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REPORT 


THIRD MEETING 


BRITISH ASSOCIATION 


FOR THE 


ADVANCEMENT OF SCIENCE; 


HELD AT CAMBRIDGE IN 1833. 


LONDON: 
JOHN MURRAY, ALBEMARLE STREET. 


1834. 


LONDON: 


PRINTED BY RICHARD TAYLOR, 
RED LION COURT, FLEET STREET. 


PREFACE, 


a et 


'T'HE Transactions of the British Association consist of 
three parts ; first, of Reports on the State of Science drawn 
up at the instance of the Association ; secondly, of Miscel- 
laneous Communications to the Meetings; and thirdly, of 
Recommendations by the Committees, having for their ob- 
jects to mark out certain points for scientific inquiry. 

It is proper to remark, that some of the Reports here 
printed are to be considered in the light of first parts of 
the intended survey of the sciences reviewed in them, the 
continuation being postponed to a future Meeting. Thus, 
the Report on Hydraulics, by Mr. G. RENNIE, will be 
completed in a second part, to be presented to the Meeting 
at Edinburgh ; the Report on the mathematical theory of 
the same science, by the Rev. Mr. Cuauuis, which is 
here restricted to problems on the common theory of Fluids, 
will be further extended to the theories which have recently 
been advanced respecting the internal constitution of Fluids 
and the state of their caloric, to account for certain phzeno- 
mena of their equilibrium and motion ; and the Report on 
Analytical Science, by the Rev. Mr. Peacock, which in the 
present volume includes Algebra, and the application of 
Algebra to Geometry, is intended to be hereafter concluded 
by a review of the Differential and Integral Calculus and the 
theory of Series. In like manner, to the Report on Botany, 
by Dr. Linpuey, which embraces only the physiological 
part of the science, that which Mr. BentHam has under- 
taken on the State and Progress of Systematic Botany will 
be supplemental ; and to the present Report, by Dr. CHARLES 
Henry, on one branch of Animal Physiology, a more 
general review of the progress of that science will be added 
by the Rev. Dr. Cuark. — / 

_ With respect to the next part of the Transactions, which 
includes the communications made to the Sections, two 
a 2 


iv PREFACE. 


rules have been adopted ; the first is, to print no oral com- 
munications unless furnished or revised by the Author him- 
self. In the former volume this rule was slightly deviated 
from, for the purpose of showing in what manner the 
Meetings were conducted. But however valuable a part of 
the proceedings of the Meetings the verbal communications 
and discussions may be, it is evidently impossible to publish 
a safe and satisfactory report of them from any minutes 
which can be taken. The second ruleis, not to print any 
of the miscellaneous communications at length ; but eitker 
abstracts of them, or notices* only, the object of the rule 
being to keep the Transactions within the bounds which 
the Association has prescribed to itself, and to prevent any 
interference with the publications of other societies. In 
the present volume, there is one paper printed at lengtht, 
which contains the results of certain experiments instituted 
expressly at the request of the Association. 

The Recommendations of various subjects for scientific 
inquiry agreed upon at Cambridge have been here incor- 
porated with those adopted at former Meetings, and the 
Suggestions which are contained in the Reports on the state 
of science, published in the present and preceding volume, 
have likewise been added; so as to present a general view of 
the desiderata in science to which attention has been invited. 
To this part of the volume are also appended those direc- 
tions for the use of observers which have proceeded from 
Committees appointed to promote particular investigations. 

To the Transactions is prefixed a brief outline of the 
General Proceedings of the Cambridge Meeting, a fuller Re- 
port of them having been rendered unnecessary by the ac- 
count which has already issued from the University press. 
The observations, however, delivered by the Rev. Mr. WuE- 
WELL on the state of science as it is exhibited in the first 
volume of the Reports of the Association, not having been 
before published, are printed at length. 


* The notices of Communications will be found in the general account of the 
Proceedings of the Sections, p. 353. 

+ ‘* Experiments on the Quantity of Rain which falls at different Heights in 
the Atmosphere.” 


eon x 


CONTENTS 


ProcEEDINGS OF THE MEETING .-...- +e sees reer certs eters 


TRANSACTIONS. 


Report on the State of Knowledge respecting Mineral Veins. By 
Joun Taytor, F.R.S., Treasurer of the Geological Society and 
of the British Association for the Advancement of Science,. &c. 


On the Principal Questions at present debated in the Philosophy 
of Botany. By Joun Linviey, Ph. D:, F.R.S., Professor of 
Botany in the University of London ..........+--++++++++- 


Report on the Physiology of the Nervous System. By WriiiiAm 
Caries Henry, M.D., Physician to the Manchester Royal In- 


Report on the present State of our Knowledge respecting the 
Strength of Materials. By Peter Bartow, F.R.S., Corr. Memb. 
Inst. France, &c. &C. 2... ee ee ees 


Report on the State of our Knowledge respecting the Magnetism 
of the Earth. By S. Hunver Curisriz, M.A., F.R.S., M.C.P.S., 
Corr. Memb. Philom. Soc. Paris, Hon. Memb. Yorkshire Phil. 
Soc.; of the Royal Military Academy ; and Member of Trinity 
College Cambridge ..... 1... +0. eee e tere ere e eee r 


Report on the present State of the Analytical Theory of Hydro- 
statics and Hydrodynamics. By the Rev. J. Cuautis, late Fel- 
low of Trinity College Cambridge... ........---+---- ee eee 


Report on the Progress and present State of our Knowledge of 
Hydraulics as a Branch of Engineering. By Grorcz RENNIE, 
AAS AER ORT gible TENG. ooh ot cca eae a eae ee 

Report on the recent Progress and present State of certain Branches 
of Analysis. By Grorce Peacock, M.A., F.R.S., F.G.S., 
F.Z.S., F.R.A.S., F.C.P.S., Fellow and Tutor of Trinity Col- 
lege Cambridge ...... 2.2.02. eee eee eee rete e eens 


TRANSACTIONS OF THE SECTIONS. 
_ I, Maruemarics anD Puysics. 
Professor CErstep on the Compressibility of Water............ 
W.R. Hamuzron on some Results of the View of a Characteristic 
Function in Optics ..........0 02s eee eee ee eee 
The Rey. H. Luoyp on Conical Refraction ..........-.-- 056: 


27 


59 


93 


131 


153 


185 


vi CONTENTS. 


Page. 
Sir Jounn F. W. Herscuet on the Absorption of Light by coloured 

Media, viewed in connexion with the undulatory Theory .... 373 
The Rey. Bapen Powe tt on the Dispersive Powers of the Media 

of the Eye, in connexion with its Achromatism ............ 374 
R. Porrer, Jun., on the power of Glass of Antimony to reflect 

i anne core mbar nec) Samim cens cs 377 

on a Phenomenon in the Interference of Light 

Hitherto! WOCdeSeribed.\).j4c00 = «fois: lm oler-uays = eheeeetehate tena eh een 378 
Sir Joun F. W. Herscuex’s Explanation of the Principle and Con- 

atroction of the Actinometer ....:. 9) ..1cjehitis : 101s bias pant 379 
M. Me ttonr’s Account of some recent Experiments on Radiant Rs 

1 SVE RRR Ae ai hr ae ARENT CID Cisio din Dalodig ico ae 381 
Joun ParpEavux on Thermo-Electricity ..2..0................ 384 
W. Syow Harris on some new Phenomena of Electrical Attrac- 

Fiemme. area Aeterna fs). This Se WG die ea coe ies Been dee 386 
The Rev. Joun G. MacVicar on Electricity ................ 390 
The Rev. J. Powrr’s Inquiry into the Cause of Endosmose and 

HS XOSMOSE ea jers s haPAE HE elses Ee cpatege vere wept eke os Seen 391 
Micuart Farapay on Electro-chemical Decomposition ........ 393 
Dr. Turner's Experiments on Atomic Weights .............. 399 
Prof. Jounnsron’s Notice of a Method of analysing Carbonaceous 

URS Se SC eg ig ha Na BY se ange hale Sate ae 400 
R. Porrzr, Jun. A Communication respecting an Arch of the 

paninigtes ESQEC ANI 5. 22015 ).15. 0 2 rele BE hss, UE IE a ee 401 


Joun Puruiipes’s Report of Experiments on the Quantities of Rain 
falling at different Elevations above the Surface of the Ground at 
Woes oe Hah cusliva wits Andie cet aes Bi eae eee 401 


II. Partosoruicat Instruments AnD Mecuanicat Arts. 


The Rev. Wm. Scorzssy on a peculiar Source of Error in Experi- 


ments: Wea the Dipping Needle... 23.42 0.01.07 2. eee 412 
The Rev. W. H. Mizzer on the Construction of anew Barometer 414 
W. L. Wuarron on a Barometer with an enlarged Scale ...... 414 
W.S. Harris on the Construction of a new Wheel Barometer .. 414 


J. Newman on a new Method of constructing a Portable Barometer 417 
The Rev. James Cummine on an Instrument for measuring the 


total heating Effect of the Sun’s Rays for a giveu time ...... 418 
———— on some Electro-magnetic Instruments 418 
Anprew Ure on the Thermostat, or Heat-governor .......... 419 
Tuomas Davison on a Reflecting Telescope ................ 420 
W.L. Wuarton on a Steam-engine for pumping Water ...... 421 
E. J. Denv on the Application of a glass Balance-spring to Chro- 

MRODACLETS 2). 2.2, 21: hcp een Re eee Macnee eer le cholo rainy tole ke Tele 421 
E. Hopexinson on the Effect of Impact on Beams...... Bo te al 421 
——————. on the direct tensile Strength of Cast Iron ...... 423 
J. I. Hawxrys’s Investigation of the Principle of Mr. Saxton’s loco- 

mouve diftercntial Pulley, be.') > 2a OPE Oe, 424 
Joun Tayior’s Account of the Depths of Mines.............. 427 


J; Owen on Naval Architecture awit see ae GEOL S 8 Es 430 


CONTENTS. vil 


Page. 
III. Naturat Historny—Anatomy—Puysio.uoey. 


Professor AGarpu on the originary Structure of the Flower, and 


the mutual Dependency of its Parts ................,..... 433 
Professor DausEny’s Notice of Researches on the Action of Light 

CEDARS GET Sipe I pia ih iS arti 8 1 Bi ol Cag 436 
Watrer Apam on some symmetrical Relations of the Bones of the 

(SS ACT lig ie OR Ra Riemer be Nig A 2 ie 437 
R. Haran on some new species of Fossil Saurians found in Ame- 

CR ee en epee RMS ae oniciol Sects Se SRE ol eco ac ease eRe ml Sees as 440 
The Rev. L. Jenyns’s Remarks on Genera and Subgenera, &c. .. 440 
J. Macartney on some parts of the Natural History of the Com- 

DKS WBE G ites ee ae an Ms PNR oA oO A ao a 44] 
J. Buackwa.t’s Observations relative to the Structure and Func- 

SUMS ARNE SSPUCLELS eet a hrs) elle nis Sor pee ie siete ee olen ooo ook ees 444 
W. Yarrect on the Reproduction of the Eel ................ 446 


C. Wittcox on the Naturalization in England of the Mytilus cre- 
natus, a native of India, and the Bate dal Heros, a native 


6) aN ietee | ee ae el Racal “bce ae er aims ete ica ng AA ac Neri Ae Soe 448 
J. Macartney’s Abstract of Observations on the Structure and 
Functions of the Nervous System ........................ 449 
H. Carurte’s Abstract of Observations on the Motions and Sounds 
Bierce Lcantere Tes i erase sean lo eins coh a eee Gehan 454 
H. Earze on the Mechanism and Physiology of the Urethra .... 460 
Burt on the Nomenclature of Clouds .................. 460 


G. H. Frexpine on the peculiar Atmospherical Phenomena as ob- 
served at Hull during April and May 1833, in relation to the 
Pm mCUICE Ol MIUCH AAS 2th 0ns Gate. Se. Bae RL ee 461 


IV. History or Science. 


Francis Baity’s short Account of some MS. Letters (addressed 
to Mr. Abraham Sharp, relative to the Publication of Mr. Flam- 
steed’s Historia Celestis,) laid on the table for the inspection of 
the Members of the Association................ 0.00 eeeeee 462 


Recommendations of the British Association for the Advancement 


2 ha ER AR, 5 Ae ae ey Ey i 4 fale ARE neh ns eas 467 
Recommendations of the Committees cat 4 \ fe gE RP, beh Oe aR 469 
Sap RETRGR ERA 2) SoG) 9. eee ys Ret ee en) Se OPN ae ey ogee, bate 484 
Prospectus of the Objects and Plan of the Statistical Society of 

CUS gees bo a0 ier, Sneath ain ena Se eR OU DR 492 
Objects and Rules of the Association’. °. o) 005.8000. 2 eee ee 497 


|G LS ee Se 2 file a ROR Ca COR UG aise AN afc clee Er 501 


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SINNOOOV SMAYASVAWL 


THIRD REPORT. 


PROCEEDINGS OF THE MERTING. 
1833. 


THE third Meeting of the British Association commenced its 
sittings at Cambridge on Monday, the 24th of June, 1833. It 
was attended by more than nine hundred Members, and. was 
honoured with the presence of several foreign philosophers. 
The extent of accommodation provided by the University, and 
by the societies of which it consists, corresponded with the 
magnitude of the Meeting. The public schools, with two 
adjoining halls, were allotted to the use of the Sections and 
Committees, and the Senate-house was appropriated to the 
reception of the general assemblies; a large proportion of the 
visitors were lodged within the walls of the Colleges, and the 
great halls of the two principal foundations were opened in 
hospitality to a concourse of guests collected from all parts by 
a common interest in scientific pursuits. 


GENERAL MEETING. 


On Monday evening, at eight o’clock, the Members assem- 
bled in the Senate-house: and a public discussion took place 
on the phenomena and theory of the Aurora Borealis. 


On Tuesday, at 1 p.m. a General Meeting was held in 
the Senate-house ; the President of the preceding year, (the 
Rey. Dr. Buckland,) resigned his office. In the course of 
his speech*, he congratulated the Meeting on the proof af- 
forded by the Report recently published, that the Association 
was pursuing a course of peculiar utility to science, whilst at the 


_ * A fuller account of the speeches delivered at the Meeting will be found 
/ annexed to the lithographed signatures, &c., published at Cambridge. 
3. b 


x THIRD REPORT—18338. 


same time it had fully redeemed its pledge of not interfering with 
the province of other Scientific Societies. 

The President (the Rev. Professor Sedgwick,) stated, in his 
opening speech, that it was the desire of the Vice-Chancellor 
and the Heads of Colleges that everything should be done on 
the present occasion to emulate, as far as circumstances per- 
mitted, the splendid reception which had been given to the 
Association by the sister University of Oxford. He dwelt on 
the advantages which such a Meeting brought with it to the 
places in which it was held, by inducing scientific foreigners to 
visit them, and expressed the delight with which he hailed such 
visits, as an omen that the great barriers which for a length 
of time had served man for man, had now been broken 
down. He described the character of the Reports which 
the Association has published ; and added that he attached so 
much value to these expositions of the state of science, that 
he had requested one of the Secretaries, (the Rev. William 
Whewell,) to present to the Meeting a fuller analysis of their 
contents. ‘The President concluded his speech with the fol- 
lowing gratifying announcement: ‘ There is a philosopher,” he 
said, ‘‘ sitting among us whose hair is blanched by time, but 
possessing an intellect stillin its healthiest vigour,—a man whose 
whole life has been devoted to the cause of truth,—my vener- 
able friend Dr. Dalton. Without any powerful apparatus for 
making philosophical’ experiments, with an apparatus, indeed, 
which many might think almost contemptible, and with very 
limited external means for employing his great natural powers, 
he has gone straight forward in his distinguished course, and 
obtained for himself in those branches of knowledge which he 
has cultivated, a name not perhaps equalled by that of any 
other living philosopher in the world. From the hour he came 
from his mother’s womb the God of nature laid his hand upon 
him, and ordained him for the ministration of high philosophy. 
But his natural talents, great as they are, and his almost 
intuitive skill in tracing the relations of material phenomena, 
would have been of comparatively little value to himself and to 
society, had there not been superadded to them a beautiful 
moral simplicity and singleness of heart, which made him go 
on steadily in the way he saw before him, without turning to 
the right hand or to the left, and taught him to do homage to 
no authority before that of truth. Fixing his eye on the most 
extensive views of science, he has been not only a successful 
experimenter, but a philosopher of the highest order; his 
experiments have never had an insulated character, but have 
been always made as contributions towards some important 


PROCEEDINGS OF THE MEETING. Xi 


end, as among the steps towards some lofty generalization. 
And with a most happy prescience of the points to which the 
rays of scattered observations were converging, he has more 
than once seen light while to other eyes all was yet in darkness ; 
out of seeming confusion has elicited order; and has thus 
reached the high distinction of being one of the greatest legis- 
lators of chemical science. 

“It is my delightful privilege this day to announce (on the 
authority of a Minister of the Crown who sits near me,*) that 
His Majesty, King William the Fourth, wishing to manifest 
his attachment to science, and his regard for a character like 
that of Dr. Dalton, has graciously conferred on him, out of 
the funds of the Civil List, a substantial mark of his royal 
favour.” 


The Rev. Wint1amM WHEWELL, being called upon by the 
President, delivered the following address :— 


The British Association for the Advancement of Science 
meets at present under different circumstances from those 
which accompanied its former Meetings. The publication of 
the volume containing the Reports applied for by the Meeting 
at York, in 1831, and read before the Meeting at Oxford last 
year, must affect its proceedings during our sittings on the 
present occasion ; and thus we are now to ‘ook for the operation 
of one part of the machinery which its founders have endea- 
voured to put in action. Entertaining the views which sug- 
gested to them the scheme and plan of the Association, they 
must needs hope that such an event as this publication will 
exercise a beneficial influence upon its future career. 

“This hope is derived, they trust, from no visionary or 
presumptuous notions of what institutions and associations can 
effect. Let none suppose that we ascribe to assembled num- 
bers and conjoined labours extravagant powers and privileges 
in the promotion of science ;—that we believe in the omnipo- 
tence of a parliament of the scientific world. We know that 
the progress of discovery can no more be suddenly accelerated 
by a word of command uttered by a multitude, than by a 
single voice. ‘There is, as was long ago said, no royal road to 
_ knowledge—no possibility of shortening the way, because he 
who wishes to travel along it is the most powerful one; and 
just as little is there any mode of making it shorter, because 
they who press forward are many. We must all start from 
our actual position, and we cannot accelerate our advance by 


* The Right Honourable T. Spring Rice. 
b2 


xii : THIRD REPORT—1833. 


any method of giving to each man his mile of the march. Yet 
something we may do: we may take care that those who come 
ready and willing for the road, shall start from the proper 
point and in the proper direction ;—shall not scramble over 
broken ground, when there is a causeway parallel to their 
path, nor set off confidently from an advanced point when the 
first steps of the road are still doubtful ;—shall not waste their 
powers in struggling forwards where movement is not progress, 
and shall have pointed out to them all glimmerings of light, 
through the dense and deep screen which divides us from the 
next bright region of philosophical truth. We cannot create, 
we cannot even direct, the powers of discovery; but we may 
perhaps aid them to direct themselves; we may perhaps 
enable them to feel how many of us are ready to admire their 
success, and willing, so far as it is possible for intellects of a 
common pitch, to minister to their exertions. 

‘< Tt was conceived that an exposition of the recent progress, 
the present condition, the most pressing requirements of the 
principal branches of science at the present moment, might 
answer some of the purposes I have attempted to describe. 
Several such expositions have accordingly been presented to 
the Association by persons selected for the task, most of them 
eminent for their own contributions to the department which 
they lad to review ; and these are now accessible to Members 
of the Association and to the public. It appears to be suitable 
to the design of this body, and likely to further its aims, that 
some one should endeavour to point out the bearing which the 
statements thus brought before it may and ought to have upon 
its future proceedings, and especially upon the labours of the 
Meeting now begun. I am well persuaded that if the President 
had taken this office upon himself, the striking and important 
views which it may naturally suggest would have been pre- 
sented in a manner worthy of the occasion: he has been 
influenced by various causes to wish to devolve it upon me, and 
I have considered that I should show my respect for the Asso- 
ciation better by attempting the task, however imperfectly, 
than by pleading my inferior fitness for it. 

‘* The particular questions which require consideration, and 
the researches which most require prosecution, in the sciences 
to which the Reports now before you refer, will be offered to 
the notice of the Sections of the Association which the subjects 
respectively concern, at their separate sittings. It is conceived 
that the most obvious and promising chance of removing 
deficiencies and solving difficulties in each subject, is to be 
found in drawing to them the notice of persons who have paid 


s 


PROCEEDINGS OF THE MEETING. xiii 


a continued and especial attention to the subject. The con- 
sideration of these points will therefore properly form a part 
of the business of the Sectional Meetings ; and all Members of 
the Association, according to their own peculiar pursuits 
and means, will thus have the opportunity of supplying any 
wanting knowledge, and of throwing light upon any existing 
perplexity. 

‘** But besides this special examination of the suggestions 
which your Reports contain, there are some more general 
reflexions to which they naturally give rise, which may perhaps 
be properly brought forward upon this first General Assembly 
of the present Meeting; and which, if they are well founded, 
may preside over and influence the aims and exertions of many 
of us, both during our present discussions and in our future 
attempts to further the ends of science. 

** There is here neither time nor occasion for any but the 
most rapid survey of the subjects to which your Reports refer, 
in the point of view in which the Reports place them before 
you. Astronomy, which stands first on the list, is not only 
the queen of sciences, but, in a stricter sense of the term, the 
only perfect science ;—the only branch of human knowledge 
in which particulars are completely subjugated to generals, 
effects to causes ;—in which the long observation of the past 
has been, by human reason, twined into a chain which binds 
in its links the remotest events of the future ;—in which we 
are able fully and clearly to interpret Nature’s oracles, so that 
by that which we have tried we receive a prophecy of that 
which is untried. The rules of all our leading facts have 
been made out by observations of which the science began 
with the earliest dawn of ‘history; the grand law of causation 
by which they are all bound together has been enunciated for 
150 years; and we have in this case an example of a science 
in that elevated state of flourishing maturity, in which all that 
remains is to determine with the extreme of accuracy the con- 
sequences of its rules by the profoundest combinations of 
mathematics, the magnitude of its data by the minutest scru- 
pulousness of observation; in which, further, its claims are so 
fully acknowledged, that the public wealth of every nation pre- 
tending to civilization, the most consummate productions of 
labour and skill, and the loftiest and most powerful intellects 
which appear among men, are gladly and emulously assigned 
to the task of adding to its completeness. In this condition of 
the science, it will readily be understood that Professor Airy, 
your Reporter upon it, has had to mark his desiderata, in no 

cases but those where some further developement of calcula- 


xiv THIRD REPORT—1880. 


tion, some further delicacy of observation, some further accu- 
mulation of exact facts, are requisite ; though in every branch 
of the subject the labour of calculation, the delicacy of obser- 
vation, and the accumulation of exact facts, have already gone 
so far that the mere statement of what has been done can 
hardly be made credible or conceivable to a person unfamiliar 
with the study. 

*< One article, indeed, in his list of recommendations to future 
labourers, read at the last Meeting of the Association, may ap- 
pear capable of being accomplished by more limited labour than 
the rest,—the determination of the mass of Jupiter by obser- 
vations of the elongations of his satellites. And undoubtedly, 
many persons were surprised when they found that on this, so 
obvious a subject of interest, no measures had been obtained 
since those which Pound took at the request of Newton. Yet 
in this case, if an accuracy and certainty worthy of the present 
condition of Astronomy were to be aimed at, the requisite ob- 
servations could not be few nor the calculation easy, when it 
is considered in how complex a manner the satellites disturb 
each other’s motions. But the Meeting will learn with pleasure 
that the task which he thus pointed out to others, he has him- 
self in the intervening time executed in the most complete 
manner. He has weighed the mass of Jupiter in the way he 
thus recommended ; and it may show the wonderful perfection 
of such astronomical measures to state, that he has proved with 
certainty, that this mass is more than 322 and less than 323 
times the mass of the terrestrial globe on which we stand. 

** Such is Astronomy: but in proceeding to other sciences, ’ 
our condition and our task are of a far different kind. Instead 
of developing our theories, we have to establish them ; instead 
of determining our data and rules with the last accuracy, we 
have to obtain first approximations to them. This, indeed, 
may be asserted of the next subject on the list, though that 
is, in its principles, a branch of Physical Astronomy ; for that 
alone of all the branches of Physical Astronomy had been al- 
most or altogether neglected by men of science. I speak of 
the science of the Tides. Mr. Lubbock terminated his Report 
on this subject, by lamenting in Laplace’s words this unmerited 
neglect. He himself in England, and Laplace in France, were 
indeed the only mathematicians who had applied themselves 
to do some portion of what was to be done with respect to this 
subject. Since our Meeting last year, Mr. Dessiou has, under 
Mr. Lubbock’s direction, compared the tides of London, Sheer- 
ness, Portsmouth, Plymouth, Brest, and St. Helena; and the 
comparison has brought to light very remarkable agreements. 


PROCEEDINGS OF THE MEETING. xv 


in the law which regulates the time of high water, agreements 
both with each other and with theory; and has at the same 
time brought into view some anomalies which will give a strong 
impulse to the curiosity with which we shall examine the re- 
cords of future observations at some of these places and at 
many others. I may perhaps here take the liberty of mention- 
ing my own attempts since our last Meeting, to contribute 
something bearing on this department. It appeared to me that 
our knowledge of one particular branch of this subject, the 
motion of the tide-wave in all parts of the ocean, was in such a 
condition, that by collecting and arranging our existing mate- 
rials, we should probably be enabled to precure abundant and 
valuable additions to them. This, therefore, I attempted to 
do; and I have embodied the result of this attempt in an 
‘ Essay towards a First Approximation to a Map of Cotidal 
Lines,’ which is now just printed in the Philosophical Transac- 
tions of the Royal Society. If the time of the Meeting allows, 
I would willingly place before you the views at which we have 
now arrived, and the direction of our labours which these 
suggest. 

*‘ In'the case of the science of Tides, we have no doubt about 
the general theory to which the phenomena are to be referred, 
the law of universal gravitation; though we still desiderate a 
clear application of the theory to the details. Im another sub- 
ject which comes under our review, the science of Light, the 
prominent point of interest is the selection of the general 
theory. Sir David Brewster, the author of our Report on this 
subject, has spoken of ‘the two rival theories of light,’ which 
are, as you are aware, that which makes light to consist in 
material particles emitted by a luminous body, and that which 
makes it to consist in undulations propagated through a sta- 
tionary ether. The rivalry of these theories, so far as they 
can now be said to be rivals, has been by no means barren of 
interest and instruction during the year which is just elapsed. 
The discussions on the undulatory theory in our scientific 
journals have been animated, and cannot, I think, be considered 
as having left the subject where they found it. The claims of 
the undulatory theory, it will be recollected, do not depend 
only on its explaining the facts which it was originally intended 
to explain; but on this;—that the suppositions adopted in 
order to account for one set of facts, fall in most wonderfully 
with the suppositions requisite to explain a class of facts en- 
tirely different ; in the same manner as in the doctrine of gra- 
vitation, the law of force which is derived from the revolutions 
of the planets in their orbits, accounts for the apparently re- 


XVi THIRD REPORT—1839. 


mote facts of the precession of the equinoxes and the tides. 
To all this there is nothing corresponding in the history of the 
theory of emission; and no one, I think, well acquainted with 
the subject, would now assert, that if this latter theory had 
been as much cultivated as the other, it might have had a simi- 
lar brilliant fortune in these respects. 

‘“‘ But if the undulatory theory be true, there must be solu- 
tions to all the apparent difficulties and contradictions which 
may occur in particular cases ; and moreover the doctrine will 
probably gain general acceptance, in proportion as these solu- 
tions are propounded and understood, and as prophecies of 
untried results are delivered and fulfilled. In the way of such 
prophecies few things have been more remarkable than the 
prediction, that under particular circumstances a ray of light 
must be refracted into a conical pencil, deduced from the theory 
by Professor Hamilton of Dublin, and afterwards verified ex- 
perimentally by Professor Lloyd. In the way of special diffi- 
culties, Mr. Potter proposed an ingenious experiment which 
appeared to him inconsistent with the theory. Professor Airy, 
from a mathematical examination of this case, asserted that the 
facts, which are indeed difficult to observe, must be somewhat 
different from what they appeared to Mr. Potter; and having 
myself been present at Professor Airy’s experiments, I can 
venture to say, that the appearances agree exactly with the 
results which he has deduced from the theory. Another gen- 
tleman, Mr. Barton, proposed other difficulties founded upon 
the calculation of certain experiments of Biot and Newton; 
and Professor Powell of Oxford has pointed out that the data 
so referred to cannot safely be made the basis of such calcula- 
tions, for mathematical reasons. There is indeed here, also, 
one question of fact concerning an experiment stated in New- 
ton’s Optics: In a part of the image of an aperture where 
Newton’s statement places a dark line, in which Mr. Barton has 
followed him, Professors Airy, Powell, and others, have been 
able to see only a bright space, as the theory would require. 
Probably the experiments giving the two different results have 
not been made under precisely the same circumstances; and 
the admirers of Newton are the persons who will least of all 
consider his immoveable fame as exposed to any shock by these 
discussions. 

“Perhaps, while the undulationist will conceive that his 
opinions have gained no small accession of evidence by this ex- 
emplification of what they will account for, those who think the 
advocates of the theory have advanced its claims too far, will 
be in some degree conciliated by having a distinct acknow- 


PROCEEDINGS OF THE MEETING. xvii 


ledgement, as during these discussions they have had, of what 
it does not pretend to explain. The whole doctrine of the 
absorption of light is at present out of the pale of its calcula- 
tions ; and if the theory is ever extended to these phenomena, 
it must be by supplementary suppositions concerning the ether 
and its undulations, of which we have at present not the slight- 
est conception. 

«There are various of the Physical subjects to which your 
Reports refer, which it is less necessary to notice in a general 
sketch like the present. The recent discoveries in Thermo- 
electricity, of which Professor Cumming has presented you 
with a review, and the investigations concerning Radiant Heat 
which have been arranged and stated by Professor Powell, are 
subjects of great interest and promise ; and they are gradually 
advancing, by the accumulation of facts bound together by 
subordinate rules, into that condition in which we may hope to 
see them subjugated to general and philosophical theories. 
But with regard to this prospect, the subjects I have mentioned 
are only the fragments of sciences, on which we cannot hope 
to theorize successfully except by considering them with refer- 
ence to their whole ; —Thermo-electricity with reference to the 
whole doctrine of electricity ; Radiant Heat with reference to 
the whole doctrine of heat. 

“If the subjects just mentioned be but parts of sciences, 
there is another on which you have a Report before you, which, 
though treated as one science, is in reality a collection of several 
sciences, each of great extent. I speak of Meteorology, which 
is reported on by Professor Forbes. ‘There is perhaps no por- 
tion of human knowledge more capable of being advanced by 
our conjoined exertions than this: some of the requisite ob- 
servations demand practice and skill; but others are easily 
made, when the observer is once imbued with sound elemen- 
tary notions; and in all departments of the subject little can 
be done without a great accumulation of facts and a patient in- 
quiry after their rules. Some such contributions we may look 
for at our present Meeting. Professor Forbes has spoken of 
the possibility of constructing maps of the sky by which we 
may trace the daily and hourly condition of the atmosphere 
over large tracts of the earth. If, indeed, we could make a 
stratigraphical analysis of the aerial shell of the earth, as the 
geologist has done of its solid crust, this would be a vast step 
for Meteorology. This, however, must needs be a difficult task : 
in addition to the complexity of these superincumbent masses, 
time enters here as a new element of variety: the strata of the 
geologist continue fixed and permanent: those of the meteoro- 


XVlii THIRD REPORT—1833. 


logist change from one moment to another. Another difficulty 
is this; that while we want to determine what takes place in 
the whole depth of the aerial ocean, our observations are neces- 
sarily made almost solely at its bottom. Our access to the 
heights of the atmosphere is more limited, in comparison with 
what we wish to observe, than our access to the depths of the 
earth. 

_ © Geology, indeed, is a most signal and animating instance of 
what may be effected by continued labours governed by common 
views. Mr. Conybeare’s Report upon this science gives you 
a view of what has been done in it during the last twenty years; 
and his ‘ Section of Europe from the North of Scotland to the 
Adriatic,’ which is annexed to the Report, conveys the general 
views with regard to the structure of Central Europe, at which 
geologists have now arrived. To point out any more recent 
additions to its progress or its prospects is an undertaking 
more suitable to the geologists by profession, than to the pre- 
sent sketch. And all who take an interest in the subject will 
rejoice that the constitution and practice of the Geological So- 
ciety very happily provide, by the annual addresses of its Pre- 
sidents, against any arrear in the incorporation of fresh acquisi- 
tions with its accumulated treasures. 

**'The science of Mineralogy, on which I had the honour of 
offering a Report to the Association, was formerly looked upon 
as a subordinate portion of Geology. It may, however, now be 
most usefully considered as a science co-ordinate and closely 
allied with Chemistry, and the most important questions for 
examination in the one science belong almost equally to the 
other. Mr. Johnston, in his Report on Chemical Science, has, 
as the subject required, dwelt upon the questions of isomor- 
phism and plesiomorphism, which I had noticed as of great im- 
portance to Mineralogy. Dr. Turner and Prof. Miller, who at 
the last Meeting undertook to inquire into this subject, have 
examined a number of cases, and obtained some valuable facts ; 
but the progress of our knowledge here necessarily requires 
time, since the most delicate chemical analysis and the exact 
measurement of 30 or 40 crystals are wanted for the satisfac- 
tory establishment of the properties of each species *. In Che- 


* Perhaps I shall not have a more favourable occasion than the present of 
correcting astatement in my Report, which is not perfectly accurate, on a point 
which has been a subject of controversy between Sir David Brewster and Mr. 
Brooke. I have noticed (p. 338.) the sulphato-tricarbonate of lead of Mr. 
Brooke, as a mineral which at first appeared to contradict Sir David Brewster’s 
general law of the connexion of crystalline form with optical structure, in as 
much as it appeared to be of the rhombohedral system, and was found to have 


PROCEEDINGS OF THE MEETING. xix 


mistry, besides the great subject of isomorphism to which I 
have referred, there are some other yet undecided questions, 
as for instance those concerning the existence and relations 
of the sulpho-salts and chloro-salts; and these are not small 
points, for they affect the whole aspect of chemical theory, and 
thus show us how erroneously we should judge, if we were to 
consider this science as otherwise than in its infancy. 

“In every science, Notation and Nomenclature are questions 
subordinate to calculation and theory. The Notation of Cry- 
stallography is such as to answer the purposes of calculation, 
whether we take that of Mohs, Weiss, or Nauman. It appears 
very desirable that the Notation of Chemistry also should be so 
constructed as to answer the same purpose. Dr. Turner in the 
last edition of his Chemistry, and Mr. Johnston in his Report, 
have used a notation which has this advantage, which that 
commonly employed by the continental Chemists does not 
possess. 

**T have elsewhere stated to the Association how little hope 
there appears at present to be of purifying and systematizing 
our mineralogical nomenclature. The changes of theory in 
Chemistry to which I have already referred, must necessarily 
superinduce a change of its nomenclature, in the same manner 
in which the existing nomenclature was introduced by the pre- 
valent theory ; and the new views have in fact been connected 
with such a change by those who have propounded them. It 
will be for the Chemical Section of the Association to consider 
how far these questions of Nomenclature and Notation can be 
discussed with advantage at the present Meeting. 

‘‘'The Reports presented at the last Meeting had a reference, 
for the most part, to physical rather than physiological science. 
The latter department of human knowledge will be more pro-. 
minently the subject of some of the Reports which are to come 
before us on the present occasion. There is, however, one of 


two axes of double refraction ; and which was afterwards found to confirm the 
law, the apparently rhombohedral forms being found by Mv. Haidinger to be 
not simple but compound. It seems, however, that the solution of the difficulty 
(for no one now will doubt that it has a solution,) is somewhat different. There 
appear to have been included under this name two different kinds of crystals 
belonging to different systems of crystallization. Some which Mr. Brooke found 
to be rhombohedral, Sir David Brewster found to have a single optical axis 
with no trace of composition; others were prismatic with two axes; and thus 
Mr. Brooke’s original determinations were probably correct. The high reputa- 
tion of the parties in this controversy does not need this explanation; but pro- 
bably those who look with pleasure atthe manner in which the apparent excep- 
tions to laws of nature gradually disappear, may not think a moment or two lost 
in placing the matter on its proper footing. 


Xe THIRD REPORT—1833. 


last year’s Reports which refers to one of the widest questions 
of Physiology; that of Dr. Prichard on the History of the 
Human Species, and its subdivision into races. The other 
lines of research which tend in the same direction will probably 
be brought before the Association in successive years, and thus 
give us a view of the extent of knowledge which is accessible 
to us on this subject. 

«In addition to these particular notices of the aspect under 
which various sciences present themselves to us as resulting 
from the Reports of last years, there is a reflexion which may 
I think be collected from the general consideration of these 
sciences, and which is important to us, since it bears upon the 
manner in which science is to be promoted by combined labour 
such as that which it is a main object of this Association to 
stimulate and organize. The reflexion to which I refer is 
this ;—that a combination of theory with facts, of general views 
with experimental industry, is requisite, even in subordinate 
contributors to science. It has of late been common to assert 
that facts alone are valuable in science ; that theory, so far as 
it is valuable, is contained in the facts; and, so far as it is not 
contained in the facts, can merely mislead and preoccupy men. 
But this antithesis between theory and facts has probably in 
its turn contributed to delude and perplex ; to make men’s ob- 
servations and speculations useless and fruitless. For it is only 
through some view or other of the connexion and relation of 
facts, that we know what circumstances we ought to notice and 
record; and every labourer in the field of science, however 
humble, must direct his labours by some theoretical views, 
original or adopted. Or if the word theory be unconquerably 
obnoxious, as to some it appears to be, it will probably still be 
conceded, that it is the rules of facts, as well as facts themselves, 
with which it is our business to acquaint ourselves. That the 
recollection of this may not be useless, we may collect from the 
contrast which Professor Airy in his Report has drawn between 
the astronomers of our own and of other countries. ‘In En- 
gland,” he says, (p. 184,) ‘‘ an observer conceives that he has 
done everything when he has made an observation.” ‘‘ In 
foreign observatories,” he adds, ‘‘ the exhibition of results and 
the comparison of results with theory, are considered as de- 
serving more of an astronomer’s attention, and demanding 
greater exercise of his intellect, than the mere observation of a 
body on the wire of a telescope.” We may, indeed, perceive 
in some measure the reason which has led to the neglect of 
theory with us. For a long period astronomical theory was 
greatly a-head of observation, and this deficiency was mainly: 


PROCEEDINGS OF THE MEETING. Xxi 


supplied by the perseverance and accuracy of English ob- 
servers. It was natural that the value and reputation which 
our observations thus acquired for the time, should lead us to 
think too disrespectfully, in comparison, of the other depart- 
ments of the science. Nor is the lesson thus taught us con- 
fined to Astronomy ; for, though we may not be able in other 
respects to compare our facts with the results of a vast and yet 
certain theory, we ought never to forget that facts can only 
become portions of knowledge as they become classed and con- 
nected ; that they can only constitute truth when they are in- 
cluded in general propositions. Without some attention to this 
consideration, we may notice daily the changes of the winds 
and skies, and make a journal of the weather, which shall have 
no more value than a journal of our dreams would have ; but 
if we can once obtain fixed measures of what we notice, and 
connect our measures by probable or certain rules, it is no 
longer a vacant employment to gaze at the clouds, or an un- 
profitable stringing together of expletives to remark on the 
weather ; the caprices of the atmosphere become steady dispo- 
sitions, and we are on the road to meteorological science. 

‘It may be added—as a further reason why no observer 
should be content without arranging his observations, in what- 
ever part of Physics, and without endeavouring at least to 
classify and connect them—that when this is not done at first, 
it will most likely never be done. The circumstances of the 
observation can hardly ever be properly understood or inter- 
preted by others; the suggestions which the observations 
themselves supply, for change of plan or details, cannot in any 
other way be properly appreciated and acted on. And even 
the mere multitude of unanalysed observations may drive future 
students of the subject into a despair of rendering them useful. 
Among the other desiderata in Astronomy which Professor 
Airy mentions, he observes, ‘‘ Bradley’s observations of stars,” 
made in 1750, “‘ were nearly useless till Bessel undertook to re- 
duce them” in 1818. “In like manner Bradley’s and Mas- 
kelyne’s observations of the sun are still nearly useless,” and 
they and many more must continue so till they are reduced. 
This could not have happened if they had been reduced and 
compared with theory at the time ; and it cannot but grieve us 
to see so much skill, labour and zeal thus wasted. The per- 
petual reference or attempt to refer observations, however nu- 
merous, to the most probable known rules, can alone obviate 
similar evils. 

_ “It may appear to many, that by thus recommending theory 
we incur the danger of encouraging theoretical speculations 


XXil THIRD REPORT—183535. 


to the detriment of observation. To do this would be indeed 
to render an ill service to science: but we conceive that our 
purpose cannot so far be misunderstood. Without here at- 
tempting any nice or technical distinctions between theory and 
hypothesis, it may be sufficient to observe that all deductions 
from theory for any other pupose than that of comparison with 
observation are frivolous and useless exercises of ingenuity, so 
far as the interests of physical science are concerned. Specu- 
lators, if of active and inventive minds, wi// form theories 
whether we wish it or no. ‘These theories may be useful or 
may be otherwise—we have examples of both results. If the 
theories merely stimulate the examination of facts, and are 
modified as and when the facts suggest modification, they may be 
erroneous, but they will still be beneficial ;—they may die, but 
they will not have lived in vain. If, on the other hand, our 
theory be supposed to have a truth of a superior kind to the 
facts; to be certain independently of its exemplification in par- 
ticular cases ;—if, when exceptions to our propositions occur, 
instead of modifying the theory, we explain away the facts,— 
our theory then becomes our tyrant, and all who work under 
its bidding do the work of slaves, they themselves deriving no 
benefit from the result of their labours. For the sake of ex- 
ample we may point out the Geological Society as a body which, 
labouring in the former spirit, has ennobled and enriched itself 
by its exertions: if any body of men should employ themselves 
in the way last described, they must soon expend the small 
stock of @ priori plausibility with which they must of course 
begin the world. 

** To exemplify the distinction for a moment longer, let it be 
recollected that we have at the present time two rival theories 
of the history of the earth which prevail in the minds of geo- 
logists ;—one, which asserts that the changes of which we trace 
the evidence in the earth’s materials have been produced by 
causes such as are still acting at the surface ; another, which 
considers that the elevation of mountain chains and the transi- 
tion from the organized world of one formation to that of the 
next, have been produced by events which, compared with the 
present course of things, may be called catastrophes and con- 
vulsions. Who does not see that all that those theories have 
hitherto done, has been, to lead geologists to study more ex- 
actly the laws of permanence and of change in the existing 
organic and inorganic world, on the one hand; and on the 
other, the relations of mountain chains to each other, and to 
the phenomena which their strata present?) And who doubts, 
that, as the amount of the full evidence may finally be, (which 


PROCEEDINGS OF THE MEETING. xxiii 


may, indeed, perhaps require many generations to accumulate,) 
geologists will give their assent to the one or the other of these 
views, or to some intermediate opinion to which both may 
gradually converge? 

“On the other hand—to take an example from a science with 
which I have had a professional concern—the theory that ery- 
stalline bodies are composed of ultimate molecules which have 
a definite and constant geometrical form, may properly and 
philosophically be adopted, so far as we can, by means of it, 
reduce to rules the actually occurring secondary faces of such 
substances. But if we assume the doctrine of such an atomic 
composition, and then form imaginary arrangements of these 
atoms, and enunciate these as explanations of dimorphism, 
or plesiomorphism, or any other apparent exception to the 
general principle, we proceed, as appears to me, unphilosophi- 
cally. Let us collect and classify the facts of dimorphism and 
plesiomorphism, and see what rules they follow, and we may 
then hope to discern whether our atomic theory of crystalline 
molecules is tenable, and what modifications of it these cases, 
uncontemplated in its original formation, now demand. 

“T will not now attempt to draw forth other lessons which 
the Report of last year may supply for our future guidance ; 
although such offer themselves, and will undoubtedly affect 
the spirit of our proceedings during this Meeting. But there 
is a reflexion belonging to what I may call the morals of science, 
which seems to me to lie on the face of this Report, and which 
I cannot prevail upon myself to pass over. In looking steadily 
at the past history and present state of physical knowledge, we 
cannot, I think, avoid being struck with this thought,—How 
little is done and how much remains to do ;—and again, not- 
withstanding this, how much we owe to the great philosophers 
who have preceded us. It is sometimes advanced as a charge 
against the studies of modern science, that they give men an 
overweening opinion of their own acquirements, of the supe- 
riority of the present generation, and of the intellectual power 
and progress of man ;—that they make men confident and con- 
temptuous, vain and proud. That they never do this, would 
be much to say of these or of any other studies ; but, assuredly, 
those must read the history of science with strange preposses- 
sions who find in it an aliment for such feelings. What is the 
picture which we have had presented to us? Among all the 
attempts of man to systematize and complete his knowledge, 
there is one'science, Astronomy, in which he may be considered 
to have been successful; he has there attained a general and 
certain theory: for this success, the labour of the most highly. 


XXIV THIRD REPORT—1835. 


gifted portion of the species for 5000 years has been requisite. 
There is another science, Optics, in which we are, perhaps, in 
the act of obtaining the same success, with regard to a part of 
the phenomena. But all the rest of the prospect is compara- 
tively darkness and chaos; limited rules, imperfectly known, 
imperfectly verified, connected by no known cause, are all that 
we can discern. Even in those sciences which are considered 
as having been most successful, as Chemistry, every few years 
changes the aspect under which the theory presents the facts 
to our minds, while no theory, as yet, has advanced beyond the 
mere horn-book of calculation. What is there here of which 
man can be proud, or from which he can find reason to be pre- 
sumptuous? And evenif the Discoverers to whom these sciences 
owe such progress as they have made—the great men of the 
present and the past—if THEY might be elate and confident 
in the exercises of their intellectual powers, who are we, that 
we should ape their mental attitudes ?—we, who can but with 
pain and effort keep a firm hold of the views which they have 
disclosed? But it has not been so; they, the really great in 
the world of intellect, have never had their characters marked 
with admiration of themselves and contempt of others. Their 
genuine nobility has ever been superior to those ignoble and 
low-born tempers. Their views of their own powers and achieve- 
ments have been sober and modest, because they have ever felt 
how near their predecessors had advanced to what they had 
done, and what patience and labour their own small progress 
had cost. Knowledge, like wealth, is not likely to make us 
proud or vain, except when it comes suddenly and unlearned ; 
and in such a case, it is little to be hoped that we shall use 
well, or increase, our ill-understood possession. 

