it more to the conventional anatomical ideas 
of his colleagues. 

As the truth of Oken's generalization began to be appreciated, it was remem- 
bered, as is usually the case, that something like it had occurred before to 
others. Autenrieth and Jean-Pierre Frank had alluded, in a general way, to 
the analogy between the skull and the vertebral column : Ulrich, reproducing, 
formally, Oken's more matured opinions on the cranial vertebrae, says, 
" Kielmeyerum praeceptorem pie venerandum quamvis vertebram tanquam 
caput integrum considerari posse in scholis anatomicis docentem audivi." 
And the essential idea was doubtless present to Kielmeyer's mind, though 
he reversed M. Dumeril's proposition, and, instead of calling the skull a ver- 
tebra, he said each vertebra might be called a skull. But these anticipations 
detract nothing from the merit of the first definite proposition of the theory. 
It would rather be an argument against its truth, if some approximative idea 
had not suggested itself to other observers of nature, who only lost the merit 
of developing it, from not appreciating its full importance. He, however, 
becomes the true discoverer who establishes the truth : and the sign of the 
proof is the general acceptance. Whoever, therefore, resumes the investiga- 
tion of a neglected or repudiated doctrine, elicits its true demonstrations, 
and discovers and explains the nature of the errors that have led to its tacit 

* Annales des Sciences Naturelles, t. iii. 1824, p. 177. 



ON THE VERTEBRATE SKELETON. 245 

or declared rejection, may calmly and confidently await the acknowledgments 
of his rights in the discovery. 

It has been unfortunate for Oken, that, with one exception — the gifted 
Bojanus — his successors in the development of the vertebral theory of the 
skull have hitherto exaggerated rather than retrenched the errors of their 
guide. Spix* lends an almost servile aid to Oken in endowing the artist's 
symbol of the cherub with all that it seems most to want, a thorax, abdomen 
and pelvis, arras, legs, hands and feet. He adopts Oken's original number 
and composition of the cranial vertebrae, and gives them new names, which 
being dissociated from Oken's peculiar idea of the essential subserviency of 
the cranial segments to certain organs of sense, are likely to be retained, 

Bojanusf seems first to have determined the true elements of the neural 
arch of the nasal vertebra; and was as happy in perceiving the pleurapo- 
physial relations of the tympanic pedicle, as Oken had been in reference to 
the palatine bone. He was less accurate in his idea of the vertebra to which 
it belonged. The analysis of Bojanus' craniovertebral system given in Table 
III. precludes the necessity of dwelling upon it in the brief historical sketch 
here attempted. 

The modifications of his original idea which Oken has introduced into his 
edition of the ' Natur-philosophie ' of ISIS, bring it into close accordance with 
that of Bojanus, excepting that Oken conceives the cranial neurapophyses to 
answer also to ribs : — " An den Seiten eines jeden Korpers liegen Fliigel- 
fortsatze, welche den Querfortsatzen der Halsvvirbel oder den Rippen ent- 
sprechen : 'die Gelenkkopfe des Hiuterhauptsbeins' (exoccipitals), 'die 
grossen ' (ali-) 'undkleinen Fliigel' (orbito-sphenoids), ' und die beiden 
Seiten des Siebbeins ' (prefrontals)," p. 304. With regard to the facial bones 
of the skull, Oken still includes the explanation of their general homology in 
his original idea, that " the head is (a repetition of ) the whole trunk with all 

its systems The encephalon is the myelon (riickenmark) ; the cranium, 

the vertebral column ; the mouth is intestine and abdomen ; the nose, lungs 
and thorax ; and the jaws, limbs (glieder)." — Op. cit. p. 300. An idea which 
vitiated his original essay, and which has had the effect of obscuring a great 
truth in nature in the smoke of a sacrifice to a false system. 

This seems the place to notice a virtual testimony to the general accuracy 
of the Okenian cranial system, published in 1816 by the present eminent 
-osteologist who holds the chair of Comparative Anatomy in the ' Jardin des 
Plantes.' In a note to his ' Prodrome d'une Nouvelle Distribution Systema- 
tique du Regne Animal,' published in the 'Bulletin des Sciences par la So- 
ciete Philomathique,' 1816, p. 105, M. de Blainville says, " J'essayerai de 
montrer(!)que la tete dans les animaux vertebres est composee, 1° d'une suite 
d'articulations ou de vertebres soudees, chacune developpee proportionnelle- 
ment au systdme nerveux particulier qu'elle renferme, comme dans le reste 
de la colonne vertebrale ; 2°, d'autant d'appendices paires qu'il y a de ces 
fausses vertebres, et pouvant avoir des usages differens " (p. 108). M. de 
Blainville does not (like Bojanus) expressly mention the general homology 
of any of these appendages to the ribs, or parial appendages of the true ver- 
tebrae ; but he leaves it to be so understood by his subsequent enumeration 
and classification of the ' appendices paires ou symmetriques,' which he de- 
scribes as being always in relation with a vertebra or median piece. He 
says, e.g. " lis peuvent etre divises en simples ou en composes, ou peut-etre 
d'apres leurs usages. Les appendices simples sont les cotes. Les appendices 
composes sont les membres, les machoires, les appareils des organes des 

* Cephalogenesis, fol. 1815. 

t Isis, 1818, and ' Parergon' in the ' Anatorae Testudinis Europaeae,' fol. 1821. 



246 REPORT — 1846. 

sens, le styloide, les branches d