“« Perhaps some of the appearance of overweening estimation 
of ourselves and our generation which has been charged against 
science, has arisen from the natural exultation which men feel 
at witnessing the successes of art. I need not here dwell upon 
the distinction of science and art; of knowledge, and the ap- 
plication of knowledge to the uses of life; of theory and 
practice. In the success of the mechanical arts there is much 
that we look at with an admiration mingled with some feeling 
of triumph; and this feeling is here natural and blameless. 
For what is all such art but a struggle,—a perpetual conflict 
with the inertness of matter and its unfitness for our purposes? 
And when, in this conflict, we gain some point, it is impossible 
we should not feel some of the exultation of victory. In all 
stages of civilization this temper prevails: from the naked in- 
habitant of the islands of the ocean, who by means of a piece 


PROCEEDINGS OF THE MEETING. XXV 


of board glides through the furious and apparently deadly line 
of breakers, to the traveller who starts along a rail-road with 
a rapidity that dazzles the eye, this triumphant joy in suc- 
cessful art is universally felt. But we shall have no difficulty 
in distinguishing this feeling from the calm pleasure which we 
receive from the contemplation of truth. And when we con- 
sider how small an advance of speculative science is implied in 
each successful step of art, we shall be in no danger of im- 
bibing, from the mere high spirits produced by difficulty over- 
come, any extravagant estimate of what man has done or can 
ir any perverse conception of the true scale of his aims and 
opes. 

te Still, it would little become us here to be unjust to prac- 
tical science. Practice has always been the origin and stimulus 
of theory: Art has ever been the mother of Science; the 
comely and busy mother of a daughter of a far loftier and 
serener beauty. And so it is likely still to be: there are no 
subjects in which we may look more hopefully to an advance in 
sound theoretical views, than those in which the demands of 
practice make men willing to experiment on an expensive scale, 
with keenness and perseverance; and reward every addition 
of our knowledge with an addition to our power. And even 
they—for undoubtedly there are many such—who require no 
such bribe as an inducement to their own exertions, may still 
be glad that such a fund should exist, as a means of engaging 
and recompensing subordinate labourers. 

**T will not detain you longer by endeavouring to follow 
more into detail the application of these observations to the 
proceedings of the General and Sectional Meetings during the 
present week. But I may remark that some subjects, circum- 
stanced exactly as I have described, will be brought under 
your notice by the Reports which we have reason to hope for 
on the present occasion. Thus, the state of our knowledge of 
the laws of the motion of fluids is universally important, since 
the motion of boats of all kinds, hydraulic machinery, the tides, 
the flowing of rivers, all depend upon it. Mr. Stevenson and 
Mr. Rennie have undertaken to give us an account of different 
branches of this subject as connected with practice ; and: Mr. 
Challis will report to us on the present state of the analytical 
theory. In like manner the subject of the strength of materials, 
which the multiplied uses of iron, stone and wood, make so inter- 
esting, will be brought. before you by Mr. Barlow. These were 
two of the portions of mechanics the earliest speculated upon, 
and in them the latest speculators have as yet advanced little 
beyond the views of the earliest. PS 

c 


XXV1 THIRD REPORT—1833. 


**I mention these as specimens only of the points to which 
we may more particularly direct our attention. I will only 
observe, in addition, that if some studies, as for instance those 
of Natural History and Physiology, appear hitherto to have 
occupied less space in our proceedings than their importance 
and interest might justly demand, this has occurred because 
the Reports on other subjects appeared more easy to obtain in 
the first instance ; and the balance will I trust be restored at 
the present Meeting. I need not add anything further on this 
subject. Among an assembly of persons such as are now met 
in this place, there can be no doubt that the most important 
and profound questions of science in its existing state will be 
those which will most naturally occur in our assemblies and 
discussions. It merely remains for me to congratulate the As- 
sociation upon the circumstances under which it is assembled ; 
and to express my persuasion that all of us, acting under the 
elevating and yet sobering thought of being engaged in the 
great cause of the advancement of true science, and cherishing 
the views and feelings which such a situation inspires, shall 
derive satisfaction and benefit from the occasions of the present 
week.” ; 


Mr. Whewell having concluded his Address, the Meeting 
adjourned, after electing by a general vote the candidates who 
had been approved by the Council and by the General Com- 
mittee. 


At eight p.m., the Members having reassembled in the Senate- 
house, Mr. Taylor read a Report on the state of our know- 
ledge respecting Mineral Veins, which was followed by a general 
discussion on the nature and origin of veins. 


On Wednesday at one P.m., the Chairmen of the Sections hay- 
ing read the minutes of their proceedings to the Meeting, the 
Rey. G. Peacock delivered a brief abstract of his Report on 
the state of the Theory of Algebra. Professor Lindley read a 
Report on the state of Physiological Botany; and Mr. G. Ren- 
nie on the state of Practical Hydraulics. Auditors were ap- 
pointed to examine the accounts. : 


On Thursday, at one p.m., the auditors reported the state 
of the accounts. The Chairmen of the Sections read the mi- 
nutes of their proceedings. Professor Christie read a Report 
on the porsent state of our knowledge respecting the Magnetism 
of the Earth. A summary of the contents of a Report on the 
state of knowledge as to the Strength of Materials, by Pro: 


PROCEEDINGS OF THE MEETING. XXVil 


fessor Barlow, was given, in the absence of the Author, by the 
Rev. W. Whevwell. 

In the evening, Mr. Whewell delivered a Lecture in the 
Senate-louse, on the manner in which observations of the Tide 
may be usefully made to serve as a groundwork for general 
views; either by observing the time of high water at different 
places on the same day, in order to determine the motion of 
the summit of the tide-wave; or by continuing the observations 
for a considerable time, and comparing them with the moon’s 
transit to obtain the semi-menstrual inequality. He observed, 
that it appears from Mr. Lubbock’s recent researches on the 
subject, that the tides of Portsmouth and Brest agree very — 
closely in the law of this inequality, and that the tides of Ply- 
mouth and London also agree; but that there is an anomaly 
which cannot at present be explained in the comparison of Brest 
with Plymouth. Professor Farish explained to the Meeting 
the advantages which he conceived would be derived from ap- 
plying the power of steam to carriages on undulating roads in 
preference to level rail-ways. 


On Friday, at one p.m., the Chairmen of the Sections havin 
read the minutes of their proceedings, the Rey. J. Challis made 
a Report on the progress of the Theory of Fluids. The Pre- 
sident stated the appropriation* to certain scientific objects of 
a portion of the funds of the Association to the amount of 
6007. Mr. Babbage, at the President’s request, explained his 
views respecting the advantages which would accrue to science 
from such a collection of numerical facts as he had formerly 
recommended under the title of ‘Constants of Nature and 
Art.” The President announced, that it had been resolved 
by the General Committee, that the Meeting of 1834 should 
take place at Edinburgh in the early part of the month of Sep- 
tember ; he read the names of the Officers and Members of the 
‘Council appointed for the ensuing year. . 

The thanks of the Meeting were then voted to the Vice- 
Chancellor and the other authorities of the University, to the 
retiring Officers and Members of the Council, to the President, 
the Secretaries for Cambridge, the Local Committee of Manage- 
ment, and the General Secretary. ae 
_ The President, in his concluding Address to the Meeting, 
explained an irregularity which had occurred in the formation 
of anew Section. In addition to the five Sections into which 
the Meeting had been divided by the authority of the General 


* Fora particular account of these appropriations, see p. Xxxvi. _ 
c2 


XXvill THIRD REPORT— 1833. 


Committee, he stated that another had come into operation, the 
object of which was to promote statistical inquiries. It had 
originated with some distinguished philosophers, but could not 
be regarded as a legitimate branch of the Association till it had 
received the recognition of the governing body ; there could be 
little doubt, however, that the new Section would obtain the 
sanction of the General Committee, with some limitation per- 
haps of the specific objects of inquiry. On this subject he 
made the following observations :— 

* Some remarks may be expected from me in reference to the 
objects of this Section, as several Members may perhaps think 
them ill fitted to a Society formed only for the promotion of 
natural science. ‘To set, as far as I am able, these doubts at 
rest, I will explain what I understand by science, and what I 
think the proper objects of the Association. By science, then, 
I understand the consideration of all subjects, whether of a pure 
or mixed nature, capable of being reduced to measurement and 
calculation. All things comprehended under the categories of 
space, time and number properly belong to our investigations ; 
and all phenomena capable of being brought under the sem- 
blance of a law are legitimate objects of our inquiries. But there 
are many important subjects of human contemplation which come 
under none of these heads, being separated from them by new 
elements ; for they bear upon the passions, affections and feel- 
ings of our moral nature. Most important parts of our nature 
such elements indeed are; and God forbid that I should call 
upon any man to extinguish them; but they enter not amon 
the objects of the Association. The sciences of morals me 
politics are elevated far above the speculations of our philosophy. 
Can, then, statistical inquiries be made compatible with our 
objects, and taken into the bosom of our Society? I think 
they unquestionably may, so far as they have to do with matters 
of fact, with mere abstractions, and with numerical results. 
Considered in that light they give what may be called the raw 
material to political economy and political philosophy ; and by 
their help the lasting foundations of those sciences may be per- 
haps ultimately laid. These inquiries are, however, it is import- 
ant to observe, most intimately connected with moral pheno- 
mena and economical speculations,—they touch the mainsprings 
of passion and feeling,—they blend themselves with the generali- 
zations of political science; but when we enter on these higher 
generalizations, that moment they are dissevered from the ob- 
jects of the Association, and must be abandoned by it, if it 
means not to desert the secure ground which it has now taken. 

“Should any one affirm (what, indeed, no one is prepared 


PROCEEDINGS OF THE MEETING. xxix 


to deny,) that all truth has one common essence, and should 
he then go on to ask why truths of different degrees should be 
thus dissevered from each other, the reply would not be dif- 
ficult. In physical truth, whatever may be our difference of 
opinion, there is an ultimate appeal to experiment and ob- 
servation, against which passion and prejudice have not a 
single plea to urge. But in moral and political reasoning, we 
have ever to do with questions, in which the waywardness of 
man’s will and the turbulence of man’s passions are among the 
strongest elements. The consequence it is not for me to tell. 
Look around you, and you will then see the whole framework 
of society put in movement bythe worst passions of our na- 
ture; you will see love turned into hate, deliberation into dis- 
cord, and men, instead of mitigating the evils which are about 
them, tearing and mangling each other, and deforming the 
moral aspect of the world. And let not the Members of the 
Association indulge a fancy, that they are themselves exempt 
from the common evils of humanity. There is that within us, 
which, if put into a flame, may consume our whole fabric,— 
may produce an explosion, capable at once of destroying all 
the principles by which we are held together, and of dissi- 
pating our body in the air. Our Meetings have been essen- 
tially harmonious, only because we have kept within our proper 
boundaries, confined ourselves to the laws of nature, and 
steered clear of all questions in the decision of which bad 
passions could have any play. But if we transgress our pro- 
per boundaries, go into provinces not belonging to us, and open 
a door of communication with the dreary wild of politics, that 
instant will the foul Damon of discord find his way into our 
Eden of Philosophy. 

‘*In every condition of society there is some bright spot on 
which the eye loves to rest. In the turbulent republics of 
ancient Greece, where men seemed in an almost ceaseless war- 
fare of mind andbody, they had their seasons of solemnity, when 
hostile nations made a truce with their bitter feelings, as- 
sembled together, for a time, in harmony, and joined in a great 
festival; which, however differing from what we now see in 
its magnitude and forms of celebration, was consecrated, like 
our present Meeting, to the honour of national genius. What- 
ever have been the bitter feelings which have so often disgraced 
the civil history of mankind, I dare to hope that they will never 
find their resting-place within the threshold where this Associa- 
tion meets; that peace and good will, though banished from 
every other corner of the land, will ever find an honoured seat 
amongst us; and that the congregated philosophers of the 
empire, throwing aside bad passion and party animosity, will, 


XXX THIRD REPORT—1833. 


year by year, come to their philosophical Olympia, to witness 
a noble ceremonial, to meet in a pacific combat, and share in the 
glorious privilege of pushing on the triumphal car of Truth. 

“ The last duty I have to perform this morning would be a 
painful one indeed, were our Assembly to be broken up into 
elements which were not again to be reunited. ‘The Association 
is not, however, dissolved ; its meeting is only adjourned to an- 
other year; and it has been a matter of great joy to me to an- 

_nounce to you, that the Committee has elected for your next 
President a distinguished soldier and philosopher; and that it 
will be your privilege to reassemble in one of the fairest capitals 
of the world,—in a city which has nursed a race of literary and 
philosophic giants,—in a land filled with natural beauties, and 
wedded to the imagination and the memory by a thousand en- 
dearing associations. 

‘** There is a solemnity in parting words, which may, I think, 
justify me (especially after what has been so well said this morn- 
ing by the Marquis of Northampton,) in passing the limits I 
have so far carefully prescribed to myself, and in treading for a 
moment on more hallowed ground. In the first place, I would 
entreat you to remember that you ought above all things to re- 
joice in the moral influence of an Association like the present. 
Facts, which are the first objects of our pursuit, are of compa- 
atively small value till they are combined together so as to 
lead to some philosophic inference. Physical experiments, con- 
sidered merely by themselves, and apart from the rest of nature, 
are no better than stones lying scattered on the ground, which 
require to be chiselled and cemented before they can be made 
into a building fit for the habitation of man. The true value 
of an experiment is, that itis subordinate to some law,—that it 
is a step toward the knowledge of some general truth. Without, 
at least, a glimmering of such truth, physical knowledge has no 
true nobility. But there is in the intellect of man an appetency 
for the discovery of general truth, and by this appetency, in 
subordination to the capacities of his mind, has he been led on to 
the discovery of general laws; and thus has his soul been fitted 
to reflect back upon the world a portion of the counsels of his 
Creator. If I have said that physical phenomena, unless con- 
nected with the ideas of order and of law, are of little worth, 
I may further say, that an intellectual grasp of material laws of 
the highest order has no moral worth, except it be combined 
with another movement of the mind, raising it to the perception 
of an intelligent First Cause. It is by help of this last movement 
that nature’s language is comprehended ; that her laws become 
pregnant with meaning; that material phenomena are instinct 
with life; that all moral and material changes become linked 


PROCEEDINGS OF THE MEETING. Xxxi 


together ; and that Truth, under whatever forms she may pre- 
sent herself, seems to have but one essential substance. ‘ 

‘I have before spoken of the distinctions between moral and 
physical science; and I need not repeat what I have said, unless 
it be once more solemnly to adjure you not to leave the straight 
path by which you are advancing,—not todesert the cause for 
which. you have so well combined together. But let no one 
misunderstand my meaning. If I have said that bad passions 
mingle themselves with moral and political sciences, and that 
the conclusions of these sciences are made obscure from the 
want of our comprehending all the elements with which we 
have to deal, I have only spoken the truth ; but still I hold that 
moral and political science is of a higher order than the physical. 
The latter has sometimes, in the estimation of man, been placed 
on a higher level than it deserves, only from the circumstance of 
its being so well defined, and grounded in the evidence of ex- 
periments appealing to the senses. Its progress is marked by 
indices the eye can follow; and the boundaries of its conquests 
are traced by landmarks which stand high in the horizon of 
man’s history. But with all these accompaniments, the moral _ 
and political sciences entirely swallow up the physical in impor- 
tance. For what are they but an interpretation of the governing 
laws of intellectual nature, having a relation in time present to 
the social happiness of millions, and bearing in their end on the 
destinies of immortal beings ? 

“Gentlemen, if I look forward with delight to our meeting 
again at Edinburgh, it is a delight chastised by a far different 
feeling, to which, had not these been parting words, I should 
not have ventured to give an utterance. It is not possible 
we should all again meet together. Some of those whose 
voices have been lifted up during this great Meeting, whose 
eyes have brightened at the presence of their friends, and 
whose hearts have beat high during the intellectual commu- 
nion of the week, before another year may not be numbered 
with the living. Nay, by that law of nature to which every 
living man must in his turn yield obedience, it is certain that 
before another festival, the cold hand of death will rest on the 
head of some who are present in this assembly. If a thought 
like this gives a tone of grave solemnity to words of parting, it 
surely ought to teach us, during our common rejoicings at the 
triumphal progress of science, a personal lesson of deep humility. 
By the laws of nature, before we can meet again, many of those 
bright faces which during the past week I have seen around me 
may be laid low, for the hand of death may have been upon 
them; but wherever we reassemble, God grant that all our 
attainments in science may tend to our moral improvement}- and 


XXXil THIRD REPORT—1833. 


may we all meet at last in the presence of that Almighty Being, 
whose will is the rule of all law, and whose bosom is the centre 
of all power!” 


SECTIONAL MEETINGS. 


The Sections assembled daily at eleven a.m., and occasion- 
ally also at half-past eight p.m., at their respective places of 
meeting, in the Schools, the Astronomical Lecture-room, and 
the Hall of Caius College. On Saturday, the Section of Na- 
tural History made an excursion to the Fens. 


Abstracts of most of the Communications which were made 
to the Sections will be found in a subsequent part of the 
volume. 

In addition to the communications of which abstracts are 
there given, notices of the following transactions appear on the 
minutes :— 

M. Quetelet described the observations which he had made 
on Falling Stars. It was suggested that such observations 
might be available in certain cases for determining differences 
of longitude. 

Mr. Potter communicated some calculations of the height 
of the Aurora Borealis, seen on the 21st of March 1833, 

Mr. Hopkins gave an abstract of a paper on the Vibration 
of Air in Cylindrical Tubes of definite length. 

Dr. Ritchie made some remarks on the Sensibility of the 
Eye, and the errors to which it is subject. 

Mr. Barton gave a view of his opinions on the Propagation 
of Heat in solid bodies. 

A letter was received from Mr. Frend regarding certain 
points in the Theory of the Tides. 

The Rev. W. Scoresby described a Celestial Compass invent- 
ed by Col. Graydon. 

Mr. R. Murphy read some remarks on the utility of observ- 
ing the Magnetic Dip in Mines. 

M. Quetelet gave an account of some observations made by 
himself and M. Necker de Saussure, which corroborate the 
statements of M. Kuppfer, respecting the inequality of magne- 
tic intensity at the top and the base of mountains. 

Professor Christie stated his views relative to the cause of 
the Magnetism of the Earth. 

Mr. A. Trevelyan read a paper on certain Vibrations of 
Heated Metals. 

Mr. Brunel exhibited and explained a Model in illustration 
of his method of constructing Bridges without centering. 


PROCEEDINGS OF THE MEETING. XXXill 


A notice of some experiments relative to Isomorphism, by 
Dr. Turner and Professor Miller, was read. 

Dr. Daubeny made a communication on the Gases given off 
from the surfaces of the water in certain thermal springs. 

The Rev. W. V. Harcourt exhibited specimens of Metal taken 
out of the crevices at the bottom of a mould in which a large 
bronze figure had been cast by Mr. Chantrey; together with 
fragments of the Bronze employed in the casting, from which 
the former specimens differed considerably in colour, frangi- 
bility, &c. 

Mr. Lowe gave an account of various chemical products 
found in the retorts and flues of Gas Works. 

Mr. Pearsall made a communication on the bleaching powers 
of Oxygen. 

Mr. J. Taylor described the character of the Ecton Mine, 
and the occurrence of the copper ore in connected cavities 
which had been explored to a depth of 225 fathoms without 
reaching the termination of them. 

Dr. Buckland described. the manner in which fibrous Lime- 
stone occurs in the Isle of Purbeck and other situations. 

_ Mr. Murchison stated, and illustrated by Maps and Sections, 
the principal results of his inquiries into the sedimentary de- 

osits which occupy the western parts of Shropshire and Here- 
fordshire, and are prolonged in a S.W. direction through the 
counties of Radnor, Brecknock, and Caermarthen, and the in- 
trusive igneous rocks which occur in certain parts of the di- 
strict. He mentioned the occurrence of freshwater Limestone 
in a detached Coal-field of Shropshire. : 

Professor Sedgwick described the leading features in the 
Geology of North Wales, the lines of elevation, the relation of 
the trap rocks to the slate system, the cleavage of the slate ; 
pointed out the relations of this tract to that examined by Mr- 
Murchison; and drew a general parallel between the slate 
formations of Wales and Cumberland. 

Mr. J. Taylor having read to the Section the concluding 
part of his Report on Veins, in the discussion which followed, 
M. Dufrénoy entered into a consideration of some phenomena of 
the igneous rocks of Britanny and Central France, viewed with 
reference to the connexion between them and the metalliferous 
veins of those districts, and remarked on the occurrence in 
Central France of mineral veins, only in the narrow zone at the 
junction of the unstratified and stratified rocks. He also made 
some remarks on the association of dolomite and gypsum, with 
the igneous rocks of the Alps and the Pyrenees. 

Professor Sedgwick gave a general account of the Red Sand= 
stones connected with the Coal-measures of Scotland, and the 


XXXIV THIRD REPORT—1833. 


Isle of Arran, with the view of showing that they are perfectly 
distinct from the similar rocks connected with the Magnesian 
‘Limestone. 2 : 

Mr. Hartop exhibited a Map and Sections to illustrate the 
series of Coal Strata in South Yorkshire, and their direction and 
varying dip in the valley of the Dun, and to the north and south 
of that river; described the characters of the strata, and the in- 
fluence of certain great dislocations on the quality of the coal. 
_. Mr. Greenough exhibited a Map of Western Europe, on 
which the relative levels of land and water were represented by 
means of colours, instead of engraving. Mr. Greenough was 
requested to permit a map on this plan to be published. 

The Rev. J. Hailstone communicated some notices relating 
to Mineral Veins. : 

Sections of the Well in the Dock Yard at Portsmouth, and 
of the Well in the Victualling Yard at Weevil, were communi- 
cated by the Rev. Mr. Leggat and Mr. Blackburn, on the part 
of the Portsmouth Philosophical Society; and a letter from 
Mr. Goodrich, explanatary of the Sections, was read. 

Mr. Mantell exhibited a perfect Femur of the Iguanodon, 
and explained its distinctive anatomical characters. 

Mr. W. C. Trevelyan exhibited specimens of Coprolites, 
and remains of Fishes, from the Edinburgh Coal-field. 
- Mr. Fox exhibited specimens of Fishes from the Magnesian 
Limestone and Marl-slate of Durham. i 

Mr. Gray made some remarks on the occurrence of Water 
in the Valves of Bivalve Shells, and exhibited a specimen of 
Spondylus varius, in which water was contained in both the 
valves. 

Mr. Ogilby gave an account of his views respecting the 
classification of Ruminating Quadrupeds, which he proposed to 
found upon the presence or absence of horns on the female sex; 
the peculiar form of the upper lip; and the presence or absence 
of the subocular and submaxillary glands. He showed the ap- 
plication of these views to the division of hollow-horned rumi- 
nating animals without horns in the female sex, which he dis- 
tributed into five new genera. 

The Rey. W. Scoresby communicated some observations on 
the adaptation of the Structure of the Cetacea to their habits 
of life and residence in the Ocean; and suggested the use 
which might be made of the peculiar forms of the Whalebone 
in their classification. 

Lieutenant Colonel Sykes exhibited a specimen of the Short- 
tailed Manis, and communicated some observations on its mode 
of progression. ! 

Mr. Brayley communicated a memoir on the laws regulating 


PROCEEDINGS OF THE MEETING. XXXV 


the distribution of the powers of producing Light and Heat 
among Animals. 

Mr. H. Strickland made some remarks on the Vipera Chersea, 
showing its specific difference from the common Viper. 

The subject of the use of the Pith in Plants, was discussed 
by Professor Burnett, Professor Henslow, Mr. Curtis, and Mr. 
Gray. 

Dr. Roupell exhibited some Drawings representing the 
effects of irritant Poisons upon the living membrane of the in- 
testinal canal of Men and Animals. 

Mr. Fisher communicated some observations on the physical 
condition of the Brain during sleep. 

Mr. Brooke made some remarks on the physiology of the 
Eye and the Ear. 

Dr. Marshall Hall gave an abstract of his views respectmg 
the reflex function of the Medulla oblongata and Medulla spi- 
nalis. 


COMMITTEES. 


~ ‘The General Committee met daily at ten a.m., and at other 
hours by adjournment, in the Hall of Trinity Hall. The Com- 
mittees of Sciences met as soon after ten as the business of the 
General Committee permitted, in the rooms of their respective 
Sections. The General Committee made the necessary arrange- 
ments for the conduct of the Meeting; formed the Sectional 
Committees of Sciences; determined the place and time of the 
next Meeting ; appointed the new Officers and Council; and 
passed the following Resolutions :— 

- 1. That thethanks of the Association be given to the Societies 
and Institutions from which it has received invitations,—in Bris- 
tol, Birmingham, Liverpool, Newcastle and Edinburgh. 

2. That Members of the Association whose subscription shall 
have been due for two years, and who shall not pay it on proper 
notice, shall cease to be Members, power being left to the Com- 
mittee or Council to reinstate them on reasonable grounds 
within one year, on payment of their arrears. 

- 3. That the number of Deputies which provincial Institutions 
shall be entitled to send to the Meetings as Members of the 
General Committee, shall be two from each Institution. 

4, That the following instructions be given to each of the Com- 
mittees of Sciences:—-- .-——- tAdait 

__To’select those points of science, which, on a review of the 
former Recommendations of the Committees, or those contained 


XXXVi THIRD REPORT—1833, 


in the Reports published by the Association, or from sugges- 
tions made at the present Meeting, they may think most fit to 
be advanced by an application of the funds of the Society, 
either in compensation for labour, or in defraying the expense 
of apparatus, or otherwise. The Committee are requested to 
confine their selections to definite as well as important objects ; 
to state their reasons for the selection, and where they may 
think proper, to designate individuals to undertake the desired 
investigations ; they are to transmit their Recommendations 
through their Secretaries to the General Committee. 

The Committees of Sciences having complied with these in- 
structions, the following Resolutions were passed by the General 
Committee: 

1. That a sum not exceeding 2001. be devoted to the dis- 
cussion of observations of the Tides, and the formation of Tide 
Tables, under the superintendence of Mr. Baily, Mr. Lubbock, 
Rev. G. Peacock, and Rev. W. Whewell. 

2. That a sum not exceeding 50/. be appropriated to the 
construction of a Telescopic Lens, or Lenses, out of Rock Salt, 
under the direction of Sir David Brewster. 

3. That Dr. Dalton and Dr. Prout be requested to ae 
experiments onthe specific gravities of Oxygen, Hydrogen, and 
Carbonic Acid; and that a sum not exceeding 50/. be appropri- 
ated to defray the expense of any apparatus which may be re- 
quired. 

4, That a series of experiments on the effects of long con- 
tinued Heat be instituted at some iron furnace, or in any other 
suitable situation; and that a sum not exceeding 50/. be placed 
at the disposal of a Sub-Committee, consisting of Professor 
Daubeny, Rev. W. V. Harcourt, Professor Sedgwick, and Pro- 
fessor Turner, to meet any expense which may be incurred *. 

5. That measurements should be made, and the necessary 
data procured, to determine the question of the permanence or 
change of the relative Level of Sea and Land on the coasts of 
Great Britain and Ireland; and that for this purpose a sum 
not exceeding 100/. be placed at the disposal of a Sub-Com- 
mittee, consisting of Mr. Greenough, Mr. Lubbock, Mr. G. 
Rennie, Professor Sedgwick, Mr. Stevenson, and Rev. W. 
Whewell ;—the measurements to be so executed, as to furnish 
the means of reference in future times, not only as to the re- 
lative levels of the land and sea, but also as to waste or exten- 
sion of the land. 


* These experiments have been instituted by Mr. Harcourt, in Yorkshire, at 
the Low Moor Iron Works, the property of Messrs. Hird and Co., and at the 
Elsecar Furnace, belonging to Earl Fitzwilliam. 


PROCEEDINGS OF THE MEETING. XXXvik 


6. That the effects of Poisons on the Animal Economy should 
be investigated and illustrated by graphic representations ; and 
that a sum not exceeding 25/. be appropriated for this object. 
Dr. Roupell, and Dr. Hodgkin were requested to undertake 
this investigation. 

7. That the sensibilities of the Nerves of the Brain should 
be investigated ; and that a sum not exceeding 25/. should be 
appropriated to this object. Dr. Marshall Hall and Mr. S. D. 
Broughton were requested to undertake these experiments. 

8. That a sum not exceeding 100/. be appropriated towards 

the execution of the plan proposed by Professor Babbage, for 
collecting and arranging the Constants of Nature and Art*. 
_ 9. That arepresentation be submitted to Government on the 
part of the British Association, stating that it would tend greatly 
to the advancement of astronomy, and the art of navigation, if the 
observations of the sun, moon and planets, made by Bradley, 
Maskelyne and Pond, were reduced ; and that.a deputation} be 
appointed to wait upon the Lords of the Treasuzy with a re- 
quest, that public provision may be made for the accomplish- 
ment of this great national object. 

Proposals for the formation of a Statistical Section were ap- 
proved. Itwas resolved, that the inquiries of this Section should 
be restricted to those classes of facts relating to communities of 
men which are capable of being expressed by numbers, and 
which promise, when sufficiently multiplied, to indicate general 
laws. 

A Committee of Statistical Science was formed t+. The Re- 
commendations § of the several Committees of Science were re- 
vised and approved. 


_ TRUSTEES OF THE ASSOCIATION. 


Charles Babbage, F.R.S. Lucasian Professor of Mathe- 
matics, Cambridge. 

R. I. Murchison, F.R.S. V.P.G.S. &c. 
_ John Taylor, F.R.S. Treas. G.S. &c. 


_ * For an abstract of Mr. Babbage’s plan, see the Appendix. ; 

+ The deputation consisted of Professor Airy, Mr. Baily, Mr. D. Gilbert and 
Sir John Herschel. The application wasimmediately complied with by the Go- 
vernment. 

. } For an account of the proceedings of this Committee, see the Appendix. 

§ These Recommendations will be found marked with an asterisk in the col- 
leguon of Recommendations and Suggestions printed in the latter part of .the 
volume, ; 


XXXVIli THIRD REPORT—1833. 


. OFFICERS. 

President.— Rev. Adam Sedgwick, F.R.S. G.S. and Wood- 
wardian Professor of Geology, Cambridge. 

Vice-Presidents.—G. B. ‘Airy, F.G.S. Plumian Pyoteddex of 
Astronomy, Cambridge. John aDolecee D.C.L. F.R.S. Instit. 
Reg. Sc. Paris. Corresp. 

President elect.—Lieut. Gen. Sir T. M. Brisbane, K.C.B. 
F.R.S. L. & E. President of the Royal Soc. Edinb. Inst. 
Reg. Sc. Paris. Corresp. 

Vice-Presidents elect.—Sir David Brewster, K.G.H. LL.D. 
F.R.S. L. & E. Rey. J. Robinson, D.D. Antconeaner Royal 
at Armagh. 

Treasurer.—Jobn Taylor, F.R.S. Treas. G. S. 

General Secretary.—Rev. W. V. Harcourt, F.R.S. G.S. 

Assistant Secretary.—John Phillips, F.R.S. G.S. Professor 
of Geology in King’s College, London. 

Secretaries for Oxford.—Charles Daubeny, M.D. F.R.S. 
L.S. Professor of Botany. Rev. B. Powell, F.R.S. Savilian 
Professor of Geometry. 

Secretaries for Cambridge.—Rev. J. 8S. Henslow, F.L.S. 
G.S. Professor of Botany. Rev. W. Whewell, F.R.S. &c.° 

Secretaries for Edinburgh.—John Robison, Sec. R.S.E. 
James D. Forbes, F.R.S. L. & E. F.G.S. Professor of Natural 
Philosophy. 

Secretary for Dublin.—Rev. Thomas Luby. 


COUNCIL. 

- Rev. W. Buckland, D.D. F.R.S. Professor of Geol. and Min; 
Oxford. W. Clift, F.R.S. Rev. T. Chalmers, D.D. Professor of 
Divinity, Edinburgh. S. H. Christie, F.R.S. Professor of Ma- 
thematics at Woolwich. Earl Fitzwilliam, F.R.S.G.S. G.B. 
Greenough, F.R.S. Pres. of the Geol. Society. T. Hodg- 
kin, M. D. London. W. R. Hamilton, Astronomer Royal 
for Ireland. W. J. Hooker, F.R.S. Professor of Botany, 
Glasgow. Robert Jameson, F. R.S. Professor of Natural Hi- 
story, Edinburgh. John Lindley, F.R.S. Professor of Botany 
in the University of London. J. W. Lubbock, Treas. R.S. 
Rev. B. Lloyd, D.D. Treas. Prov. of Trin. Coll. Dublin. 
R. I. Murchison, F.R.S. &c. Patrick Neill, M.D. F.R.S.E. 
Engh George Rennie, F.R.S. Rey. W. Ritchie, LL.D. 

E.R.S. Professor of Nat. Philosophy in the University of Lon- 
don. J. S. Traill, M.D. W. Yarrell, F.L.S. &c... Ex officio 

members,—The Prasees and Officers of the Association. _ 

Secretaries.—Edward Turner, M.D. F.R.S. Sec. G.S. Rey. 
James Yates, F.L.S. G.S. 


PROCEEDINGS OF THE MEETING. XKXI1X 


COMMITTEES OF SCIENCES. 


J. Mathematics and General Physics. 


-Chairman.—Sir D. Brewster, FE.R.S. &c. 

Deputy Chairman.—Rev. G. Peacock, F.R.S. 

Secretary.—Professor Forbes. 

Viscount Adare, F.R.S. Professor Airy. Professor Bab- 
bage. Francis Baily, V.P.R.S. John Barton, F.R.S. Rev. 
J. Bowstead. Sir'T’. M. Brisbane, F.R.S. Professor Christie. 
Rev. H. Coddington, F.R.S. E. J. Cooper. Dr. Corrie, 
E.R.S. G. Dollond, F.R.S. Lieut. Drummond. Davies 
Gilbert, D.C.L. F.R.S. Rev. R. Greswell, F.R.S. Pro- 
fessor W. R. Hamilton. Hon. C. Harris, F.G.S. G. Harvey, 
E.R.S. Sir John F. W. Herschel, F.R.S. E. Hodgkinson. 
W. Hopkins. John Hymers. Rev. Professor I’. Jarratt. 
Rev. Dr. Lardner, F.R.S. Rev. Dr. Lloyd. Professor Lloyd. 
J. W. Lubbock, Treas. R.S. R. Murphy, F.R.S. ——Phil- 
pott. R. Potter, jun. Professor Powell. Professor Quetelet. 
Professor Rigaud. Rev. Dr. Robinson. Rev. R. Walker, 
E.R.S. W. L. Wharton. C. Wheatstone. Rev. W. Whewell, 
F.R.S. Rev. R. Willis, F.R.S. 


Il. Chemistry, Mineralogy, $c. 


Chairman.—J. Dalton, D.C.L. F.R.S. 

Deputy Chairman.—Rev. Professor Cumming. 

Secretary.—Professor Miller. 7 

Professor Daniell. Professor Daubeny. M. Faraday, 
D.C.L. Rev. W. Vernon Harcourt, F.R.S. W. Snow Harris, 
E.R.S. W. Hatfeild, F.G.S. J. F. W. Johnston, A.M. Rev. 
D. Lardner, LL.D. E.R.S. Rev. B. Lloyd, LL.D. Bods 
Pearsall. Dr. Prout, F.R.S. Professor W. Ritchie. Rev. W. 
Scoresby, F.R.S. W. Sturgeon. Professor Turner. 


Ill. Geology and Geography. 


_—-Chairman.—G. B. Greenough, E.R.S. Pres) GS. 
- Deputy Chairmen.—Rev. Dr. Buckland, F.R.S. G.S. RAE 
Murchison, F.R.S. V.P.G.S. < Ww 
- Seeretaries.—W. Lonsdale, F.G.S. John Phillips, E.R.S. 
G.S. : >. RM 
“Dr. Boase. James Bryce, jun. F.G.S. Joseph Carne, 
F.R.S. G.S. Major Clerke, C.B. F.R.S. M. Dufrénoy. Sir 
Philip Malpas de Grey Egerton, E.R.S. G.S. Dr. Fitton, F.R.S. 
G.S. Rev. J. Hailstone, E.R.S. G.S. Professor Harlan. 
G. Mantell, F.R.S. G.S. Lieut. Murphy, R. E. Marquis of 


xl THIRD REPORT—1833. 


Northampton, F.R.S. G.S. Rev. Professor Sedgwick. Colonel 
Silvertop, F.G.S. W. Smith. John Taylor, F.R.S. Treas. 
G.S. W. C. Trevelyan, F.G.S. H. T. M. Witham, F.G.S. 
Rey. J. Yates, F.G.S. 


IV. Natural History. 


Chairman.—Rev. W. L. P. Garnons, F.L.S. 

Deputy Chairman.—Rev. L. Jenyns, F.L.S. 

Secretaries —C. C. Babington, F.L.S. D. Don, F.L.S. 

Professor Agardh. G. Bentham, Sec. Hort. Soc. F.L.S. 
J. Blackwall, F.L.S. W. J. Burchell. Professor Burnett. 
W. Christy, F.L.S. Allan Cunningham, F.L.S. J. Curtis,F.L.S. 
E. Forster, F.R.S. Treas. L.S. G. T. Fox, F.L.S. J. E. 
Gray, F.R.S. Rev. Professor Henslow. Rey. Dr. Jermyn. 
Rev. W. Kirby, F.R.S. L.S. Professor Lindley. W. Ogilby, 
F.L.S. Dr. J. C. Prichard, F.R.S. J. F. Royle, F.L.S. 
J. Sabine, F.R.S. L.S. P. J. Selby, F.L.S. J. F. Stephens, 
F.L.S. H. Strickland. Colonel Sykes, F.R.S. L.S. Richard 
Taylor, F.L.S..G.S. W. G. Werscow. J. O. Westwood, 
F.L.S. W. Yarrell, F.L.S. 


V. Anatomy, Medicine, Sc. 


Chairman.—Dr. Haviland. 

Deputy Chairman.—Drx. Clark. 

Secretaries.—Dr. Bond. Mr. Paget. ~ 

Dr. Alderson. S. D. Broughton, F.R.S. W. Clift, F.R.S. 
G.S. Dr. Dugard. H. Earle, F.R.S.. Dr. Marshall Hall, 
F.R.S. Dr. Hewett. Dr. Malcavey. Dr. Macartney. Pro- 
fessor Mayo. Dr. Paris, F.R.S. Dr. Prout, F.R.S. Dr. 
Roget, F.R.S. G.S. Dr. Thackeray. Dr. D. Thorp. 


VI. Statistics. 


‘Chairman.—Professor Babbage. 

Secretary.—J. E. Drinkwater, M.A. 

H. Elphinstone, F.R.S. W. Empson, M.A. Earl Fitz- 
william, F.R.S. H. Hallam, F.R.S. E. Halswell, F.R.S. 
Rey. Professor Jones. Sir C. Lemon, Bart. F.R.S. J. W: 
Lubbock, Treas, R.S. Professor Malthus. Capt. Pringle. 
M. Quetelet. Rev. E. Stanley, F.L.S. G.S. Colonel Sykes; 
F.R.S. F.L.S..G.S. Richard Taylor, F.L.S. G.S. di 


TRANSACTIONS. 


Report on the State of Knowledge respecting Mineral Veins. 

- By Joun Taytor, F.R.S., Treasurer of the Geological So- 
ciety and of the British Association for the Advancement of 
Science, Fc. Fe. 


I HAVE found it very difficult to execute the task proposed to 
me in a manner Satisfactory to myself, as we have at this time 
no digested account of the views entertained by geologists of 
the present day upon this interesting subject. The most per- 
fect treatise is that of Werner, which deserves much attention 
for the observation of facts which it displays; but as it was 
written to propound a theory, and as that theory depended 
upon views of the structure of the crust of the earth which 
modern geology has at least thrown much doubt upon, so his 
work cannot be taken as an outline of our present state of 
knowledge. 

Since his time but little has been attempted respecting vein 
formations; and the subject has been, I think, rather neglected 
by geologists, who have advanced other branches of the science 
with extraordinary skill, industry and success. Detached pa- 
pers have, indeed, appeared by English authors, among which 
that on the veins of Cornwall, by Mr. Joseph Carne, holds a 
distinguished place. 

As some proof that the subject of veins has not been much 
attended to, I would remark, that in the Second Series of the 
Transactions of the Geological Society of London, consisting 
now of the first and second volumes complete, and two Parts of 
the third volume, no paper expressly on veins is to be found. 
In the First Series there are two papers, one by the late Mr. 
W. Phillips, giving an outline of facts more generally observed 
with respect to veins in Cornwall, from observations made 
principally in the year 1800. Another is by Dr. Berger, on 
the physical structure of Devon and Cornwall, from observa- 
tions made in 1809. The writeradopts the Wernerian theory, 
and mentions cases which he thinks confirmatory of its truth. 

In the four volumes of the Transactions of the Royal Geo- 
ert: Society of Cornwall, we shall find this subject more 

: B 


2 ' THIRD REPORT—1833. 


attended to, and there are several communications relating to 
it: among the authors are Dr. Boase, Mr. Carne, Dr. Davey, 
Mr. R. W. Fox, and Mr. John Hawkins. One of the papers 
by Mr. Carne is that to which I have before alluded. 

One of the most recent works by foreign writers is that of 
the late M. Schmidt of Siegen. He was an experienced prac- 
tical miner, and wrote chiefly with a view to his art, describing 
the various derangements in mineral veins, and tracing the best 
rules to be observed in pursuing researches in difficult circum- 
stances. He adopts the Wernerian theory of formations, and 
refers to the author of it as the great master of the subject. 

Though no general theory has of late been produced in re- 
gular form, yet with the great attention that has been given to 
geology by so many eminent men, an extended field of observa- 
tion has taken place, leading to a very general change of opi- 
nion on most important points; many conjectures respecting the 
formation of veins have sprung up, and which, when the facts 
are more investigated, and they shall have been recorded and 
classified, may form the groundwork for a more enlarged and 
rational theory, by which their phenomena and structure may 
be explained, and the causes of their formation, the manner of 
filling up, and the circumstances of the varied derangements 
and dislocations, may be traced and be better understood. 

The subject is of threefold importance: first, as it relates to 
science, wherein a better knowledge of veins generally must 
very materially contribute to sound investigations as to the 
structure of the rocks that inclose them: secondly, as it is much 
owing to the pursuit of the minerals which are deposited in veins 
that we have acquired and may yet extend our knowledge of 
geology in general: thirdly, in relation to the question some- 
times proposed as to the usefulness of geological science, the 
most ready answer may be given, if it be considered that this 
inquiry will relate to subjects of practical utility, in which man- 
kind are universally and largely interested. 

Before I proceed to any account of the opinions as to the 
formation of veins, I would offer some definition descriptive of 
their character and structure, that in proceeding with our sub- 
ject we may clearly understand what is meant to be treated on. 

Werner lays it down, ‘‘ That veins are particular mineral re- 
positories, of a flat or tabular shape, which in general traverse 
the strata of mountains, and are filled with mineral matter dif- 
fering more or less from the nature of the rocks in which they 
oceur. 

_ “Veins cross the strata, and have a direction different from 
theirs. Other mineral repositories, such as particular strata or 


$$ 
rr 


REPORT ON MINERAL VEINS. _ 3 


beds, of whatever thickness they occur, have, on the contrary, 
a similar direction with the strata of the rock, and instead of 
crossing, run parallel with them: this forms the characteristic 
difference.” 

. Playfair says: ‘‘ Veins are of various kinds, and may in ge- 
neral be defined, separations in the continuity of a rock, of a 
determinate width, but extending indefinitely in length and 
depth, and filled with mineral substances different from the 
rock itself. The mineral veins, strictly so called, are those filled 
with sparry or crystallized substances, and containing the me- 
tallic ores.” 

Mr. Carne says: “ By a true vein I understand the mineral 
contents of a vertical or inclined fissure, nearly straight, and 
of indefinite length and depth. 'These contents are generally, 
but not always, different from the strata or the rocks which the 
vein intersects. ‘Irue veins have regular walls, and sometimes 
a thin layer of clay between the wall and the vein; small 
branches are also frequently found to diverge from them on 
both sides.” 

Mr. Carne mentiens other veins, which he distinguishes from 
the true ones as being shorter, crooked, and irregular in size ; 
he considers these to have formed in a different manner: but 
this will be discussed hereafter. 

These definitions seem to me to be sufficient for our pur- 

pose; but it may be advantageous here to introduce some 
further description of circumstances connected with veins, and 
to explain the terms’ usually employed to describe them. 
\ Being tabular masses, generally of no great width, any one 
will, whether vertical or inclined, present at its intersection 
with the surface a line nearly straight: this may be from north 
to south, or from east to west, or in any intermediate course. 
This is usually called the direction; by miners frequently the 
run of the vein, or the course of the vein, and is denoted by the 
points of the compass it may cross. 

The length, as Werner states, is indefinite, it being doubtful 
whether any vein has been pursued to a perfect termination. 

The tabular mass, again, may be either vertical to the plane 
of the earth’s surface, or may deviate from this position by in- 
clining to one side or the other of the perpendicular. ‘This 
deviation is called the inclination of the vein; by the Cornish 
miners the underlie. It is measured by the angle made with 
the perpendicular; and as the dip will be to one side of the 
direction, the latter being known, the other is easily expressed. 

The depth to which veins descend into the earth is unknown, 
as well as the length, and for the same reason. 

BR 


4 THIRD REPORT—1833. 


The only dimension we can ascertain is that across from oné 
side to the other of the tabular mass, and is measured from one 
wall to the other, which is the term used in England for the 
cheeks or sides presented by the inclosing rock. This dimen- 
sion is called the width, or frequently the size of the vein. 

The width varies considerably in the same vein. In Europe 
a vein containing ore is considered to be a wide one if it ex- 
ceeds five or six feet. In Mexico the width of veins is gene- 
rally greater. 

In metalliferous veins the deposits of ore are extremely irre- 
gular, forming masses of very diversified form and extent, and 
are separated from each other by intervening masses of vein- 
stone or matrix, either entirely devoid of ore, or more or less 
mixed with it. It is rare to find a vein entirely filled with ore 
in any part. 

In this respect they differ from most beds, where, as in those 
of coal, the whole is a uniform mass. 

The layer of clay, which, as Mr. Carne says, is frequent in 
such veins, will deserve particular notice when we consider 
their general structure and the theories of their formation: 
this is called Saal-bande by the Germans, and jflookan by the 
Cornish miners. 

The clearest idea of a vein will be obtained by imagining a 
crack or fissure in the rocks, running in nearly a straight line, 
extending to great and unknown length and depth, and filled 
with various substances. 

I do not intend by this description to convey any theoretic 
opinion as to the manner in which such fissures may have been 
formed, or as to the mode of their being furnished with their 
present contents. These are subjects on which the greatest 
diversity of opinion has existed in former times, and this diver- 
sity is continued to the present period. It is the main business 
of this Report to state these opinions, and to describe our pre- 
sent state of knowledge of this difficult subject. I feel great 
distrust of my power to do it justice; but I am encouraged by 
the idea that a feeble sketch may induce abler hands to pursue 
the design, and throw more and more light upon this interesting 
branch of geology. 

It would be of little use to go into details of the conjectures 
of ancient authors, or into the mysteries with which this sub- 
ject was enveloped in the age of alchemy. 

The earliest writer who is worthy to be consulted is Agricola 
(whose proper name was Bauer): he resided in the Saxon Erz- 
gebirge, and died in the middle of the sixteenth century.. He 
has been called the father of mineralogy, and of the science of 


REPORT ON MINERAL VEINS. 5 


mining. He had the rare merit of emerging from the mists and 
clouds of an absurd school of philosophy, which had till then 
obscured the objects which it pretended to illustrate; and he 
first subjected them to inquiries prompted by sound reason and 
just views of nature. 

His writings were numerous, and’in such pure Latin that 
they are said to be entitled to a place among the classics. He 
treats of veins in a work called Bermannus, but more particularly 
in the third book of his great work De Re Metallica. 

Agricola being held to be the first who has written anything 
certain onthe formation of veins, and his theory of the manner 
of their being filled up having, with some modifications, been 
for a long period generally received, and in part even adopted. 
by Werner, I shall commence from his time the notice of the 
opinions promulgated by various writers antecedent to Werner 
and Hutton. 

Some have maintained, That veins and their branchings are 
to be considered as the branches and twigs of an immense trunk 
which exists in the interior of the globe : 

That from the bowels of the earth metallic particles issued 
forth in the form of vapours and exhalations through the rents, 
in the same manner as sap rises and circulates in vegetables. 

This speculation was proposed by Von Oppel, captain-ge- 

neral of the Saxon mines, who wrote in 1749. He was a skilful 
miner and an accurate observer; and it is singular that this opi- 
nion is not consistent with most that he has elsewhere said on 
the subject, which generally rather agreed with the views which 
were adopted by Werner and others. 
- Henkel, who wrote in the early part of the seventeenth cen- 
tury, and who has been held to be the father of mineralogical 
chemistry, first attributed the formation of the contents of veins 
to peculiar exhalations: he supposed the basis of each metal 
and mineral to have existed in the substance of the rock, and 
to have been developed by a peculiar process of nature. 

Becher about the same time supported very similar views. 
Stahl, who commented upon the writings of Becher, had ad- 
vanced a somewhat similar opinion; but he afterwards rejected 
this theory, and considered veins, as well as the substances of 
which they are composed, as having been formed at the same 
time with the earth itself. 
~ Zimmerman, chief commissioner of mines in Saxony, who 
died in 1747, had an idea that the variety of minerals contained 
in veins had been produced by a transformation of the sub- 
stance of the rock. 

_ Charpentier, in 1778, supported nearly similar opinions, and 


6 THIRD REPORT—1833. 


combated strenuously against the theory which considers veins 
to have been rents that were afterwards filled up by different 
mineral substances. 

This is the theory, however, which, from the time of Agricola 
to the present day, has been most generally received, namely, 
that veins were fissures which have been since filled up by de- 
grees with mineral matters. 

The causes of such fissures, and the mode of their contents 
being deposited, have been variously stated, and have given rise 
to much conjecture; and allowing for these differences, the main 
proposition has been supported by many writers. Among these 
I would name Agricola; Balthazar Roésler, an eminent miner of 
Freyberg, who died in 1673; Hoffman, a commissioner of mines 
at the same place, in 1746; Von Oppel, before mentioned, who, 
though he had indulged in other speculations, distinctly lays 
down in his Introduction to Subterranean Geometry, (Dres- 
den, 1749,) that veins were formerly fissures, open in their su- 
perior part, and that they traverse and intersect the strata. 

Bergman entertained opinions very similar, which were also 
supported by Delius, an author on mining, of considerable ce- 
lebrity, who wrote about 1770. 

Gerhard, in his Essay on the History of the Mineral King- 
dom, (Berlin, 1781,) gives a collection of interesting facts con- 
cerning veins, and considers them to have originally been rents, 
which were afterwards filled up with mineral substances. 

To this list may be added Lasius, in his Observations on the 
Mountains of the Hartz,in 1787; and Linnzus is stated ‘‘to have 
wondered at the nature of that force which split the rocks into 
those cracks; and adds, that probably the cause is very familiar, 
—that they were formed moist, and cracked in drying *.” 

In England we have testimony to the same opinion from 
Dr. Pryce, who wrote his Mineralogia Cornubiensis in 1778, 
He says, ‘‘ When solid bodies were separated from fluid, certain 
cracks, chinks and fissures in various directions were formed, 
and as the matter of each stratum became more compact and 
dense by the desertion of moisture, each stratum within itself 
had its fissures likewise, which, for the most part, being in- 
fluenced by peculiar distinct laws, were either perpendicular, 
oblique,” &c. 

He afterwards adds, that those very fissures are the wombs 
or receptacles of all metals, and most minerals. He assigns the 
derangements of veins to the effect of fracture by violence, and 
quotes subsidence as one of the probable causes of such dislo- 
cations. He says there can be no doubt that many alterations 

* Hill. 


REPORT ON MINERAL VEINS. 7 


have happened to various parts of the earth before, at, and 
after the Flood, from inundations, earthquakes, and the dis- 
solvent powers of subterranean fire and water, which variety of 
causes and circumstances must infallibly have produced many 
irregularities in the disposition and situation of circumjacent 
strata and lodes*. 

He describes twelve kinds of lodes or veins in Cornwall, 
naming them from their chief contents. But the most remark~- 
able observation of Dr. Pryce is respecting the relative age of 
veins, of which he seems to have given the first intimation. 
Werner, long after, states this as a discovery of his own, and 
as an essential part of his theory. His translator, however, 
(Dr. Anderson,) does Pryce justice, and remarks that his ob- 
servations must have been unknown to Werner, who showed 
much anxiety in all cases to confer on every writer the merit 
which was due to him. 

Dr. Anderson quotes the passage as one of much importance. 

‘‘ Because the cross gossans or cross flookans run through 
all veins of opposite directions, without the least interruption 
from them, but, on the contrary, do apparently disjoint and 
dislocate all of them, it seems reasonable to conclude, that the 
east and west veins were antecedent to cross veins, and that 
some great event, long after the Creation, occasioned those 
transverse clefts and openings. But how or when this should 
come to pass, we cannot presume to form any adequate idea.” 

Kirwan supports the doctrine that some veins were originally 
open, as appears from the rounded stones and petrifactions 
found in them. Thus, in the granitic mountain of Pangel in 
Silesia there is a-vein filled with globular basalt. So also in 
veins of wacken, in Joachimstahl in Bohemia, trees and their 
branches have been found. 

But he deems it improbable that all veins were originally 
open to day, and filled from above. He inclines to the theory 
of veins being filled by the percolation of solutions of the me- 
tals and earths. 

Having now taken a cursory view of the opinions held before 
Werner published his Theory of Veins, and seen something of 
the state of knowledge relating to this subject, we may bear 
in mind the materials which he had to work with, and take 
into account his well-known views as to the origin of rocks from 
aqueous deposition, and we shall comprehend the system which 
he developed, with respect to veins, in the only work, I believe, 
which proceeded from his own hand, and which was published 


* ‘Lode’ is the term used in Cornwall for a metalliferous vein. 
+ Mineralogia Cornubiensis, p. 101. . 


8 THIRD REPORT—1833. 


at Freyberg in 1791. Werner adopts, in the first place, the 
proposition that the spaces now occupied by veins were origi- 
nally rents formed in the substance of rocks, and states that 
this is not a new opinion. 

He claims the merit of having ascertained in a more positive 
manner the causes which have produced these rents, and of 
having brought forward better proofs of it than had formerly 
been done. 

He admits that rents may be produced by many different 
causes, but he assigns the greater part to subsidence. He lays 
it down, that when the mass of materials of which the rocks 
were formed by precipitation in the humid way, and which was 
at first soft and moveable, began to sink and dry, fissures must 
of necessity have been formed, chiefly in those places where 
mountain chains and high land existed. He adds, that rents 
and fissures are still forming from time to time in mountains 
which haye a close resemblance to those spaces now occupied 
by veins, and that this happens in rainy seasons and from 
earthquakes. 

He adduces as a proof of his assertions, that veins, in respect 
of their form, situation and position, bear a strong resemblance 
to rents and fissures which are formed in rocks and in the 
earth; that is to say, both have the same tabular figure, and 
the deviations which they make from their general direction 
are few in number and very inconsiderable; and he remarks, 
that all the veins of a mining district, more particularly when 
they are of the same formation, have a similar direction, which 
shows them to have been produced by the same general cause. 

But what Werner claimed as altogether new, and what he 
challenges as his own particular discovery is, 

1. To have determined and described in a more particular 
manner the internal structure of veins, as well as the formation 
of the different substances of which they are composed, and to 
have settled the relative age of each. 

2. To have given the most accurate observations and most 
perfect knowledge of the meetings and intersections of veins, 
and to have made these observations subservient to the deter- 
mining their relative ages. 

3. 'To haye determined the different vein formations, parti- 
cularly metalliferous veins, as well as their age. 

4, ‘To have been the first who entertained the idea that the 
spaces which veins occupy were filled by precipitations from 
the solutions, which at the same time formed by other precipi- 
tations the beds of mountains, and to haye furnished proofs of 
this: and, 


REPORT ON MINERAL VEINS. 9 


5. To have determined the essential differences that are 
found between the structure of veins and that of beds. 

Werner illustrates his propositions by many observations, 
which his intimate acquaintance with the extensive mining di- 
stricts in which he was engaged gave him the power of observin 
and recording ; and it must be conceded, at least, that his state- 
ment of facts, and his arrangement of them, give him a manifest 
superiority over most writers upon this subject. Every one 
who has had opportunity to see much of these storehouses of 
nature will be struck with the accuracy of most of his descrip- 
tions, whether they admit the theory by which they are ex- 
plained, or not. 

He allows that the enrichment of veins, or their being filled 
with ores or metals, may have taken place by, 

1, a. A particular filling up from above. 

6. By particular internal canals. 
ec. By infiltration across the mass of the vein. 

2. A metallic vein may be increased by the junction of a new 
metalliferous vein. 

3, Though rarely, the richness of a vein may be the effect of 
an elective attraction or affinity of the neighbouring rock. 

The mode assigned by Werner for the formation of the 
spaces now occupied by veins is still further demonstrated, in 
his opinion, by the relation which veins have to one another ; as, 

_ Their intersecting one another. 
Their shifting one another. 
Their splitting one another into branches. 
Their joining and accompanying one another. 
Their cutting off one another. 

All these peculiarities, he remarks, are produced by the ef- 
fects of a new fissure upon one that is older. 

Subsidence having been the cause of fissures he thinks is 
proved by the difference in the level in the parts of the same 
stratum or bed in which a vein is inclosed; and this throwing 
up or down, as the miners term it, bears a proportion to the 
size of the vein. 

The interior structure of many veins is quoted to show that 
the fissures had been originally open, and which had been af- 
terwards filled by degrees. 

Such veins are composed of beds, arranged in a direction pa- 
rallel to their sides; their crystallizations are supposed to show 
these beds to have been deposited successively on each other, 
and that those next the walls have been first formed. A cir- 
cumstance much relied on, also, is the existence of rolled masses 
or water-borne stones, fragments of the adjacent rock, some- 


10 THIRD REPORT—1833. 


times forming a breccia, remains or impressions of organic bo- 
dies, coal and rock salt substances of recent formation, and other 
matters, which should appear to have come in from above. 

This theory obtained considerable attention, and was very 
generally adopted from the time of its being made known; and 
it has, I believe, many adherents at this day, particularly among 
miners or those who have much opportunity of actual observa- 
tion. 

Hutton’s Theory of the Earth was published afterwards, in 
1795; and as his views regarding the operations employed in 
the formation of the structure of the rocks differed entirely 
from those who assigned to them an aqueous origin, so it will 
readily be supposed that he would promulgate a new explana- 
tion of the formation of veins. 

According to Playfair, this theory embraced the following 
propositions :— : 

It allowed that veins are of a formation subsequent to the 
hardening and consolidation of the strata which they traverse, 
and that the crystallized and sparry structure of the substances 
contained in them shows that these substances must have con- 
creted from a fluid state. 

It assumes that this fluidity was simple like that of fusion by 
heat, and not compound like that of solution in a menstruum. 

It is inferred that this is so from the acknowledged insolu- 
bility of the substances that fill the veins in any one menstruum, 
and from the total disappearance of the solvent, if there was 
any, it being argued that nothing but heat could have escaped 
from the cavities. 

It is further maintained, that as the metals generally appear 
in veins in the form of sulphurets, the combination to which 
their composition is owing could only have taken place by the 
action of heat. And, further, that metals being also found na~- 
tive, to suppose that they could have been precipitated pure 
and uncombined from any menstruum, is to trespass against all 
analogy, and to maintain a physical impossibility. 

It is therefore inferred, that the materials which fill the mi- 
neral veins were melted by heat, and forcibly injected in that 
state into the clefts and fissures of the strata. 

The fissures must have arisen, not merely from the shrinking 
of the strata while they acquired hardness and solidity, but 
from the violence done to them when they were heaved up and 
elevated in the manner which the theory has laid down. 

Slips or heaves of veins, and of the strata inclosing them, 
are to be explained from the same violence which has been 
exerted. 


ee 


REPORT ON MINERAL VEINS. 11 


It is admitted as interesting to remark, that in the midst. of 
the signs of disturbance which prevail in the bowels of the 
earth, there reigns a.certain symmetry and order, which indi- 
cates a force of incredible magnitude, but slow and gradual in 
its effects. 

Further, that as a long period was required for the elevation 
of the strata, the rents made in them are not all of the same 
date, nor the veins all of the same formation. <A vein that forces 
the other out of its place, and preserves its own direction, is 
evidently the more recent of the two. 

The parallel coats lining the walls or sides of the vein, which 
are attributed by Werner and others to aqueous deposition, are 
ascribed to successive injections of melted matter. 

Veins have been considered as traversing only the stratified 
parts of the globe. They do, however, occasionally intersect 
the unstratified parts, particularly the granite; the same vein 
often continuing its course across rocks of both kinds without 
suffering material change. 

It is asserted that all the countries most remarkable for their 
mines are primary, and that Derbyshire is the most considera- 
ble exception to this rule that is known. 

This preference which the metals appear to give to the pri- 
mary strata, is considered as consistent with Dr. Hutton’s 
theory ; and particularly as these strata, being the lowest, have 
also the most direct communication with those regions from 
which the mineral veins derive all their riches. 

In arguing further upon this theory, it is assumed that no- 
thing of the substances which fill the veins is to be found any- 
where at the surface; and that, contrary to the allegation of 
some that mineral veins are less rich as they go further down, 
it is stated that this is not generally so, and that the mines in 
Derbyshire and Cornwall are richest in depth, as they would 
be if filled with melted matter from below. 

Again, it is said that if veins were filled from above, and by 
water, the materials ought to be disposed in horizontal layers 
across the vein; and that this opinion is sufficiently refuted by 
the fact that rarely any metallic ore is found out of the vein, or 
in the rock on either side of it, and least of all where the vein 
is richest. i 
_ The foregoing seem to be the most important. allegations in 
support of the Huttonian theory; and I have taken them nearly 
in the order in which they are given in Professor Playfair’s il- 
lustrations of this celebrated system. 

There is yet another doctrine regarding the formation of 
veins, which, though it is not of modern date, and has had but 


12 THIRD REPORT—1833. 


few supporters among writers upon the subject, has yet claims to 
be considered, and particularly as it has of late been urged upon 
our notice, and by some whose observations have been made in 
districts where veins of various order are abundant. . 

This theory is, in short, That veins were formed at the same 
time with the rocks themselves; that the whole was a contem- 
poraneous creation; and that there have been neither fissures 
subsequent to the consolidation of the mass, nor filling up from 
above or below, or disturbances to produce the heaves or shifts 
which we see. 

When this hypothesis was first proposed I do not know, but 
that it was long since we may infer, as Agricola regards the 
opinion which supposes veins such as we now see them to have 
been formed at the same time with our globe, to be at variance 
with fact, and he calls it the opinion of the vulgar. The same 
hypothesis was indeed supported by Stahl; but he seems to 
have adopted it rather on account of the difficulties attendant 
on any other explanation that had been proposed, than for any 
good reason that he had to give. 

Such are, however, but assertions, to be received with doubt 
by any one who inquires freely and without prejudice. Partial 
evidence may appear for some such formations; but it is another 
affair to attribute all veins to such an origin, and thus to sweep 
away at once the difficulty of explaining many complicated ap- 
pearances. 

The doctrine of a contemporaneous formation of veins has 
lately found an advocate in Dr. Boase, in his paper on the 
geology of Cornwall. After commenting on the division into 
different orders, which Mr. Carne had indicated as to veins, 
according to certain appearances in their direction and the 
character of the substances with which they are filled, he says 
he cannot detect any characters which are not common to all 
the Cornish veins; and since some of them are generally ac- 
knowledged to be contemporaneous with the rock, he concludes 
that they have all the same origin. . 

Dr. Boase, however, candidly sets out by stating that he had 
purposely refrained from making inquiries at the mines con- 
cerning the phenomena of veins, and that his experience is 
therefore principally confined to those which occur in cliffs, 
quarries, and natural sections that are exposed to open view. 

Lest this admission should create surprise, he remarks that 
such sources of information are invaluable as the only ones 
easily available to exercise the senses on the nature of veins; 
for, unless to those much accustomed to descend into mines, 
they may as well be visited blindfold. 


REPORT ON MINERAL VEINS. 13 


He remarks, however, as to the veins of Cornwall, that their 
great irregularity in size and in form, their frequent ramifica- 
tions, their similarity of composition and intimate connexion 
with the rocks which they traverse, and, above all, the large 
masses of slate which they envelop, are all circumstances to 
disprove their origin from fissures, and to support their con- 
temporaneous origin. 

Dr. Boase suggests that veins follow the arrangement of the 
joints of the rocks, and that it may thus be explained why the 
different series of veins cross each other, and why the veins of 
each series are respectively parallel. 

And he thinks that thus we may suppose how veins which 
are crossed may seem to abut or terminate against those that 
are opposed thereto; having, when in the same line, that pecu- 
liar appearance that has been attributed to intersection, and 
the appearance of being heaved when on the opposite sides of 
the cross vein, they are not on the same line, but occur in the 
parallel joints of distant layers. . 

The latter occurrence, he remarks, although very common, 
is not however universal ; for, in some instances, the part of the 
vein supposed to have been intersected has never been found. 

As Mr. Carne had observed, that when contemporaneous 
veins meet each other in a cross direction, they do not exhibit 
the heaves and interruptions of true veins, but usually unite. 

Dr. Boase says that this statement is opposed to his obser- 
vations, and that the phenomenon of intersection is common to 
all kinds of veins. Further, he expresses a doubt whether 
heaves in veins are not after all rather apparent than real, but 
explains that he does not mean to assert that they do not ex- 
hibit these phenomena, but that this arrangement, as in the 
case of small veins, only gives the appearance of being moved 
from the original positions. 

I have now stated the opinions which, as far as I know, have 
been generally received on the subject of the formation of veins, 
from which it will appear that there are three leading hypo- 
theses. 

Ist. That which supposes them to have been open fissures, 
caused. by disruption, and occasioned principally by subsidence 
of parts of the rocks, which fissures were afterwards filled up 
with various matters by deposits from aqueous solution, chiefly 
from above. 

» Modifications of this theory are, That such rents in the earth 
may have been caused in other ways, such as earthquakes, or 
certain great convulsions, as well as by subsidence: 

That they may have been filled by the infiltration of solu- 


14 THIRD REPORT—1833. 


tions, which deposited the substances with which they were 
charged in the veins, or by the process of sublimation from 
below. 

The second theory allows that veins were formed subse- 
quently to the consolidation of the rocks; but the cause prin- 
cipally assigned for such fissures is the violence done to the 
strata by the elevation or upheaving of other rocks from 
below. 

And it is an essential part of this theory that the materials 
which fill the veins were forcibly injected upwards in a state of 
complete fusion by heat. 

The third theory is that denying any subsequent processes 
which might either cause rents and fissures, or might fill them 
with matter which differs from the rocks which inclose them: 
the whole formation was contemporaneous with the rocks them- 
selves, the mineral substances which we find in veins having 
separated and arranged themselves into the forms in which we 
now see them to exist. 

The advocates of these theories have each zealously asserted 
the truth of his own system, and refused to admit of causes or 
explanations which appeared to militate against it; and thus a 
boundary has been set, as it appears to me, to that freedom of 
inquiry which is so desirable in such cases, and a limit drawn 
round the reasoning faculties of man upon evidence which may 
come before him. 

It will appear, from what has already been said, that veins 
have very different characters and appearances ; and this might 
be made more clear, if it were here the proper place to enlarge 
upon the subject and point out the distinctions.’ For our pur- 
pose, however, it may be sufficient to remark upon two or three 
principal varieties. First, then, are those which have beyond 
all comparison been most explored and examined, on account 
of the stores which they contain,—the metalliferous veins. As 
these have been penetrated in all directions to the greatest ex- 
tent that human power and ingenuity have been able to effect, 
so their structure is better known and more accurately ob- 
served. 

Similar to these, and occurring with them, and therefore well 
known, are others, which, though barren of metals, are yet 
often called true veins; and these, as well as the first, come 
pretty fully under the view of the miner. 

Next there are veins, regular in their structure to a great 
extent, filled with matter which has the character of being de- 
rived from igneous origin, such as are usually called dykes of 
trap, whinstone, &c., &c.; to which would be added by most 


REPORT ON MINERAL VEINS. 15 


geologists of the present day, the veins of granite, porphyry, 
quartz, &c. , 

Some of these have been examined below the surface, where 
they pass through coal-fields, or other deposits of useful mine- 
rals, but containing in themselves nothing to reward the toil of 
exploring them: little has been seen of their contents and con- 
figuration, and our knowledge of them is more limited. 

Lastly, there are tortuous and irregular veins or ramifications 
in most rocks, extending to limited distances, as far as our ob- 
servations permit us to judge, seldom offering a valuable return 
for any effort to explore them, and of which, therefore, our 
knowledge is but superficial. 

Such veins, according to Mr. Carne, have been usually di- 
stinguished from true veins by their shortness, crookedness, 
and, irregularity of size, as well as by the similarity of the con- 
stituent parts of the substances which they contain to those of 
the adjoining rocks, with which they are generally so closely 
connected as to appear a part of the same mass. Two other 
distinctive marks may be added; one is, that when they cross 
they do not exhibit the heaves of true veins, but usually unite; 
the other is, that when there is an apparent heave it is easy to 
perceive that what appear to be separate parts of the same vein 
are different veins terminating at the cross vein. 

Such may be, probably, of contemporaneous formation; and 
there may he deposits of ore also which it would be difficult to 
refer the structure of to any other hypothesis, particularly such 
as contain ores so intimately mixed with the rocks as to form a 
constituent part of them. 

L would suggest, that if from any one of these classes we were 
to form a judgement as to the whole, error would probably be 
the consequence, or, at any rate, the view would be a narrow 
and contracted one, and our decisions would be defective in 
many important respects. 

_ To have conducted the inquiry in this manner seems to me 
to have been the error in many who have preceded us in for- 
warding the state of knowledge on vein formations. Nor do I 
mean to detract from the great merit of many of them on this 
account; the field of observation is too vast to become fully 
acquainted with it; it extends over the most rugged parts of 
the earth’s surface, and its boundaries are not reached in the 
deep recesses of its bowels. It is no wonder that in the earlier 
stages of such inquiries men should be strongly impressed with 
what lay immediately before them, and should view with dis- 
trust what they might only learn from description. 

- Such impressions may be traced in looking at the authors of 


16 THIRD REPORT—1833. 


the systems which we have reviewed. Werner expressly tells 
us, that we are indebted to miners for the theories which he 
deemed most worthy of acceptation, and he names as such 
Agricola, Résler, Henkel, Hoffman, Von Oppel, Charpentier, 
and Trebra. We may add his own name and that of Dr. Pryce, 
in our own country, as intimately acquainted with mining. Now 
all such men would be more acquainted with the metalliferous 
veins and such as accompany them; and from these they would 
derive much evidence in favour of the opinions which they ad- 
vocated; at least, partaking, as I probably do, in the same pre- 
judices, so it would appear to me, if by the labour of other 
inquirers I did not know that there were other facts requiring 
a different explanation. 

Again, Dr. Hutton and his commentators had largely ob- 
served veins which may fairly be attributed to injection; they 
had found dykes of trap passing through coal-beds, and con- 
verting them into cinder. Such evidence of the effects of heat 
and of a filling up by matter in fusion is not to be resisted ; but 
when we look at what is said of the metalliferous veins by some 
of the writers on this side of the question, we observe great 
want of practical knowledge and many errors, arising out of 
the attempt to make all bend to a single method of solving the 
problem. 

For the third hypothesis of contemporaneous formation there 
is this to be said,—that some veins exist which seem to admit of 
no other explanation; and that this being allowed to such as 
will have but one theory, this is at once the easiest, because it 
gets rid of many difficulties without further trouble; but we 
can hardly be satisfied to adopt it as universal upon experience 
that has been principally confined to sections in quarries and 
in cliffs, or to such as are exposed to open view. 

Our present state of knowledge as to the formation of veins 
should therefore, in my opinion, be allowed to admit that most 
of the causes which have been stated have operated at various 
periods and through a long succession of time, some prevailing 
at one epoch, and some at another, modified by circumstances 
which we can but imperfectly comprehend or explain. 

In this view we may allow of a classification of veins accord- 
ing to their probable mode of origin; and such a classification 
has been thought of by some of our ablest geologists of the 
present day, and was indeed propounded in one of our sections 
at Oxford last year by our present learned President, who ex- 
pressed his opinion that there were three different sets of veins: 
—l. Those which have been plainly mere fissures or cracks, 
and which have been subsequently filled; 2. Those of injec- 


REPORT ON MINERAL VEINS. 17 


tion; 3. The contemporaneous veins, which might more aptly 
be termed veins of segregation. 

~ Here I might close this Report, which is’ already much too 
tedious, were it not that I may be expected to notice briefly 
some of the facts adduced by the advocates of the respective 
theories, and, by comparing them, show how far they are enti- 
tled to be considered as objections on one side, or as proofs 
on the other, with the confidence which has been assigned to 
them. f 

Werner and Hutton agree in allowing that rents took place 
subsequently to the consolidation of the rocks, or at the time of 
their consolidation. They differ as to the cause of the rents: 
Werner ascribes it to subsidence, or to sinking and shrinking 
of the solid materials of our globe ; Hutton, to violent upheaving 
of matter from below, breaking up the superinjacent strata. 

Either of these causes seems adequate to the effect, and in 
either case corresponding strata might be found having different 
levels of position on opposite sides of the fissure, as is constantly 
the case. This by miners in the North of England is called the 
throw of the vein; and it is clear that one side may as well be 
thrown up as the other thrown down. Mr. Fox and Dr. Boase 
urge the great irregularity of the width of veins, the difficulty 
of supposing the sides to be supported, and some other objec- 
tions to the hypothesis of open fissures. Irregularity of width 
is but a comparative term; and taking into consideration the 
immense extent of their dimensions in length and depth, it 
amounts in my opinion to but little. 

~The other objections are in a great degree anticipated and 
answered by Werner; and, after all, difficulties can hardly be 
urged against the positive testimony of some veins having been 
open, which is afforded by the substances found in them, such 
as rolled pebbles, petrifactions, &c. 

The parallelism of veins of one formation is insisted upon by 
‘Werner as a proof of his view of the subject ; and I confess that 
there appears to me to be considerable difficulty in explaining 
this, on the supposition that fissures were caused by a mass 
protruded upwards through strata already formed. From such 
a cause one should expect not to have a number of cracks pa- 
rallel to each other, but rather to see them radiating from the 
centre of the greatest disturbance. In the metalliferous veins 
‘we may certainly observe this parallelism to a great extent. 
Mr. Carne has beautifully illustrated this in Cornwall, and has 
shown how the productive veins generally have an east and 
west course ; how, as they differ in their contents, they differ 
a _ their direction, each class being, however, parallel in 

. ic 


18 THIRD REPORT—1833. 


itself; and how these facts illustrate relative ages of forma- 
tion. 

This tendency to an east and west direction of the metallife- 
rous veins may be observed not only in Cornwall but in the 
stratified parts of England, in the mining districts of Europe, 
and in the range of the great veins of Mexico. 

Mr. Robert Fox, having discovered galvanic action to ensue 
by the connexion of an apparatus, constructed to detect it, with 
portions of metalliferous veins, suggests whether some analogies 
may not be traced between electro-magnetic currents and the 
directions of veins: nothing upon which any hypothesis can be 
built seems, however, as yet to have been proposed ; and it may 
be doubted whether, when this test is applied to masses of ore, 
the experiment is not liable to many objections. A principal 
one seems to be, that by the very act by which we gain access 
to the vein, we lay it open to atmospheric action, and conse- 
quently to decomposition. Chemical agency commences, and 
with it, very naturally, galvanic influences are excited. 

Veins containing ores little subject to decomposition have, I 
apprehend, been found to give little or no indications of this 
nature. 

It may, however, be that this general direction of metallife- 
rous veins may not obtain as to veins of injection; and in that 
ease we shall have additional reason to admit more causes than 
one to have been in operation. 'This is a matter deserving ex- 
tensive observation. 

Other veins have been stated to cross the metalliferous veins: 
they are generally filled in a different manner. If they contain 
any ores, they are frequently of different metals from those in 
the former. ‘They pass through or traverse the other veins, 
cutting them through, and suffering a disturbance to take place 
in their linear direction, or what the miners significantly term 
a heave. 

This fact is relied upon as proving that veins are of different 
ages, as first asserted by Pryce, much insisted upon by Werner, 
and allowed by Hutton and Playfair. 

Those who dispute this inference, therefore, are the advo- 
cates for the sole operation of contemporaneous causes: they 
object that rules which have been proposed for ascertaining 
the exact tendency of such disturbances having béen found to 
be subject to exceptions, the proof of dislocation is wanting, or 
that dislocation has taken place without motion. The latter 
proposition, at any rate, appears to me to be very difficult to 
understand ; and I think if any part of this intricate subject 1s 
clear and intelligible, it is that the relative age of veins is made 


REPORT ON MINERAL VEINS. 19 


out by these facts, even although we may not yet be able to 

apply rules for every case,—a subject which has been con- 
sidered as highly important in its practical application to the 
art of mining. 

The greatest controversy, however, relates to the mode in 
which ‘veins have been filled. Here, again, we must remark, 
how the opinions of observers have been influenced by the facts 
coming under their immediate observation. 

Werner, and the mining authors on whom he relies, drew 
their inferences from metalliferous veins. Hutton and his fol- 
lowers regarded chiefly those of another class; and this great 
author and his commentator Professor Playfair were evidently 
ill informed as to metalliferous veins. 

That certain veins have been filled by injection from below, 
and with matter in igneous fusion, seems to be rendered certain 
by evidence, which is clearer than most we possess on such sub- 
jects, and must be admitted at once. ‘Thus, when we see a 
trap dyke traversing a bed of coal and charring the combusti- 
ble matter, and affecting the rock itself with visible effects of 
great heat, we must assent to the cause assigned ; and when we 
see matter of igneous origin not only filling the veins, but over- 
flowing on the surface, or insinuating itself between adjacent 
beds, the case is plainer than most that occur in geological re- 
search. 

But though one class of theorists have proposed this as the 
universal cause of the filling up of veins, ought we to admit this 
to be true, when we find so many in which no similar appear- 
ances are to be traced? j 

Why, for instance, if the ores were forced from below, did 
the power which injected them just limit itself to raising them _ 
within a short distance of the surface,—for where shall we find 
an instance of their being protruded above it? 

If the metallic contents of veins were injected from below, we 
ought to be able to trace something like the direction of the 
currents in which the matter flowed; we ought to see some 
continuity in the operation, and some connexion between the 
masses of ore which occur in veins; whereas the contrary of 
each is notoriously evident to every observer. 

__ It would seem also to be very probable, if the enrichment was 
from below, and the matter was forced in from those regions 
whence their treasures are supposed to be derived, that by a 
nearer approach to the depths of the earth we should find the 

riches more abundant. 
Professor Playfair admits this inference, and disposes of the 
difficulty by arguing that it is so; and says, that though mines 
cz ; 


20 THIRD REPORT—1833. 


in Mexico and Peru are said to be less rich as they descend 
further, those of Derbyshire and Cornwall exhibit the very 
contrary. 

He is unfortunate in this allegation, and the facts will not 
bear him out, as every one of common experience must know ; 
and thus, as I have before observed, we have hypotheses sup- 
ported by a limited knowledge of the facts. 

The theory of the filling up of veins by precipitation from 
aqueous solutions, is defective in not being able to show what 
menstruum could render such substances soluble in water ; 
and this difficulty must remain an important one, unless en- 
larged knowledge should hereafter afford the means of ex- 
plaining it. 

But when we are told that the supposition is absurd, that 
water cannot arrange its deposits in planes highly inclined, that 
no appearance of stalactites is to be found in veins, nor can 
we see in them any substance like those on the earth’s surface, 
which aqueous action has removed,—it must be recollected that 
we know silex is soluble in water at high temperatures; that 
crystals do arrange themselves on the sides of vessels in planes 
highly inclined; that stalactites of chalcedony, of quartz, and of 
iron pyrites, have been found deep in the veins in Cornwall, and 
that much of the substance of the surrounding rocks, and such 
as we see on the surface, and adjoining and inclosing the veins 
themselves, is found in them, occupying much of their space, 
previously having been worn down into fragments, into loose 
sand, and into clay or mud, the latter of which is so common 
that, as I have before observed, it is relied on by the miner as 
a distinguishing character of regular veins*. 

The action of water may, I think, be as fairly assumed as 
that of fire ; and we may consider what their joint powers might 
be, when compelled, as it were, to act together, under circum- 
stances that immense pressure might produce. 

But in examining the contents of veins, we are, I think, likely 
to be struck, not only by the appearance of a complication of 
causes, but by evidence of their succession, admitting the pro- 
bability not only of different agents having been employed, but 
of their having done their work separately as well as conjointly, 


* Mr. Weaver describes the contents of the great vein of Bolafios in Mexico 
thus: ‘ The chief mass of this vein may be said to consist of the detritus of the 
adjacent rocks, more or less consolidated, and generally hard; nay, in places, 
it is actually composed of a conglomerate. Proper vein-stones, such as fiuor 
or cale spar, are, comparatively speaking, casualties. In this basis the finer 
delicate silver ores and native silver are dispersed, in common with the harder 


and coarser ores of blende, iron, and copper, besides lead ores.” 


REPORT ON MINERAL VEINS. 21 


—of having operated at different periods, and of one having 
produced effects for which another was inadequate. 

As we cannot easily conceive how the metallic ores can have 
been deposited from solution in water, and appearances are 
much against their having been injected in a state of fusion, 
there is another supposition which, though not free from diffi- 
culties, has yet probability enough in its favour to have gained 
it many supporters,—which is, that these and some other sub- 
stances have been raised from below by sublimation. This is 
not a new opinion, for though the older writers expressed it in 
an indistinct manner, and spoke of metallic vapours and exha- 
lations,—and thus we shall find it proposed by Becher, Stahl, 
Henkel, and others,—yet their meaning evidently was, that sub- 
stances had been volatilized by heat, and assumed their places 
in veins by condensation, or by combining with other materials. 

We know for certain that some of the metallic sulphurets 
may be so volatilized, and will reassume their form and be 
produced in a crystallized state ; and so far nothing is assumed 
beyond our knowledge: but as we find these sulphurets, which 
compose by far the greater part of the metallic contents of veins, 
in insulated masses, surrounded on all sides by other substances, 
which we can hardly conjecture to have been sublimed, we en- 
counter much difficulty in explaining how the process can have 
taken place; and it becomes even more difficult when we see 
how very much these different classes of substances are incor- 
porated, and how they completely, in most instances, envelop 
and inclose each other. 

The hypothesis of filling up by sublimation would also seem 
to require that the deepest portions of veins should be richer, 
especially considering the very small extent to which after all 
they have been perforated; but yet, shallow as our workings 
into the earth really have been, there is much appearance of 
their having in many instances gone below the richest deposits 
of the metals. 

This seems to have been the case in some of the deepest 
mines in Mexico, and in several in our own country. It is im- 
possible, indeed, to say that greater deposits may not exist still 
lower down; and though veins have not been traced to their 
termination, they. have in many instances been pursued until 
the indications of metallic produce have become faint and hope- 
less. And these unfavourable appearances have increased very 
commonly with increasing depth, which is as much, perhaps, as 
we are likely to know about it, as the operations of the miner 
are thus arrested, and the inducement to further experiment is 
taken away. 


22 THIRD REPORT—1833. 


The agency of sublimation has lately been advocated by Pro- 
fessor Necker of Geneva, in a paper read before the Geological 
Society of London*; and he has extended an ingenious hypo- 
thesis of Dr. Boué, who would bring under a general law the 
relation of metalliferous veins and deposits to those crystalline 
rocks which, by the majority of modern geologists, are consi- 
dered to have been produced by fire; and thus to lead to the 
inference that the metals were deposited in the former by sub- 
limation from the latter. 

M. Necker inquires, 1. Whether there is near each of the 
known metalliferous deposits any unstratified rock ? 

2. If none is to be found in the immediate vicinity, is there 
no evidence which would lead to the belief that an unstratified 
rock may extend under the metalliferous district ? 

3. Do there exist metalliferous deposits entirely disconnected 
from unstratified rocks ? 

Professor Necker answers these questions by showing that in 
various countries there are such relations as he supposes, and 
admits, in reply to the last, that there are cases where the depo- 
sits seem to be unconnected with any trace of unstratified rock. 

If metalliferous deposits are commonly in crystalline rocks 
which are attributed to igneous origin, it must be allowed also 
that there are others abundantly rich where no apparent con- 
nexion is to be traced. M. Necker mentions the mountain 
limestone as such; but he does not seem aware of the extent 
of those deposits, which, with the beds of grit and shale which 
alternate with it, present numberless regular veins abounding 
with certain ores. 

As this fact is indisputable, it seems necessary to show not 
only that unstratified rocks may be under them—which there is 
little doubt about,—but that there should be some connexion 
between the veins which contain the metals and similar chan- 
nels or passages in the rocks below. No such evidence, I be- 
lieve, at present exists; and J am not aware of any veins having 
yet been found to penetrate from the stratified rocks into those 
upon which they rest. 

This supposition must therefore, like many others, be taken 
as a mere probability to account for some appearances in certain 
places, but not to explain all the phenomena. 

There is one point which, before I conclude, I would endea- 
vour to press on the attention and consideration of future ob- 
servers, because, in the first place, it does not appear to have 
been much regarded by writers on the subject; and next, be- 


* March 28th, 18382. 


REPORT ON MINERAL VEINS. 23 


cause, though it seems to offer objections to some received 
theories, it may, when better understood, assist in developing 
the truth. 

This is the relation that the contents of a vein bear to the 
nature of the rock in which the fissure is situated. 

Thus in the older rocks, we see the same vein intersecting 
clay-slate and granite: it is itself continuous, and there is no 
doubt of its identity; and yet the contents of the part inclosed 
by the one rock shall differ very much from what is found in 
the other. In Cornwall, a vein that has been productive of 
copper ore in the clay-slate, passing into the granite becomes 
xicher, or, what is more remarkable, furnishes ores of the same 
metal differently mineralized. If we pursue it further into the 
granite, the produce of metal frequently is found to diminish. 

Veins in some cases cut through the elvan courses, as well as 
the clay-slate inclosing these porphyries: the ores are rich and 
abundant in the latter; in other instances they fail altogether. 

Less striking differences in the structure of the rock seem to 
affect the contents of the veins; and appearances as to the tex- 
ture and formation of the strata are often regarded by miners 
with more anxiety than the indications presented by the vein 
itself; and a change of ground is relied upon with an assu- 
rance, derived from experience, as a more certain basis to au- 
gur upon, for better or for worse, than almost any other which 
the difficult art of mining has to offer. 

‘Numberless facts might be collected and adduced to show 
that this is not mere speculation; but it will nowhere appear 
more clearly than if we examine the various beds of limestone 
grit, &c., in the great lead mines in the North of England. 

Here we shall find a series of stratified rocks, and that por- 
tion of the series which has been most productive of lead ores, 
occupying a thickness of nearly 280 yards. It is divided into 
55 distinct beds, which are accurately described in Mr. West- 
garth Forster’s section, each having its name known to the 
miners of the country. Nine of these beds are of limestone, 
about 18 are of gritstone or siliceous sandstone, and the re- 
al are plate or black shale, with thin beds of imperfect 
coal. 

Now the lead veins pass through all these beds, and have 
been worked more or less into all of them; and it has thus been 
proved, that though the fissure is common to all, yet lead ore is 
only found abundantly in particular beds, and those very much 
the same, if we examine the immense number of mines which 
are working in this district. 

Where the veins pass through the shale, little or no ore is to 


24 THIRD REPORT—1833. 


be found in them;- where they are inclosed by the gritstones, 
there they become more productive; but it is in one of the beds 
of limestone, and one only, that the great deposit of lead ore is 
to be found. 

In the great mining field of Alstone Moor, this bed is called: 
the great limestone, and yet its thickness is only about 23 
yards out of the 280 which the series of lead measures occupy; 
and notwithstanding this, four-fifths of all the lead ore found 
in the district is derived from such parts of the veins as are 
inclosed by this particular stratum. 

The veins equally passing through the other beds, and traced 
by imumerable workings through them, are yet only rich in 
metallic treasure where they repose in this favoured stratum *. 

Though perhaps few cases are so striking as this, yet it is 
evident that the same thing takes place to a certain extent 
with all the metals, in all rocks and im all countries. 

If it is a fact and correctly stated, it must be considered in 
reference to the theories propounded to us, and it seems directly 
opposed to the doctrine of forcible injection; but it may admit 
of probable explanation by calling in certain affinities, either by 
the advocates of precipitation from water, or by those who may 
contend that sublimed vapours might be attracted to particular 
spots. 


* To illustrate the comparative bearing of the different beds in the manor of 
Alstone Moor, Mr. Thomas Dickinson, the Moor Master for Greenwich Hospi- 
tal, extracted for me an exact account of the ore produced from each bed in all 
the mines of the manor in the year 1822, which gave the following results :— 


Limestone Beds.—Great limestone .......s.+-0+0000 20,827 bings. 
Tittle limestone — .........eccessees 287 
Four-fathom limestone ......... 91 
Scar limestone! NUN 90 
Tyne bottom limestone ......... 393 
— 21,688 
Gritstone Beds.—High slate sill .........s.0.0eeeeeee 107 
Lower slate sill:........0.cescencsse 289 
Firestorie “2232.0. 2 Sea: 262 
Pattingonis silliseosd es Sees 290082 259 
High Goal ‘sills 2.0. cofavsl gc sabvante 327 
ow, coalaswle.cceaee-erratasa seca 154 
WiUit’ 4 oceans cntsecuesmeenaraeeenes 306 
Quarry Nazel. wccasaceeerscccvese ess 44 
Nattrass Gill hazel ............0.. 21 
. Six-fathom hazel ...............6+. 576 
Dlaty hazehsot. ssddinantadeepnnaiee a 18 
Hazel under scar limestone...... 2 
= 2,000 


Whole produce of the mines of the manor..........+. 24,053 bings. 


REPORT ON MINERAL VEINS. 25 


That metallic ores are found to repose in rocks which seem 
congenial to them, and that their combinations are modified by 
changes in the rocks, will not I think be disputed by practised 
miners, or by those who have most narrowly searched into the 
hidden recesses of the earth. 

Facts must be observed and compared, effects must be traced 
to probable causes, and difficulties must be explained or can- 
didly admitted, if we would enlarge and generalize our know- 
ledge of vein formations. There are obstacles to the progress 
of this knowledge; for,as Dr. Boase has remarked, it is not easy 
for a person unaccustomed to it to use his eyes with much ad- 
vantage, in the places where the study can best be pursued. 

It is the miners’ business, however, not only to see clearly, but 
to consider all the intricate appearances that veins exhibit; and 
I would exhort them not to be satisfied merely with the obser- 
vations their art may seem to require, but to extend them toa 
larger view of the subject, and to contribute, as many of their 
eminent predecessors have done, to the common stock of general 
science. 

If the imperfect view which I have thus endeavoured to give 
of prevalent opinions should assist in such endeavours, or 
should stimulate any persons in undertaking a further pursuit 
of the subject, it would be to mea source of great gratification; 
as the desire of promoting such inquiries must be my apology 
for attempting the task which I have undertaken, 


pakspo ani icy’ tt ine ncninehle oF 

sypere tecassaairc: ND ams wnnddeady } ; 

ar Peas shes pab wei dad ibe dor coli Geglvents 
. ; — coon g ' 


ry 


PTET) Dae 


Lupa 
slid Doane’ oe 


jae Le cate arena cpa asm Aig 
i _ ae iain ade veeselll) § usenet 


ie saroilch ih: ase cre 


-ccauepont 
babe deatea Pe 


_ Be ee | 
Z ee a 
No ERE inertia 

Eepearty ees tab on tr 


Pe ov A ie bees th 8 
niga pa Pos cn nh eae 


Weed es) 
Masaya 
' MBayiipann, 


Uf baa ada she iyo 
siete Ped Frag Pag Ia) Lesijabiccs bE haggtes 
‘ Nimans - Abt! ' M4 me kein bese ol Paha 
as Henly WhPenr’ rele le seater if rans ha 
—_ a ee : ase at sie 
Poynwled ori 
r mts yi {sux P% “hh 


. a 


et strt eink Dr ol . 

f Thee a De ot: °t lhe 4 stag Pi Ate sme finn 7 

- ri el arpa F be, OF hey meni yah. Dah epee, i kloe yes Soe 
es | eer ake af 9 ont id cena ninny ered aia e 
Pie pete will Sree ty ib sped i reno one 8 " 

u Arwirot) I i if TKS iW or a : Bar nih chew: ; 

a" Satine y HA re hk beth OF ovr wm albesil us om 
Pa Pores tien Been earn nh ween a ae | 
Myt-iltns, PPR AT oh yt ators! Le this and + 

ee tel At ee ype Wend qRer* 4 


wm Meh 3 Paes voc leoteltad idee aioe eas? 
‘J ; > Tair ie | OTe ives fa whe syne! igile abies enact i % 
t. # pele: te fet ety Sevee Ra slices 
ve 4 


On the Principal Questions at present debated in the Philoso- 
phy of Botany. By Joun Linvtey, Ph. D., F.RS., §c., 
Professor of Botany in the University of London. 


Ir we compare the state of Botany at the end of the last cen- 
tury with its present condition, we shall find that it has become 
so changed as scarcely to be recognised for the same science. 
Improvements in the construction of the microscope, the disco- 
veries in vegetable chemistry, the exchange of artificial methods 
of arrangement for an extended and universal contemplation of 
natural affinities, the reduction of all classes of phanomena to 
general principles, and, above all things, the adoption of the 
philosophical views of Gothe, together with the recognition of 
an universal unity of design throughout the vegetable world, 
are undoubtedly the principal causes to which this change is 
to be ascribed. 

As the general nature of recent discoveries, and a sufficient- 
ly definite notion of the present state of botanical science, may 
be collected from the introductory works which have appeared 
in this country within the last three years, it is presumed that 
the object of the British Association will be attained if the 
present Report is confined to the most interesting only of those 
subjects upon which botanists have been recently occupied, 
and to an indication of the points to which it is more particularly 
desirable that inquiries should now be directed. I have also 
excluded everything that relates to mere systematic botany, in 
the hope that some one will take that subject as the basis of a 
separate Report. 

Elementary Organs.—This country has, till lately, been re- 
markably barren of discoveries in vegetable anatomy, since the 
time of Grew, who was one of the fathers of that branch of 
science. Whatever progress has been made in the determina- 
tion of the exact nature of those minute organs, by the united 
powers of which the functions of vegetation are sustained, it 
has been chiefly in foreign countries that it has taken place: 
the names of Mirbel, Moldenhauer, Kieser, Link and Amici, 
stand alone during the period when their works were published ; 
and it has only been within a very few years that those of 
Brown, Valentine, Griffith and Slack have entered into com- 
petition with the anatomists of Germany and France. 

By the researches of these and other patient inquirers, we 


28 THIRD REPORT—18383. 


have already reduced our knowledge of the exact internal struc- 
ture of plants to a state of very considerable precision; although 
it must be confessed that vegetable anatomy is still the field 
where the greatest discoveries may be expected. 

It is now generally agreed that the old opinions, that the tis- 
sue of plants is either a membrane doubled together in endless 
folds, or a congeries of cavities formed in solidifiable mucus 
by the extrication of gaseous matter, are equally erroneous, 
and that it really consists of distinct sacs or cells, pressed to- 
gether and adhering to each other by the sides where they are 
in contact. 

It is considered that this is proved by the following circum- 
stances. 1. By the action of some powerful solvent, such as 
nitric acid, the cells may be artificially separated from each 
other. 2. In parts which become succulent, the cells separate 
spontaneously, as in the receptacle of the strawberry, the berry 
of the privet, &c. 3. When the parts are young, their tissue 
may be easily separated by pressure in water. 4. It is con- 
formable to what has been observed in the growth of plants. 
Amici found that the new tubes of Chara appear like young buds 
from the points or axille of pre-existing tubes ; an observation 
that has been confirmed by Mr. Henry Slack*. It has been 
distinctly proved by M. Mirbel +, that the same thing occurs in 
the case of Marchantia polymorpha. 'That learned botanist, in 
the course of his inquiries into the structure of this remarkable 
plant, may be said to have been present at the birth of its cel- 
lular tissue; and he found that in all cases one tube or utricle 
generated another, so that sometimes the young masses of tis- 
sue had the appearance of knotted or branched cords. He satis- 
fied himself, by a beautifully connected series of observations, 
that new parts are not formed by the adhesion of vesicles origi- 
nally distinct, as many have asserted, but by the generative 
power of one first utricle, which engenders others endowed with 
the same property. 

It appears that when first formed the sacs are completely 
closed up, so that there is no communication between the one 
and the other, excepting through the highly permeable mem- 
brane of which they are composed. This, indeed, is not con- 
formable to the observations of those who have described and 
represented pores or passages of considerable magnitude pierced 
in the sides of the sacs; but it has been satisfactorily shown by 
Dutrochet, that the spaces supposed by such observers to be 


* Transactions of the Society of Arts, vol. xlix. 
+ “Recherches Anatomiques et Physiologiques sur le Marchantia polymor- 
pha,” in Nouv. Ann. du Muséum, vol. i. p. 98. 


— 


REPORT ON THE PHILOSOPHY OF BOTANY. 29 


pores are nothing more than grains of amylaceous matter stick- 
ing to the sides of the sacs; for he found that by immersing the 
latter in hot nitric acid, the supposed pores became opake, and 
by afterwards moistening them with a weak solution of caustic 
potash, they recovered their transparency: we also find that 
the supposed pores are readily detached from the sides of the 
sacs to which they adhere; and I think it may be added, that 
our microscopes are now alone sufficient to show what they are. 

The question as to the perceptible porosity of vegetable tis- 
sue may therefore be considered, I think, disposed of as a 
general fact; for the objection that Dr. Mohl has taken to this 
explanation *,—namely, that in a transverse section we ought to 
find such grains projecting from the sides of the cells like little 
eminences,—cannot surely be entitled to much weight, if we op- 
pose to this negative observation the positive evidence already 
mentioned, and especially if we consider that it is next to im- 
possible for the keenest knife to make a section of such delicate 
parts without carrying away such particles upon its edge. There 
are, nevertheless, cases in which the point is still open to in- 
vestigation. 

Thus Mirbel, in his second memoir on the Marchaniia j, 
positively declares that the curious cells which line the anther 
of the common gourd, are continuous membranes till just be- 
fore the expansion of the flower, when they very suddenly en- 
large, and their sides divide into the narrow ribands or threads 
which give their name to what we call fibrous cells. In this, 
and the multitudes of similar cases with which Purkinje has 
made us acquainted, there can be no doubt that the sides of 
the cells consist ultimately of nothing but openwork; but still 
it seems certain that during the principal part of their existence 
they were completely closed up. 

It is also probable that in other cases the sides of the cells 
or vessels ultimately give way and slit; but this rending seems 
to be a phenomenon attendant upon the cessation of the ordi- 
nary functions of tissue, and independent of their original con- 
struction. 

In coniferous plants the wood is in a great measure com- 
posed of closed tubes, tapering to each end, the sides of which 
are marked with circles, containing a smaller circle in their cen- 
tre. These-circles have long been considered undoubted pores, 
and it does not appear possible to prove them otherwise by any 
of the tests already mentioned. 


* Ueber die Poren des Pflanzen-Zellgewebes, p. 11. Tubingen, 1828. 
t Archives de Botanique, vol. i. 


30 THIRD REPORT—1833. 


I have endeavoured to show * that they are glands of a pecu- 
liar figure, which stick to the sides of the tubes; and I have 
ascertained that the large round holes that are certainly found 
in coniferous tissue are caused by the dropping or rubbing off 
of such supposed glands. But a very different opinion is en- 
tertained by Dr. Mohl}, whose observations have been con- 
firmed by Dr. Unger{. In the opinion of the former of these 
botanists the supposed glands of coniferous tissue are circular 
spaces where the membrane of the tube becomes abruptly ex- 
tremely thin; and it is said that transverse slices of coniferous 
wood, made at an angle of forty-five degrees, demonstrate the 
fact. Dr. Mohl is also of opinion, as has been already said, 
that the porous appearances above mentioned, and ascribed to 
the adhesion of amylaceous matter to the sides, are of a similar 
nature. 

It has been shown by Mr. Griffiths, that in the kind of tissue 
called the dotted duct, the suspicion of Du Petit Thouars that 
this form of tissue is composed of short cylindrical cells placed 
end to end, and opening into each other, is correct; their com- 
munication, however, is not by means of an organic perfo- 
ration, but is produced by the absorption and rupture of the 
ends which come in contact. Mr. Slack has also stated, 
in a very good paper upon Vegetable Tissue§, that in other 
cases the vessels of plants open into each other where they 
come in contact; as, for example, at the conical extremities, 
where ducts join each other ; but he represents this to be owing 
to the obliteration of their membrane at that point; the internal 
fibre, of which they are in part composed, remaining like a 
‘grating stretched across the opening where the enveloping 
membrane has disappeared. 

In a short paper, published in the Journal of the Royal In- 
stitution in December 1831, I have endeavoured to show that 
membrane and fibre are to be considered the organic elements 
of vegetable tissue, contrary to the more usual opinion that 
membrane only is its basis: this was attempted to be proved, 
not only by the fact that the simple cells of the testa of Mau- 
randia, &c., are apparently formed by a fibre twisted spirally 
in the inside of their membrane, but also by the elastic spires I 
had discovered on the outside of the seed of Collomia, in which 
it is plain that no membrane whatever is generated. 


* Introduction to Botany, p. 16. t. 2. f. 7. 

+ Ueber die Poren des Pflanzen-Zeligewebes. 

{ Botanische Zeitung, October 7, 1832. 

§ Transactions of the Society of Arts, vol. xlix. 


| 


REPORT ON THE PHILOSOPHY OF BOTANY. 31 


It would, however, appear from the researches of Mirbel*, 
that the presence of a twisted fibre within a cell is not always 
the cause of the spiral or fibrous character so common in tissue. 
He finds, as has been already stated, that the cells that line 
the anther of a gourd are at first membranous and closed, and 
that they continue in this state till just before the bursting of the 
anther, when they suddenly divide in such a way as to assume 
the appearance of delicate threads, curved in almost elliptical 
rings, which adhere to the shell of the anther by one end; these 
rings are placed parallel with each other in each cell, to which 
they give an appearance like that of a little gallery with two 
rows of pilasters, the connecting arches of which remain after 
the destruction of the roof and walls. He also watched the 
development of the curious bodies called elaters in the Mar- 
chantia, which he describes to the following effect. At first 
they are long slender tubes, pointed at each end; at a subse- 
quent stage their walls thicken, and become less transparent, 
and are marked all round through their entire length with two 
parallel, very close, spiral streaks ; later still the tubes enlarge, 
and their streaks become slits, which divide the walls all round, 
from one end to the other, into two filaments; and, finally, the 
circumvolutions of the filaments separate, assume the appear- 
ance of a corkscrew, acquire a rust colour, and the elater is 
complete. These elaters he considers organically identical with 
the spiral vessel, and hence he concludes that every description 
of vessel is a cell, differing from ordinary cells in being larger. 

Upon the general accuracy’of these observations I am dis- 
-posed to place great confidence; and I would even add, that the 
theory of pierced or open cellular tissue being produced by 
the spontaneous rending of its membrane, is apparently con- 
nected with an observation of my own‘, that in some plants 
simple vegetable membrane will tear more readily in one direc- 
tion than another. It is nevertheless to be observed, that the 
theory of fibre being one of the organic elements of tissue does 
not seem to have occurred to the experienced physiologist to 
whose observations I am referring, and that some of the ap- 
pearances he mentions at a stage preceding transformation are 
very like those of the development of an internal fibre. 

The opinion of the organic identity of all the forms of tissue 
has also been maintained by Mr. Slack, in the paper already 
referred to, and by Dr. Mohl, in his memoir on the comparative 
anatomy of the stem of Cyeadee, Conifere, and Tree Ferns. 


* Archives de Botanique, vol. i. 
+ Introduction to Botany, p. 2. 


32 THIRD REPORT—1833. 


The latter considers that the dotted tubes of Cycadee un- 
doubtedly pass directly into the vessels called by the Germans 
vasa scalariformia; but my own observations do not confirm 
this statement. : 

Circulation Whether or not plants have a circulation ana- 
logous to that of animals, is a topic that was more open to con- 
jecture at a time when the real structure of the former was un- 

nown, than it can be at the present day. Knowing, as we 
now do, that a tree is more analogous to a Polype than toa 
simple animal; that it is a congeries of vital systems, acting 
indeed in concert, but to a great degree independent of each 
other, and that it has myriads of seats of life, we cannot expect 
that in such productions anything absolutely similar to the mo- 
tion of the blood of animals from and to one common point 
should be found. The idea of circulation existing in plants 
must therefore be abandoned; but that a motion of some kind 
is constantly going on in their fluids was sufliciently proved by 
the well-known facts of the flow of the sap, the bleeding of the 
vine, the immense loss plants sustain by evaporation, and by 
similar phenomena. ‘The motion was for the first time beheld 
by Amici, the Professor at Modena, who discovered it in the 
Chara. He found that in this plant the cylindrical cells of the 
stem are filled with fluid, in which are suspended grains of 
green matter of irregular form and size. ‘These grains were 
distinctly seen to ascend one side of each tube, and descend the 
other, after the manner of a jack-chain, and to be continually 
in action, in the same manner, as long as the cell retained its 
life; the motion of the grains was evidently due to the ascend- 
ing and descending current in the fluid contained within the 
tube-like cell. It could not be ascertained that any kind of 
communication existed between the cells, but each was seen to 
have a motion of its own. 

The observations of Amici have been verified in this country 
chiefly upon species of Nitella ; and from the investigations of 
Mr. Solly, Mr. Varley, and Mr. Slack*, the nature of the phz- 
nomenon has been determined with considerable precision. 

Among other things, it has been ascertained that in Nétella 
the currents have always a certain relation to the axis of growth, 
the ascending current uniformly passing along the side of the 
cell most remote from the axis, and the descending current 
along the side next the axis. 

Similar motions have been seen in several other plants. In 
the cells of Hydrocharis Morsus-Rane the fluid has been ob- 


* Transactions of the Society of Arts, vol. xlix. 


REPORT ON THE PHILOSOPHY OF BOTANY. 30 


served to move round and round their sides in a rotatory man- 
ner, which, however, has not been seen to follow any particular 
law. In the joints of the hairs of Tradescantia virginica 
several currents of a similar nature exist; and in the hair of 
the corolla of a species of Pentstemon, Mr. Slack has observed 
several currents taking various directions, some continuing to 
the summit of the hair, whilst others turn and descend in va- 
rious places, two currents frequently uniting in one channel. 

It may hence, possibly, be assumed that in the cells of plants, 
when filled with fluid, there is a very general rotatory move- 
ment, which is confined to each particular cell. ‘This, it is ob- 
vious, can form no part of the general circulation of the system, 
which must often occur with great rapidity, and which must 
take place from the roots to the extremities. The rotatory 
motion may perhaps be considered a sort of motion of di- 
gestion, and connected with the chemical changes which matter 
undergoes in the cells from the united action of light, heat, 
and air. 

What has been supposed to be a discovery of the universal 
motion of sap has been made by Professor Schultz of Berlin, 
who remarked two torrents, one of which was progressive, and 
the other retrogressive, in what he calls the vital vessels (ap- 
parently the woody fibre) in the veins of Chelidoniwm majus, 
and in the stipule of Ficus elastica. 

His observations have been repeated by a Commission of the 
Tnstitute, composed of MM. Mirbel and Cassini, who have 
reported* that they have also seen the motion described by 
Professor Schultz; and I have myself witnessed it as is repre- 
sented by those observers. But it appears probable, from se- 
veral circumstances, that the motion that has been seen has 
either been owing merely to the vessels in which it was re- 
marked having been cut through, and emptying themselves of 
their contents, as Mr. Slack has suggested, or else was nothing 
but the common rotatory motion imperfectly observed. 

Structure of the Axis.—From the period when M. Desfon- 
taines first demonstrated the existence of two totally distinct 
modes of increase in the diameter of the stems of plants, it has 
been received as a certain fact that monocotyledonous plants 
increase by addition to the centre of their stem, and dicotyle- 
donous by addition to the circumference. Nothing has yet 
arisen to throw any doubt upon the exactness of this notion in 
wegard to dicotyledonous plants; but Dr. Hugo Mohl has 
endeavoured to show} that monocotyledonous stems are not 

* Annales des Sciences, vol. xxii. p. 89. 

; + Mohl, “De Palmarum Structura,” in Martius’s Genera et Species Palmarum. 
Ovv. D 


34 THIRD REPORT—1833. 


formed in the manner that has been supposed. According to 
him, the new matter from which the wood results is not a mere 
addition of new matter to the centre, but consists of bundles 
of wood, which, originating at the base of the leaves, take first 
a direction towards the centre, and then a course outwards 
towards the circumference, forming a curve; so that the stem 
of a Palm is, in fact, a mass of woody arcs intersecting each 
other, and having their extremities next the circumference of 
the trunk. I regret that I have not been able to consult Dr. 
von Martius’s splendid work on Palms since this Report was 
commenced, and that I am therefore unable to state upon what 
evidence Dr. Mohl has rested his theory. 

The same writer has stated* that Cycadee—that singular 
tribe, which is placed, as it were, on the boundary line between 
cellular and vascular plants,—are not in a great measure desti- 
tute of vessels as is commonly supposed, but, on the contrary, 
are composed exclusively of spiral vessels and their modifica- 
tions, without any mixture of woody fibre. I have already ad- 
verted to this hypothesis in speaking of the same author's state- 
ment, that the dotted tubes of Cycadee are a slight modi- 
fication of vasa scalariformia. Dr. Mohl is also of opinion 
that Cycadee are not exogenous in their mode of growth, as 
seems to be indicated by their appearance when cut, and by 
their dicotyledonous embryo, but that they are more like 
Palms in their manner of forming their wood, which is essen- 
tially endogenous. He asserts that the stem of Cycadee, in 
regard to its anatomical condition, must be considered inter- 
mediate between that of Tree Ferns and Conifere, just as their 
leaves and fructification undoubtedly are. He states that in 
Cycadee a body of wood is gradually formed of the fibres con- 
nected with the central and terminal bud; that so long as this 
original wood is soft, and capable of giving way to the fibres 
that are continually passing downwards, no second cylinder of 
wood is formed; but in time the original wood becomes hard- 
ened, and then the new fibres find their way outward and down- 
ward, collecting into a second cylinder on the outside of the 
original wood. It is obvious that this explanation is not so sa- 
tisfactory as could be desired; for, in the first place, such a 
distinction between Cycadee and Exogene as that which Dr. 
Mohl states to exist, is verbal rather than real, since he admits 
that the second cylinder of wood is formed externally to the 
first; and secondly, it is obvious that if that structure which is 
represented in the 21st plate of the third volume of the Hortus 


* Ueber den Bau des Cycadeen Stammes und sein Verhaltniss xu den Stamm 
der Coniferen und Baumfarrn. 4to. Munich, 1882. 


REPORT ON THE PHILOSOPHY OF BOTANY. 35 


Malabaricus be correct, where the stem of Cycas circinalis is 
shown to have several concentric zones, precisely as in other 
exogenous trees, it must follow that Dr. Mohl’s explanation 
would be still more inadmissible; accordingly, this author dis- 
credits the fact of the stem of Cycas cércinalis haying numer- 
ous concentric zones. It is, however, certain, from the speci- 
mens brought to England by Dr. Wallich, that the structure 
of this Cycas is really such as is shown in the Hortus Malaba- 
ricus. It is nevertheless extremely well worth further inquiry 
whether there is not some important but as yet undiscovered 
peculiarity in the mode of forming their stem by Cycade@; for 
it must be confessed that growth by a single terminal bud, 
after the manner of Palms, is not what we should expect to 
meet with in exegenous trees. 

Professor Schultz of Berlin has indicated* the existence of a 
group of plants, the structure of whose stems he considers at 
variance with all the forms at present recognised; and to this 
group he refers Cycadee : but the assemblage of orders which 
he collects under what he calls the same plan of growth is so 
extremely incongruous as to lead to no other conclusion than 
that subordinate modifications of internal structure are of no 
general importance, but are merely indicative of individual pe- 
culiarities. 

Dr. Mohl further states, that Cryptogamic plants of the 
highest degree of organization, such as Ferns, Lycopodiacea, 
Marsileacee, and Mosses, in ail which a distinct axis is found, 
have a mode of growth neither exogenous nor endogenous, but 
altogether of a peculiar nature. In these plants, when once 
the lower part of the stem is formed it becomes incapable of 
any further alteration, but hardens, and the stem continues to 
grow only by its point, which lengthens merely by the progres- 
sive development of the parts already formed, without sending 
downwards any fibrous or woody bundles, as both in exoge- 
nous and endogenous plants. 

M. Lestiboudois, the Professor of Botany at Lille, distin- 
guishes Monecotyledons from Dicotyledons, upon principles 
different from those generally adnfitted. According to this 
writer, dicotyledonous trees have two systems, one, the central, 
consisting of the medullary sheath and the wood; the other, 
the cortical, composing the bark. These two systems increase 
separately, so that in Dicotyledons there are two surfaces of 
increase, that of the central system, which adds to its outside, 
and that of the cortical system, which adds to its inside: but 

* Naturliches System des Pflanxenreichs nach seiner inneren Organization. 
8vo. Berlin, 1832. 

D2 


1853. 


36 THIRD REPORT 


in the stem of Monocotyledons there is only one surface of in- 
crease, namely, that on the inside; and hence he concludes 
that such plants have only a cortical system, and consist of 
bark alone. It must be obvious that there are too many ana- 
tomical objections to this theory to render it deserving of any 
other than this incidental notice *. 

The cause of the formation of wood has always been a sub- 
ject upon which physiologists have been unable to agree; and 
if the opinions held by the writers of the last century have been 
disproved, it cannot be added that those of the present day are 
by any means settled. It is now, indeed, admitted on all hands 
that wood is a deposit in some way eee ie with the action 
of leaves ; for it has been proved beyond all question that the 
quantity of wood that is formed is in direct proportion to the 
number of leaves that are evolved, and to their healthy action, 
and that where no leaves are formed, neither is wood deposited. 
But it is a subject of dispute whether wood is actually or- 
ganized matter generated by the leaves, and sent downwards 
by them, or whether it is a mere secretion, which is deposited 
in the course of its descent from the leaves to the roots. The 
former opinion has been maintained in different forms by De 
la Hire, Darwin, Du Petit Thouars, Poiteau, and myself, and 
would perhaps have been more generally adopted if it had not 
been too much mixed up with hypothetical statements, to the 
reception of which there are in the opinion of many persons 
strong objections. For example, it has been asserted that the 
wood of trees is an aggregation of the roots of myriads of buds 
in a state of action, and that consequently a tree is an asso- 
ciation of individuals having a peculiar organic adhesion and 
a common system of growth, but each its own individual life. 
To this view it is no doubt very easy to raise objections, some 
of which it may be difficult, in the present state of our know- 
ledge, to answer; and therefore it is better for the moment to 
leave this part of the proposition out of consideration, and to 
confine it to the simple statement that wood is organized matter, 
generated by the leaves, and sent downwards by them. In 
support of this it is argued? Ist, That an anatomical examination 
of a plant shows that the woody systems of the leaf and stem 
are continuous: 2ndly, That this is not only the fact in exogenous 
plants, but in all endogenous and cellular plants that have 
been examined ; so that it may be considered a universal law: 
3rdly, That in the early spring, and for some time after plants be- 
gin to grow, the woody matter is actually to be seen and traced 


5 a) U - . . 
* Achille Richard, Nouveaux Elémens de la Botanique, 5me edit. p. 119. _ 


Y 


ee ee eee 


REPORT ON THE PHILOSOPHY OF BOTANY. 37 


descending in parallel tubes from the origin of the leaves, and 
from no other place: 4thly, That in all cases where obstacles 
are presented to the descent of such tubes, they turn aside, and 
afterwards resume their parallelism when the obstacle has been 
_ passed by: 5thly, That in endogenous plants, such as Palms, and 
in some exogenous trees, such as Lignum Vite, they cross and 
interlace each other in a manner which can only be accounted 
for by their passing downwards, the one over the other, as the 
leaves are developed: and, finally, That the perfect organization 
of the wood is incompatible with a mere deposit of secreted 
matter. To all which the following evidence has been added 
by M. Achille Richard. He states* that he saw in the pos- 
session of Du Petit Thouars a branch of Robinia Pseudacacia, 
on which Robinia hispida had been grafted. The stock had 
died, but the scion had continued to grow, and had emitted 
from its base a sort of plaster, formed of very distinct fibres, 
which surrounded the extremity of the branch to some distance, 
and formed a sort of sheath; thus demonstrating incontestibly 
that fibres do descend from the base of the scion, to overlay 
the stock. 

To this several objections have been taken, the most im- 
portant of which are the following. If wood were really or- 
ganized matter, emanating from the leaves, it must necessarily 
happen that in grafted plants the stock ought in time to acquire 
the nature of the scion, because its wood would be formed en- 
tirely by the addition of new matter, said to be furnished by 
the leaves of the scion; so far, however, is this from being the 
fact, that it is well known that in the oldest grafted trees there 
is no action whatever exercised by the scion upon the stock, 
but that, on the contrary, a distinct line of organic demarcation 
separates the wood of the one from the other, and the shoots 
emitted from the stock by wood said to have been generated 
by the leaves of the scion, are in all respects of the nature of 
the stock. Again,—if a ring of bark from a red-wooded tree 
is made to grow in the room of a similar ring of bark of a white- 
wooded tree, as it easily may be made, the trunk will increase 
in diameter, but all the wood beneath the ring of red bark will 
be red, although it must have originated in the leaves of the 
tree which produces white wood. It is further urged, that in 
grafted plants the scion often overgrows the stock, increasing 
much the more rapidly in diameter, or that the reverse takes 
place, as when the Pavia lutea is grafted upon the common 
Horse-chestnut,—and that these circumstances are inconsistent 


i 
* Nouveaux Elémens de la Botanique, Sme edit. p. 105. 


on 


38 THIRD REPORT—18335. 


with the supposition that the wood is organic matter engendered 
by leaves. ‘To these statements there is nothing to object as 
mere facts, for they are true; but they certainly do not warrant 
the conclusions that have been drawn from them. One most 
important point is overlooked by those who employ these argu- 
ments, namely, that in all plants there are two distinct simul- 
taneous systems of growth, the cellular and the fibro-vascular, 
of which the former is horizontal, and the latter vertical. The 
cellular gives origin to the pith, the medullary rays, and the 
principal part of the cortical integument ; the fibro-vascular, to 
the wood and a portion of the bark; so that the axis of a plant 
may be not inaptly compared to a piece of linen, the cellular 
system being the woof, the fibro-vascular the warp. It has alse 
been proved by Mr. Knight* and M. De Candolle + that buds 
are exclusively generated by the cellular system, while roots are 
evolved from the fibro-vascular system. Now if these facts are 
rightly considered, they will be found to offer an obvious expla- 
nation of the phenomena produced by those botanists who think 
that wood cannot be matter generated in an organic state by the 
leaves. ‘The character of wood is chiefly owing to the colour; 
quantity, size, and distortions of the medullary rays, which be- 
long to the horizontal system; it is for this reason that there is 
so distinct a line drawn between the wood of the graft and 
stock, for the horizontal systems of each are constantly pressing 
together with nearly equal force, and uniting as the trunk in- 
creases in diameter. As buds from which new branches elon- 
gate are generated by cellular tissue, they also belong to the 
horizontal system; and hence it is that the stock will always 
produce branches like itself, notwithstanding the long super- 
position of new wood which has been taking place in it from 
the scion. 

The case of a ring of red bark always forming red wood be- 
neath it, is precisely of the same nature. After the new bark 
has adhered to the mouths of the medullary rays of the stock, 
and so identified itself with the horizontal system, it is gradually 
pushed outwards by the descent of woody matter from above 
through it: but in giving way it is constantly generating red 
matter from its horizontal system, through which the wood de- 
scends, which thus acquires a colour that does not properly 
belong to it. With regard to the instances of grafts over- 
growing their stock, or vice versd, it is obvious that these are 
susceptible of explanation upon the same principle. If the hori- 
zontal system of both stock and scion has an equal power of 


* Philosophical Transactions, 1805, p. 257. 
+ Physiologie Végétale, p. 158. 


REPORT ON THE PHILOSOPHY OF BOTANY. 39 


lateral extension, the diameter-of each will remain the same; 
but if one grows more rapidly than the other, the diameters 
will necessarily be different: where the scion has a horizontal 
system that develops more rapidly than that of the stock, the 
latter will be the smaller, and vice versd. It iss however, to be 
observed, that in these cases plants are altogether in a morbid 
state, and will not live for any considerable time. 

Those who object to the theory of wood being generated by 
the action of leaves, either suppose—Ist, that liber is developed 
by alburnum, and wood by liber; or, 2ndly, that ‘the woody and 
cortical layers originate laterally from the cambium furnished 
by preexisting layers, and nourished by the descending sap*.” 
The first of these opinions appears to be that of M. ‘Turpiny, 
as far as can be collected from a long memoir upon the grafting 
of plants and animals; but I must fairly confess that 1 am not 
sure I have rightly understood his meaning, so much are his 
facts mixed up with gratuitous hypothesis and obscure specu- 
lations upon the action of what he calls globuline. The second 
is the opinion commonly entertained in France, and adopted 
by M. De Candolle in his latest published work. 

The objections to the views of M. Turpin need hardly be 
stated in a Report like this, where conciseness is so much an 
object. Those which especially bear upon the view taken by 
M. De Candolle are, that his theory is not applicable to all 
parts of the vegetable kingdom, but to exogenous plants only; 
that it is inconceivable how the highly organized parallel tubes) 
of the wood, which can be traced anatomically from the leayes, 
and which are formed with great rapidity, can be a lateral de- 
posit from the liber and alburnum; that they are manifestly 
formed long before it can be supposed that the leaves have 
commenced their office of elaborating the descending sap; and, 
finally, that endogenous and cryptogamic plants, in which there 
is no secretion of cambium, nevertheless have wood. 

Such is the state of this subject at the time I am writing. To 
use the words of M. De Candolle, ‘‘' The whole question may 
be reduced to this,—Either there descend from the top of a tree 
the rudiments of fibres, which are nourished and developed by 
the juices springing laterally from the body of wood and bark, 
or new layers are developed by preexisting layers, which are 
nourished by the descending juices formed in the leaves {.” 

As this is one of the most curious points remaining to be 
settled among botanists, and as it is still as much open to dis- 


* De Candolle, Physiologie Végétale, p. 165. 
+ See Annales des Sciences, vols. xxiv. and xxv., particularly vol. xxv. p. 43. 
}De Candolle, Physiologie Végétale, p. 157. 


40 THIRD REPORT—1833. 


cussion as ever, I have dwelt upon it at an unusual length, in 
the hope that some Member of the British Association may 
have leisure to prosecute the inquiry. Perhaps there is no 
mode of proceeding to elucidate it which would be more likely 
to lead to positive results, than a very careful anatomical exami- 
nation of the progressive development of the Mangel Wurzel 
root, beginning with the dormant embryo, and concluding with 
the perfectly formed plant. 

Arrangement of Leaves.—It has for a long time been thought 
that the various modes in which leaves, and the organs which 
are the result of them, are arranged upon a stem might be re- 
duced to the spiral, and that all deviations from this law of 
arrangement are to be considered as caused by the breaking of 
spires into verticilli. In the Pine Apple, for instance, the Pine 
Cone, the Screw Pine, and many other plants, the spiral arrange- 
ment of the leaves is so obvious that it cannot be overlooked ; 
in trees with alternate leaves this same order of arrangement 
may be discovered if a line is drawn from the base of one leaf 
to that of another, always following the same direction; even 
in verticillate plants we not unfrequently see that the whorls 
are dislocated by the preternatural elongation of their axis, and 
then become converted into a spire; and the same phenome- 
non is of common occurrence among the verticilli of leaves in 

»the form of calyx, corolla, stamens, and carpella, which com- 
pose the flower. This will be the more distinctly apparent if we 

Weonsider that, as M. Adolphe Brongniart has shown*, what we 
¢e horls in a flower often are not so, strictly speaking, but 
only a series of parts placed in close approximation, and at dif- 
ferent heights, wpon the short branch that forms their axis. 

Dr. Alexander Braun has endeavoured} to prove mathema- 
tically that the spiral arrangement of the parts of plants is not 
only universal, but subject to laws of a very precise nature. 
His memoir is of considerable length, and would be wholly un- 
intelligible without the plates that illustrate it. It is therefore 
only possible on this occasion to mention the results. Setting 
out with a contemplation of the manner in which the scales of 
a Pine Cone are placed, to which a long and ingenious method 
of analysis was applied, he found that several different series of 
spires are discoverable, between which there invariably exist 
peculiar arithmetical relations, which are the expression of the 
various combinations of a certain number of elements disposed 
in aregular manner. All these spires depend upon the posi- 


* Annales des Sciences, vol. xxiii. p. 226. 
+ Vergleichende Untersuchung tiber die Ordnung der Schuppen an den Tan- 
nenzxapfen. Ito. 1830, 


. 


REPORT ON THE PHILOSOPHY OF BOTANY. Al 


tion of a fundamental series, from which the others are devia- 
tions. The nature of the fundamental series is expressed by a 
fraction, of which the numerator indicates the whole number 
of turns required to complete one spire, and the denominator 
the number of scales or parts which constitute it: thus 4 in- 
dicates that eight turns are made round the axis before any 
seale or part is exactly vertical to that which was first formed, 
and the number of scales or parts that intervene before this 
coincidence takes place is twenty-one. 

It does not appear that this inquiry has as yet led to any 
practical application, although one might have expected that 
as the natural affinities of plants are determined, in a great de- 
gree, by the accordance that is observable in the relative posi- 
tion of their parts, the spires of which those parts are composed 
might have had something in common which would be suscep- 
tible of being expressed by numbers. If any practical applica- 
tion can be made of Dr. Braun’s fractions, it seems likely to be 
confined to the distinction of species. His observations seem, 
however, to have established the truth of the doctrine that, be- 
ginning with the cotyledons, the whole of the appendages of 
the axis of plants,—leaves, calyx, corolla, stamens, and car- 
pella,—form an uninterrupted spire, governed by laws which 
are almost constant. - a 

Structure of Leaves.—The leaves of plants have been found” 
by M. Adolphe Brongniart to be not merely expansions of_ 
the cellular integument of stems, traversed by veins originating” 
in the woody system, but to be organs in which the int , 
parenchyma is arranged with beautiful uniformity, in the man- 
ner most conducive to the end of exposure to light and air, and 
of elaboration, for which the leaves are chiefly destined. In 
their usual structure leaves have been found by this observer 
either to consist of two principal layers,—of which the upper, 
- into which the ascending sap is first introduced, is formed of 
compact cells, more or less perpendicular to the plane of the cu- 
ticle, and the under, into which the returning sap is propelled, 
is formed of very lax cavernous tissue, more or less parallel with 
the cuticle of the lower surface,—or else of two layers perpen- 
dicular to the cuticle, with a central parallel stratum. 

The observations of Drs. Mohl and Meyen generally confirm 
this; but at the same time the latter instances several cases in 
which the texture of the leaf has been found to be nearly the 
same throughout. 

Dutrochet* states, in addition, that the interior of the leaf 


* Annales des Sciences, vol. xxv. p. 245. 


42 THIRD REPORT—1833. 


is divided completely by a number of partitions caused by the 
ribs and principal veins, so that the air cavities have not actually 
a free communication in every direction through the parenchy- 
ma, but are to a certain extent cut off from each other. This 
is conformable to what M. Mirbel has described in Marchantia, 
who finds the leafy expansions of that plant separated by par- 
titions into chambers, between which he is of opinion that 
there is no other communication than what results from the per- 
meability of the tissue *. j 

The statement of M. Adolphe Brongniart, that all leaves in- 
tended to exist in the air are furnished with a distinct cuticle 
on their two faces, while those which are developed under 
water have no cuticle at all, has not been disproved, unless in 
the case of Marchantia}, whose under surface can scarcely be 
said to have a distinct cuticle; but this plant, which can only 
exist in humid shady places, is*perhaps rather a proof of the 
accuracy of the theory of M. Brongniart than an exception 
to it. 

That the stomata in all cases open into internal cavities in 
the leaf, where the tissue is extremely lax and cavernous, ap- 
pears also extremely probable, It was especially found to be 
the case by M. Mirbel in his so often quoted remarks upon 
Marchantia. 

With regard to the stomata themselves, no one appears yet 
to have confirmed the observation of Dr. Brown}, that their 

“apparent orifice is closed up bya membrane. On the contrary, 
the observations of M. Mirbel on Marchaniia, if they are to 
be taken as illustrative of the usual structure of those singular 
organs, go to establish the accuracy of the common opinion 
that the stomata are apertures in the cuticle. That most skil- 
ful physiologist, while watching the development of Marchantia, 
remarked the very birth of the stomata, which he describes as 
taking place thus:—The appearance of a little pit in the middle 
of four or five cells placed in a ring is a certain indication of 
the beginning of a stoma. The pit evidently increases by the 
enlargement and separation of the surrounding cells. If the 
nascent stoma consists of five cells, of which one is surrounded 
by four others, then the central one is destroyed; but if it con- 
sists of three or four cells adjusted so as to form a disk, then 
the stoma is caused by the separation of their sides in the cen- 
tre, by which means a sort of star is created. It is true that 


* “Recherches Anatomiques et Physiologiques sur le Marchantia polymor- 
pha,” in Nouveaux Annales du Muséum, vol. i. p. 7. 

+ Ibid. p. 93. 

} Suppl. primum Prédromi Flore Nove Hollandia, p. 3. 


REPORT ON THE PHILOSOPHY OF BOTANY. 43 


the stomata of Marchantia are in some respects different from 
what are found upon flowering plants; yet I think we can hardly 
doubt that the plan upon which they are all formed is essen- 
tially the same. 

Dutrochet also confirms * the statement of Amici, that the 
stomata are perforations; for he finds that when leaves are de- 
prived of their air by the air-pump, it is chiefly on the under 
side, where the greatest number of stomata is found, that little 
air bubbles make their appearance; and that itis through the 
stomata that water rushes into the cavernous parenchyma to 
supply the loss occasioned by the abstraction of air. 

Anther, §¢c.—Some curious remarks upon the nature of the 
tissue that lines the cells of the anther have been published by 
Dr. John E. Purkinje, Professor of Medicine at Breslau. His 
researches are chiefly directed to the determination of the na- 
ture of the tissue that is in immediate contact with the pollen; 
and he has demonstrated in an elaborate Essay}, that the opi- 
nion emitted by Mirbel in 1808}, that the cause of the dehis- 
cence of the anther is its lining, consisting of cellular tissue cut 
into slits and eminently hygrometrical, is substantially true. 
He shows that this lining is composed of cellular tissue chiefly 
of the fibrous kind, which forms an infinite multitude of little 
spriiigs, that when dry contract and pull back the valves of the 
anthers by a powerful accumulation of forces which are indivi- 
dually scarcely appreciable: so that the opening of the anther 
is not a mere act of chance, but the admirably contrived result 
of the maturity of the pollen,—an epoch at which the surround- 
ing tissue is necessarily exhausted of its fluid by the force of 
endosmosis exercised by each particular grain of pollen. 

That this exhaustion of the circumambient tissue by the en- 
dosmosis of the pollen is not a mere hypothesis, has been 
shown by Mirbel in a continuation of the beautiful memoir I 
have already so often referred to§. He finds that, on the one 
hand, a great abundance of fluid is directed into the utricles, 
in which the pollen is developed a little before the maturity of 
the latter, and that by a dislocation of those utricles the pollen 
loses all organic connexion with the lining of the anther; and 
that, on the other hand, these utricles are dried up, lacerated, 
and disorganized, at the time when the pollen has acquired its 
full development. 

* Aninales des Sciences, vol. x¥V. p. 247. 

+ De Cellulis Antherarum fibrosis. 4to. Wratislavie, 1830. 

¢ “Observations sur un Systéme d’Anatomie Comparée de. Végétaux, fondés 
sur l’'Organization de la Fleur,” in A¢émoires de l'Institut, 1808, p. 331. 


§ “Complément des Observations sur le Marchantia polymorpha,” in dr- 
chives de Botanique, vol. i. 


4A THIRD REPORT—1833. 


The Origin of the Pollen, connected as it intimately is with 
the singular phenomena of vegetable sexuality, has naturally 
been of late an object of some inquiry. To the important dis- 
coveries of the younger Brongniart and of Dr. Robert Brown, 
M. Mirbel has added some observations*, detailed with that 
admirable clearness and precision which give so great a value 
to all his writings, and which are the more interesting as they 
serve to explain what was before obscure, and to correct what 
appears to have been either inaccurately or imperfectly de- 
cribed. ‘This he has been enabled to do by beginning his in- 
quiry at the very earliest period when the organization of the 
anther can be discovered: his subject was the common Gourd. 
At a very early time the whole tissue of the anther is of the 
most perfect uniformity, consisting of cellules, the transverse 
section of which represents nearly regular hexagons and penta- 
gons. In every cell, without even excepting those which com- 
pose the superficies of the anther, are found little loose bodies, 
so exceedingly minute that a magnifying power of 500 or 600 
diameters is scarcely sufficient to examine them: they may be 
compared to transparent, nearly colourless vesicles, more or 
less round, and of unequal size. At a stage but little more ad- 
vanced, you may observe on each side of the medial line of a 
transverse section of a lobe of an anther, a collection of cellules 
rather larger than the remainder: it will afterwards be seen 
that it is here that the pollen is engendered; such cells are 
therefore called pollen-cells.. Ina bud, a line and a half or two 
lines in diameter, some remarkable alterations were found to 
have taken place; the pollen-cells had enlarged and their gra- 
nules had so much increased in number, that they nearly filled 
the cells in opake masses. ‘These granules and pollen-cells 
formed together a greyish mass, connected with the rest of the 
tissue by the intervention of a cellular membrane, which, not- 
withstanding its organic continuity with the surrounding parts, 
is at once distinguishable; for while the cells of the surrounding 
parts elongate parallel to the plane of the surface, and to the 
plane of the base of the anther, those of the cellular membrane 
elongate from the centre to the circumference. In more ad- 
vanced anthers, the sides of the pollen-cells, from being thin 
and dry, had changed to a perceptible thickness, and their sub- 
stance, gorged with fluid, resembled a colourless jelly. When 
the buds were three or four lines long, an unexpected phzno- 
menon presented itself. At first the thick and succulent walls 
of each pollen-cell dilated so as to leave an empty space between 
the inner face and the granules, not one of which separated 


* “Complement des Observations,” Sc., as above quoted. 


REPORT ON THE PHILOSOPHY OF BOTANY. 45 


from the general mass, which showed that some power kept 
them united. Shortly after, four appendages, like knife-blades, 
developed at equal distances upon the inner face of the cell, 
and gradually projected their edges towards the centre, till at 
last they divided the granular mass into four little triangular 
bodies; when the appendages had completely united at their 
edges, they divided the cavity of the pollen-cells into four di- 
stinct boxes, which then began to rounden, and finally became 
little spherical masses. Each of these was the rudiment of a 
grain of pollen, subsequently acquired a membranous integu- 
ment, hardened, became yellow, and thus arrived at maturity. 

What is perhaps most important in these observations is the 
demonstration of the original organic continuity of all the parts 
of the anther, against the statement of M. Adolphe Brongniart, 
and also against what appears to be the opinion of Dr. Brown, 
as far as can be collected from the manner in which he speaks 
of the evolution of the pollen in Tradescantia virginica*. 

Although it is not directly shown by these observations whe- 
ther the perfect grain of pollen has one or two integuments,—a 
question that may still be said to be unsettled,—it nevertheless 
appears from other instances that M. Mirbel admits the exist- 
ence of an outer not distensible coat, and of an inner highly 
extensible lining. A curious paper upon this point+ has been 
published by a Saxon botanist named Fritzsche. By means of: 
a mixture of two parts by weight of concentrated sulphuric 
acid, and five parts of water, he found that the grains of pollen 
can be rendered so transparent as to reveal their internal struc- 
ture, and that the whole process of the emission of the pollen- 
tubes can be distinctly traced. He describes the universal pre- 
sence of two coatings to the grains of pollen; and he also finds 
that the pollen contains a quantity of oily particles in addition 
to the moving corpuscles,—a fact which has also been noticed 
by Dr. Brown. 

Although the generalizations in this work are less satisfactory 
than could be desired, it must nevertheless be considered a 
most valuable collection of facts, and as containing the best 
arrangement that has as yet appeared of the various forms un- 
der which the pollen is seen. 

Fertilization. —'The road which some years since was so 
happily opened by Amici to the discovery of the exact manner 
in which vegetable fertilization takes effect, is every day be- 

coming more and more direct. The doubts of those who could 
not discern the tubes that are projected into the style by the 


* Observations upon Orchidez and Asclepiadee, p. 21. 
+ Beitrtige zur Kenntniss des Pollen. 4to. Berlin, 1833. 


46 THIRD REPORT—1833. 


pollen, have been removed; the important demonstration by 
Dr. Brown of the universal presence of a passage through the 
integuments of the ovulum at the point of the nucleus has been 
extended and confirmed by M. Mirbel in a paper of the high- 
est interest *; the fact that it is at the point of the nucleus (where 
this passage exists,) that the nascent embryo makes it appear- 
ance, is now undisputed; the passage of the contents of the 
pollen down the pollen-tubes, and the curious discovery of a 
power of motion in the granules that are thus emitted, are also 
recognised: it now only remains to be proved that the pollen- 
tubes come in contact with the nucleus, and the whole secret of 
fertilization is revealed. A few remarkable contributions to 
this part of the subject have lately been made. 

Some plants have the passage or foramen in their ovulum so 
remote from any part through which the pollen-tubes can be 
supposed to convey their influence, as to have thrown consider- 
able difficulty in the way of the supposition that actual contact 
between the point of the nucleus and the fertilizing tissue is 
indispensable. 

The manner in which, notwithstanding the apparent difficulty 
of such contact taking place, this happens in Statice Armeria, 
was long since made out by Dr. Brown, in whose possession I 
several years ago saw drawings illustrating this phenomenon ; 
it has since been explained by M. Mirbel. Another case, pre- 
senting similar apparent difficulties, occurs in Helianthemum. 
In plants of that genus the foramen is at that end of the ovulum 
which is most remote from the hilum; and although the ovula 
themselves are elevated upon cords much longer than are usually 
met with, yet there are no obvious means of their coming in con- 
tact with any part through which the matter projected into the 
pollen-tubes can be supposed to descend. It has, however, 
been ascertained by M. Adolphe Brongniart+, that at the time 
when the stigma is covered with pollen, and fertilization has 
taken effect, there is a bundle of threads, originating from the 
base of the style, which hang down in the cavity of the ovarium, 
and, floating there, are abundantly sufficient to convey the in- 
fluence of the pollen to the points of the nuclei. So again in 
Asclepiadee. In this tribe, from the peculiar conformation of 
the parts, and from the grains of pollen being all shut up in a 
sort of bag, out of which there seemed to be no escape, it was 
supposed that this tribe must at least form an exception to the 
general rule. But before the month of November 1828}, the 

* Nouvelles Recherches sur la Structure de U Ovule Végétal et sur ses Déve- 


loppements. ‘Also Additions aux ‘Nouvelles Recherches,’ §c. 
+ Annales des Sciences, vol. xxiv. p. 123. t Linnea, vol. iv. p. 94. 


———— — 


REPORT ON THE PHILOSOPHY OF BOTANY. 47 


celebrated Prussian traveller and botanist Ehrenberg had dis- 
covered that the grains of pollen of Asclepiadee acquire a sort 
of tails which are all directed to a suture of their sac on the side 
next the stigma, and which at the period of fertilization are 
lengthened and emitted; but he did not discover that these 
tails are only formed subsequently to the commencement of a 
new vital action connected with fertilization, and he thought 
that they were of a different nature from the pollen-tubes of 
other plants; he particularly observed in Asclepias syriaca 
that the tails become exceedingly long and hang down. 

In 1831 the subject was resumed by Dr. Brown* in this 
country, and by M. Adolphe Brongniart + in France, at times so 
nearly identical, that it really seems to me impossible to say 
with which the discovery about to be mentioned originated : it 
will therefore be only justice if the Essays referred to are spoken 
of collectively instead of separately. ‘These two distinguished 
botanists ascertained that the production of tails by the grains 
of pollen was a phenomenon connected with the action of ferti- 
lization; they confirmed the existence of the suture described 
by Ehrenberg ; they found that the true stigma of Asclepia- 
dee is at the lower part of the discoid head of the style, and 
so placed as to be within reach of the suture through which the 
pollen-tubes or tails are emitted; they remarked that the latter 
insinuate themselves below the head of the style, and follow its 
surface until they reached the stigma, into the tissue of which 
they buried themselves so perceptibly that they were enabled to 
trace them, occasionally, almost into the cavity of the ovarium; 
and thus they established the highly important fact, that this 
family, which was thought to be one of those in which it was 
impossible to suppose that fertilization takes place by actual 
contact between the pollen and the stigma, offers the most 
beautiful of all examples of the exactness of the theory, that it 
is at least owing to the projection of pollen-tubes into the sub- 
stance of the stigma. In the more essential parts these two 
observers are agreed: they, however, differ in some of the de- 
tails; as, for instance, in the texture of the part of the style 
which I have here called stigma, and into which the pollen- 
tubes are introduced. M. Brongniart both describes and figures 
it as much more lax than the contiguous tissue, while on the 
other hand Dr. Brown declares that he has in no case been able 
‘to observe “the slightest appearance of secretion, or any dif- 


* Observations on the Organs and Mode of Fecundation of Orchidew and 
Asclepiadez. London, October 1831. 

+ Annalesides Sciences for October and November 1831 ; from observations 
made in July, August and September of that year. 


oOo 


48 THIRD REPORT—18338. 


ferences whatever in texture between that part and the general 
surface of the stigma” (meaning what I have described as the 
discoid head of the style): but this is not the place for entering 
into the discussion of these subordinate points. 

Orchidee are another tribe in which similar difficulties have 
been found in reconciling structure with the necessity of con- 
tact between the pollen and stigma in order to effect impregna- 
tion. Indeed it seems in these plants as if every possible pre- 
caution had been taken by nature to prevent such contact. 
Nevertheless it is represented by M. Adolphe Brongniart, in a 
paper read before the Academy of Sciences of Paris in July 
1831 *, that contact is as necessary in these plants as in others, 
and that in the emission of pollen-tubes they do not differ from 
other plants. These statements have been followed up by Dr. 
Brownj, in en elaborate Essay upon the subject, in which the 
results that are arrived at by our learned countryman are es- 
sentially to the same effect. To these there is at present no- 
thing equally positive to oppose; but as the indirect observa- 
tions of Mr. Bauer}, and the general structure of the order, are 
very much at variance with the probability of actual contact 
being necessary, and especially as Dr. Brown is obliged to have 
_ recourse to the supposition that the pollen of many of these 
plants must be actually carried by insects from the boxes in 
which it is naturally locked up,—it must be considered, I think, 
that the mode of fertilization in Orchidec is still far from being 
determined. I must particularly remark that the very proble- 
matical agency of insects, to which Dr. Brown has recourse in 
order to make out his case, seems to be singularly at variance 
with his supposition§ that the insect forms, which in Ophrys 
are so striking, and which he finds resemble the insects of the 
countries in which the plants are found, are intended rather to 
repel than to attract. It may be true, as Dr. Brown observes, 
that there is less necessity for the agency of insects in such 
flowers as the European Ophryde@; but what other means 
than the assistance of insects can be supposed to extricate the 
pollen from the cells in the insect flowers of such plants as 
Renanthera Arachnites, the whole genus Oncidium, Tetramicra 
rigida, several species of Epidendrum, Cymbidium tenuifolium, 
Vanda peduncularis, and a host of others ? 

* Annales des Sciences, vol. xxiv. p. 113. 

+ Observations upon the Organs and Mode of Fecundation of Orchidex and 
Asclepiadez. 

t Illustrations of the Genera and Species of Orchideous Plants. Part II. 
“‘ Fructification,” tabb. 5. 12. 13. 14. 


§ “ Proceedings of the Linnean Society,”’ June 5, 1832, as given in the Lon- 
don and Edinburgh Philosophical Magaxine and Journal. 


’ 


REPORT ON THE PHILOSOPHY OF BOTANY. 49 


Origin of Organs.—There is no part of vegetable physiology 
so obscure as that which relates to the origin of organs. We 
find a degree of simplicity that is perfectly astonishing in the 
fundamental structure of the whole vegetable kingdom; we 
are able to prove by rigorous demonstration that every one of 
the appendages of the axis is a modification of a leaf, to which 
there is a constant tendency to revert; we see that in some cases 
a part which usually performs one function assumes another, 
as in the Alstrémerias, whose leaves by a twist of their petiole 
turn their under surface upwards: but we are entirely ignorant 
of the causes to which these changes are owing. An impor- 
tant step in elucidating the subject has been lately taken by 
M. Mirbel, in his memoir upon the structure of Marchantia po- 
lymorpha. The young bulbs by which this plant is multiplied 
are originally so homogeneous in structure, that there is no 
apparent character in their organization to show which of their 
faces is destined to become the upper surface, and which the 
under. For the purpose of ascertaining whether there existed 
any natural but invisible predisposition in the two faces to un- 
dergo the changes which subsequently become so apparent, 
and by means of which their respective functions are performed, 
or whether the tendency is given by some cause posterior to 
their first creation, the following experiments were instituted. 
Five bulbs were sown upon powdered sandstone, and it was 
found that the face which touched the sandstone produced 
roots, and the opposite face formed stomata. It was, however, 
possible that, the five bulbs might have all accidentally fallen 
upon the face which was predisposed to emit roots; other 
experiments of the same kind were therefore tried, first with 
eighty and afterwards with hundreds of little bulbs,—and._ the 
result was the same as with the five. This proved that either 
face was originally adapted for producing either roots or sto- 
mata, and that the tendency was determined merely by the po- 
sition in which the surfaces were placed. The next point to 
ascertain was, whether the tendency once given could be after- 
wards altered; some little bulbs, that had been growing for 
twenty-four hours only, had emitted roots; they were turned, ~ 
so that the upper surface touched the soil, and the under was 
exposed to light. : In twenty-four hours more the two faces 
had both produced roots; that which had originally been the 
under surface went on pushing out new roots; that which had 
originally been the upper surface had also produced roots: 
but in a few days the sides of the young plants began to rise 


from the soil, became erect, turned over, and finally recovered 
1833. E Ba 


50 THIRD REPORT—1833. 


in this way their original position, and the face which had ori- 
ginally been the uppermost, immediately became covered with 
stomata. It, therefore, appears that the impulse once given, 
the predisposition to assume particular appearances or func- 
tions is absolutely fixed, and will not change in the ordinary 
course of nature. ‘This is a fact of very high interest for 
those who are occupied in researches into the causes of what is 
called vegetable metamorphosis, an expression which has been 
justly criticised as giving a false idea of the subject to which 
it relates. : 

_ Morphology.—When those who first seized upon the im- 
portant but neglected facts out of which the modern theory of 
morphology has been constructed, asserted that all the appen- 
dages of the axis of a plant are metamorphosed leaves, more 
was certainly stated than fhe evidence would justify; for we 
cannot say that an organ is a metamorphosed leaf, which in 
point of fact has never been a leaf. What was meant, and that 
which is supported by the most conclusive evidence, is, that 
every appendage of the axis, whether leaf, bractea, sepal, petal, 
stamen, cr pistillum, is originally constructed of the same ele- 
ments, arranged upon a common plan, and varying in their 
manner of development, not on account of any original differ- 
ence in structure, but on account of special and local predis- 
posing causes: of this the leaf is taken as the type, because it 
is the organ which is most usually the result of the develop- 
ment of those elements,—is that to which the other organs 
generally revert, when from any accidental disturbing cause 
they do not assume the appearance to which they were originally 
predisposed,—and, moreover, is that in which we have the most 
complete state of organization. 

This is not a place for the discussion of the details upon 
which the theory of morphology is founded; it is sufficient to 
state that it has become the basis of all philosophical views of 
structure, and an inseparable part of the science of botany. Its 
practical importance will be elucidated by the following circum- 
stance. Fourteen or fifteen years ago I was led to take a 
view of the structure of Reseda very difterent from that usually 
assigned to the genus; and when a few years afterwards that 
view was published, it attracted a good deal of attention, and 
gained some converts among the botanists of Germany and 
France. It was afterwards objected to by Dr. Brown upon 
several grounds; but I am not aware that they were considered 
sufficiently valid to produce any change in the opinions of those 
who had adopted my hypothesis. Lately, however, Professor 


REPORT ON THE PHILOSOPHY OF BOTANY. 51 


Henslow has satisfactorily proved*, in part by the aid of a 
monstrosity in the common Mignonette, and in part by a severe 
application of morphological rules, that my hypothesis must 
necessarily be false ; and I am glad to have this opportunity of 
expressing my full concurrence in his opinion. 

It has long been known that the ligulate and tubular corollas 
of Composite are anatomically almost identical, and that their 
difference consists only in the five petals of the tubular corolla 
all separating regularly for a short distance from their apex, 
while the five petals of the ligulate corolla adhere up to their 
very points, except on the side next the axis of inflorescence, 
where two of them are altogether distinct except at their base. 
M. Leopold von Buch explains this circumstance in the follow- 
ing manner. He states that these ligulate corollas when unex- 
panded bear at their point a little, white, and very viscid body 
or gland, which is a peculiar secretion that dries up when it 
comes in contact with the atmosphere. The adhesion of this 
gland is too powerful to be overcome by the force of the style 
and stamens pressing against it from within. The corollas, which 
are gradually curved outwards by the growth of those in the 
centre of the inflorescence, at the same time bend down the 
style, which consequently presses up against the line of union 
of the two petals nearest the axis: although the style cannot 
overcome the adhesion of the viscid gland at the point of the 
corolla, it is able in time to destroy the union of the two inte- 
rior petals, which finally give way and allow the stamens and 
style toescape. As soon as this takes place, the corolla can no 
longer remain erect, but falls back towards the circumference 
of the capitulum, and thus contributes to the radiating character 
of this sort of inflorescence. When the viscid body is either 
not at all, or very imperfectly produced at the point of the co- 
rolla, as sometimes happens in the genus Hieracium, especially 
HZ. bifurcum, tubular corollas are produced instead of ligulate 
ones. 

The ovulum is the organ where the greatest difficulty has 
occurred in reducing the structure to anything analogous to 
that of other parts. It is true that Du Petit Thouars regarded 
it as analogous to aleaf bud; but his view appears to have been 
purely hypothetical, for I am not aware that he had any distinct 
evidence of the fact. Some years ago M. Turpin, in showing 
the great similarity that exists between the convolute bractez 
of certain Marcgraviacee and the exterior envelope of the 
ovulum, took the first step towards proving that the hypothesis 


* Transactions of the Philosophical Society of Cambridge, vol. v. Part I. 
EZ 


52 THIRD REPORT—1833. 


of Du Petit Thouars was susceptible of demonstration ; it was’ 
more distinctly shown by the interesting discovery of Professor 
Henslow, that the leaves of Malaxis paludosa had on their 
margins what no doubt must be considered buds, but what in 
structure are an intermediate state between buds and ovula; 
and it has been recently asserted by Engelmann’*, still, how- 
ever, without the production of any proof, that “ovula are buds 
of,a higher kind, their integuments leaves, and their funiculus 
the axis, all which, in cases of retrograde metamorphosis, are in 
fact converted into stem and green leaves.” The nearest ap- 
proach to a demonstration that has yet been afforded of ovula 
being buds is in a valuable paper by Professor Henslow, just 
printed inthe Transactions of the Philosophical Society of Cam- 
bridge}, in which it is shown that in the Mignonette the ovula 
are in fact transformed occasionally into leaves, either solitary 
or rolled together round an axis, of which the nucleus is the 
termination. ; 

M. Dumortier has endeavoured to prove t that the embryo 
itself is essentially the same as a single internodium of the stem 
with its vital point or rudimentary bud attached to it. Although 
the author’s demonstration is a failure, and his paper a series 
of confused and illogical reasoning, yet there can be little doubt 
that the hypothesis itself is a close approximation to the truth. 

Dr. George Engelmann has recently attempted § to classify 
the aberrations from normal structure, which throw so much 
light upon the real origin and nature of the organs of plants. 
He has collected a very considerable number of cases under 
the following heads. 1. Retrograde metamorphosis (Regressus), 
when organs assume the state of some of those on the outside 
of them, as when carpella change to stamens or petals, hypo- 
gynous scales to stamens, stamens to pétals or sepals, sepals to 
ordinary leaves, irregular structure to regular, and the like. 
2. Foliaceous metamorphosis (Virescentia), when all the parts 
of a flower assume more or less completely the state of leaves. 
3. Disunion (Disjunctio), when the parts that usually cohere 
are separated, as the carpella of a syncarpous pistillum, the 
filaments of monadelphous stamens, the petals of a monopeta- 
lous corolla, &c. 4. Dislocation (Apostasis); in this case the 
whorls of the flower are broken up by the extension of the . 
axis. 5. Viviparousness (Diaphysis), when the axis is not only 
elongated, but continues to grow and form new parts, as in those 


* De Antholysi Prodromus, p. 61. + vol. v. Part I. 
{ Nova Acta Academie Nature Curiosorum, vol. xvi. p. 245, 
§ De Antholysi Prodromus. 


REPORT ON THE PHILOSOPHY OF BOTANY. 53 


instances where one flower grows from within another. And 
finally, 6. Proliferousness (Ecblastesis), when buds are deve- 
loped in the axille of the floral organs, so as to convert a sim- 
ple flower into a mass of inflorescence. A very considerable 
number of instances are adduced in illustration of these divi- 
sions, and the work will be found highly useful as.a collection 
of curious or important facts. 

- The doctrines of morphology, and the evidence in support 
of them, may now be considered so far settled as to require but 
little further illustration for the present. This is, however, 
only true of flowering plants: in the whole division of flower- 
less plants there has been scarcely any attempt to discover the 
analogy of organs, and to reduce their structure to a correspond- 
ing state of identification. 1 some time since* endeavoured to 
excite attention to this subject, by hazarding some speculations 
which had at least the merit of novelty to recommend them ; 
but I cannot discover that any one has since turned his atten- 
tion to the inquiry, although it must be confessed that the com- 
parative anatomy of flowerless plants is among the most inter- 
esting topics still remaining for discussion, and that it is rather 
discreditable to Cryptogamic botanists that the elucidation of 
so very curious a matter should be postponed to the compara- 
tively unimportant business of distinguishing or dividing genera 
and species. 

Gradual Development.—The theory of the gradual deve- 
lopment of the highest class of organic bodies, in consequence 
of a combination and complication of the phenomena attendant 
upon the development of the lowest classes, has acquired so 
great a degree of probability among animals, that it has become 
a question of no small interest whether traces of the same, or a 
similar law, cannot be found among plants. In an inquiry of 
such a nature, it seems obvious that attention should in the first 
instance be directed to a search after positive and incontestable 
facts, and that mere hypotheses should in the beginning be to- 
tally rejected. The only circumstances that occur to me as 
bearing directly upon this point are the following. It has been 
ascertained by M. Mirbel, in his memoir on the Marchantia, 
that the sporule of that very simple plant is a single vesicle, 
which, when it begins to grow, produces other vesicles on its 
surface, which go on propagating in the same manner, every 
new vesicle engendering others; and that different modifica- 
tions of this process produce the different parts that the per- 
fect plant finally develops. 


* Outlines of the First Principles of Botany, p. 533, &c. Introduction to the 
Natural System of Botany, p. 313, &c. , 


54 THIRD REPORT—1830. 

The same principle of growth appears to obtain in Conferve, 
and probably is found in other vegetables of the lowest grade. 

This is analogous to what takes place in the formation of the 
embryo of Vasculares. In the opinion of Dr. Brown and of 
Mirbel, the first rudiment of a plant far more complicated than. 
Marchantia, consists also of a vesicle, but suspended by a 
thread to the summit of the cavity of the ovulum; and the dif- 
ference between the one case and the other is, that while in the 
Marchaniia the original vesicle, “‘ as soon as it is formed, pos- 
sesses all the conditions requisite for developing a complete 
plant on the surface of the soil; on the other hand, that of 
flowering plants must, on pain of death, commence its deve- 
lopment in the interior of the ovulum, and cannot continue it. 
further until it has produced the rudiments of root, stem, and 
cotyledons *. 

Beyond this I do not think that any attempt has been made 
to elucidate the question. 

Trritability.—Dyr. Dutrochet has published} the result of 
some experiments with the air-pump upon the pneumatic system 
of plants. Independently of confirming the fact, already gene- 
rally known, of plants having the means of containing a large 
quantity of air, he arrived at the unexpected result, that the 
sleep of plants and their irritability are certainly dependent upon 
the presence of air within them. A sensitive plant, left in the 
vacuum of an air-pump for eighteen hours, indicated no sign 
whatever of the accustomed collapse of its leaflets on the ap- 
proach of night, nor when it was restored to the air could it be. 
stimulated by the smartest shocks; but in time it recovered its 
irritability. When flowers that usually close at night were 
placed in a vacuum while expanded, they would not close; and 
when flowers already closed. were placed in the same situation, 
they would not unfold at the return of morning; whence Dr. 
Dutrochet infers that the internal air of plants is indispensably 
necessary to the exercise of their alternate motions of sleeping 
and waking, and in general to the existence of the faculty they 
possess of indicating by their movements the influence of ex- 
ternal exciting causes. 

Action of Coloured Light.—Professor Morren, of Ghent, has 
mentioned { the result of some experiments upon the action of 
the coloured rays upon germination; and he has found that 
while those rays in which the illuminating power is the most 
feeble were, as might have been expected, the most favourable 
to germination, their power of decomposing carbonic acid, and 


* Archives de Botanique, vol.i. + Annales des Sciences, vol. xxv. p. 243. 
{ Annales des Sciences, vol. xxvii. p. 201. 


~ 


REPORT ON THE PHILOSOPHY OF BOTANY. 55 


producing a green deposit in the parenchyma, is in proportion 
to their illuminating property; that no decomposed rays effect 
this so rapidly as white light; and that the yellow ray possesses 
the greening power in the highest degree, the orange in a very 
slight degree, and violet, red and purple not at all. 
Colours.—Nothing can be named in the whole range of bo- 
tany upon which information is so much wanted as the cause of 
the various colours of plants. It was, indeed, long since sus- 
pected by Lamarck that the autumnal colouring of leaves and 
’ fruits was a morbid condition of those parts; and it has subse- 
quently been ascertained that all colours are owing to the pre- 
sence of a substance, called ehromule by De Candolle, which 
fills the parenchyma, assuming different tints. Green has also 
been clearly made out to be connected with exposure to light, 
and has been considered to be in all probability owing to the 
deposition of the carbon left upon the decomposition of car- 
bonic acid. Some botanists have also observed the connexion 
of red colour with acidity ; but still we had scarcely any positive 
knowledge of the cause of the production of any colour except 
green, till M. Macaire of Geneva* remarked, that just before 
leaves begin to change colour in the autumn, they cease parting 
with oxygen in the day, although they go on absorbing it at 
night; whence he concluded that their chromule is oxygenated, 
by which a yellow colour is first caused, and then a red,—for he 
found that in all cases a change to red is preceded by a change 
to yellow. He also ascertained that the chromule of the red 
bractez and calyx of Salvia splendens is chemically the same 
as that of autumnal leaves. Coupling this with the fact that 
petals do not part with oxygen, it would seem as if their colour, 
if yellow or red, may also be owing to a kind of oxygenation. 
But according to M. Theodore de Saussure}, coloured fruits 
part with their oxygen; so that, if this be true, red and yellow 
cannot always be ascribed to such a cause. M. De Candollet 
has some excellent observations upon this subject in his recent 
admirable digest of the laws of vegetable physiology ; in which 
he concludes, from the inquiries hitherto instituted, that all co- 
lours depend upon the degree of oxygenation. When oxygen 
is in excess, the colour seems to tend to yellow or red; and 
when it is deficient, or when the chromule is more carbonized, 
which is the same thing, it has a tendency to blue. Local ad- 
ditions of alkaline matters are also called in aid of an explanation 
of the various shades of colour that flowers and fruits present. 


* Mémoires de la Société Physique de Généve, vol. iv. p. 50. 
+ Ibid. vol. i. p. 284. » + Physiologie Végétale, p. 906. 


56 THIRD REPORT—1833. 


Dr. Dutrochet is of opinion* that the whitish spots we some- 
times see in leaves, and the paler tint that generally character- 
izes the under side of the same organs, are owing to the presence 
of air beneath the cuticle. He finds that the arrow-head shaped 
blotch on the upper side of the leaf of Trifolium pratense, and 
the whitish spots on Pulmonaria officinalis, disappear when the 
leaves are plunged in water beneath the exhausted receiver of 
the air-pump, and that the lower surface of leaves acquires the 
same depth of colour as the upper under similar circumstances. 
This he ascribes to the air naturally found in the leaves being 
abstracted, and its place supplied with water; a conclusion 
which agrees with what might be inferred from the anatomical 
structure of the parts in question. 

Excretions.—It has long been known that some plants are 
incapable of growing, or at least of remaining in a healthy state, 
in soil in which the same species has previously been cultivated. 
For instance, a new apple orchard cannot be made to succeed 
on the site of an old apple orchard, unless some years inter- 
vene between the destruction of the one and the planting of 
the other: in gardens, no quantity of manure will enable one 
kind of fruit-tree to flourish on a spot from which another tree 
of the same species has been recently removed ; and all farmers 
practically evince, by the rotation of their crops, their expe- 
rience of the existence of this law. 

Exhaustion of the soil is evidently not the cause of this, for 
abundant manuring will not supersede the necessity of the usual 
rotation. The celebrated Duhamel long ago remarked, that the 
Elm parts by its roots with an unctuous dark-coloured substance ; 
and, according to De Candolle, both Humboldt and Plenck 
suspected that some poisonous matter is secreted by roots; 
but it is to M. Macaire, who at the instance of the first of these 
three botanists undertook to inquire experimentally into the 
subject, that we owe the discovery of the suspicion above al- 
luded to being well founded. He ascertained} that all plants 
part with a kind of fecal matter by their roots, that the nature 
of such excretions varies with species or large natural orders: 
in Cichoracee and Papaveracee he found that the matter was 
analogous to opium, and in Leguminose to gum; in Graminee 
it consists of alkaline and earthy alkalies and carbonates, and 
in Euphorbiacee of an acrid gum-resinous substance. These 
excretions are evidently thrown off by the roots on account of 
their presence in the system being deleterious ; and it was found 
by experiment, that plants artificially poisoned parted with the 


* Annales des Sciences, vol. xxv. p. 246. 


¢ De Candolle, Physiologie Végétale, p. 249. 


REPORT ON THE PHILOSOPHY OF BOTANY. 57 


poisonous matter by their roots. For instance, a plant of Mer- 
curialis had its roots divided into two parcels, of which one was 
immersed in the neck of a bottle filled with a weak solution of 
acetate of lead, and the other parcel was plunged into the neck 
of a corresponding bottle filled with pure water. In afew days 
the pure water had become sensibly impregnated with acetate 
of lead. ‘This, coupled with the well known fact that plants, 
although they generate poisonous secretions, yet cannot absorb 
them by their roots without death, as, for instance, is the case 
with Atropa Belladonna, seems to prove that the necessity of 
the rotation of crops is more dependent upon the soil being 
poisoned than upon its being exhausted. 

‘This is a part of vegetable physiology of vast importance to 
an agricultural country like England, and may possibly cause 
a total revolution in our system of husbandry. 

All that M. Macaire can be said as yet to have done, is to 
have discovered the fact and to have pointed out certain strong 
examples of it; but if the discovery is to be converted toa 
practically useful purpose, we require positive information upon 
the following points :— 

1. The nature of the fecal excretions of every plant culti- 
vated by the farmer. 

2. The nature of the same excretions of the common weeds 
of agriculture. 

3. The degree in which such excretions are poisonous to the 
plants that yield them, or to others. 

4. The most ready means of decomposing those excretions 
by manures or other means. ‘ 

It would be superfluous to point out what the application 
would be of such information as this; but I cannot forbear ex- 
pressing a hope that a question upon which so many deep inter- 
ests are involved may be among the first to occupy the atten- 
tion of the chemists of the British Association. 


med anni paren 
Sa fe ashe “fyb bitte teat toa, ¢ ietay ati copa 
abe aahe Regan ine y scree tigarvanry: sails ro 
Sy soa ar eave teD Hise Debit ih 


BE ppibenty’ at 
Sar vitae tO 
es ae ncne eee 
(ey riviera ce ; 
ah La rt. Sf a eR a aR 
-Biape tieetl ae etme sree a 
Pa epee hed fu 
canted esi wt iw. 
wisi moat 0 
wee serie era.'S veneer 
ahepttawre' one Tag 
ha gia he Fetes asst AS 
pre dooky cocrtlive' alee peeing? os piped ster gen cyt bhedea 
> —gppeanaaiedn oF rvaterncey Bat pnts stad donatteetent 
eat epost e ephicceyaa honda setter matters eAY photo” wee | 
toh ay iw gH tae et al rertbaeb giverceot gonna de abort me | 
iio siamiainanel vocal ot pines ee 
oN nh ThE ea fe bi ] ee Ca tala e * 
At RU, RA i yi oe 
ayers Sas ett | be ities oe Syria ‘M ; ! : 
. Pea eronr RG ee eae oi Es NE Ta Feo eee | 
ary wnat sth ids A Hih gh ett valle gi Cola gah og BR soy t penetnteag +: ul 
> Rc i al Rape: ae ete a 
at wea 42, % adr, wetter | 7 
piitial eraeaikest mere oF pio a pll te 
pe Ag dis > ates Ae y Aa ieee 
Pegi = \ 
bed 4 y 


[ 59 J 


Report on the Physiology of the Nervous System. By Wit- 
-uiam Cuartes Henry, M.D., Physician to the Manchester 
Royal Infirmary. 


Introduction—Tux science of Physiology has for its object to 
ascertain, to analyse, and to classify the qualities and actions 
which are peculiar to living bodies. 'These vital properties re- 
side exclusively in organized matter, which is characterized by 
a molecular arrangement, not producible by ordinary physical 
attractions and laws. Matter thus organized consists essen- 
tially of solids, so disposed into an irregular network. of laminz 
and filaments, as to leave spaces occupied by fluids of various 
natures. ‘Texture’ or ‘tissue’ is the anatomical term by which 
such assemblages are distinguished. Of these the cellular, or 
tela cellulosa, is most elementary, being the sole constituent of 
several, and a partial component of all tissues and. systems. 
Thus the membranes and vessels consist entirely of condensed 
cellular substance ; and even muscle and nerve are resolvable, 
by microscopic analysis, into globules deposited in attenuated 
cellular element. , 

But though the phenomena, which are designated as vital, 
are never found apart from organization, and have even by 
some naturalists been regarded as identical with it, yet in the 
order of succession vital actions seem necessarily to stand to 
organized structures in the relation of antecedents; for the 
production of even the most rudimentary forms and textures 
implies the previous operation of combining tendencies or ‘ vital 
affinities’. ‘The origin and early development of these vital 
tendencies, and of organized structures, are beyond the pale of 
exact or even of approximative knowledge. But it is matter of 
certainty, that life is the product only of life; that every new 
plant or animal proceeds from some pre-existent being of the 
same form and character; and thus that the image of the great 
Epicurean poet, ‘‘ Quasi cursores vitai lampada tradunt,” pos- 
sesses a compass and force of illustration which, as a supporter 
of the doctrine of fortuitous production, he could not have him- 
self contemplated. 

The popular notions respecting life are obscure and indeter- 
minate; nor are the opinions even of philosophers characterized 
by much greater distinctness or mutual accordance. Like other 
complex terms, ‘life’ can obviously be defined only by an enume- 


60 THIRD REPORT—1833. 


ration of the phenomena which it associates. This enumera- 
tion will comprehend a greater or a smaller number of particu- 
lars, according to the station in the scale of living beings which 
is occupied by the object of survey. In its simplest manifesta- 
tion, the principle of life may be resolved into the functions of 
nutrition, secretion and absorption. It consists, according to 
Cuvier, of the faculty possessed by certain combinations of 
matter, of existing for a certain time and under a determinate 
form, by-attracting unceasingly into their composition a part of 
surrounding substances, and by restoring portions of their own 
substance to the elements. ‘This definition comprehends all 
the essential phenomena of vegetable life. Nutritive matter is 
drawn from the soil by the spreading fibres of the root, through 
the instrumentality of spongioles or minute turgid bodies at 
their extremities, which act, according to Dutrochet, by a power 
which he has called ‘endosmosis.’ The same agency raises the 
nutrient fluid through the lymphatic tubes to the leaves, where 
it seems to undergo a kind of respiratory process, and becomes 
fit for assimilation. These changes, and the subsequent pro- 
pulsion of the sap to the different parts and textures, plainly 
indicate independent fibrillary movements, which are repre- 
sented in animal life by what Bichat has termed ‘the pheno- 
mena of organic contractility’. ‘The power residing in each 
part of detecting in the circulating fluid, and of appropriating, 
matters fitted to renovate its specific structure, is designated in 
the same system by the term ‘organic sensibility’. 

Ascending from the vegetable to the animal kingdom, the 
term ‘life’ advances greatly in comprehensiveness. ‘The exist- 
ence of a plant is limited to that portion of space in which acci- 
dent or design has inserted its germ; while animals are for 
the most part gifted with the faculties of changing their place, 
and of receiving from the external world various impressions. 
Along with the general nutritive functions, the higher attri- 
butes of locomotion and sensation are therefore comprised in 
the extended compass of meaning which the term ‘life’ acquires 
with the prefix ‘animal’. The nutritive functions, too,emerging 
from their original simplicity, are accomplished by a more com- 
plex mechanism, and by agencies further removed from those 
which govern the inanimate world. 

Locomotion is effected either by means of a contractile tissue, 
or of distinct muscular fibres. ‘These fibres have been said to 
consist of globules resembling, and equal in magnitude to, those 
of the blood, disposed in lines, in the elementary cellulosity, which 
by an extension of the analogy is compared to serum. But the 
latest microscopical observations of Dr. Hodgkin are opposed 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 61 


to this globular constitution of the contractile fibre. ‘‘ Innu- 
merable very minute but clear and fine parallel lines or strie 
may be distinctly perceived, transversely marking the fibrillz.” 
Irritability, or the faculty of contracting on the application of a 
stimulant, is a property inherent in the living fibre. It is an 
essential element of all vital operations, except of those which 
have their seat in the nervous system, such as sensation, voli- 
tion, the intellectual.states, and moral affections. All the phe- 
nomena of life, in the higher animals, may then be ultimately 
resolved into the single or combined action of these two ele- 
mentary properties,—irritability and nervous influence, each 
residing in its appropriate texture and system. 

These preliminary remarks are designed to unfold the prin- 
ciples to be followed in classifying the vital functions. In ge- 
neral or comparative physiology, a strictly scientific arrangement 
would contemplate first the phenomena of the most elementary 
life, and would successively trace the more perfect development 
of those simple actions and their gradual transition into more 
complex processes, as well as the new functions, superadded 
in the ascending scale of endowment. But such a mode of 
classification is wholly inapplicable to the particular physiology 
of man and of the more perfect animals, viewed by itself and 
without reference to inferior orders of beings; for the nutri- 
tive functions of this class, which correspond with the elemen- 
tary actions of the simplest vegetable life, are effected by a 
complex system of vessels and surfaces, deriving their vital 
powers from contractile fibres, and controlled, if not wholly 
governed, by nervous influence. It is then manifest, that in 
the higher physiology the general laws of contractility and ‘in- 
nervation’ must precede the description of the several functions, 
which all depend on their single or united agency. The parti- 
cular functions will afterwards be classed, as they stand in more 
immediate relation to one or other of the two essential princi- 
ples of life. 

In the present state of physiological knowledge, it is impos- 
sible to determine absolutely, and without an opening to con- 
troversy, whether the functions of muscle or those of nerve are 
entitled to precedency. If each were equally independent of 
the other in the performance of their several offices, the question 
of priority would resolve itself into one of simple convenience. 
The actions of the nervous system, if contemplated for the short 
interval of time during which they are capable of persisting 
without renovation of tissue, are entirely independent of the 
contractile fibre. But it is certain that the cooperation of 
nerve is required in most, if not in all, the actions of the mus- 


62 THIRD REPORT—1833. 


cular system. Thus the voluntary muscles in all their natural 
and sympathetic contractions receive the stimulant impulse of 
volition through the medium of nerve; and though the mode, 
in which the motive impression is communicated to the invo- 
luntary muscles, is still matter of controversy, there seems suffi- 
cient evidence * to sanction the conclusion that nerve is in this 
case also the channel of transmission ;—“ that the immediate 
antecedent of the contraction of the muscular fibre is univer- 
sally a change in the ultimate nervous filament distributed to 
that fibre.” If this be correct, the physiological history of 
muscle’cannot be rendered complete without reference to that 
of nerve. 

In the higher manifestations of life, nervous matter is in- 
vested with the most eminently vital attributes. It is the ex- 
clusive seat of the various modes of sensation, and of all the 
intellectual operations; or, rather, it is the point of transition, 
where the physical conditions of the organs, which are induced 
by external objects, pass into states of mind, becoming per- 
ceptions; and where the mental act of volition first impresses 
a change on living matter. These two offices of conducting 
motive impressions from the central seat of the will to the mus- 
cles, and of propagating sensations from the surface of the body 
and the external organs of sense to the sensorium commune, 
have been of late years shown to reside in distinct portions of 
nervous substance. 

The honour of this discovery, doubtless the most important 
accession to physiological knowledge since the time of Harvey, 
belongs exclusively to Sir Charles Bell. It constitutes, more- 
over, only a part of the new truths, which his researches have 
unveiled, regarding the general laws of nervous action, and the 
offices of individual nerves. His successive experiments on 
function, guided always by strong anatomical analogies in struc- 
ture, in origin, or in distribution, have led to the entire remo- 
delling of nervous physiology, and to the formation of a system 
of arrangement, based on essential affinities and on parity of 
intimate composition, instead of on apparent sequence or prox- 
imity of origin. Among the continental anatomists, MM. Ma- 
gendie and Flourens have contributed most largely to our 
knowledge of this part of physiology; the former by repeating 
and confirming the experiments of Bell, as well as by various 
original inquiries; the latter by his important researches into 
the vital offices of the brain and its appendages. Much light, 


_ * See “A Critical and Experimental Enquiry into the Relations subsisting 
between Nerve and Muscle,” in the 37th vol. of the Edinburgh Medical and 
Surgical Journal. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 63 


too, has been thrown on the functions of several of the ence- 
phalic nerves, and especially of those supplying the face and 
its connected cavities, by Mr. Herbert Mayo, who has analysed 
their anatomical composition, and pursued their course with 
singular precision, and has thus been enabled to correct some 
errors of detail in the system of Sir Charles Bell. 

Nervous System.—In man, and in other vertebrated animals, 
the nervous system consists of the cerebrum, cerebellum, me- 
dulla oblongata, medulla spinalis, and of the encephalic, spinal, 
and ganglionic nerves. It seems most natural to observe this 
order of anatomical sequence in recording what is known of 
nervous functions. . 

Cerebrum, or Brain-proper.—The physiology of the brain 
has received of late years very considerable accessions, and its 
vital offices, viewed as an entire organ, have now probably been 
ascertained with sufficient precision. Some portion of this newly 
acquired knowledge has been gathered from experiments on 
living animals, but the greater and more valuable part has 
flowed from the study of comparative development. In this 
latter field of inquiry, 'Tiedemann’s elaborate history of the pro- 
gressive evolution of the human brain during the period of 
foetal existence, with reference to the comparative structure of 
that organ in the lower animals, merits an early and detailed 
notice. It had been discovered by Harvey, that the foetus in 
the human species, as well as in inferior animals, is not a pre- 
cise facsimile of the adult, but that it commences from a form 
infinitely more simple, and passes through several successive 
stages of organization before reaching its perfect development. 
In the circulatory system, these changes have been minutely 
observed and faithfully recorded*. ‘Tiedemann has traced a 
similar progression in the brain and nervous system, and has 
moreover established an exact parallel between the temporary 
states of the foetal brain in the periods of advancing gestation, 
and the permanent development of that organ at successive 
points of the animal scale. The first part of his work is simply 
descriptive of the nervous system of the embryo at each suc- 
cessive month of foetal life. It constitutes the anatomical ground- 
work upon which are raised the general laws of cerebral forma- 
tion, and the higher philosophy of the science. In the second 
part, Tiedemann has established, by examples drawn from all 
the grand divisions of the animal kingdom, the universality of 
the law of formation, as traced in the nervous system of the 


* See an excellent Essay on the Development of the Vascular System in the 
Feetus of Vertebrated Animals, by Dr. Allen Thomson. 


64 THIRD REPORT—1833. 


human foetus, and the existence of one and the same funda- 
mental type in the brain of man and of the inferior animals. 

The facts which have been unfolded by the industry of Tiede- 
mann, besides leading to the universal law of nervous develop- 
ment, throw important light upon nervous function: for it is 
. observed that the successive increments of nervous matter, and 
especially of brain, mark successive advances in the scale of 
being ; and, in general, that the development of the higher in- 
stincts and faculties keeps pace with that of brain. Thus, in 
the zoophyta, and in all living beings destitute of nerves, no- 
thing that resembles an instinct or voluntary act is discovera- 
ble. In fishes the hemispheres of the brain are small, and. 
marked with few furrows or eminences. In birds they are 
much more voluminous, more raised and vaulted than in rep- 
tiles; yet no convolutions or anfractuosities can be perceived 
on any point of their surface, nor are they divided into lobes. 
The brain of the mammalia approaches by successive steps to 
that of man. That of the rodentia is at the lowest point of 
organization. ‘Thus the hemispheres in the mouse, rat, and 
squirrel are smooth and without convolutions. In the carnivo- 
rous and ruminating tribes, the hemispheres are much larger 
and marked by numerous convolutions. In the ape tribe the 
brain is still more capacious and more convex; it covers the 
cerebellum, and is divided into anterior, middle, and posterior 
lobes. It is in. man that the brain attains its greatest magni- 
tude and most elaborate organization. Sommerring has proved 
that the volume of the brain, referred to that of the spinal mar- 
row as a standard of comparison, is greater in man than in any 
other animal. 

Various attempts have been made of late years, chiefly 
by the French physiologists, to ascertain the functions of the 
brain by actual experiment. It will appear from a detailed 
survey of their labours, that little more than a few general 
facts respecting the function of its /arger masses and great na- 
tural divisions have flowed from this mode of research. The 
offices of the smaller parts of cerebral substance cannot with 
any certainty be derived from the phenomena that have been 
hitherto observed to follow the removal of those parts, since 
the most practised vivisectors have obtained conflicting results. 
Nor is it difficult, after having performed or witnessed such 
experiments, to point out many unavoidable sources of fallacy. 
In operations on living animals, and especially on so delicate 
an organ as the brain, it is scarcely possible for the most skilful 
manipulator to preserve exact anatomical boundaries, to restrain 
hemorrhage, or prevent the extension to contiguous parts, of 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 65 


the morbid actions consequent upon such serious injuries, and 
to distinguish the secondary and varying phenomena, induced 
‘by the pressure of extravasated blood, or the spread of an in- 
flammatory process, from those which are essential and pri- 
mary. The ablation of small and completely insulated portions 
of brain must, then, be classed among the “ agenda” of experi- 
mental physiology. 

The most decisive researches, that have been hitherto insti- 
tuted on the functions of the brain, are those of M. Flourens. 
His mode of operating was to remove cautiously successive thin 
slices of cerebral matter, and to note the corresponding changes 
of function. He commenced with the hemispheres of the brain, 
which he found might be thus cut away, including the corpora 
striata and thalami optici, without apparently occasioning any 
pain to the animal, and without exciting convulsive motions. 
Entire removal of the cerebrum induces a state resembling coma; 
the animal appears plunged in a profound sleep, being wholly 
lost to external impressions, and incapable of originating mo- 
tion; it is deprived, too, according to Flourens, of every mode 
of sensation. Hence the cerebrum is inferred to be the organ in 
which reside the faculties of perception, volition and memory. 
Though not itself sensible, in the ordinary acceptation of the 
word,—that is, capable, on contact or injury, of propagating sen- 
sation,—yet it is the point where impressions made on the ex- 
ternal organs of sense become objects of perception. This ab- 
sence of general sensibility observed in the brain has also been 
experimentally demonstrated in the nerves dedicated to the func- 
tions of sight, of smell and of hearing, and constitutes, perhaps, 
one of the most remarkable phenomena that have been disclosed 
by interrogating living nature. Flourens appears, however, to 
have failed in proving that ali the sensations demand for their 
perception the integrity of the brain. He has himself stated 
that an animal deprived of that organ, when violently struck, 
‘‘ has the air of awakening from sleep,” and that if pushed for- 
wards, it continues to advance after the impelling force must 
have been wholly expended. Cuvier has therefore concluded, 
in his Report to the Academy of Sciences upon M. Flourens’ 
paper, that the cerebral lobes are the receptacle in which the 
impressions made on the organs of sight and hearing only, be- 
come perceptible by the animal, and that probably there too 
all the sensations assume.a distinct form, and leave durable im- 
pressions,—that the lobes are, in short, the abode of memory. 
The lobes, too, would seem to be the part in which those mo- 
tions which flow from spontaneous acts of the mind haye their 
i But a power of effecting regular and combined move~ 

. F 


66 THIRD REPORT—1833. 


ments, on external stimulation, evidently survives the destruc- 
tion of the cerebral hemispheres. 

A very elaborate series of experiments on the functions of 
the brain in general, and especially on those of its anterior por- 
tion, have been since performed by M. Bouillaud*. That ob- 
server concurs with Flourens in viewing the cerebral lobes as 
the seat of the remembrance of those sensations which are fur- 
nished to us by sight and hearing, as well as of all the intel- 
lectual operations to which these sensations may be subjected, 
such as comparison, judgment and reasoning. But he proves 
that the ordinary tactual sensibility does not require for its 
manifestation the presence of the brain. For animals entirely 
deprived of brain were awakened by being struck, and gave 
evident indications of suffering when exposed to any cause of 
physical pain. Bouillaud observes, too, that the iris continues 
obedient to the stimulus of light, after ablation of the hemi- 
spheres, and on this ground calls in question the loss of vision 
asserted by Flourens. Nor are the lobes (he contends,) the 
only receptacle of intelligence, of instincts and of volition: for 
to admit this proposition of Flourens would be to grant that an 
animal which retains the power of locomotion, which makes 
every effort to escape from irritation, which preserves its appro- 
priate attitude, and executes the same movements after as be- 
fore mutilation, may perform all those actions without the agency 
of the will or of instinct. Another doctrine of Flourens, which 
has been experimentally refuted by Bouillaud, is, “that the 
cerebral lobes concur as a whole in the full and entire exercise 
of their functions; that when one sense is lost, all are lost; 
when one faculty disappears, all disappear;” in short, that a 
certain amount of cerebral matter may be cut away without ap- 
parent injury, but that when this limit is passed, all voluntary 
acts and all perceptions perish simultaneously. Bouillaud, on 
the contrary, has described several experiments which show 
that animals, from whom the anterior or frontal part of the brain 
had been removed, preserved sight and hearing, though de- 
prived of the knowledge of external objects, and of the power 
of seeking their food. 

The second part of M. Bouillaud’s researches is entirely de- 
voted to the functions of the anterior lobes of the brain. These 
were either removed by the scalpel, or destroyed by the actual 
cautery, in dogs, rabbits and pigeons. Animals thus mutilated 
feel, see, hear and smell; are easily alarmed, and execute a 
number of voluntary acts, but cease to recognise the persons 


* Magendie, Journal de Physiologie, tom. x. p. 36. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS sysTEM. 67 


or objects which surround them. They no longer seek food, 
or perform any action announcing a combination of ideas. Thus 
the most docile and intelligent dogs lost all power of compre- 
hending signs or words which were before familiar to them, 
became indifferent to menaces or caresses, were no longer 
amenable to authority, and retained no remembrance of places, 
of things, or of persons. They saw distinctly food presented 
to them, but had ceased to associate with its external qualities 
all perception of its relations to themselves as an object of de- 
sire. The anterior or frontal part of the brain is hence inferred 
to be the seat of several intellectual faculties. Its removal oc- 
casions a state resembling idiotism, characterized by loss of the 
power of discriminating external objects, which, however, co- 
exists with the faculties of sensation. 

It will be unnecessary to describe fully in this place the ex- 
periments of Professor Rolando of Turin, performed in 1809, and 
published in Magendie’s Journal, tom. iii., 1823, since the more 
important of his facts have reference, not to the brain-proper, 
but to the cerebellum. His paper certainly contains some cu- 
rious anticipations of phenomena, since more accurately ob- 
served by Flourens and Magendie; yet as regards the brain, 
properly so called, his results are vague and inconclusive. 
Accident, rather than a well matured design, seems to have 
directed what parts of the brain he should remove; and from 
having comprehended in the same injury totally distinct anato- 
mical divisions, he has rendered it impossible to arrive at the 
precise function of any one part. Thus we are told that injury 
of the thalami optici and tubercula quadrigemina in a dog was 
followed by violent muscular contractions. Now all subsequent 
experimenters agree, that irritation of the thalami is incapable 
of inducing convulsive motions ; and Flourens has proved that 
this property has its beginning in the tubercula,—an important 
fact, which Rolando, with a little more precision in anatomical 
manipulation, could scarcely have failed to discover. 

Magendie has described* some curious experiments on the 
corpora striata, which, though closely analogous in their results 
to those on the cerebellum, have their proper place in this 
section. Removal of one corpus striatum was followed by no 
remarkable change; but when both had been cut away, the 
animal rushed violently forwards, never deviating from a recti- 
linear course, and striking against any objects in its way. In his 
lecture of February 7, 1828, Magendie, in the presence of his 
class, removed both corpora striata from a rabbit. The animal 


* Journal de Physiologie, tom. iii. p. 376. 
F2 


68 THIRD REPORT—1833. 


attempted to rush forwards, and, if restrained, appeared rest= 
less, continuing in the attitude of incipient progression. One 
thalamus opticus was then cut away from the same animal. The 
direction of its motion was immediately changed from a straight 
to a curved line. It continued for some time to run round in 
circles, turning towards the injured side. When the other 
thalamus was removed, the animal ceased its motions and re- 
mained perfectly tranquil, with the head inclined backwards. 
These experiments, it may be observed, furnish no support to 
the opinions of MM. Foville and Pinel Grandchamps, who have 
assigned the anterior lobes and corpora striata as the parts 
presiding over the movements of the inferior extremities, and 
the posterior lobes and thalami as regulating the superior. 

Cerebellum.—It may be regarded as nearly established by 
modern researches, that the cerebellum is more or less directly 
connected with the function of locomotion. The precise nature 
and extent of its control over the actions of the voluntary 
muscles are, however, far from being clearly determined. In 
the higher animals, the mental act of volition probably has its 
commencing point, as productive of a physical change, in the 
brain-proper; though it must be confessed that some of the 
experiments of Flourens, and all of those of Bouillaud, indicate 
the persistence of many instinctive, and even of some automatic 
motions, after destruction of the brain. But there does appear 
sufficient evidence to prove that those volitions which have 
motion as their effect, whatever be their origin, whether in the 
cerebrum, cerebellum, or medulla oblongata*, require for their 
accomplishment the cooperation of the cerebellum. This evi- 
dence has been mainly supplied by the same inquirers whose 
researches on the cerebrum have been already analysed. 

In the order of time, though not of importance, the experi- 
ments of Professor Rolando stand foremost. Injuries of the 
cerebellum, he observed, were always followed by diminished 
motive power; and this partial loss of power was always in 
direct proportion to the amount of injury. <A turtle survived 
upwards of two months the entire removal of the cerebellum, 
continuing sensible to the slightest stimulus; but when irritants 
were applied, it was totally unable to move from its place. 
M. Flourens has since arrived at similar, but more definitive 
results. He removed in succession thin slices from the cere- 
bellum. After the first two layers had been cut away, a slight 
weakness and want of harmony and system in the automatic 
movements were noticed. When more cerebellic substance had 


* Flourens, Mémoires de l’ Académie, tom. ix. 


wan tl 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 69 


been removed, great general agitation became ‘apparent. ‘The 
pigeon which was the subject of operation retained, as at first, 
the senses of sight and hearing, but was capable of executing 
‘only irregular unconnected muscular efforts. It lost by degrees 
the power of flying, of walking, and even of standing. Removal 
of the whole cerebellum was followed. by the entire disappear- 
ance of motive power. The animal, if laid upon its back, tried 
in vain to turn round; it perceived and was apprehensive of 
blows, with which it was menaced, heard sounds, seemed aware 
of danger, and made attempts to escape, though ineffectually, 
—in short, while it preserved, uninjured, sensation and the ex- 
ercise of volition, it had lost all power of rendering its muscles 
obedient to the will. The cerebellum is hence supposed by 
Flourens to be invested with the office of ‘‘ balancing, regu- 
lating or combining separate sets of muscles and limbs, so as 
to bring about those complex. movements depending on simul- 
taneous and conspiring efforts of many muscles, which are ne- 
cessary to the different kinds of progressive motion.” Bouil- 
laud, who has successfully disputed several of the opinions of 
Flourens respecting the functions of the cerebrum, fully concurs 
with him as to those of the cerebellum. 

Yet, it must be admitted, that there exists also conflicting 
experimental testimony on this subject. M. Fodera* states 
that he has found the removal of a part of the cerebellum to 
be followed, in all cases, either by motion backwards, or by 
that position of the body which precedes retrograde movement. 
The head is thrown back, the hind legs separated, and the 
fore legs extended forwards, and pressed firmly against the 
ground. More complete destruction of the cerebellum occa- 
sions the animal to fall on its side; but the head is still inclined 
rigidly backwards, and the anterior extremities agitated with 
convulsive movements, tending to cause retrograde motion of 
the body. Injuries of one side of the cerebellum were observed 
to produce paralysis of the same side of the body; as might, 
indeed, have been anticipated from the direct course, without 
decussation, of the restiform columns which ascend to form the 
cerebellum. Magendie has described} precisely the same re- 
sults.. A duck, whose cerebellum had. been destroyed, could 
swim only backwards. In the course of his experimental lec- 
tures, Magendie, having removed the cerebellum in several rab- 
bits, demonstrated to his class the phenomena of retrograde 
movement, exactly as they have been recorded by Fodera. It 
is, then, impossible to regard the conclusions of Flourens as 


* Journal de Physique, July 1823. + Ibid. tom. iii. p. 157. 


70 THIRD REPORT—1833. 


fully established, opposed as they are by those of so skilful an 
experimenter as Magendie. Indeed, while Flourens conceives 
the cerebellum to preside over motion, MM. Foville and Pinel 
Grandchamps attribute to it the directly opposite function of 
sensation: and this doctrine seems to derive some support from 
anatomical disposition; for it has been proved by Tiedemann 
that the cerebellum is nothing more than an expansion or pro- 
longation of the corpora restiformia, and posterior columns of 
the spinal medulla, which columns have been shown by Sir 
Charles Bell to have the office of conveying sensations. But 
it is not the less true that all recent experiments, even those of 
Fodera and Magendie, point to some connexion between the 
cerebellum and the power of voluntary motion. In the present 
state of our knowledge it would be unsafe to contend for more 
than the probable existence of some such general relation. 
This, then, is all that seems deserving of confidence respect- 
ing the functions of the cerebellum itself. But there are some 
singular phenomena which, though residing in other structures 
more or less near to the cerebellum, are so analogous to those 
already described as to call for notice in this place. Magendie 
has described * the results of injury to the crura cerebelli of a 
rabbit. Complete division of the right crus was followed by 
rapid and incessant rotation of the body upon its own axis, from 
left to right. This singular motion having continued two hours, 
Magendie placed the rabbit ina basket containing hay. On 
visiting it the following day he was surprised to find the animal 
still turning round as before, and completely enveloped in hay. 
The eyes were rigidly fixed in different lines; that of the injured 
side being directed forwards and downwards, that of the other 
side backwards and upwards. If both crura were divided, no 
motion followed. Magendie hence concluded that these ner- 
vous cords are the conductors of impulsive forces which coun- 
terbalance one another, and that from the equilibrium of these 
two forces result the power of standing, and even of maintaining 
a state of rest, and of executing the different voluntary motions. 
The inquiry naturally presented itself, whether these forces 
are inherent in the crura themselves, or emanate from the cere- 
bellum or some other source. To determine this question, 
portions of substance were removed from both sides of the 
cerebellum, but unequally, so as to leave intact # on the left 
side and } only on the right. The animal rolled towards the 
right side, and its eyes were fixed in the manner already de- 
scribed. But the left crus being divided, the animal rolled to 


* Journal de Physiologie, tom. iv. 399. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 71 


the left side. Hence it appears that section of the crus has 
more influence over the lateral rotation of the body than injury 
of the cerebellum itself; and that the impulsive force does not 
belong (at least exclusively) to the cerebellum. When the cere- 
bellum was divided precisely in the median line, the animal 
seemed suspended between two opposing forces, sometimes in- 
clining towards one side, as if about to fall, and again thrown 
suddenly back to the opposite side. Its eyes were singularly 
agitated, and seemed about to start from the orbits. Similar 
movements followed division of the continuous fibres in the 
pons Varolii. Serres has described a case of similar rotatory 
motion occurring in the human subject. A shoemaker ha- 
bituated to excess in alcoholic liquors, after great intemperance 
was seized with an irresistible disposition to turn round upon 
his own axis, and continued to move so till death ensued. On 
inspecting the brain, one of the crura cerebelli was found much 
diseased, and this was the only alteration of structure visible 
in any part of the nervous system. 

M. Flourens has published in a recent volume of the Mé- 
moires de l Académie des Sciences* a description of some 
strikmg abnormal motions which followed the division of the 
semicircular canals of the ears of birds. ‘Though these organs 
have no anatomical relation to the cerebrum or cerebellum, the 
altered motions resulting from their division are so analogous 
to those observed by Magendie after lesions of the corpora 
striata and crura, that they may be most conveniently described 
in the same section. ‘Two of the semicircular canals are ver- 
tical, and one horizontal. Division of the horizontal canals on 
each side occasioned a rapid horizontal movement of the head 
from right to left, and back again, and loss of the power of 
maintaining an equilibrium, except when standing, or when 
perfectly motionless. There was also the same singular rota- 
tion of the animal round its own axis which follows injury of 
the crura cerebelli. Section of the inferior vertical canal on 
both sides produced violent vertical movements of the head, 
with loss of equilibrium in walking or flying. There was in this 
case no rotation of the body upon itself, but the bird fell back- 
wards, and remained lying on its back. When the superior 
vertical canals were divided, the same phenomena were ob- 
served as in section of the inferior, except that the bird fell 
forward on its head, instead of backward. All the canals, both 
vertical and horizontal, having been divided, in another pigeon, 
violent and irregular motions in all directions ensued. When, 


* tom. ix. p. 454. 


72 THIRD REPORT—1833. 


however, the bony canals were so cautiously divided as to leave 
their internal membranous investment uninjured, these ab- 
normal motions were not produced. It is, therefore, in these 
membranes, or rather in the expansion of the acoustic nerve 
which overspreads them, that the cause of this phenomenon 
must reside. No explanation is proposed by Flourens of the 
control thus exercised by a nerve supposed to minister exclu- 
sively to the sense of hearing, over actions so entirely opposite 
in character. It is remarkable that the irregular movements 
should observe the same direction in their course as the canals, 
by the section of which they are induced. ‘Thus the direction 
of the inferior vertical canal is posterior, that of the superior 
is anterior, corresponding perfectly with the directions of the 
abnormal motions. 

Medulla Oblongata.—The medulla oblongata, or “‘bulbe 
rachidien,” is reducible into six columns, or three pairs, viz. 
two anterior or pyramidal, which partially decussate, two mid- 
dle or olivary, and two posterior or restiform, which proceed 
forwards without crossing. It is continuous in structure with the 
spinal marrow, and enjoys, by virtue of this relation, the same 
function of propagating motion and sensation. But it is distin- 
guished from the spinal medulla by special and higher attributes, 
being endowed with the faculty of originating motions, as well 
as with that of regulating and conducting them. ‘The medulla 
oblongata, with the cerebrum and cerebellum, constitute, in short, 
according to Flourens*, those portions of the nervous system 
which exercise their functions “spontaneously or primordial- 
ly,” and which originate and preside over the vital actions of 
the subordinate parts. To this latter order of parts, which re- 
quire an exciting or regulating influence, belongs the spinal 
medulla. In the superior class, Flourens seems to assign even 
a higher place to the medulla oblongata than to the cerebrum 
or cerebellum. For the cerebrum, he observes, may act without 
the cerebellum; and this latter organ continues to regulate the 
motions of the body after removal of the cerebrum. But the 
functions of neither cerebrum nor cerebellum survive the destruc- 
tion of the medulla oblongata, which seems to be the common 
bond and central knot combining all the individual parts of the 
nervous system into one whole. 

The medulla oblongata was regarded by Legallois as the 
mainspring or ‘‘ premier mobile” of the inspiratory movements. 
He repeated before a Commission of the Institute of France the 
leading experiments on which his opinion rested}. Ina rabbit 


* Mémoires de l’ Académie des Sciences, tom. ix. p. 478. 
+ CGwores de Legallois, tom. i. p. 247. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 73 


five or six days old, the larynx was detached from the os hy- 
oides and the glottis exposed to view. The brain and cere- 
bellum were then extracted without arresting the inspirations, 
which were marked by four simultaneous motions,—a gaping of 
the lips, an opening of the glottis, the elevation of the ribs, 
and the contraction of the diaphragm. Legallois next removed 
the medulla oblongata, when all these motions ceased together. 
In a second rabbit, instead of extracting at once the entire me- 
dulla, it was cut away in successive thin slices. The four in- 
spiratory movements continued after the removal of the three 
first slices, but ceased after the fourth. It was found that the 
fourth had reached the origin of the eighth pair of nerves. If, 
instead of destroying the part in which this motive influence 
resides, it be simply prevented from communicating with the 
muscles which are subservient to inspiration, a similar effect 
ought to be produced. Now it is obvious that the medulla 
oblongata must transmit its influence to the muscles which 
raise the ribs, through the medium of the intercostal nerves, 
and therefore of the spinal marrow, and to the diaphragm 
through the phrenic nerves, and to these through the spinal 
marrow. In another rabbit, therefore, the medulla spinalis was 
cut across about the level of the seventh cervical vertebra. 
The effect of this operation was to arrest the elevation of the 
ribs, the other three inspiratory motions still continuing. A 
second section was made near the first cervical vertebra, and 
consequently above the origin of the phrenic, with the effect of 
suspending the contraction of the diaphragm. The par vagum 
was next divided in the neck, and the opening of the glottis 
ceased. ‘There remained then, of the four inspiratory move- 
ments, only the gaping of the lips, which, however, was suffi- 
cient to attest that the medulla oblongata still retained the 
power of producing them all. This power had ceased to call 
forth the other three motions, only because it no longer had 
communication with their organs. 

M. Flourens, in a recent memoir already referred to*, has 
confirmed and extended the views first announced by Legal- 
lois. He has distinctly traced the comparative action of the 
medulla spinalis and oblongata, on respiration, in the four classes 
of vertebrated animals. In birds, he found that all the lumbar 
and the posterior dorsal medulla might be destroyed without 
impeding the respiratory function, though it was arrested by 
removal of the costal medulla. In the mammalia the costal also 


* Mémoires de l Académie, tom. ix. 1830. 


74 THIRD REPORT—1833. 


might be removed, for though the raising of the ribs ceased, 
the action of the diaphragm continued as long as the origin of 
the phrenic nerve remained uninjured. In frogs, all the spinal 
medulla may be destroyed, except the portion, whence spring the 
nerves supplying the hyoideal apparatus. Every part of the 
spinal marrow may be removed in fishes without affecting re- 
spiration; for all the nerves distributed to the respiratory organs 
of fishes have their origin in the medulla oblongata. It is 
hence apparent that the spinal marrow exercises only a variable 
and relative action on the respiratory function, in the different 
classes of vertebrated animals. In descending from the higher 
to the lower points of this scale, the spinal marrow is seen pro- 
gressively to disengage itself from cooperation in these move- 
ments, while the medulla oblongata tends more and more to 
concentrate them in itself, till in fishes the proper functions of 
the two medullz show themselves completely distinct, the spinal 
ministering to locomotion and sensation, and the oblongata to re- 
spiration. ‘The medulla oblongata is, then, the ‘‘ premier mo- 
teur” or the exciting and regulating principle of the inspiratory 
movements in all classes of vertebrated animals; besides par- 
ticipating, by virtue of its continuity with the spinal marrow, in 
the proper functions of that organ. From a second series of 
experiments, M. Flourens concludes that there exists a point 
in the nervous centres at which the section of those centres 
produces the sudden annihilation of all the inspiratory move- 
ments; and that this point corresponds with the origin of the 
eighth pair of nerves, commencing immediately above, and ending 
a little below, that origin,—a result precisely agreeing with that 
obtained by Legallois. 

Spinal Marrow.—It is apparent, that the functions of the 
three grand divisions of the nervous system, already described, 
have not yet been distinctly and fully ascertained. Our know- 
ledge of those, which next fall under survey, is more definite 
and substantial. The vital offices of the spinal medulla—re- 
garded by Legallois as the mainspring of life, and as alone re- 
gulating the actions of the heart and nobler organs,—are now 
reduced to conveying to the muscles the motive impulse of voli- 
tion, and to propagating to the sensorium commune, impressions 
made on the external senses. Itis not invested with the power 
possessed by the cerebrum and cerebellum, and perhaps by the 
medulla oblongata, of spontaneously originating muscular mo- 
tions. It is mainly, if not exclusively, a conductor; a medium 
of communication between the brain and the external instru- 
ments of locomotion and sensation. Flourens, indeed, conjec- 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SysTEM. 75 


tures that it also has the office of associating the partial con- 
tractions of individual muscles into ‘‘mouvemens d’ensemble,” 
necessary to the regular motions of the limbs. 

_ Before recording what is known of the spinal cord itself, it 
will be proper to advert to some recent experiments of Magen- 
die on the serous fluid in which it is immersed. It would 
appear that a quantity of liquid, varying from two to five 
ounces in the human subject, is always interposed between 
the arachnoid tunic and the pia mater, or proper membrane of 
the cord. The intermembranous bag, occupied by this fluid, 
communicates with the ventricular cavities at the calamus scrip- 
torius by a round aperture, often large and patent in hydroce- 
phalic subjects. _Magendie has therefore named this serous 
liquid ‘cerebro-spinal’. In living animals, it issues in a stream 
from a puncture of the arachnoid. Its removal occasions great 
nervous agitation, and symptoms resembling those of canine 
madness. The sudden increase of its quantity induces coma. 
Its presence seems essential to the undisturbed and natural ex- 
ercise of the nervous functions; and this influence probably is 
dependent upon its pressure, ‘temperature and chemical con- 
stitution, since any variation of these conditions is followed by 
the phenomena of nervous disorder. 

‘The great medullary cord is divided by a double furrow into 
two lateral halves; and each of these is again subdivided by the 
insertions of the ligamenta dentata into two columns, one pos- 
terior and one anterior. It has been long known that section 
of-any part of the spinal marrow excludes from intercourse with 
the brain all those parts of the body, which derive their nerves 
from the cylinder of medulla below the pot of injury. The 
muscles, so supplied, are no longer obedient to the control of 
the will, and the tegumentary membranes similarly situated en- 
tirely lose their sensibility. This interruption of the relations 
which subsist between the central seat of volition and sensation, 
and the rest of the body, whether due to direct injury of the 
great nervous masses or communicating nerves, or produced by 
the pressure of extravasated fluids, by morbid growths, or by 
various poisonous matters, constitutes the condition known by 
the name ‘ paralysis’. In cases of this kind it is frequently ob- 
served that the powers of sensation and locomotion are simulta- 
neously impaired or destroyed. But examples are not want- 
ing, even in the earliest clinical records, of the total loss of one 
of those faculties with perfect integrity of the other. Such facts 
naturally suggested the belief that the power of propagating 
sensations, and that of conveying motive impressions, resided 
in distinct portions of the nervous system. This opinion, how- 


76 THIRD REPORT—1833. 


ever, remained mere matter of conjecture until a recent period, 
‘when it was unequivocally established by Sir Charles Bell. 
From the original experiments of that most distinguished phy- 
siologist, repeated and confirmed by Magendie, it follows that 
the faculty of conducting sensations resides exclusively in the 
two posterior columns of the medulla, while that of communica- 
ting to the muscular system the motive stimulus impressed by 
volition is the attribute of the two anterior columns. The same 
limitation of function is found in the nervous roots which spring 
from these separate columns. Thus each spinal nerve is fur- 
nished with a double series of roots, one set of which have their 
origin in the anterior medullary column, and one in the pos- 
terior. ‘The spinal nerves are, in consequence of this anatomi- 
cal composition, nerves of twofold function, containing in ‘the 
same sheath distinct continuous filaments from both columns, 
and exercising, in the parts to which they are distributed, the 
double office of conductors of motion and sensation. It will 
afterwards appear, in our history of individual nerves, that all 
those which spring from the brain, except the fifth and eighth 
pairs, possess only a single function. 

Sufficient experimental proof of the foregoing propositions 
has been furnished by Sir Charles Bell and by M. Magendie. 
Thus, division of the posterior roots of the spinal nerves is uni- 
formly followed by total absence of feeling in the parts of the 
body to which the injured nerves are distributed, while their 
motive power remains undiminished. Magendie has further 
observed, that if the medullary canal be laid open, and the two 
posterior cords be touched or pricked slightly, there is instant 
expression of intense suffering; whereas, if the same or a greater 
amount of irritation be applied to the anterior columns, there 
are scarcely any signs of excited sensibility. ‘The central parts 
of the medulla seem also nearly impassable*. They may be 
touched, and even lacerated, according to Magendie, without 
exciting pain, if precautions are taken to avoid the surrounding 
medullary substance. In general, the properties of the spinal 
marrow, and especially its sensibility, seem to reside mainly on 
its surface ; for slight contact, even of the vascular membranes 
covering the posterior columns, caused acute pain. 

The first experiment of Sir C. Bell consisted in laying open 
the spinal canal of a living rabbit, and dividing the posterior 
roots of the nerves that supply the lower limbs. The animal 
was able to crawl. In his second trial he first stunned the 
rabbit, and then exposed the spinal marrow. On irritating the 


* Annales de Chimie et de Physique, tom: xxiii. p. 436. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 77 


posterior roots, no motion was induced in any part of the mus= 
cular frame ; but on grasping the anterior roots, each touch of 
the forceps was followed by a corresponding contraction of the 
muscles supplied by the irritated nerve. Magendie has de- 
scribed * the following experiments, which he has since declared. 
were made without any knowledge of the prior ones of Sir C. 
Bell. The subjects chosen for the operation were puppies 
about six weeks old; for in these it was easy to cut with a sharp 
scalpel through the vertebrz and to expose the medulla. In the 
first, the posterior roots of the lumbar and sacral nerves were 
divided, and the wound closed: violent pressure, and even prick- 
ing with a sharp instrument, awakened no sensation in the limb. 
supplied by the nerves which had been cut; but its motive power 
was uninjured. A second and a third trial gave the same re- 
sults. Magendie then divided in another animal, though with 
some difficulty, the anterior roots of the same nerves on one 
side. The hind limb became flaccid and entirely motionless, 
though it preserved its sensibility. Both the anterior and pos- 
terior roots were cut in the same subject with destruction of 
motion and sensation. Ina second paper; Magendie has re- 
lated the following additional facts. ‘The introduction of nux 
vomica into the animal economy is well known to give rise to 
violent tetanic convulsions of the whole muscular system. This 
property was made available as a test of the functions of the 
separate orders of nervous roots. It was found that, while all 
the other muscles of the body were agitated, when under the 
influence of this poison, by violent spasmodic contractions, the 
limb, supplied by nerves whose anterior roots had been pre- 
viously divided, remained supple and motionless. But when 
the posterior roots only had been cut, the tetanic spasms were 
universal. It would seem, however, that the seats of the two 
faculties of conducting motion and sensation are not strictly 
insulated by exact anatomical lines, but that they rather pass 
into each other with rapidly decreasing intensity. Thus irri- 
tation of the anterior roots, when connected with the medulla, 
gives birth, along with motive phenomena, to some evidences 
of sensibility ; and, vice versd, stimuli applied to the posterior 
roots, also undivided, occasion slight muscular contractions. 
In this last case it is, indeed, probable that the irritation tra- 
velled from the posterior roots upwards to the brain in the ac- 
customed channel, and gave rise to a perception of pain, which 
prompted the muscular effort. Indeed, after division of the 
posterior nervous roots, ordinary stimulants, applied to the 


* Journal de Physiologie, tom, ii. p. 276, August 1822. + Ibid, tom. ii. p. 366, 


78 THIRD REPORT—1833. 


ends not connected with the medulla, produced no apparent 
effects; though the galvanic fluid directed upon either order 
of roots gave rise to muscular contractions. ‘These were more 
complete and energetic when the anterior roots were the sub- 
jects of the experiment. . 

Besides the evidence tiius obtained by direct experiments on 
living animals, several important facts have been gathered from 
the pathology of the nervous system in man. These consist of 
cases of insulated paralysis of either motion or feeling, referred 
to the changes in structure observed after death. Sir Charles 
Bell has himself recorded several examples of this kind strongly 
confirming his experimental results ; and others of similar ten- 
dency are scattered through the successive volumes of Magen- 
die’s Journal*. But it must be admitted, that evidence of this 
kind is seldom distinct and conclusive. The structural changes, 
induced by disease, are rarely so circumscribed in seat and 
extent as to represent adequately the operations of the scalpel; 
and often when they are thus isolated within anatomical bound- 
ing lines, they affect, by pressure, or by the spread of the same 
morbid process, in a degree too slight to leave decided traces, 
the functions of contiguous parts, thus clouding the judgments 
of the best pathologists, and invalidating their mferences. 
There is, however, a very remarkable case described by Pro- 
fessor Royer Collard, to which these objections do not apply. 
Sprévale, an invalided soldier, was upwards of seventeen years 
the subject of medical observation in the Maison de Santé of 
Charenton. This individual remained for the last seven years 
of his life with the legs and thighs permanently crossed, and 
totally incapable of motion, though retaining their sensibility. 
On opening after death the spinal canal, there was found the 
pultaceous softening (ramollissement) of the whole anterior part 
of the medulla, and of almost the whole of the fibrous cords 
which form it. The anterior roots of the spinal nerves had 
also lost their accustomed consistency; while the posterior sur- 
face of the spinal cord, and its investing membrane, were healthy. 
Several of the cases observed by Sir Charles Bell furnish also 
unequivocal proof of the soundness of the views developed by 
experiment. 

There exist, indeed, few truths in physiology established 
on so wide and solid a basis of experimental research and pa- 
thological observation, as those deduced by Sir Charles Bell, 
the original discoverer, and by Magendie, his successor in the 
path of inquiry, respecting the offices of the spinal medulla. 


* See in particular Dr. Rullier’s case, tom. iii. p. 173; and Dr. Koreff’s, 
tom. iv. p. 376. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SysTEM. 79 


This organ may now be regarded as mainly, if not solely, a 
medium of intercourse between the external world and the brain, 
and again between the brain and the voluntary muscles, its two 
anterior columns being subservient to motion, its two posterior 
to sensation. In the present state of our knowledge it would 
be fruitless to try to penetrate into the minute philosophy of 
these actions: but it seems probable, from recent discoveries 
on the ultimate anatomy of tissue, that these actions are mole- 
cular, having their place in the globular elements, into which all 
living textures are resolvable by microscopic analysis ;—that 
the physical changes, e. g. impressed by external objects on the 
delicate net-work of nerve which invests the tegumentary mem- 
branes and open cavities, are propagated thence, from particle 
to particle, along the continuous filaments, to their origins in the 
posterior spinal columns, and thence to the central point, where 
they become objects of perception;—and that the motive sti- 
mulus of volition is similarly transmitted down the anterior co- 
lumns and nerves, to the organs of locomotion. Indeed, it is a 
legitimate inference from Sir Charles Bell’s discoveries, that a 
simple nervous filament, or medullary column, can only propa- 
gate an impression in one line of direction, viz. either towards 
or from the central seat. of perception and of will; and this cu- 
rious law of nervous actions would. seem to point at some in- 
sensible molecular motion as their essential condition. 

It remains to. investigate the arguments which have been 
supposed to prove the residence in the spinal marrow of the 
power of originating and controlling the actions of the heart. 
This question has been matter of eager controversy, from its 
bearing upon the general relations of nerve and muscle. With- 
out prejudging this latter topic, it may simplify its future con- 
sideration, and will at the same time be more consistent with 
strict arrangement, to state here merely the facts which have 
reference to the spinal medulla. 

The work of Legallois, entitled ‘‘ Eapériences sur le Prin- 
cipe de la Vie, notamment sur celui des Mouvemens du Coeur 
et sur le Siége de ce Principe*,” was the first remarkable essay 
on the relations between the heart and the spinal cord. It will, 
however, be sufficient to allude in general terms to the conclu- 
sions of Legallois, since they have been entirely subverted by 
the subsequent researches of Dr. Wilson Philip and M. Flou- 
rens. Legallois’s main doctrine was, that the principle which 
animates each part of the body resides in that part of the spinal 
medulla whence its nerves have their origin; and that it is also 


* Cures de Legallois, tom. i. pp. 97, 99, &c. 


80 THIRD REPORT—1833. 


from the spinal cord that the heart derives the principle of its 
life and its motion*. ‘The experimental proof supposed to 
establish these propositions consisted in destroying in different 
rabbits portions of the cervical, dorsal and lumbar medulla. 
Cessation of the heart’s action was affirmed to be the constant 
result of the operation; but even in some of Legallois’s own ex- 
periments +, the motions of the heart continued after consider- 
able injury had been inflicted on the spinal cord, and especially 
on its lower divisions. Still more unequivocal is the evidence 
that has been advanced by Dr. Wilson Philip, in his Inguiry 
into the Laws of the Vital Functions. His experiments, which 
were very numerous and judiciously varied, show that the cir- 
culation continues long after entire removal of the spinal mar- 
row, and that by artificially maintaining respiration, the motions 
of the heart may be almost indefinitely prolonged. Flourens, 
in the 10th vol. of the Mém. de l Académie}, has lately con- 
firmed Dr. Philip’s views : he has shown that the circulation is 
entirely independent of the spinal marrow. ‘The influence ap- 
parently exerted is only secondary, being due to the suspension 
of the respiratory movements. ‘Thus all those portions of the 
spinal marrow which can be destroyed in the different classes 
of animals without arresting respiration, may be removed with- 
out affecting the circulation. In fishes and frogs the entire 
spinal cord may be destroyed without checking the heart’s mo- 
tions, because in these classes the medulla oblongata presides 
exclusively over the respiratory function. 

Nerves.—The classification of nerves, which is most conve- 
nient to the physiologist, is based upon their vital properties or 
functions. Such an arrangement would distribute them into— 
1, nerves of motion; 2, nerves both of motion and sensation; 
3, the nerves ministering to the senses of sight, smell and hear- 
ing; and 4, the ganglionic system, or, according to Bichat, 
nerves of organic life. Sir Charles Bell has added a fifth class, 
comprising nerves which he supposes are dedicated to the 
respiratory motions. But it will afterwards appear, that the 
existence of an exclusive system of respiratory nerves is not 
supported by sufficient evidence. 

The first class of nerves exercising the single office of con- 
veying motion comprehends the 3rd, 4th, 6th, portio dura of 
the 7th, the 9th, and perhaps two divisions of the 8th, viz. the 
glossopharyngeal and spinal accessory. Mr. H. Mayo’s expe- 
riments detailed in his Anatomical and Physiological Commen- 
taries, No. 11. (and Journal de Physique, tom. iii.) throw much 


* p. 259. + pp. 100, 101, 105. { p. 625. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 81 


light on the functions of several of these nerves. The motions of 
the iris, he shows, require the integrity of the third pair, division 
of these nerves being always followed by full dilatation of the 
pupils, which cease to be obedient to the stimulus of light. If the 
divided end of the nerve communicating with the eye be pinched 
by the forceps, theiris contracts. Hence it is apparent that dimi- 
nution of the aperture of the pupil is the result of action, and 
dilatation of the pupil the result of relaxation, of the iris. Flou- 
rens has shown that complete extirpation of the tubercula quad- 
rigemina also paralyses the iris, and that irritation of those bo- 
dies excites its contractions. The same effect is noticed by Mayo 
to arise from division or irritation of the optic nerve. He divided 
the optic nerves within the cranium of a pigeon immediately 
after decapitation. "When the end of the nerve connected with 
the ball of the eye was seized in the forceps, no action ensued; 
but when the end attached to the brain was irritated, the iris 
immediately contracted. 'These several experiments clearly in- 
dicate the dependence of the iris upon the optic nerve, upon 
the tubercula from which one root of that nerve springs, and 
upon the third pair. The stimulus of light impinges upon the 
retina, is conveyed along the optic nerve through the tubercle 
to the sensorium, whence the motive impression is propagated 
to the iris by the third encephalic nerve. 

It is not so easy to define the precise mode of action of the 
pathetici, or fourth pair ofnerves. Sir Charles Bell * supposes 
that they are destined ‘‘to provide for the insensible and in- 
stinctive rolling of the eyeball, and to associate this motion of 
the eyeball with the winking motions of the eyelids.” He even 
conjectures that “‘the influence of the fourth nerve is, on cer- 
tain occasions, to cause a relaxation of the muscle to which it 
goes.” It is certain, however, from its exclusive distribution to 
the superior oblique muscle, that the fourth is a nerve of motion. 
The sixth nerve is also a nerve of voluntary motion, and is sent 
to the rectus externus of the eyeball. 

Sir Charles Bell has placed the portio dura of the seventh 
pair among his respiratory nerves. There is, however, no doubt 
that it is simply a motive nerve, and that it is indeed the only 
nerve of motion, which supplies all the muscles of the face, 
except those of the lower jaw and palate. Division of this 
nerve occasions no expression of pain, according to Bell; but 
Mayo’s experience is opposed to this absence of sensibility +. 
“The motion of the nostril of the same side instantly ceased, 


* Natural System of Nerves, p. 358. 

+ See Mr. H. Mayo’s Anatomical and Physiological Commentaries, Part I.; 
and Outlines of Human Physiology, 2nd edit., p. 334, 

1833. G 


82 THIRD REPORT—1833. 


after its section in an ass*, and that side of the face remained 
at rest and placid during the highest excitement of the other 
parts of the respiratory organs.” ‘These and similar observa- 
tions are all consistent with the opinion, that the seventh is 
simply a nerve of voluntary motion. It will afterwards appear 
that it has no claim to any further endowment. 

Mr. Herbert Mayo infers from his experiments, that the three 
divisions of the eighth pair are all. nerves both of motion and 
sensation. Thus the glossopharyngeus is a nerve of motion to 
the pharynx, and perhaps of sensibility to the tongue. He 
observed that “on irritating the glossopharyngeal nerve in an 
animal recently killed, the muscular fibres about the pharynx 
acted, but not those of the tongue.” Irritation of the spinal 
accessory produced both muscular contractions and pain. The 
par vagum, he conceives, bestows sensibility on the membrane 
of the larynx, besides conveying the motive stimulus to its 
muscles. ‘This nerve has been the subject of experiment from 
the earliest times, and Legallois has minutely described the 
results obtained by successive inquirerst. These were singu- 
larly discordant, and gave origin to the most opposite theories 
of the mode of action of the par vagum. In the greater number 
of experiments, section of this nerve was followed, after a longer 
or shorter interval, by death. Piccolhomini contended that the 
division of the nerve was fatal from its arresting the move- 
ments of the heart, and after him Willis supported the same 
doctrine. By Haller, on the contrary, the cause of death was 
sought in disturbance of the digestive functions. Bichat and 
Dupuytren seem to have been the first to obtain a glimpse of the 
true seat of injury. The former remarked that the respiration 
became very laborious after section of the nerve, and Dupuytren 
distinctly traced death to asphyxia. Legallois has established 
by numerous experiments the accuracy of this last view. He 
has shown that in very young animals death is the immediate 
consequence of the operation of cutting either the par vagum 
or its recurrent branch, and that the suddenness of the effect 
is due to the narrowness of the aperture of the glottis in early 
age. In adult animals, the asphyxia is induced by the effusion 
of serous fluids and ropy discoloured mucus into the bronchial 
tubes and air-cells. More recently, Dr. Wilson Philip has prac- 
tised the section of the par vagum with an especial reference 
to its influence upon digestion. He divided the nerve below 
the origin of the inferior laryngeal branch, as in this ease the 


* pp. 106, 107. + Outlines of Human Physiology, 2nd edit., p. 337. 
+ Cuovres, p. 154 et seq. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 89 


dyspnea is much less considerable than when the wound is in- 
flicted on the higher portion*. It was found, in all these trials, 
that food introduced into the stomach after the operation re- 
mained wholly undigested. Hence Dr. Philip infers the de- 
pendence of secretion upon nervous influence, a conclusion, it 
has been remarked by Dr. Alison, not logically deducible from 
the experimental data +. 

The par vagum cannot then, it is obvious, be included in the 
class of nerves subservient solely to motion; and it is even 
doubtful whether the other two divisions of the eighth pair are 
not also endowed with sensibility. Respecting the function of 
the ninth, or lingual, there is, however, no place for hesitation. 
It has been experimentally proved by Mr. Mayo to supply the 
muscles of the tongue; though he also asserts that pinching it 
with the forceps excited pain. Three of these nerves, the third, 
sixth, and ninth, arise, it was first remarked by Sir Charles 
Bell, from a tract of medullary matter continuous with the an- 
terior column of the spinal marrow: and hence their exclusive 
office of conducting motive impressions. 

If. There are thirty-two pairs of nerves of similar anatomical 
origin and composition, which possess the twofold office of com- 
municating motion and sensation. Of these, all excepting one 
(the fifth pair of the cerebral nerves) spring from the spinal 
marrow. These thirty-one pairs are precisely analogous in 
formation, being all constituted of two distinct series of roots, 
one from the anterior column, and one from the posterior column 
of the spinal marrow. The posterior funiculi collected together 
form a ganglion, seated just before this root is joimed by the 
anterior root. It has been already stated that the power of 
propagating sensation resides in the posterior column, and in 
the nervous roots arising from it, and that the motive faculty 
has its seat in the anterior column and roots. The evidence, 
also, supplied by Bell and Magendie, that the spinal nerves are 
hence nerves of double office, has been fully detailed. It re- 
mains, then, to establish the title of the fifth pair of cerebral 
nerves to be included in the same class with the spinal nerves. 

The analogy in structure and mode of origin between the 
fifth pair and the nerves of the spine has been long matter of 
observation. . Prochaska has thus distinctly noticed it in a pas- 
sage of his Essay De Structuré Nervorum, published in 1779, 
first pointed out to me by my friend Dr. Holme: “ Quare 
omnium cerebri nervorum, solum quintum par post ortum suum 


* Experimental Inquiry, 3rd edit., p. 109. 
+ Dr. Alison, Journal of Science, vol. ix. p. 106. 
G2 


84 THIRD REPORT—1833. 


more nervorum spinalium, ganglion semilunare dictum, facere 
debet? sub quo peculiaris funiculorum fasciculus ad tertium 
quinti paris ramum, maxillarem inferiorem dictum, properat, 
insalutato ganglio semilunari, ad_similitudinem radicum ante- 
riorum nervorum spinalium?” Sommerring has also pointed 
out with equal clearness the resemblance in distribution be- 
tween the smaller root of the fifth and the anterior roots of the 
spinal nerves. But Sir Charles Bell was the first to establish 
the identity of their functions, and to arrange them prominently 
in the same natural division. His experiment consisted in 
exposing the fifth pair at its root, in an ass, the moment the 
animal was killed. ‘On irritating the nerve, the muscles of the 
jaw acted, and the jaw was closed with a snap. On dividing 
the root of the nerve in a living animal, the jaw fell relaxed.” 
In another experiment the superior maxillary branch of the 
fifth nerve was exposed. ‘‘'Touching this nerve gave acute 
PRS hibit the side of the lip was observed to hang low, and 
it was dragged to the other side.” Sir Charles Bell concluded 
that the fifth nerve and its branches are endowed with the attri- 
butes of motion and sensation. This, though correct as regards 
the nerve itself, viewed as a whole, is strictly true only of the 
lowest of its three divisions, viz. the inferior maxillary. The 
ophthalmic and the superior maxillary, the subject of the last 
experiment, are nerves simply of sensation. Mr. Herbert Mayo 
in the Essay already referred to, has pointed out this error, 
and has defined with minute precision the relative offices of the 
fifth and seventh nerves. By a careful dissection of the fifth 
nerve he found that the anterior branch, or smaller root, which 
goes, as Prochaska was aware, entirely to the inferior maxillary, 
is distributed exclusively to the circumflexus palati, the ptery- 
goids, and temporal and masseter muscles. He observed that sec- 
tion of the supra and infra orbitar branches, and of the inferior 
maxillary, near the foramina, whence they emerge, induces loss 
of sensation in the corresponding parts of the face. It may then 
be regarded as fully proved that the trigeminus or fifth pair is 
the nerve which bestows sensation on the face and its appen- 
dages, and motion only on the muscles connected with the lower 
jaw. The other muscles of the face derive their motive power 
from the portio dura of the seventh nerve. 

M. Magendie has also published several memoirs on the 
functions of the fifth pair. In these he attempts to prove 
that the olfactory nerve is not the nerve of smell; that the op- 
tic is but partially the nerve of vision; and that the auditory is 
not the principal nerve of hearing. It is in the fifth pair that 
he supposes all these distinct and special endowments to reside. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 85 


But the experimental proof will be found to be singularly in- 
conclusive. The olfactory nerves were entirely destroyed in a 
dog. After the operation it continued sensible to strong odours, 
as of ammonia, acetic acid, or essential oil of lavender ; and the 
introduction of a probe into the nasal cavity excited the same 
motions and pain as in an unmutilated dog. The fifth pair was 
then divided in several young animals, the olfactory being left 
entire. All signs of the perception of strongly odorous sub- 
stances, as sneezing, rubbing the nose, or turning away the head, 
entirely disappeared. From these facts Magendie infers that 
the seat of the sensations of smell is in the fifth, and not in the 
first pair of nerves. It is obvious that Magendie has con- 
founded two modes of sensation, which are essentially distinct 
in their nature and in their organic seat, viz. the true percep- 
tions of smell, and the common sensibility of the nasal passages. 
‘The phenomena, which he observed to cease after the section 
of the fifth nerve, are the results of simple irritation of the pi- 
tuitary membrane, and are manifestly wholly unconnected with 
the sense of smelling, since they are producible by all powerful 
chemical agents, even though inodorous, as, for example, by 
sulphuric acid. No proof has been given that the true olfac- 
tory perceptions do not survive the destruction of the fifth pair. 
Indeed, in a subsequent paper, Magendie confesses that the 
loss of sensibility in the nasal membrane, after section of the 
fifth, does not prove the residence of the sense of smell in the 
branches of that nerve; but merely that the olfactory nerve re- 
~ quires, for its perfect action, the cooperation of the fifth pair, 
ur that it possesses only a special sensibility to odorous parti- 
cles. | 

_ There is even less ground for supposing that the fifth pair is 
in any degree subservient to the senses of sight and hearing. 
After cutting this nerve on one side, the flame of a torch was 
suddenly brought near the eye, without inducing contraction of 
the pupil; but the direct light of the sun caused the animal to 
close its eyelids. Thus the sensibility of the retina, though 
somewhat impaired, was not destroyed by division of the fitth 
pair. But section of the optic nerves was immediately followed 
by total blindness. In another rabbit Magendie divided. the 
fifth pair on one side, and the optic nerve on.the other. The 
animal, he states, was completely deprived of sight, though the 
eye, in which the fifth pair only had been cut, remained suscep- 
tible to the action of the solar rays. No evidence, however, is 
offered to show that the animal was entirely blind: on the con- 
trary, the only change observed, on approaching a torch to an 
vninjured eye, was contraction of the iris; and this we are told 


86 THIRD REPORT—1833. 


was actually observed in the eye of the side, on which the fifth 
nerve had been divided. 

Magendie has assigned another singular function to the fifth 
pair, viz. to preside over the nutrition of the eye. Twenty-four 
hours after section of this nerve, incipient opacity of the cornea 
was observed, which gradually increased till the cornea became 
as white as alabaster. ‘There was also great vascularity of the 
conjunctiva extending to the iris, with secretion of pus, and for- 
mation of false membranes in the anterior chamber. About the 
eighth day, the cornea began to detach itself from the sclerotica, 
the centre ulcerated, and the humours of the eye finally escaped, 
leaving only a small tubercle in the orbit. In this experiment, 
the nerve had been divided in the temporal fossa, but when cut 
immediately after leaving the pons Varolii, the morbid changes 
were less marked, the movements of the globe of the eye were 
preserved, the inflammation was limited to the superior part of 
the eye, and the opacity occupied only a small segment of the 
circumference of the cornea. After division of the nerve near 
its origin in the medulla, no traces of disease were discoverable 
in the eye till the seventh day, and these symptoms never be- 
came very prominent. Several cases have been since recorded 
of structural disease of this nerve in the human subject, with 
the concomitant symptoms. ‘That of Lainé, described by Serres 
in the 4th vol. of Magendie’s Journal, furnishes strong support 
to the views of Magendie*. 

A different explanation of this fact and of others which have 
a tendency to refer secretion and nutrition to the control of 
the nervous system has been proposed by Dr. Alison. Mucous 
surfaces are protected from the contact of air and foreign bo- 
dies by a copious secretion, which is evidently regulated in 
amount by their sensibility, since it is increased by any unusual 
irritation. This is especially true of the membrane of the eye. 
Now section of the fifth pair is known to paralyse the sensibi- 
lity of that organ, and the contact of air or other irritating body 
upon the insenstble membrane, instead of inducing an aug- 
mented mucous discharge, will excite the inflammatory process 
described by Magendie. The disorder of the digestive func- 
tion+, which followed division of the par vagum in the experi- 
ments of Dr. Wilson Philip, and the ulceration of the coats of 
the bladder after injury of the lower part of the spinal marrow, 
are attributed by Dr. Alison to the same cause. — 

The class of nerves which comprehends the fifth pair and 


* See also a case of destruction of the olfactory nerves, tom. v. 
+ Outlines of Physiology, p. 71. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 87 


the thirty-one pairs of spinal nerves, becomes, after the union 
of their roots, invested with a twofold endowment, and conti- 
nues so throughout their entire course and final distribution to 
the muscular tissue. It would appear, indeed, from a later 
paper of Sir Charles Bell*, that nerves of sensation, as well as 
of motion, are necessary to the perfect action of the voluntary 
muscles. ‘‘ Between the brain and the muscles there is a circle 
of nerves; one nerve conveys the influence from the brain to 
the muscle, another gives the sense of the condition of the 
muscle to the brain.” In the case of the spinal nerves this 
circle of intercourse is at least probable; but proof of its ne- 
cessity must be obtained, from observing the habitudes of those 
encephalic nerves, which minister exclusively to motion. Now 
it is found, on minute dissection, that the muscles of the eye- 
ball, which are supplied by the third, fourth and sixth motive 
nerves, also receive sensitive filaments from the ophthalmic 
branch of the fifth; and that the muscles of the face, to which 
the portio dura is distributed, are also furnished with branches 
of sensation from the fifth. Sir Charles Bell has further shown 
that the muscles of the lower jaw, to which the motive im- 
pression is propagated by the muscular branch of the inferior 
maxillary, draw nervous supplies also from the ganglionic or 
sensitive branch of that division of the fifth pair. This com- 
plicated provision has its origin, he supposes, in its being ‘ ne- 
cessary to the governance of the muscular frame that there 
should be consciousness of the state or degree of action of the 
muscles.” 

Il. The olfactory, auditory and optic nerves are gifted with 
a special sensibility to the objects of the external senses, to 
which they respectively minister. Magendie seems to have 
been the first to prove, experimentally, that they do not also 
share the common or tactile sensibility. He exposed the olfac- 
tory nerves, and found that, like the hemispheres of the brain 
from which they spring, they are insensible to pressure, prick- 
ing, or even laceration. Strong ammonia was dropped upon 
them without eliciting any signs of feeling. The optic nerve, 
and its expansion on the retina, participate with the olfactory 
in this insensibility to stimulants. This was proved by Ma~- 
gendie in the human subject as well as in animals. In perform- 
ing the operation of depressing the opaque lens, he repeatedly 
touched the retina in two different individuals without awaken- 
ing the slightest sensation. The portio mollis, or acoustic nerve, 
was also touched, pressed, and even torn without causing pain. 


* Philosophical Transactions, 1826, p. 163. 


88 THIRD REPORT—1833. 


IV. The functions of the ganglia, of the great sympathetic 
nerve, and its intricate plexuses and anastomotic connexions, are 
matter, at present, of conjecture. Dr. Johnstone, in an Essay 
on the Use of the Ganglions, published in 1771, has described 
a few inconclusive experiments on the cardiac nerves. He 
supposes that ‘ ganglions are the instruments by which the 
motions of the heart and intestines are rendered uniformly in- 
voluntary,”—a notion which Sir Charles Bell has shown to be to- 
tally unsound. The best history of opinions, to which indeed 
our knowledge reduces itself, will be found in the physiological 
section of Lobstein’s work, De Nervi Sympathetic: Fabrica, 
Usu, et Morbis*. 

In the earliest of his communications to the Royal Society, 
as well as in his last work on the Nervous System}, Sir Charles 
Bell has maintained the existence of a separate class of nerves, 
subservient to the regular and the associated actions of respira-. 
tion. The origins of these nerves { ‘‘are in a line or series, and 
from a distinct column of the spinal marrow. Behind the corpus 
olivare, and anterior to that process, which descends from the 
cerebellum, called sometimes the corpus restiforme, a convex 
strip of medullary matter may be observed. From this tract of 
medullary matter, on the side of the medulla oblongata, arise, in 
succession from above downwards, the portio dura of the seventh 
nerve, the glossopharyngeus nerve, the nerve of the par vagum, 
the nervus ad par vagum accessorius, and, as I imagine, the 
phrenic and the external respiratory nerves.” ‘The fourth pair 
is also received into the same class. 

This doctrine of an exclusive system of respiratory nerves, 
associated in function by virtue of an anatomical relation of 
their roots, has not, as Sir Charles Bell seems himself aware §, 
received the concurrence of many intelligent physiologists of 
this country or of the Continent. Mr. Herbert Mayo, in the ad- 
mirable Essay already referred to, was the first to point out the 
true relations of the fifth and seventh nerves. He has shown 
that the muscles of the face, excepting those already enumer- 
ated, which elevate the lower jaw, receive their motive nerves 
exclusively from the seventh, and consequently that this nerve 
must govern a// their motions, voluntary as well as respiratory. 
But Dr. Alison, in his very elaborate paper || ‘‘On the Physiolo- 
gical Principle of Sympathy,” has cast considerable doubts on 


* Paris 1823. + 4to, 1830. 
t The Nervous System of the Human Body, p. 129. 4to, 1830. 
§ Op. cit., p. 115. 


| Fransactions of the Medico-chirurgical Sociely of Edinburgh, 1826, vol. ii. 
p. 165. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 89 


the soundness of this part of Sir Charles Bell’s arrangement, as 
respects not only the individual nerves thus classed together, but 
even the general principle on which the entire system rests. 
The reasoning of Dr. Alison consists, first, in referring the 
phenomena of natural and excited respiration to the compre- 
hensive order of sympathetic actions. In these “the pheno- 
menon observed is, that on an irritation or stimulus being 
applied to one part of the body, the voluntary muscles of an- 
other, and often distant part, are thrown into action.” Now 
it has been long since fully established by Dr. Whytt, that 
these associations in function cannot be referred to any con- 
nexions, either in origin or in course, of the nerves supplying 
remote organs so sympathizing; and that a sensation is the 
necessary antecedent of the resulting muscular aetion. Thus 
it is known that these actions cease in the state of coma; 
are not excited when the mind is strongly impressed by any 
other sensation or thought; and that the same muscular con- 
tractions may be induced by the irritation of different parts of 
the body, provided the same sensation be excited. Dr. Alison 
has, however, failed to show* that the essential acts of inspira- 
tion, viz. the contractions of the diaphragm and intercostals, 
require the intervention of a sensation. ‘Their continuance in 
the state of coma, and in the experiments of Legallois and 
Flourens after the entire removal of the brain, and their di- 
stinct reference by these two inquirers to the medulla oblon- 
gata, which has never been supposed to be the seat of sensa- 
tion, prove them to be independent of the will and of perception. 
But this is true only of the essential, not of the associated 
respiratory phenomena. . 

Dr. Alison proceeds to show that there is equal reason for 
classing almost all the nerves of the brain, and many more of 
the spinal nerves, with those exclusively named respiratory by 
Sir Charles Bell. Thus the lingual nerve governs an infinite 
number of motions strictly associated with respiration: the in- 
ferior maxillary “‘ moves the muscles of the lower jaw in the 
action of sucking,—an action clearly instinctive when first per- 
formed by the infant, frequently repeated voluntarily during 
life, and always in connexion with the act of respiration.” 
Again, the sensitive branches of the fifth pair cooperate in the 
act of sneezing. But if these nerves be admitted into the 
system, the fundamental principle of that system, viz. origin 
in a line or series, is at once violated. Nor is this connexion 
in origin more than matter of conjecture, as regards two of the 


* p. 176, and note. 


90 THIRD REPORT—1833. 


most important of the nerves, classed by Sir Charles Bell himself 
as respiratory,—the phrenic and the external respiratory. These 
two nerves branch from the cervical or regular double-rooted 
series. Moreover, the circumstance of rising in linear suc- 
cession is not found to associate nerves in function. ‘‘ Be- 
tween the roots of the phrenic nerve and those of the inter- 
costals, there intervene in the same series the origins of the 
three lowest cervical nerves, and the first dorsal, which go 
chiefly to the axillary plexus and to the arm, and which are 
not respiratory nerves.” 


In recapitulation, the following facts are among the most 
important that have been fully ascertained in the physiology of 
the nervous system. 

1. One universal type has been followed in the formation of 
the nervous system in vertebrated animals. The brain of the 
human foetus is gradually evolved in the successive months of 
uterine existence; and these stages of progressive develop- 
ment strictly correspond with permanent states of the adult 
brain at inferior degrees of the animal scale. 

2. These successive increments of cerebral matter are found 
to be accompanied by parallel advances in the manifestation of 
the higher instincts and of the mental faculties. 

3. That the brain is the material organ of all intellectual 
states and operations, is proved by observation on comparative 
development, as well as by experiments on living animals, and 
by the study of human pathology. But there does not exist 
any conclusive evidence for referring separate faculties, or 
moral affections, to distinct portions of brain. 

4, Certain irregular movements are produced by injuries of 
the corpora striata, thalami optici, crura cerebelli, and semi- 
circular canals of the internal ear. 

5. The tubercula quadrigemina preside over the motions 
of the iris, and their integrity seems essential even to the func- 
tions of the retina. They are also, according to Flourens, the 
points, at which irritation first begins to excite pain and mus- 
cular contractions. 

6. The cerebellum appears to exercise some degree of con- 
trol over the instruments of locomotion; but the precise na- 
ture and amount of this influence cannot be distinctly defined. 

7. The cerebrum, cerebellum and medulla oblongata possess 
the faculty of acting primordially, or spontaneously, without 
requiring foreign excitation. The spinal cord and the nerves 
are not endowed with spontaneity of action, and are therefore 
termed subordinate parts. 


REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 91 


8. The medulla oblongata exercises the office of originating 
and regulating the motions essential to the act of respiration. 
By virtue of its continuity with the spinal marrow, it also par- 
ticipates in the functions of that division of nervous matter. 

9. The function of the spinal cord is simply that of a con- 
ductor of motive impulses, from the brain to the nerves supply- 
ing the muscles, and of sensitive impressions from the surface 
of the body to the sensorium commune. These two vital 
offices reside in distinct portions of the spinal medulla,—the 
propagation of motion in its anterior columns, the transmission 
of sensations in its posterior columns. There is no necessary 
dependence of the motions of the heart, and the other invo- 
luntary muscles, on the spinal marrow. 

10. The nerves are comprehended in the four following 
classes :—I. Nerves simply of motion; II. Of motion and sen- 
sation; III. Of three of the senses; IV. The ganglionic sy- 
stem. 

I. The nerves of motion are the third, fourth, sixth, portio 
dura of the seventh, and the ninth. It is not ascertained whe- 
ther the glossopharyngeal and spinal accessory nerves belong to 
this or to the second class. 

II. The function of ministering both to motion and sensation 
is possessed by the fifth pair of cerebral nerves, and by the 
spinal nerves, which agree precisely in anatomical composition. 

The par vagum, however, which is one of the irregular nerves, 
has also a twofold endowment. 

III. This division comprises the first and second pairs, and 
the portio mollis of the seventh pair. These nerves are insen- 
sible to ordinary stimulants, and possess an exclusive sensibility 
to their respective objects,—viz. odorous matter, light, and aérial 
undulations. 

IV. The system of the great sympathetic nerve, and its as- 
sociated plexuses and ganglia. 


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[ 93] 


Report on the present State of our Knowledge respecting the 
Strength of Materials. By Peter Bartow, Esq., F.R.S., 
Corr. Memb. Inst. France, Sc. §c. 


Tur theory of the strength of materials, considered merely as 
a branch of mechanical or physical science, must be admitted 
to hold only a very subordinate rank ; but in a country in which 
machinery and works of every description are carried to a great 
extent, it certainly becomes a subject of much practical im- 
portance ; and it was no doubt viewing it in this light which led 
the Committee of the British Association, at their last Meeting, 
to do me the honour to request me to furnish them with a 
communication on the subject. In drawing my attention to this 
inquiry, the Committee have subdivided it into the following 
heads :—1. Whether, from the experiments of different authors, 
we have arrived at any general principles? 2. What those 
principles are? 3. How modified in their application to dif- 
ferent substances? And what are the differences of opinion 
which at present prevail on those subjects ? 

To these questions, without a formal division of the Essay, 
I shall endeavour to reply in the following pages, by drawing 
a concise sketch of the experimental and theoretical researches 
which have been undertaken with reference to these inquiries. 

The subject of the strength of materials, from its great prac- 
tical importance, has engaged the attention of several able 
men, both theoretical and practical, and much useful informa- 
tion has been thereby obtained. As far as relates to the me- 
chanical effects of different strains, everything that can be 
desired has been effected; but the uncertain nature of mate- 
rials generally, will not admit of our drawing from experiment 
such determinate data as could be wished. ‘Two trees of the 
same wood, grown in the same field, having pieces selected 
from the same parts, will frequently differ from each other very 
considerably in strength, when submitted to precisely the same 
strain. The like may be said of two bars of iron from the 
same ore, the same furnace, and from the same rollers, and 
even of different parts of the same bar; and so likewise of 
two ropes, two cables, &c. We must not, therefore, in ques- 
tions of this kind, expect to arrive at data so fixed and deter- 
minate as in many other practical cases ; but still, within cer- 
tain limits; much important information has been obtained for 


94 THIRD REPORT—1833. 


the guidance of practical men; and by tabulating such results 
in a subsequent part of this article, I shall endeavour to answer 
the leading questions of the Committee of the British Associa- 
tion, as far, at least, as relates to experimental results. In re- 
ference to theory, it must also be admitted that some. uncer- 
tainty still remains; but this likewise is in a great measure to 
be referred to the nature of the materials, which is such as to 
offer resistances by no means consistent with any fixed and 
determinate laws. 

Hence some authors have assumed the fibres or crystals 
composing a body to be perfectly incompressible, and others 
as perfectly elastic; whereas it is known that they are strictly 
neither one nor the other, the law of resistance being differ- 
ently modified in nearly every different substance ; and as it is 
requisite theoretically to assume some determinate law of action, 
it necessarily follows that some doubt must also hang over this 
branch of the subject. It is, however, fortunate that whatever 
may be the uncertainty on these points, the relative: strength 
of different beams or bolts of the same material, of similar forms’ 
and submitted to similar strains, is not thereby affected; so that 
whatever may be the law which the fibres or particles of a 
body observe in their resistance to compression or extension, 
still, from the result of a well conducted series of experiments, 
the absolute resisting force of beams of similar forms, of the 
same materials, of any dimensions, submitted to similar strains, 
may, as far as the mean strength can be depended upon, be 
satisfactorily deduced. An examination of these different views 
taken of the subject by different writers will, it is hoped, be 
found to furnish a reply to the other queries of the Committee. 
The first writer who endeavoured to connect this inquiry with 
geometry, and thereby to submit it to calculations, was the ve- 
nerable Galileo, in his Dialogues, published in 1633. He there 
considers solid bodies as being made up of numerous small 
fibres placed parallel to each other, and their resistance to se- 
paration to a force applied parallel to their length, to be pro- - 
portional to their transverse area,—an assumption at once ob- 
vious and indisputable, abstracting from the defects and irre- 
gularities of the materials themselves. He next inquired in 
what manner these fibres would resist a force applied perpen- 
dicularly to their lencth: and here he assumed that they were 
wholly incompressible; that the fibres under every degree of 
tension resisted with the same force, and, consequently, that 
when a beam was fixed solidly in a horizontal position, with one 
end in a wall or other immoveable mass, the resistance of the 
integrant fibres was equal to the sum of their direct resistances 


REPORT ON THE STRENGTH OF MATERIALS. 95. 


multiplied by the distance of the centre of gravity of their sec~ 
tion from the lowest point; about which point, according to 
this hypothesis, the motion must necessarily take place. 

The fallacy of these assumptions was noticed, but not cor- 
rected, by several subsequent authors. Leibnitz objected to 
the doctrine of the fibres resisting equally under all degrees 
of tension, but admitted their incompressibility, thereby still 
making the motion take place about the lowest point of the sec- 
tion; but he assumed for the law of resistance to extension that 
it was always proportional to the quantity of extension. Ac- 
cordingly as the one or the other of these hypotheses was 
adopted, the computed transverse resistance of a beam, as de- 
pending on the absolute strength of its fibres, varied in the 
ratio of 3 to 2; and many fanciful conclusions have been drawn 
by different authors relative to the strength of differently formed 
beams, founded upon the one or the other of these assumptions, 
which, however, it will be unnecessary to refer to more parti- 
cularly in this article. 

We have seen that each of these distinguished philosophers 
supposed the incompressibility of the fibres; but James Ber- 
noulli rejected this part of Leibnitz’s hypothesis, and considered 
the fibres as both compressible and extensible, and that the 
resistance to each force was proportional to the degree of ex- 
tension or compression. Consequently, the motion instead of 
taking place, as hitherto considered, about the lowest point of 
the section, was now necessarily about a point within it; and 
his conclusion was, that whatever be the position of the axis of 
motion, or, as it is now commonly called, the neutral axis, the 
same force applied to the same arm of a lever will always pro- 
duce the same effect, whether all the fibres act by extension 
or by compression, or whether only a part of them be extended, 
and a part compressed. Dr. Robison, in an elaborate article 
on this subject, also assumes the compressibility and exten- 
sibility of the fibres, and as a consequence, assumes the centre 
of compression as a fulcrum, about which the forces to exten- 
sion are exerted, and the resistance of both forces to be directly 
proportional to the degree of compression or extension to which 
they are exposed; that is, he assumed each force, although 
not necessarily offering equal power of resistance, to be indivi- 
dually subject to the law of action appertaining to perfectly 
elastic bodies. In carrying on the experiments which laid the 
foundation of my Essay on the Strength of Timber, §c., in 1817, 
I was led by several circumstances I had observed to doubt 
whether, in the case of timber, this assumption of perfect elas- 
ticity was admissible. And as some of the specimens used in 


96 THIRD REPORT—18338. 


my experiments showed very distinctly after the fracture the 
line about which the fracture took place, I thought of availing 
myself of this datum, and that which gave the strength of direct 
cohesion, in order to deduce the law of resistance from actual 
experiment, instead of using any assumed law whatever. 

The result of this investigation implied that the resistance 
was nearly as first assumed by Galileo, and although very dif- 
ferent from what I had anticipated, yet, as an experimental re- 
sult, I felt bound to abide by it, attributing the discrepance to 
the imperfect elastic properties of the material. Mr. Hodgkin- 
son, however, in a very ingenious paper read at the Manchester 
Philosophical Society in 1822, has pointed out an error in. 
my investigation, by my having assumed the momentum of 
the forces on each side the neutral axis as equal to each other, 
instead of the forces themselves; consequently the above de- 
duction in favour of the Galilean hypothesis fails. This paper 
did not come to my knowledge till the third edition of my Essay 
was nearly printed off, and the correction could not then be 
made; but being made, it proves that the law of actual resistance 
approaches much nearer to that of perfect elasticity than from 
the nature of the materials there could be any reason to expect; 
so that in cases where the position of the neutral axis is known, 
and also its resistance to direct cohesion, a tolerably close ap- 
proximation may be made to the transverse strength of a beam 
of any form, by assuming the resistance to extension to be pro- 
portional to the quantity of extension, and the centre of com- 
pression as the fulcrum about which that resistance is exerted. 
But Ihave before observed, and beg again to repeat, that by 
far the most satisfactory data will always be obtained by ex- 
periments on beams of the like form (however small the scale,) 
and of the same material as those to be employed, because then 
the law of resistance forms no part of the inquiry, and does not 
necessarily enter into the calculation, the ultimate strengths 
being dependent on the dimensions only, whatever may be the 
absolute or relative resistance of the fibres to the two forces 
we have been considering. . 

At present I have only considered the resistance-of a beam 
to a transverse strain; but there is another mode of application, 
in which, again, the law of resistance necessarily enters, and 
has led to many curious and even mysterious conclusions. This 
is when a force of compression is applied parallel to the length. 
In the case of short blocks, the resistance of the material to a 
crushing force is all that is necessary to be known; and in the 
Philosophical Transactions for 1818 we have a highly valuable 
table of experimental results on a great variety of materials, by 


REPORT ON THE STRENGTH OF MATERIALS. 97 


George Rennie, Esq., which contains nearly all the information 
on this subject that can be desired. But when a beam is of con- 
siderable length in comparison with its section, it is no longer 
the crushing force that is to be considered: the beam will bend 
and be ultimately destroyed by an operation very similar to 
that which breaks it transversely; and the investigation of these 
circumstances has called forth the efforts of Euler, Lagrange, 
and some other distinguished mathematicians. 

When a cylindric body considered as an aggregate of pa- 
rallel fibres is pressed vertically in the direction of its length, 
it is difficult to fix on data to determine the point of flexure, 
since no reason can be assigned why it should bend in one way 
rather than in another; still, however, we know that practically 
such bending will take place. And it is made to appear, by the 
investigations of Euler and Lagrange, that with a certain weight 
this ought theoretically to be the case, but that with a less 
weight no such an effect is produced,—an apparent interruption 
of the law of continuity not easily explained, which exhibits 
itself, however, analytically, by the expression for the ordinate 
of greatest inflection being imaginary till the weight or pressure 
amounts to a certain quantity. Another mysterious result from 
these investigations is, that while the column has any definite di- 
mensions, and is loaded with a certain weight, inflection as above 
stated takes place; but if the column be supposed infinitely 
thin, then it will not bend till the weight is infinitely great. 
These investigations of two such distinguished geometers are 
highly interesting as analytical processes, but the hypothesis 
on which they are founded, namely, that of the perfect elas- 
ticity of the materials, is inconsistent with the nature of bodies 
employed in practice: they form, therefore, rather an exercise 
of analytical skill than of useful practical deductions. There 
is, however, one useful result to be drawn from these processes, 
which is, that the weight under which a given column begins 
to bend is directly as its absolute elasticity; so that, having de- 
termined experimentally the weight which a column of given 
elasticity will support safely, or that at which inflection would 
commence, we may determine the weight which another column 
of the same dimensions, but of different elasticity, may be 
charged with without danger. 

M. Gerard, a.member of the Institute of France, aware of 
the little practical information to be drawn from investigations 
wholly hypothetical, has given the detail of a great number of 
actual experimental results connected with this subject on oak 


erp beams of considerable dimensions, carried on at the ex- 
. H , 


1 
4 
98 THIRD REPORT—18835. 


pense of the French Government, from which he has drawn the 
following empirical formulz, viz.— 


Pf? _ 11784451 (f+ :03)ah* 
age 1-3 

Ls aes 8 a h?: 

ay Ne 8161128 ah?; 

where P = half the weight in kilogrammes, a the less, and h 
the greater sides of the section, f half the length of the column, 
and 6 the versed sine of inflection, the dimensions being all in 
metres *. 

How far these formule are to be trusted in practical con- 
structions is, however, I consider, rather doubtful, because they 
are drawn from a number of results which differ very greatly 
from each other; and in one case in particular the result, as 
referred to the deflection of beams, has been satisfactorily shown 
to be erroneous by Baron Charles Dupin, in vol. x. of the Jour- 
nal de l’E'cole Polytechnique, as also by a carefully conducted 
series of experiments in my Essay on the Strength of Timber, 
§c. I conceive it, therefore, to be very desirable that a set of 
experiments on this application of a straining force on vertical 
columns should be undertaken, and it is, perhaps the only 
branch of the inquiry connected with the strength of materials 
in which there is a marked deficiency of practical data; at the 
same time it is one in which both timber and iron are being con- 
stantly employed. We see every day in the metropolis houses 
of immense height and weight being built, the whole fronts of 
which, from the first floors, are supported entirely by iron or 
wooden columns; and all this is done without any practical rule 
that can be depended upon for determining whether or not 
these columns are equal to the duty they have to perform. 

I say this with a full knowledge that Mr. Tredgold has fur- 
nished an approximate rule for this purpose ; but the principle 
on which it is founded has no substantial basis. The extra- 
ordinary skill which Mr. Tredgold possessed in every branch 
of this subject, and the great ingenuity he has displayed in in- 
vestigating and simplifying every calculation connected with 
architectural and mechanical construction, certainly entitle his 
opinion to high consideration; but still on a subject of such 
high importance, it would be much more satisfactory to be pos- 
sessed of actual experimental data. The supposition he ad- 
vanced was made entirely as a matter of necessity, and I am 


1. In oak beams 


2. In fir beams 


* See Traité Analytique de la Résistance des Solides. 


| Ped 


REPORT ON THE STRENGTH OF MATERIALS. 99 


confident that no one would have been more happy than him- 
self to have been enabled to substitute fact for hypothesis, had 
he possessed the means of adopting the former. But unfor- 
tunately such a series of experiments are too expensive and 
laborious to be undertaken by an individual situated as he was, 
having a family to maintain by his industry, and whose close 
and unremitting application to these and similar inquiries, in all 
probability shortened his valuable life*. 

At present I have referred principally to experiments made 
with a view of. determining the ultimate strength of materials ; 
and with data thus obtained practical men have been enabled 
to pursue their operations with safety, by keeping sufficiently 
within the limits of the ultimate strain the materials would bear, 
or rather with which they would just break, some working to a 
third, others to a fourth, &c., of the ultimate strength, according 
to the nature of the construction, or the opinion of the con- 
structor. 

But it is to be observed, that although we may thus ensure 
perfect safety as far as relates to absolute strength, there are 
many cases in which a certain degree of deflection would be 
very injurious. It is therefore highly necessary to attend also 
to this subject, particularly as the deflection of beams and their 
ultimate strength depend upon different principles, or are at 
least subject to different laws. Hence most writers of late date 
give two series of values, one exhibiting the absolute or relative 
strength, and the other the absolute or relative elasticities. 
These values will of course be found to differ in different au- 
thors, on account of the uncertainty in the strength of the ma- 
terials already referred to, but amongst recent experiments the 
difference is not important: they will also be found differently 
expressed, in consequence of some authors deducing these 
numbers from experiments differently made. Some, for ex- 
ample, have drawn their formule for absolute strength from 
experiments made on beams fixed at one end and loaded at the 
other, using the whole length; some, again, from experiments 
on beams supported at each end and loaded in the middle, 
using the half length. Some. take the length in feet, and the 
section in inches; others all the dimension in inches; and a 
similar variety occurs in estimating the elasticity. Also, in the 
latter case, some authors employ what is denominated the mo- 
dulus of elasticity, in which latter case the weight of the beam 


* Mr. Tredgold’s Principles of Carpentry, and his Treatise on the Strength 
of Iron, ought to be in the possession of every practical builder; besides which 
two works, he published many separate articles on the same subject in different 
numbers of the Philosophical Magazine. 


H2 


100 THIRD REPORT—1833. 


itself, and consequently its specific gravity, enters. These va- 
rieties of expressions, however, are not to be understood as 
arising from any difference of opinion amongst the authors from 
whom they proceed, but merely as different modes of expressing 
the same principles: indeed, in reply to that inquiry of the 
Committee with reference to this point, I may, I think, venture 
to say there is not at present any difference of opinion on any 
of the leading principles connected with the strength of mate- 
rials, with the exception of such as are dependent entirely upon 
the imperfect nature of the materials themselves, and which, as 
we have seen, will give rise to different results in the hands of 
the same experimenter and under circumstances in every re- 
spect similar. 

As I distinguish the doctrine of the absolute resistance or 
strength of materials, which is founded on experiment, from 
that which relates to the amount and resolution of the forces or 
strains to which they are exposed, which is geometrical; and 
as I confine myself to the former subject only in this Essay, it 
is not, I conceive, necessary to extend the preceding remarks 
to any greater length. I shall therefore conclude by giving 
a table of the absolute and relative values of the ultimate 
strength and elasticity of various species of timber and other 
materials, selected from those results in which I conceive the 
greatest reliance may be placed. 

Formule relating to the ultimate Strength of Materials in 
cases of Transverse Strain.—Let 1, b, d, denote the length, 
breadth and depth in inches in any beam, w the experimental 
breaking weight in pounds, then will = = 8 be a constant 
quantity for the same material, and for the same manner of ap- 
plying the straining force; but this constant is different in dif- 
ferent modes of application. Or, making S constant in all cases 
for the same material, the above expression must be prefixed 
by a coefficient, according to the mode of fixing and straining. 

1. When the beam is fixed at one end, and loaded at the 
other, 


lw _ 
ie i 
2. When fixed the same, but uniformly loaded, 
1 lw 
Ce 0 ae 
3. When supported at both ends, and loaded in the middle, 
1 lw 7 


REPORT ON THE STRENGTH OF MATERIALS. 101 


4, Supported the same, and uniformly loaded, 


1 lw 
B MR 
5. Fixed at both ends, and loaded in the middle, 
1 lw 
6 x bae ish 
6. Fixed the same, but uniformly loaded, 
1 lw 
12 * 5a? = * 


7. Supported at the ends, and loaded at a point not in the 
middle. Then, x m being the division of the beam at the point 
of application, 

nm _ tw 
ola ais 


: 1 
Some authors state the coefficients for cases 5 and 6 as 3 


and 7 but both theory and practice have shown these numbers 


to be erroneous. 

By means of these formule, and the value of S, given in the 
following Table, the strength of any given beam, or the beam 
requisite to bear a given load, may be computed. This column, 
however, it must be remembered, gives the ultimate strength, 
and not more than one third of this ought to be depended upon 
for any permanent construction. 

Formule relating to the Deflection of Beams in cases of 
Transverse Strains.—Retaining the same notation, but repre- 
senting the constant by E, and the deflection in inches by 8, 
we shall have, 


S22 wile 5 3 w 
Case 1. aT > 4 bd2s => ° Case 4, my x bdss = E. 
12 Pw 2 ew 
Roem Pt BS ane 
1 lI? w 5 lw 
es  balea ag eS are 


Hence, again, from the column marked E in the following 
Table, the deflection a given load will produce in any case may be 
computed ; or, the deflection being fixed, the dimension of the 
beam may be found. Some authors, instead of this measure of 


- 102 THIRD REPORT—1833. 


3 
elasticity, deduce it immediately from the formula 3b dzs : aa 5 = E, 
substituting for w the height in inches of a column of the ma- 
terial, having the section of the beam for its base, which is equal 
to the weight w, and this is then denominated the modulus of 
elasticity. It is useful in showing the relation between the 
weight and elasticity of different materials, and is accordingly 
introduced into the following Table. 

The above formule embrace all those cases most commonly 
employed in practice. There are, of course, other strains con- 
nected with this inquiry, as in the case of torsion in the axles 
and shafts of wheels, mills, &c., the tension of bars in suspen- 
sion bridges, and those arising from internal pressure in cylin- 
ders, as in guns, water-pipes, hydraulic presses, &c.; but these 
fall rather under the head of the resolution of forces than that 
of direct strength. It may just be observed, that the equation 
due to the latter strain is . 


t(c —n) =nR, 


where ¢ is the thickness of metal in inches, c the cohesive power 
in pounds of a square inch rod of the given material, » the 
pressure on a square inch of the fluid in pounds, and R the in- 
terior radius of the cylinder in inches. Our column marked C 
will apply to this case, but here again not more than one third 
the tabular value can be depended upon in practice. 


ha oe 
ee 


REPORT ON THE STRENGTH OF MATERIALS. 103 


Table of the Mean Strength and Elasticity of various Materials, as 
deduced from the most accurate Experiments. 


Cc. lw 
Mean S=a@ Bw 


E=—333 
strength of | Constants 6d33 | Modulus of Remarks, 
sta- | cohesionon | for trans- | Constants | Elasticity. 
+ |aninch sec-| verse for deflec- 


strains. tions. 


1800 | 4609000) 3739000\of English growth. 
2026 6580000) 4988000 ditto. 
1560 | 5417000) 4457000 ditto. 
Birch, Common 700 1900 | 6570000} 5406000 ditto. 
, American Black} 750 1500 | 5700000} 3388000) American. 


2650 |10512000| 5878000)|Berbice. 

2500 | 7437000) 4759000\ditto. 
11000 1550 | 6350000} 5378000 
11000 1730 | 6420000} 6268000 
5780 1030 | 2803000) 3007000/English. 
Fir, New England ... 12000 1100 | 5967000) 6249000 
—, Riga 750} 12600 1130 | 5314000) 4080000 
1140 | 3400000) 2797000)\Scotland. 
2700 |10620000) 6118000|Berbice. 
1120 | 4200000) 4480000 
3400 767000) 4649000) America, South. 


Mahogany 
Norway Spars 1470 | 5830000) 5789000 

: from} 70 1200 | 3490000) 2872000) \ Results very va 
Oak, English { to...| 900 2260 | 700000047020000 \ riable. 


—.,, African 2000 | 9500000)55830000 
iati 1380 | 3880000) 2257000 
1760 | 8590000) 5674000 
1450 | 4760000) 3607000 


2200 | 6760000) 6488000|East Indies. 
1630 | 5000000) 4364000 
1340 | 7360000) 6423000 
2460 | 9660000) 7417000|East Indies. 
2700 |10620000) 5826000|Berbice. 


8100 69120000) 5530006 : 
Mean of Englis 
9000 eis 6770000) ( and Foreign. 


oot ssa RGB Et ee mare 

pa amales- alga crager 

Rate | ry 5 2 Poe 4 
eta at Ne: serena 


% 


re tHe . 
ME ) Cas | Rey er fi 
tebe Hes i 
is 


4 er 
mab 


hai ¢ . A VARs :. 
ra ; 
mas Ay 
s Atal 
; iy ee) tee | ‘ Vr 
Near sulrbav ince Named aay > lige WEG So vce. tean 
27 ons a "one “Thue f an bi ke RP ‘4 


SBE REY ii reap eed eer 
RAPES sonia fps ¢ oe ae 
ae Hed cue AERA f5 ‘ 
> Rua any ‘ 
“lie Ait a nih 4 


‘er fi 3 \ 
fame a 3! re re dts 
he aerated oes 
ei. | rafal Ay 
7 ¢ \ s Wile a 
TWIN? ae As 2 
hr pay “tif fri ' } ; } WHDE T . cay 
OH THe TRAD Pa A Hier 
ET UGE: tei “hy ; abet 
: ee et. ces Se 
ae BS ae y , ( 
ays5 xi pan At in ib, ; pst ¥ baal ; 


PRES. iy . Ta ; ‘i mt 
ut « ayy » oe ] { , SPs Means 
ss Giese = | eae r : iy ae gin Maia 
eh Sa et ee ee ‘W hii eis f + Seger Ne : ene 

her stl raat ox 


ees ge 


Bai vite febsdisee dgacane:, ifn a aries 
ath ber \ Meee 'D cikaes ie 
: ’ % ; epota a py 


_ “rey Johettrtichinipna lens Pa ee RE 


[ 105 ] 


Report on the State of our Knowledge respecting the Magnetism 
of the Earth. By S. Hunter Curistiz, Esq., M.A., FBS. 
M.C.P.S., Corr. Memb. Philom. Soc. Paris, Hon. Memb. 
Yorkshire Phil. Soc.; of the Royal Military Academy ; and 
Member of Trinity College, Cambridge. 


Hap the discovery of the loadstone’s directive power been made 
by a philosopher who at the same time pointed out its import- 
ance to the purposes of navigation, we might expect that his 
name would have been handed down to posterity as one of the 
greatest benefactors of mankind. The discovery was, however, 
most likely made by one so engaged in maritime enterprise that, 
in his eyes, this application constituted its whole value; and it 
is not improbable that, being for some time kept secret, it may 
have been the principal cause of the success of many enterprises 
attributed to the superior skill and bravery of the leaders. The 
knowledge of this property of the magnet, though gradually 
diffused, would long be guarded with jealousy by those who 
justly viewed it as of the highest advantage in their predatory 
or commercial excursions; and this is, perhaps, the cause of the 
obscurity in which the subject is veiled. If the discovery is 
European, there is no people, from the character of their early 
enterprises, and, I may add, from the nature of the rocks of 
their country, more likely to have made it than the early Nor- 
wegians; and as there is reason for believing that they were 
acquainted with the directive property of the loadstone at least 
half a century earlier than its use is supposed to have been 
known in other parts of Europe, it may be but justice to allow 
them the honour of having been the discoverers. Whether the 
discovery was made in Asia or in Europe, in the North or in 
the South, I am not, however, now called upon to decide, but 
to point out the consequences which have followed that disco- 
very by unveiling gradually phenomena, though less striking, 
yet equally interesting, and some even more difficult of expla- 
nation. 

These phenomena are, the variation of the magnetic needle, 
with its annual and diurnal changes; the dip of the needle; and 
the intensity of the magnetic force of the earth; which are, how- 
ever, all comprised under two heads,—The Direction and the 
Intensity of the terrestrial magnetic force. 


106 THIRD REPORT—1833. 


I. The Direction of the Terrestrial Magnetic Force. 


1. The Variation of the Needle.—For some centuries after 
the directive property of the loadstone was discovered, it was 
generally supposed that the needle pointed correctly towards 
the pole of the heavens. It has however been said, on the 
authority of a letter by Peter Adsiger, that the variation of 
the needle was known as early as 1269; and if we fully admit 
the authenticity of this letter, we must allow that the writer 
was at that date not only aware of the fact, but that he had 
observed the extent of the deviation of the needle from the 
meridian*. It is possible that such an observation as this 
may have been made at this early period by an individual de- 
voting his time to the examination of magnetical phenomena; 


* This curious and highly interesting letter, dated the 8th of August 1269, 
is contained in a volume of manuscripts in the Library of the University of 
Leyden, and we are indebted to Cavallo for having published extracts from it. 
The variation is thus referred to: “Take notice that the magnet (stone), as well 
as the needle that has been touched (rubbed) by it, does not point exactly to 
the poles; but that part of it which is reckoned to point to the south declines a 
little to the west, and that part which looks towards the north inclines as much 
to the east. The exact, quantity of this declination I have found, after numer- 
ous experiments, to be five degrees. However, this declination is no obstacle 
to our guidance, because we make the needle itself decline from the true south 
by nearly one point and an half towards the west. A point, then, contains five 
degrees.” (Letter of Peter Adsiger, Cavallo On Magnetism, London 1800, p. 317.) 
It is certainly extraordinary, if so clear an account of the deviation of the needle 
from the meridian as this, was communicated to any one by the person who had 
himself observed that deviation, that for more than two centuries afterwards we 
should have no record of a second observation of the fact. This alone would 
throw doubt on the authenticity of the letter, and the estimate given of the 
variation may appear to confirm these doubts; for, according to the period of 
change which best agrees with the observations during more than two hundred 
years, the variation, if observed, would have been found to be westerly instead 
of easterly in 1269. It may however be urged, that as the whole period of 
change has not yet elapsed since observations were made, we are not in pos- 
session of a sufficient number of facts to authorize us to draw conclusions re- 
specting the variation at such an early date; and also, that if the letter be spu- 
rious, or the original date have been altered to that which it bears, this or the 
fabrication can only have been for the purpose of founding claims in consequence 
of the contents of this letter; and as no such claims have been advanced, there 
appears no motive either for fabrication or alteration. In a preceding part of 
the letter the author gives methods for finding the poles of a loadstone; and 
certainly the direction of the axis could not be determined to within five degrees 
by either of these; so that, as regards the loadstone, we may, I think, conclude 
that the author did not make the observation. As a matter of curious history 
connected with magnetism, it is desirable that either the authenticity of this 
letter should be clearly established, or reasons given for doubting it, by those 
who have an opportunity of consulting the original. 


REPORT ON THE MAGNETISM OF THE EARTH. 107 


and as it is probable that for some time subsequent to the dis- 
covery of the directive property of the needle the deviation in 
Europe was not of sufficient magnitude to have been easily de- 
tected by means of the rude instruments then in use, it may 
very likely be owing to this circumstance that we have not 
earlier records of the variation*. 'That Columbus, the most 
scientific navigator of his age, when he commenced his career 
of discovery, and undertook to show the western route to India, 
was not aware of it, is clear, since the discovery during his first 
voyage has been attributed to him. However, although Co- 
lumbus may have noticed that the needle did not in every situa- 
tion point due north, and Adsiger, long before him, may even 
have rudely obtained the amount of its deviation, the first ob- | 
servations of the variation on which any reliance can be placed 
appear to have been made about the middle of the sixteenth 
century, and shortly afterwards it was well known that the va- 
riation is not the same in all places f. 

2. Change in the Direction of the Needle-——When it was 
first determined by observation, about 1541, that the needle 
did not point to the pole of the earth, it was found that this vari- 
ation from the meridian, at Paris, was about 7° or 8° towards 
the east. In 1550 it was observed 8° or 9° east; and in 
1580, 114° east. Norman appears to have been the first who 
observed the variation with any degree of accuracy in Lon- 
don. He states that he observed it to be 11° 15’ east t, but he 
was not aware that it does not remain constant in the same 
place§. In 1580, Burough found the variation at Limehouse 
to be 113° or 114° east||, and his observations appear to be 


* * Another reason why the variation was not earlier observed may be that the 
natural magnet was first used for the purposes of navigation, and its directive 
line was that which pointed to the pole star. As it was therefore considered 
that the natural magnet indicated the direction of the meridian, and it was 
found that a needle touched by it had the directive power, when the needle was 
introduced it was assumed that this also pointed in the meridian. 

+ The New Attractive, by Robert Norman, chap. ix. London 1596. 

t Ibid. No date is given for this observation ; but from the circumstance of 
Burough referring to Norman’s book in the preface to his Discourse of the Va- 
riation of the Compasse, dated 1581, (the copy of this to which I have access 
was printed in 1596, but the Bodleian Library contains one printed in 1581,) 
it would appear that there must have been an earlier edition of Norman’s book 
than that of 1596, and that his observations must have been made before 1581. 
Bond, Philosophical Transactions, vol. viii. p. 6066, gives 1576 as the date of 
Norman’s observations. 

§ “ And although this variation of the needle be found in travaile to be divers 
and changeable, yet at anie land or fixed place assigned, it remaineth alwaies 
one, still permanent and abiding.” New Attractive, chap. ix. 

|| The mean of his observations, which do not differ 20', is 11° 19! east. 


108 THIRD REPORT—1833. 


entitled to much confidence ; but he was of the same opinion 
as Norman with respect to the constancy of the variation*. 
Gunter, in 1612, found the variation in London to be 5° 36/ 
east; and Gellibrand, in 1633, observed it 4° 4’ east. Dr. Wal- 
lis considers Gellibrand to have been the discoverer of “ the 
variation of the variation};” but if Gunter had any confidence 
in his own observations and those of Burough, he must have 
been aware of the change in the variation. In 1630, Petit 
found the variation at Paris to be 44° east, but suspected, at 
the time, that the earlier observations there had been incorrect; 
and it was not until 1660, when he found the variation to be 
only 10’ east, that he was satisfied of the change of the varia- 
tion. About ten years later, Azaut, at Rome, where the va- 
riation had been observed 8° east, found it to be more than 2° 
west; and Hevelius, who at Dantzick in 1642 had found it to 
be 3° 5/ west, now found it to be 7° 20! west. 

3. Diurnal Change in the Variation.—This was discovered 
in 1722 by Graham, to whose talents and mechanical skill 
science is so deeply indebted. He found that with several 
needles, on the construction of which much care had been be- 
stowed, the variation was not always the same; and at length 
determined that the variation was different at different hours 
in the day, the greatest westerly variation occurring between 
noon and four hours after, and the least about six or seven 
o'clock in the eveningt. Wargentin at Stockholm in 1750, 
and Canton in London from 1756 to 1759, more particularly 
observed this phenomenon; and the latter determined that 
the time of minimum westerly variation in London was between 
eight and nine in the morning, and the time of maximum be- 
tween one and two in the afternoon. Canton likewise deter- 
mined in 1759, that the daily variation was different at different 
times in the year, the maximum change occurring about the end 
of June, and the minimum in December §. Cassini, during more 
than five years and a half, namely, from May 1783 to January 
1789, carefully observed, at particular hours, the direction of a 
needle suspended in the Observatory at Paris; and although 
he does not correctly state the course of the daily variation, 
overlooking altogether the second maximum west, and the pro- 
gress of the needle towards the east in the early part of the 


* © For considering it remayneth alwaies constant without alteration in every 
severall place, there is hope it may be reduced into method and rule.” Dis- 
course, chap. x. 

+ Philosophical Transactions, 1701, vol. xxii. p. 1036. 

¢ Lbid. 1724, vol. xxxiii. p. 96. 

§ Ibid. 1759, vol. xli. p. 398. 


REPORT ON THE MAGNETISM OF THE EARTH. 109 


morning*, yet his observations and reniarks are of great value 
as pointing out the annual oscillations of the needle}. Since 
this, the diurnal variation has been very generally observed, 
but by no one with greater care and perseverance than by the 
late Colonel Beaufoy f. 

In order to determine whether the course of the diurnal va- 
riation is influenced by the elevation of the place of observation, 
the zealous and indefatigable De Saussure undertook a series 
of observations on the Col du Géant, nearly 11,300 feet above 
the level of the sea. This series, after incurring much personal 
inconvenience and even risk in that region of snow and of storms, 
he completed; and he has compared the results with observa- 
tions which he made immediately before and after at Chamouni 
and Geneva. From this comparison it appears that the course 
of the diurnal variation was nearly the same on one of the 
highest mountains, in a deep and narrow valley at its foot, and 
in the middle of a plain or of a large valley. The times of the 
maxima, east and west, are in each case nearly those previously 
determined by Canton, these maxima occurring rather later on 
the Col du Géant than at the other stations. Excluding in all 
cases the results where extraordinary causes appear to have 
operated, the extent of the diurnal variation at Chamouni ex- 
ceeds that at Geneva and also that on the Col, the two latter 
being very nearly the same. The observations, however, are, 
as Saussure very justly remarks, much too limited to give cor- 
rect means §. 

5. The Dip of the Magnetic Needle.—Norman having found 
with different needles, and with one in particular on the con- 
struction of which he had bestowed much pains, that although 
perfectly balanced on the centre previously to being touched 
by the magnet, after this operation the north end always de- 
clined below the horizon, devised an instrument by which he 


* Journal de Physique, Mai 1792, tom. xl. p. 345. + Ibid. p. 348. 

+ Many of the results of Colonel Beaufoy’s observations are published in the 
Edinburgh Philosophical Journal, vols. i. ii. iii. iv. and vii. 

§ Saussure, Voyages dans les Alpes, tom. iv. p. 302 au p. 312. As Saussure 
does not give the mean results, I insert them here. 


Time of absolute mmaximum./Time of second maximum.!py tent - Elevation 
| diurnal | above the 
East. West. East. West. | change.| — sea. 
hom ee nn one Ree 
Geneva ...... 7 56 a.m.| 1 09 P.m.\6 26 p.m.\11 17 p.m.J15 42}) 1305 
Chamouni ...| 7 34 1 41 7 44 10 46 17 06| 34538 
Col du Géant| 8 09 2 00 5 51 10 17 15 43} 11274 


110° THIRD REPORT—1833. 


could determine the inclination of the needle to the plane of the 
horizon*. The figure given of the instrument is sufficiently 
rude, but the principles of its construction, as stated by Nor- 
man, are correct. With this instrument he found the inclina- 
tion of the needle to the horizon in London to be about 71° 50’, 
but gives no date to the observation, though Bond assigns 1576 
as the time}. Although in a theoretical point of view it would 
be desirable to have so early a record of the dip, particularly as 
subsequent observations lead us to suppose that the dip attained 
its maximum after this time, yet, considering the uncertainty 
attending such observations, even with the present improved 
instruments, we cannot place much confidence in this result, 
however we may rely upon the author having used every pre- 
caution in his power to ensure accuracy. Having determined 
the dip of the needle in London, Norman states that this de- 
clining of the needle will be found to be different at different 
places on the earth}, though he does not take a correct view 
of the subject, for he considers that the needle will always be 
directed towards a fixed point. 

5. Variation of the Dip.—Subsequent observations by Bond, 
Graham, Cavendish, and Gilpin, and the more recent ones in 
our own time, have shown that the inclination of the needle to 
the horizon at the same place, like the angle which it makes 
with the meridian, is subject to change; but the diurnal oscil- 
lations of the direction are of too minute a character to have 
been ascertained with the imperfect instruments which we 
possess. 

This is an outline of the phenomena hitherto observed, de- 
pending upon the direction of the forces acting upon the needle. 
Various attempts have been made to account for those obser- 
vable at fixed points on the earth’s surface at different periods, 
and also to connect those depending on the different positions 
of the places of observation, but hitherto with only very partial 
success. It is not my intention to enter into a detailed history 
of these attempts, but I may briefly notice some of the most 
remarkable. 

To Gilbert we are indebted not only for the first clear views 
of the principles of magnetism, but of their application to the 
phznomenon of the directive power of the needle; and indeed 
we may say that, with the exception of the recent discoveries, 
all that has been done since, in magnetism, has for its foundation 
the principles which he established by experiment§. He con- 

* New Attractive, chap. iii. iv. 


+ Philosophical Transactions, 1673, vol. viii. p. 6066. 
t New Attractive, chap. vii. § Gilbert, De Magnete, §c. Lond. 1600. 


REPORT ON THE MAGNETISM OF THE EARTH. 111 


sidered that the earth acts upon a magnetized bar, and upon 
iron, like a magnet, the directive power of the needle being due 
to the action of magnetism of a contrary kind to that at the 
end of the needle directed towards the pole of the earth. He 
applied the term “pole” to the ends of the needle directed 
towards the poles of the earth, according to the view he had 
taken of terrestrial magnetism, designating the end pointing 
towards the north, as the south pole of the needle, and that point- 
ing towards the south, as its north pole*. It is to be regretted 
that some English philosophers, guided by less correct views, 
have since his time applied these terms in the reverse sense, 
which occasionally introduces some ambiguity, though now they 
are used in this country, as on the Continent, in the sense ori- 
ginally given to them by Gilbert. ) 

In 1668 Bond published a Table of computed variations in 
London, for every year, from that time to the year 1716+. The 
variations in this Table agree nearly with those afterwards ob- 
served for about twenty-five years, beyond which time they 
differ very widely; and I only notice this Table as the first em- 
pirical attempt at the solution of a problem which is, as yet, 
unsolved. Bond afterwards proposed to account for the change 
in the variation and dip of the needle by the motion of two 
magnetic poles about the poles of the earth. He professed not 
only to give the period of this motion, but to be able to point 
out its cause, and even proposed to determine the longitude 
by means of the dip{. He, however, did not make public either 
his methods or his views; but with regard to the longitude, it 
is probable they were the same as those afterwards adopted by 
Churchman. 

Halley considered that the direction of the needle at different 
places on the earth’s surface might be explained on the suppo- 
sition that the earth had four magnetic poles§, and that the 
change in the direction at the same place was due to the motion 
of two of these poles about the axis of the earth, the other two 
being fixed. He does not enter into any calculations to show 
the accordance of the phenomena with such an hypothesis, but 
conjectures that the period of revolution of these poles is about 
700 years ||. : 

Since this time, calculations have been made by various au- 
thors, both on the hypothesis of two magnetic poles and on 
that of four, with the view of comparing the results of these 


. * Gilbert, De Magnete, §c., lib. i. cap. iv. 
+ Philosophical Transactions, 1668, vol. iii. p. 789. 
t Ibid. 1673, vol. viii. p. 6065. § Ibid. 1683, vol. xiii. p. 208. 
|| Zbid. 1692, vol. xvii. p. 563. 


112 THIRD REPORT—1833. 


hypotheses with actual observation. ‘The most recent attempt 
of this kind is that by Professor Hansteen. He adopts Halley's 
hypothesis of four magnetical poles, but considers that they all 
revolve, and in different periods, the northern poles from west 
to east, and the southern ones from east to west. The results 
calculated on this hypothesis agree pretty nearly with the ob- 
servations with which they are compared; but as considerable 
uncertainty attends magnetical observations, excepting those 
of the variation made at fixed observatories, and especially the 
early ones of the dip and variation, on which the periods of the 
poles and their intensities must so much depend, it would cer- 
tainly be premature to say that such an hypothesis satisfactorily 
explains the phenomena of terrestrial magnetism. If we admit 
that the progressive changes which take place in the direction 
of the needle are due to the rotation of these poles, we must 
look to the oscillations of the same poles for the cause of the 
diurnal oscillation of the needle. Any hypothesis which by 
means of two or more magnetic poles will thus connect the 
phenomena of magnetism, is of great advantage, however un- 
able we may be to give a reason for the particular positions of 
the poles, or for their revolution. Hansteen refers these to 
the agency of the sun and moon. 

Without assigning any cause either for the direction of the 
needle, or for the progressive change of that direction, attempts 
have been made to account for its diurnal oscillations. But 
before taking a review of these, it is necessary that I should 
state more particularly the precise nature of the phenomenon. 
This I cannot do better than by referring to the results de- 
duced from Canton’s observations*. From these it appears 
that in London, during the twenty-four hours, a double oscilla- 
tion of the needle takes place, the absolute maximum west 
happening about half-past one in the afternoon, and the abso- 
lute maximum east, that is, the minimum west, about nine in 
the morning; besides which there was another maximum east 
about nine in the evening, and a maximum west near midnight 
or very early in the morning, the two latter maxima being small 
compared with the absolute maxima. Colonel Beaufoy’s very 
extensive series of observations, made when the variation was 
between 24° and 25° west, (Canton’s having. been made when 
it was 19°,) give nearly the same results, the absolute maxima 
happening somewhat earlier, and the second maxima west 
about eleven in the evening. 

Canton explained the westerly motion of the needle in the 


* Philosophical Transactions, 1759, p. 398, and 1827, pp. 333, 334, 


REPORT ON THE MAGNETISM OF THE EARTH. 113 


latter part of the morning, and the subsequent easterly motion, 
by supposing that the heat of the sun acted upon the northern 
parts of the earth as upon a magnet, by weakening their in- 
fluence, but offered no explanation of the morning easterly mo- 
tion of the needle. 

Oersted’s discovery of the influence of the closed voltaic 
circuit upon the magnetic needle, and the consequent discoveries 
of Davy, Ampére and Arago, immediately led to the considera- 
tion, whether all the phenomena of terrestrial magnetism were 
not due to electric currents; and the discovery of Seebeck, that 
electric currents are excited when metals having different 

owers of conducting heat are in contact,—which discovery 
with but few holds the rank to which it is eminently entitled,— 
pointed to a probable source for the existence of such currents. 
At the conclusion of a highly interesting paper on the develop- 
ment of electro-magnetism by heat, Professor Cumming re- 
marks that ‘‘magnetism, and that to a considerable extent, it 
appears, is excited by the unequal distribution of heat amongst 
metallic, and possibly amongst other bodies. Is it improbable 
that the diurnal variation of the needle, which follows the 
course of the sun, and therefore seems to depend upon heat, 
may result from the metals, and other substances which com- 
pose the surface of the earth, being unequally heated, and con- 
sequently suffering a change in their magnetic influence?” And 
in the second part of a paper, detailing some thermo-magnetical 
experiments, read before the Royal Society of Edinburgh, 
Dr. Traill considers “that the disturbance of the equilibrium 
of the temperature of our planet, by the continual action of the 
sun’s rays on its intertropical regions, and of the polar ices, 
must convert the earth into a vast thermo-magnetic apparatus: ” 
and “that the disturbance of the equilibrium of temperature, 
even in stony strata, may elicit some degree of magnetism*.” 
Previous to this, I had adopted the opinion that temperature, 
_ if not the only cause, is the principal one of the daily variation. 
It did not, however, appear to me, that any of the experiments 
hitherto made bore directly on the subject, since the metals 
producing electric currents by their unequal conduction of heat 
were only in contact at particular parts, and in no case had 
such currents been excited by different metals having their 
surfaces symmetrically united throughout. I in consequence 
instituted a series of experiments with two metals so united, 
and found that electric currents were still excited on the 


* Transactions of the Philosophieal Society of Cambridge, vol. ii. p. 64. 
+ Philosophieal Transactions, 1823, p. 392. 
1833. I 


114 ; THIRD REPORT—1838. 


application of heat, the phenomena corresponding to magneti¢ 
polarization in a particular direction with reference to the place 
of greatest heat*. From these experiments I drew the con- 
clusion that one part of the earth, with the atmosphere, being 
more heated than another, two magnetic poles, or rather elec- 
tric currents producing effects referrible to such poles, would _ 
be formed on each side of the equator, poles of different names 
being opposed to each other on the contrary sides of the equa- 
tor; and that different points in the earth’s equator becoming 
successively those of greatest heat, these poles would be carried 
round the axis of the earth, and would necessarily cause a de- 
viation in the horizontal needle +. On comparing experimentally 
the effects that would result from the revolution of such poles 
with the diurnal deviations at London, as observed by Canton 
and Beaufoy, also with those observed by Lieut. Hood at Fort 
Enterprise, and finally with the late Captain Foster’s at Port 
Bowen, I found a close agreement in all cases in the general 
character of the phenomena, and that the times of the maxima 
east and west did not differ greatly in the several cases. The 
double oscillation of the needle, to which I have referred in 
Canton’s and Beaufoy’s observations, clearly resulted from this 
view of the subject. Some of the experiments to which I have 
referred showed that when heat was applied to a globe, the 
electric currents excited were such, that on contrary sides of 
the equator the deviations of the end of the needle of the 
same name as the latitude were at the same time always in the 
same direction, either both towards east or both towards west. 
No observations having at that time been made on the diurnal 
variation of the needle in a high southern latitude, I considered 
“that the agreement of the theoretical results with such ob- 
servations would be almost decisive of the correctness of the 
theory.” Captain Foster’s observations at Cape Horn, South 
Shetland, and the Cape of Good Hope, show most decidedly 
that in the southern hemisphere the diurnal deviations of the 
south end of the needle correspond very precisely with those of 
the north end in the northern hemisphere ; and most fully bear 
me out in the view which I had taken. These valuable obser- 
vations have been placed in my hands by His Royal Highness 
the President, and the Council of the Royal Society, and I in- 
tend, when I have sufficient leisure, rigidly to compare them, 
and likewise those to which I have already referred in the 
northern hemisphere, with the diurnal deviations that would 


Traniactions, 1827, pp. 321, 326. 
} Ibid. pp. 827, 328. 


* “Theory of the Diurnal Variation of the Magnetic Needle,” Philosophical 


REPORT ON THE MAGNETISM OF THE EARTH. 115 


result at the corresponding places on the earth’s surface, on 
the supposition that such electric currents as I have supposed 
are excited on contrary sides of the equator, in consequence of 
different parts on the earth’s surface becoming successively the 
places of greatest heat, during its revolution upon its axis. 
Should there be found in these results that accordance which I 
have reason to expect, there will, I think, be no doubt that the 
diurnal deviation of the needle is due to electric currents excited 
by the heat of the sun. 

I have already adverted to the hypotheses of two or more 
poles, by means of which attempts have been made to explain 
the phenomena of terrestrial magnetism, and I may now re- 
mark, that if we admit the existence of such poles, we must be 
careful not to consider the magnetic meridians as great circles: 
they are unquestionably curves of double curvature. Nor must 
we consider these poles to be, like the poles of a magnet, cen- 
tres of force not far removed from the surface. If such centres 
of force exist for the whole surface of the earth, the experi- 
mental determinations of the magnetic force at different places, 
to which I shall shortly advert, at least show that they cannot 
be far removed from the centre of figure. 

In the delineation of magnetic charts, more attention has 
hitherto been paid to the Halleyan lines, or lines of equal varia- 
tion, than to any others; and I am not disposed to undervalue 
charts where such lines alone are exhibited : to the navigator they 
are of the greatest value; but they throw little light on the phe- 
nomena in general. Ifthe meridians were correctly represented, 
they would at least indicate clearly their points of convergence, if 
such in all cases exist ; but the lines that would be most likely to 
guide us to a true theory of terrestrial magnetism, are the nor- 
mals to the direction of the needle. | If, as is highly probable, 
the direction of the needle is due to electric currents circulating 
either in the interior or near the surface of the earth, these 
normals would represent the intersection of the planes of the 
currents with the surface of the earth ; and, by their delineation, 
we should have exhibited in one view the course of the currents 
and the physical features by which that course may be modified, 
so that any striking correspondences which may exist, would 
_ be immediately seized, and lead to important conclusions. 

Changes of temperature I consider to be the principal cause of 
the diurnal changes in the direction of the needle: and if any 
connexion exist between these electric currents and climate, 
we are to expect that the curvature of these normal lines will 
be influenced by the forms, the extent and direction of the con- 
tinents or seas over which they pass, and also by the height, 

I 


116 THIRD REPORT—1838. 


direction and extent of chains of mountains, and probably by 
their geological structure. 

‘These normal lines may, to a certain extent, agree with the 
lines of equal dip, which have already been delineated upon 
some charts. In Churchman’s charts they are represented in 
the positions they would have on Euler’s hypothesis of the earth 
haying two magnetic poles. The only use, however, of such 
hypothetical representations is, that by comparison with actual 
observation they become tests of the correctness of the theory,. 
or they may point out the modifications which it requires, in 
order that it may accord with observation. In Professor Hans- 
teen’s chart the lines of equal dip are projected from observa- 
tions reduced to the year 1780. Considering how very deficient 
we are, even now, in correct observations of the dip, I should 
not be disposed to place much reliance upon the accuracy of 
these lines, particularly where they cross great extents of sea 
affording no poiuts of land necessary for observations of the dip. 
Of these lines of equal dip the most important is the magnetic: 
equator, or that line on the earth at which the dipping needle 
would be horizontal. The observations giving this result can 
of course be but few, and are therefore very inadequate for the 
correct representation of this line. In order to obviate this’ 
difficulty, M. Morlet made use of all observations not very re- 
mote from the equator, determining the distance of that line 
from the place of observation by means of the law, that the 
tangent of the magnetic latitude is half the tangent of the dip, 
which is derived from the hypothesis of two magnetic poles near 
to. the centre of the earth. By this means the position of the 
equator was determined throughout its whole extent; and a 
surprising agreement was found between the determinations of 
each point by means of different observations, which shows 
that, within certain limits near the equator, the hypothesis very 
correctly represents the observations. ‘This line exhibits in- 
flections in its course which have been attributed, and probably 
with justice, to the physical constitution of the surface in their 
vicinity *. It has been considered also that a general resem= 
blance exists between the isothermal lines and the lines of equal 
dip on the surface of the earth +. . 

All the lines to which I have here referred have been hitherto 
represented on a plane, either on the stereographical, the glo- 
bular, or Mercator’s projection. Mr. Barlow has, however, 
very lately represented the lines of equal variation on a globe, 
from a great mass of the most recent documents connected with 

* Biot, Traité de Physique. 
+ Hansteen, Edinburgh Philosophical Journal, vol. iii. p. 127. 


REPORT ON THE MAGNETISM OF THE EARTH. 117 


the variation, furnished to him by the Admiralty, the East India 
Company, and from other sources. If to the lines of equal .va- 
riation were added the magnetic meridians and their normals, 
the isodynamic lines, with those of equal dip, such a globe would 
form the most complete representation of facts connected with 
terrestrial magnetism that has ever been exhibited, and might 
indicate relations which have hitherto been overlooked. 
Having discovered that a peculiar polarity is imparted to iron 
by the simple act of rotation, I was led to consider whether the 
principal phenomenon of terrestrial magnetism is not, ina great 
measure, due to its rotation. ‘The subsequent discovery by 
Arago, that analogous effects take place during the rotation of 
all metals, and Faraday’s more recent discovery, that electrical 
currents are not only excited during the motion of metals, but 
that such currents are transmitted by them, render such an 
opinion not improbable. It is, however, to be remarked, that, 
in all these cases, motion alone is not the cause of the effects 
produced ; but that these effects are due to electricity induced 
in the body by its motion in the neighbourhood of a magnetized 
body. If, then, electrical currents are excited in the earth in 
consequence of its rotation, we must look to some body exterior 
to the earth for the inducing cause. ‘The magnetic influence 
attributed by Morichini and Mrs. Somerville to the violet ray, 
and the effect which I found to be produced on a magnetized 
needle when vibrated in sunshine, and which appeared not to 
admit of explanation without attributing such influence to the 
sun’s rays, might appear to point to the sun as the inducing 
body. ‘The experiments, however, of Morichini and Mrs. So- 
merville, have not succeeded on repetition ; and in a recent re- 
petition of my own experiments, in a vacuum, by Mr. Snow 
Harris, the effects which I observed were not detected. I had 
found that the effects produced on an unmagnetized steel needle 
differed from those produced. on a similar needle when magnet- 
ized, and therefore considered that the idea of these effects 
being independent of magnetism was precluded ; but Mr. Har- 
ris’s results may possibly be considered to indicate that they 
were due solely to currents of air excited by the sun’s rays. 
These circumstances render it doubtful whether the sun’s rays 
possess any magnetic influence independent of their heating 
power; but besides this, supposing such influence to exist, if 
electric currents were induced in the earth during its rotation, 
they would be nearly at right angles to the equator, and would 
therefore cause a magnetized needle to place itself nearly per- 
pendicular to the meridians, or parallel to the equator. 
Although it would therefore appear that: the rotation of the 


118 " THIRD REPORT—1833. 


earth is not the cause of its magnetism, yet it is highly pro- 
bable, from Mr. Faraday’s experiments*, that, magnetism ex- 
isting in the earth independently of it, electrical currents may 
be produced, not only by the earth’s rotation, but by the motion 
of the waters on its surface, and even by that of the atmosphere; 
so that the direction and intensity of the magnetic forces would 
be modified by the influence of these currents. 

This subject is at present involved in obscurity: still, consi- 
dering how many have been the discoveries made within a few 
years,—all bearing more or less directly upon it, though none 
afford a complete explanation of the phanomena,—it does not 
appear unreasonable to expect that we are not far removed from 
a point where a few steps shall place us beyond the mist in which 
we are now enveloped. 

Mr. Fox, having observed effects attributable to the electri- 
city of metalliferous veins, appears disposed to refer some of 
the phenomena of terrestrial magnetism to electrical cur- 
rents existing in these veins}; but although we should not be 
warranted in denying the existence of these currents, indepen- 
dently of the wires made use of in Mr. Fox’s experiments, or 
even their influence on the needle, yet I think we should be 
cautious in drawing conclusions from these experiments f. 


Il. Intensity of the Terrestrial Magnetie Foree. 


I have as yet said little on the intensity of the terrestrial mag- 
netic forces. Graham, after having discovered the daily varia- 
tion of the needle, suspected that the force which urges it varies 
not only in direction, but also in intensity. He made a great 
variety of observations with a dipping needle, but drew no ge- 
neral conclusion from his results. Indeed, with the instruments 
then in use, he was not likely to determine that which has al- 
most escaped detection with instruments of more accurate con- 
struction, for the diurnal variation of the whole magnetic force 
may perhaps still be considered doubtful. Later observations, 
particularly those of Professor Hansteen, have shown that the 
time of vibration of a horizontal needle varies during the day, 
from which it was inferred that the horizontal force also varies. 
Professor Hansteen, by this means, found that the horizontal 
intensity of terrestrial magnetism has a diurnal variation, de- 


* Philosophical Transactions, 1832, p. 176. + Ibid. 1830, -p 407. 

+ Mr. Henwood informs me that he has repeated the experiments of Mr. 
Fox in from forty to fifty places not before experimented on, and that he pro~ 
poses greatly extending them. As far as he can yet see, he considers that his 
results go to confirm Mr. Fox’s deductions,—I suppose with regard to the elec- 
tricity of metalliferous veins. 


REPORT ON THE MAGNETISM OF THE EARTH. 119 


creasing, at Christiana, until ten or eleven o'clock in the morn- 
ing, when it attains its minimum, and then increases until four 
or five o'clock after noon, when it appeared to reach its maxi- 
mum*. By observing, at different times of the day, the direc- 
tion of a horizontal needle thrown nearly at right angles to the 
meridian, by the action of two powerful magnets, placed in the 
meridian, passing through its centre, after correcting the ob- 
servations for the effect of changes of temperature on the in- 
tensity of the force of the magnets, I found that at Woolwich 
the terrestrial horizontal intensity decreased until 10° 30™ a.m., 
when it reached its minimum, and increasing from that time, 
attained its maximum about 75 30" p.m.}. ‘This agreement, in 
results obtained by totally independent methods, removes all 
doubt respecting the diurnal variation of the horizontal force. 
The difference in the time of the maximum in the two cases 
may be accounted for, independently of the difference in the 
variation at the two places of observation, by the circumstance 
that no correction for the effect of temperature on the time of 
vibration is made in Professor Hansteen’s observation. As no 
such correction had hitherto been made, it must have been con- 
sidered that differences in the temperature at which observations © 
were made had little influence on the intensity of the vibrating 
heedle ; but in the communication containing these observations, 
I pointed out the necessity of such a correction{; and since 
then, in deducing the terrestrial intensity from the times of vi- 


* Edinburgh Philusophical Journal, vol. iv. p. 297. 

+ Philosophical Transactions, 1825, pp.50 & 57. An inconvenience attending 
the method which I employed is, that the observations require a correction for 
temperature which is not very readily applied, as will be seen by referring to my 
paper ; but this might in a great measure be obviated, by rendering the tempera- 
ture of the magnets employed always the same previous to observation. If, how- 
ever, in order to retain the needle in its position nearly at right angles to the me- 
ridian, torsion were applied instead of the repulsive forces of magnets, the correc- 
tion for temperature would be nearly reduced to that due to the effects produced. 
on the intensity of the needle itself by changes of temperature. But even this 
method is not without objection; for the sensibility of the needle depending 
upon the number of circles of torsion requisite to bring it into the proper posi- 
tion, if a wire were employed, unless very long, its elasticity would be impaired 
by more than two or three turns; and it is doubtful whether a filament of glass 
of moderate length would bear more than this without fracture. I had pro- 
posed to the late Captain Foster, previous to his last voyage, that he should de- 
termine the horizontal intensity at different stations, and also its diurnal changes 
by this method, and had a balance of torsion constructed for him for the purpose ; 
but as the instrument is extremely troublesome in its adjustments, I consider 
that the many other observations which he had to make did not allow him time 
for the extensive use of this instrument which he had proposed. It is, however, 
very desirable that it should be ascertained how far this method is applicable. 

{Philosophical Transactions, 1825. 


« 


120 THIRD REPORT—18383. 


bration of a needle, it has been customary to apply a correction 
for differences in the temperatures at which the observations 
may have been made. 

‘The horizontal intensity varying during the day, it becomes 
a question whether this arises from a change alone in the direc- 
tion of the force, or whether this change of direction is not 
accompanied by a change in the intensity of the whole force. 
In a communication to the Philosophical Society of Cambridge *, 
I suggested that deviations, from whatever cause, in the direc- 
tion of the horizontal needle, were referrible to the deviations 
which, under the same circumstances, would take place in the 
direction of the dipping needle. Adopting these views, Captain 
Foster infers, from observations made by him at Port Bowen, 
on the corresponding times of vibration of a dipping needle, 
supported on its axis and suspended horizontally, that the diur- 
nal change in the horizontal intensity is due principally, if not 
wholly, to a small change in the amount of the dip. ‘The observa- 
tions, however, do not indicate that the force in. the direction of 
the dip is constant. Captain Foster's observations at Spitzber- 
gen7} show, more decidedly, the diurnal variation of this force: 
there, its maximum intensity appears to have occurred at about 
3" 36™ a.m., and the minimum at 2" 47™ p.M. ; its greatest change 
amounting to 5 of its mean value. The maximum horizontal 
intensity appears to have occurred a little after noon, and the 
minimum nearly an hour after midnight; but there is consider- 
able irregularity in the changes which it undergoes. It would, 
however, appear, from these observations, that the variations in 
the absolute intensity were in opposition to those in the hori- 
zontal resolved part of it; so that the principal cause of the 
latter variations must have been a change in the dip itself. 
Captain Foster considers “ that the times of the day when these 
changes are the greatest and least, point clearly to the sun as 
the primary agent in the production of them; and that this 
agency is such as to produce a constant inflection of the pole 
towards the sun during the twenty-four hours.” ‘This is in per- 
fect accordance with the conclusions I had previously drawn 
from the experiments on which I founded the theory of the di- 
urnal variation of the needle {, as I had shown that if the diur- 
nal variation of the needle arise from the cause which I have 
assigned for it, the dip ought to be a maximum, in northern la- 
titudes, nearly when the sun is on the south magnetic meridian, 
and a minimum when it has passed it about 130°. 


* Transactions of the Philosophical Sogiety of Cambridge, 1820. 
+ Philosophical Transactions, 1828. t Ibid, 1827, pp. 345, 349. 


REPORT ON THE MAGNETISM OF THE EARTH. 121 


_, Humboldt was the first who determined that the intensity of 
the whole magnetic force is different at different positions on 
the earth’s surface. Having made observations on the times of 
vibration of the same dipping needle, at various stations in the 
vicinity of the equator, and approaching to the northern pole, 
he found that the intensity of the terrestrial force decreases in 
approaching the equator; but no precise law, according to which 
the intensity depends upon the distance from the equator, can 
be determined from these observations. _Numberless observa- 
tions have since been made in both hemispheres, with every 
precaution to ensure accuracy in the results, but they do not in 
general accord with the theoretical formule with which they 
have been compared. « . 

“On the hypothesis of two magnetic poles not far removed 
from the centre of the earth, if 3 represent the dip, 4 the mag- 
netic latitude of the place of observation, I the intensity of the 
force in the direction of the dip, and m a constant, then 


weer m 
~ (4. — 3 sin? 3)’ 
tan’ = 2 tan A; 
and therefore, 


I= * V (3 sin? A + 1); 


or ifz is the angular distance from the magnetic pole, or the 
complement of the latitude, 


I= Zw (Bcos?é + 1). 


By comparing his own observations with the first of these 
formulz, Captain Sabine came to the conclusion that they were 
“‘ decisive against the supposed relation of the force to the ob- 
served dip, and equally so against any other relation whatso- 
ever, in which the respective phenomena might be supposed to 
vary in correspondence with each other.” Comparing them, 
however, with the last formula, he concludes that “ the accord- 
ance of the experimental results with the general law proposed 
for their representation, cannot be contemplated as otherwise 
than most striking and remarkable.” How the same set of 
observations should be in remarkable accordance with the one 
formula and at variance with the other, when these formule are 
‘dependent on each other, it is difficult to conceive; but the 
conclusion drawn by Captain Sabine from his observations, at 
least shows the danger of relying upon any single set of obser- 
vations as confirmatory or subversive of theoretical views. I 


122 THIRD REPORT—1839. 


have not yet compared with these results of theory the numer- 
ous observations made by Captain Foster, both in the northern 
and in the southern hemispheres ; but it is my intention to do 
this as soon as I can determine what correction ought to be 
made for the differences of temperature at the several stations: 
I do not, however, anticipate any very close accordance. 

In Captain Sabine’s observations, the observed intensities, 
compared with those deduced from the preceding formule, are 
in excess near the equator, and in defect near the pole; and it 
is not improbable that, as Mr. Barlow has suggested, this in- 
crease of magnetic action near the equator above that which 
the theory gives, is due to the higher temperature in the equa- 
torial regions *. I am, however, disposed.to assign even a more 
powerful influence than this to difference of temperature ; for I 
think it very possible, and indeed not improbable, that this may 
be the primary cause of the polarity of the earth, although its 
influence may be much modified by other circumstances. At 
the conclusion of the paper on the diurnal variation}, to which 
I have already referred, I have suggested an experiment which 
I think might throw much light on this subject. I have pro- 

osed that a large copper sphere, of uniform thickness, should 
be filled with bismuth, the two metals being in perfect contact 
throughout, and that experiments should be made with it simi- 
lar to those which I had made with one of smaller dimensions, 
but from which I was unable to obtain any very definite results, 
in consequence of the want of uniformity in the thickness of the 
copper and in the contact of the two metals. On heating the 
equator of such a sphere, the parts round the poles being cooled 
by caps of ice—which might not unaptly represent the polar ices, 
—we may expect that currents of electricity would be excited ; 
in which case the direction of those currents would decide whe- 
ther the experiment were illustrative of the principal phenome- 
non of terrestrial magnetism, or not. Should these currents of 
electricity be in the direction of the meridians,—which is impro- 
bable, since in this case opposing currents would meet at the 
poles, and there would be no means of discharge for them,—I 
think we might then conclude that the magnetism of the earth 
cannot be due to the difference in the temperature of its polar 
and equatorial regions; but if, on the contrary, the currents 
should be in a direction parallel to the equator,—in which case 
their action upon a magnetized needle would be to urge it in 
the direction of the meridians,—I should then say that, in order 
to account for the terrestrial magnetic forces, and the diurnal 

* Edinburgh New Philosophical Journal, July 1827. 
+ Philosophical Transactions, 1827, p. 354. 


REPORT ON THE MAGNETISM OF THE EARTH. 123 


changes in their direction and intensity, it would only be re- 
quired to show, that electrical phenomena may be excited, in 
such bodies as the earth and the atmosphere, by a disturbance 
in their temperature when in contact. As I consider that if 
such an experiment were carefully made it must give conclusive 
results, I would strongly suggest to the Council of the British 
Association the importance of having it made. 

It has been a question whether the intensity of terrestrial 
magnetism is the same at the surface of the sea and at heights 
above that surface to which we can attain. MM. Gay-Lussac 
and Biot, in their aérostatic ascent, could detect no difference 
at the height of 4000 metres*. Saussure had, however, con- 
cluded, from the observations which he made at Geneva, Cha- 
mouni, and on the Col du Géant, that the intensity was consi- 
derably less at the latter station than at either of the former, 
the difference in the levels being in the one case about 10,000 
feet, in the other about 7800+. 

M. Kupffer t also considers that his observations in the vi- 
cinity of Elbours, in which the difference of elevation of his two 
stations was 4500 feet, show clearly that the horizontal intensity 
decreases as we ascend above the surface; and he accounts for 
this decrease not having been observed by MM. Biot and Gay- 


* Biot, Traité de Physique. 

+ Voyages dans les Alpes, tom. iv. p. 313.—I take for granted that, admit- 

ting the accuracy of Saussure’s observations, they warranted the conclusions 
he drew from them ; but some unaccountable errors must have crept in, either in 
transcribing or in printing them; for not only the means which he deduces do not 
result from the observations, but the numbers which he employs contradict his 
conclusions. I transcribe the passage from the only edition I can consult, pub- 
lished at Neufchatel, 1796. ‘“ A‘ Généve ces vingt oscillations employerent 
5m 2°; 4m 50°; 5m; 4m 40®; dont la moyenne étoit 5™ 0*4; le thermometre 
étant 46 dégrés. A‘ Chamouni 5™ 33"; 5" 34°; moyenne 5" 33°*:5; thermometre 
12 dég. Au Col du Géant 5™ 30°3; 5™30°5 ; 5™31°-4; 5™ 34*-6, moyenne 
5™ 32°-45; thermometre 12:4 dégrés.”’ 
_ “Or les forces magnétiques sont, inversement comme les quarrés des tems. 
Mais, 4 Généve, le tems étoit 5™ 0*-4 ou 300°.4, dont le quarré = 111155°56 ; 
a Chamouni 5™ 33°-5 = 333%-5, dont le quarré = 111223. Au Géant 5m 
32°45 = 332*-45, dont le quarré = 11523-0025; d’ou il suivroit que la plus 
grande force étoit dans la plaine, et la plus petite sur la plus haute montagne, 
a peu pres d’une cinquieme: observation bien importante, si elle étoit confirmée 
par des expériences répétées, et faites 4 la méme température.” 

The means of the above observations are 4™ 53° = 293°, 5™ 33°-5 = 333*-5, 
and 5™ 31*-7 = 331°-7; and the squares of these numbers are 85849, 111222°25, 
110024:89. So that, according to this, the force was greatest at Geneva, and 
least. at Chamouni. Taking Saussure’s numbers, 300*-4, 333°-5, 332°-45, their 
Squares are 90240:16, 111222-25, 110523-0025; so that still the general con- 
clusions are the same. 

} Voyage dans les Environs du Mont Elbronz. Rapport fait @ U Académie 
Impériale des Sciences de St. Petersbourg, p. 88. 


124 THIRD REPORT—18338. 


Lussac, by its having been counteracted by the increase of in- 
tensity, arising from the diminution of temperature. Mr. Hen- 
wood informs me that he has made corresponding observations, 
consisting of two series, each of 3900 vibrations at each place; 
on Cairn Brea Hill, 710 feet above the level of the sea; at the 
surface of Dolcoath mine, 570 feet above the sea; and at a depth 
of 1320 feet beneath the surface in Dolcoath mine, or 950 feet 
below the level of the sea; and that, after clearing the results 
from the effects of temperature, the differences are so minute 
that he cannot yet venture to say he has detected any difference 
in the magnetic intensity at these stations. If, notwithstanding 
these results, we are to admit the correctness of M. Kupffer’s 
conclusions, I think we must infer that the diminution of hori- 
zontal intensity at his higher station was due to an increase in 
the dip, which element would not probably be so much affected 
by a change of elevation in a comparatively level country, like 
Cornwall, as on the flank of such a mountain mass as Elbours. 

Before dismissing the subject of the terrestrial intensity, I 
should mention that attempts have been made to delineate on 
charts the course of isodynamic lines. Professor Hansteen has 
published a chart in which this is done for the year 1824... Of 
all observations, however, requisite for graphic exhibitions con- 
nected with terrestrial magnetism, those on the authority of 
which such lines must be drawn are fewest in number and least 
satisfactory in their results; we should, therefore, be very cau- 
tious in drawing conclusions from such delineations. 

Hitherto I have only referred to such changes in the direction 
of the magnetic force, and in its intensity, as appear to depend 
upon general causes; but, besides these, sudden and sometimes 
considerable irregular changes occur. ‘These have very gene- 
rally been attributed to the influence of the aurora borealis, 
whether visible or not at the place of observation; and I think 
it not improbable that some may be due to a peculiar electrical 
state of the atmosphere, independent of that meteor. ‘The in- 
fluence of the aurora borealis on the magnetic needle has, how- 
ever, been denied by some, principally because, during the 
occurrence of that meteor at Port Bowen, Captain Foster did 
not observe peculiar changes in the direction of the needle, al- 
though, from his proximity to the magnetic pole, the diurnal 
change sometimes amounted to 4° or 5°; and, under such cir- 
cumstances, it was considered that these changes ought to have 
been particularly conspicuous. In a paper inserted in the se- 
cond volume of the Journal of the Royal Institution, 1 have, 
however, shown that Captain Foster’s Port Bowen observations 
do not warrant the conclusions which have been drawn from 


REPORT ON THE MAGNETISM OF THE EARTH. 125 


them, and have pointed out circumstances which may, in this 
ease, have rendered the effect of the aurora upon the horizon- 
tal needle less sensible than might have been expected. ‘That 
changes in the direction and intensity of the terrestrial forces 
are simultaneous with the aurora borealis I feel no doubt, for I 
have seen the changes in the direction of the needle to accord 
so perfectly with the occurrence of this meteor, and to such an 
extent, that in my mind the connexion of the phenomena be-. 
came unquestionable*. As, however, the magnetic influence 
of the aurora borealis has been doubted, I shall here point out 
the manner in which I consider the effects may be best ob- 
served. 

If the magnetic forces brought into action during an aurora 
are in the direction of the magnetic meridian, they will affect a 
dipping needle adjusted to the plane of that meridian, but the 
direction of an horizontal needle will remain unchanged: on 
the other hand, if the resultant of these forces makes an angle 
with the meridian, the direction of the horizontal needle will be 
changed, but the dipping needle may not be affected. In order 
to determine correctly the magnetic influence of the aurora by 
means of an horizontal needle, it is therefore necessary not only 
to have regard to those forces which influence its direction, but 
likewise to those which affect the horizontal intensity. The 
effects of the former are the objects of direct observation, but 
those of the latter are not'so immediately observable. As, du- 
ring an aurora, the intensity may vary at every imstant,—and it is 
these changes which are to be detected,—the method of deter- 
mining the intensity by the time of vibration of the needle can- 
not here be applied, and other means must be adopted. The 
best method «appears to me to be that which I employed for 
determining the diurnal variation of the horizontal intensity, 
the needle being retained nearly at right angles to the meridian 
by the repulsive force of a magnet, or by the torsion of a fine 
wire or thread of glass. For the purpose, then, of detecting 
in all cases the magnetic influence of the aurora, I consider that 
two horizontal needles should be employed; one, adjusted in 
the meridian, for determining the changes which may take place 
in the direction of the horizontal force, and the other at right 
angles to the meridian, to determine the changes in the inten- 
sity of that force, arising principally from new forces in the 
plane of the meridian, and which would affect the direction of 
the dipping needle alone. Both these needles should be deli- 


* For the observations to which I here particularly refer, see the Journal of 
the Royal Institution, yol. ii. p. 272. ‘ 


126 THIRD REPORT—1833. 


eately suspended, either by very fine wire, or by untwisted 
fibres of silk. In order to render the changes in the direction 
of the needle in the meridian more sensible, its directive force 
should be diminished by means of two magnets north and south 
of it, and having their axes in the meridian. These magnets 
should- be made to approach the needle until it points about 
30° on either side of the meridian, and they should be so ad- 
justed that the forces acting upon the needle will retain it zn 
equilibrio with its marked end at about 30° to the east and 30° 
to the west of north, and also at south. The needle is to be 
left with its marked end pointing south, for the purpose of ob- 
serving the changes occurring in its direction. If magnets are 
employed to retain the second needle nearly at right angles to 
the meridian, they should be made to approach its centre until 
the points of equilibrium are at about 80° east, 80° west and 
south, the observations being made with the needle at 80° east 
or 80° west. An objection to this method of adjusting this 
needle by means of magnets, and to which I have already re- 
ferred in a note, is that any change in their temperature will 
have a very sensible effect on the direction of the needle in this 
position; and should such change take place during the ob- 
servations, corrections must be applied to the results before 
any accurate conclusions can be drawn from them. As, how- 
ever, an aurora is not generally of long continuance, any change 
in the temperature of the magnets during the observations is 
much more easily guarded against than where the observations 
have to be continued during successive days and at different 
seasons of the year. I have before remarked that this incon- 
venience will be, in a great measure, obviated by employing the 
torsion of a fine wire, or a very fine filament of glass, to retain 
the needle at about 80° from the meridian. In this case, the 
ratio of the force of torsion to the terrestrial force acting upon 
the needle having been determined, a measure will be obtained 
of the changes which take place in the intensity of the terres- 
trial force during the occurrence of an aurora. It is very de- 
sirable that it should be ascertained whether the effects on the 
needle are simultaneous with any particular class of phenomena 
connected with the aurora; whether these effects are dependent 
on the production of beams and corruscations, or on the forma- 
tion of luminous arches ; or whether any difference exists in the 
effects produced by these. In order to determine this, it is ne- 
cessary that the times of the occurrence of the different pha- 
nomena, and also of the changes in the directions of the needles, 
should be accurately noted; and for such observations, three 
observers appear to be indispensable. 


REPORT ON THE MAGNETISM OF THE EARTH. 127 


Whether the direction of the needle may be influenced by 
_ the electrical state of the clouds, is much more doubtful than 
the influence of the aurora. I am not aware of any extended 
series of observations made with a view to determine this point. 
Having adjusted, in a particular manner, a needle between two 
magnets, so that the directive force was considerably diminished, 
I found that the changes in the positions of electric clouds was 
accompanied by changes in the direction of the needle; but, 
although the observations indicate that the needle was thus 
affected, they are of too limited a nature to draw any general 
conclusion from*. Some observations of Captain Sir Everard 
Home, however, indicate the same kind of influence. Ina con- 
versation which I had with him last year, having referred to the 
effect I had observed to be produced by the sun’s rays, of bring- 
ing a vibrating needle to rest, it brought to his mind a similar 
effect which he observed during a thunder-storm. He has fa- 
voured me with his observations, and from these it appears 
that, in two instances, a needle came sooner to rest during a 
thunder-storm than it had previous or subsequent to it. The 
are at which the vibrations ceased to be counted is not recorded, 
but the number of vibrations was reduced in one case from 100 
to 40, and in another from 200 to 120. I have, in consequence 
of these observations, requested Lieutenant Barnett of the Royal 
Navy, who is engaged in the survey of the southern coast of the 
Gulf of Mexico, to make similar observations, should he have 
an opportunity; and as thunder-storms are so frequent, and of 
such intensity on that coast, I think he may obtain some im- 
portant results as connected with the influence of the electric 
state of the atmosphere upon the vibrations and direction of 
the needle. 

- Upon a review of all the phenomena of terrestrial magnetism, 
and considering the intimate relation which has been established 
between magnetism and electricity, by which it appears that, if 
not identical, they are only different modifications of the same 
principle, there can, I think, be little doubt that they are due 
to electric currents circulating round the earth. How these 
currents are excited, whether by heat, by the action of another 
body, or in consequence of rotation, we are not at present able 
to determine; but however excited, they must, though not 
wholly dependent upon them, be greatly modified by the phy- 
sical constitution of the earth’s surface. We are, therefore, not 
to expect that symmetry in their course which would be the 
_ * Philosophical Transactions, 1823, p.354. The arrangements which I have 


just described for determining the influence of the aurora borealis are well 
adapted for deciding this point. , 


128 THIRD REPORT—1833. 


consequence of a symmetrical constitution of that surface. But 
even if such symmetry did exist, the action of all the currents 
at different stations on the surface could scarcely be referred 
to the same two points as centres of force; and without this 
symmetry, it would be absurd to expect it. The hypothesis, 
therefore, of only two poles, as explanatory of the phenomena, 
must be rejected; and if we are to refer these phenomena to 
centres of action, we must, besides two principal ones, admit 
the existence of others depending, upon local causes. 

It has been said that if we refer the magnetism of the earth 
to another body, we only remove the difficulty, and gain little 
by the supposition*. It, however, appears to me, that if we 
could show that the magnetism of the earth is due to the action 
of the sun, independent of its heat,—which, however, I think 
the more probable cause,—the problem would be reduced to 
the same class as that of accounting for the light of the sun, the 
heating and chemical properties of its rays: we only know the 
facts, and are not likely to know more. 

If difficulties meet us at every step when we attempt to ex- 
plain the general phenomena of terrestrial magnetism, these 
difficulties become absolutely insurmountable when we come to 
the cause of their progressive changes. Here, at least, we 
must for the present be satisfied with endeavouring to discover 
whether these changes are governed by any general laws: 
should they be so, their cause may possibly be discovered. 
Diligent and careful observation is the only means by which 
we can hope to attain this end, and indeed is that on which we 
must principally rely for gaining a more correct knowledge of 
all the phenomena, and of their causes ; and, consequently, im- 
provements in the methods of observation, and in the instru- 
ments to be employed, become of the highest importance. 

This Report has already so far exceeded the limits within 
which I wished to have confined it, that I must restrict the re- 
marks on this part of the subject to a few points. 

In the observations of Humboldt, in those of M. Rossel, of 
Captain Sabine, and of Captain Foster, the terrestrial magnetic 
intensity had been determined by the vibrations of a dipping 
needle in the plane of the magnetic meridian; but as there is 
by this means, in consequence of the friction upon the axis, a 
difficulty in obtaining a sufficient number of vibrations to ensure’ 
accuracy, and a dipping instrument is besides ill adapted for 
carriage, Professor Hanstegn proposed to determine the same 
by means of a small needle suspended horizontally by a few 


* Hansteen’s Inquiries concerning the Magnetism of the Earth. 


P 
REPORT ON THE MAGNETISM OF THE EARTH. 129 


untwisted fibres of silk, The advantages, however, attending 
this method of Professor Hansteen, I consider to be more ap- 
parent than real; for without determining the dip, the hori- 
zontal force, deduced from the vibrations of the horizontal 
needle, cannot be reduced to the force in the direction of the 
dip; and if the dip is determined, two instruments become ne- 
cessary where, before, only one was requisite. 

In order to obviate the inconveniences attending each of these 
methods, I have proposed a construction for a dipping needle, 
by means of which the observations which determine the di- 
rection of the terrestrial force will also give a measure of its 
intensity. The general principle of the construction is simply, 
that the centre of gravity of the needle should not be in its 
centre of figure, but in a line drawn from that centre at right 
angles, both to its axis of motion and to its magnetic axis; so 
that, by two observations, one with the centre of gravity up- 
wards, and the other with it downwards, the dip, and likewise 
the relation which the static momentum of its weight bears to 
that of the terrestrial magnetic force acting upon the magnetism 
of the needle, may be determined. The principles on which 
these determinations depend, and. the advantages which I pro- 
pose from the adoption of this construction, are fully described 
in a paper read before the Royal Society, and which will appear 
in the Philosophical Transactions of this year. 
_ Professor Gauss has proposed a method of determining the 
intensity and the changes it undergoes, by which he hopes to 
reduce magnetical observations to the accuracy of astronomical 
ones. By the vibrations of a magnetized bar he determines the 
product of the terrestrial magnetic intensity by the static mo- 
mentum of its free magnetism. By introducing a second bar, 
and by observing at different:distances the joint effects of the 
first; and of the terrestrial magnetism on this, he determines 
the ratio of the terrestrial intensity to the static momentum of 
the free magnetism of the first.. Eliminating this last from the 
two equations, he obtains an absolute measure of the terres- 
trial magnetic intensity, independent of the magnetism of the 
bar. This is a most important result, for we shall thus be en- 
abled to. determine the changes which the terrestrial intensity 
- undergoes:in long intervals of time. It is, however, to be ob 
served, that it is only the horizontal intensity which is thus: 
determined, and that, in order to determine the intensity of the 
whole force, another element, namely, the dip, must also be: ob- 
served; and I fear much that the introduction of this element 
will, in a great measure, counteract that accuracy of which the 
merged proposed for determining the times of vibration appear 

; K 


130 THIRD REPORT—1833. 


capable. This must be an objection, even where the observa- 
tions are made in a fixed observatory ; but where an apparatus 
has to be moved from one station to another, I think the method 
could scarcely be applied successfully, principally on account 
of the delicacy of the preliminary observations, and of the time 
requisite for making them, in addition to that required for the 
observations by which the terrestrial intensity and its variations 
are to be determined. However greatly I may admire the saga- 


city which Professor Gauss has shown in devising means for. 


the determination of an absolute measure of the horizontal in- 


tensity, I cannot avoid seeing the difficulties which may occur . 


in its practical application. 

The method which Professor Gauss proposes, and has prac- 
tised, of observing the course of the daily variation, and of de- 
termining the time of vibration, by means of a plane mirror 
fixed on the end of the needle, perpendicularly to its axis, and 
observing the reflected image of the divisions of a scale by 
means of a theodolite fixed at a distance, appears to admit of 
the greatest possible precision, and will probably supersede 
other methods of observing the daily variation. 

I have adverted to the necessity of careful and diligent ob- 
servation of all the phenomena of terrestrial magnetism, as the 
surest means of arriving at a knowledge of their causes: it is 
with reluctance I state it, but I believe it to be a fact, that this 
is the only country in Europe in which such observations are not 
regularly carried on ina national observatory. Such an omission 
is the more to be regretted, seeing that no one has, I believe, 
carried on a regular series of observations on the diurnal va- 
riation, since the valuable ones by Colonel Beaufoy were inter- 
rupted by his death, this interruption happening at a time when 
it was peculiarly desirable that the series should be unbroken. 
At this time the needle near London had begun to show a re- 
turn towards the true meridian; but whether this was one of 
those oscillations which have occasionally been observed, or 
that, having really attained its maximum of westerly deviation, 
it was returning in the contrary direction, is, I believe, undecided 
at. the present moment. Of all the data requisite for deter- 
mining the laws which govern the phenomenon of the variation, 
the time of the maxima and their magnitude are the most im- 
portant. I trust that ere long the important desideratum will 
be supplied of a regular series of magnetical observations in 
the national Observatory of Great Britain. 


Royal Military Academy, 
_ 22nd June, 1833. 


 ———— 


£1816] 


Report on the present State of the Analytical Theory of H: ydro- 
statics and Hydrodynamics. By the Ruy. J. Cuauuis, late 
Fellow of Trinity College Cambridge. 


Tue problems relating to fluids, which have engaged the atten- 
tion of mathematicians, may be classed under two heads,—thosé 
which involve the consideration of the attractions of the con- 
stituent molecules, and the repulsion of their caloric; and those 
in which these forces are not explicitly taken’ account of. In 
the latter class the reasoning is made to depend on some pro- 
perty derived from observation. For instance, water is observed 
to be very difficult of compression; and this has led to the 
assumption of absolute incompressibility, as the basis of thé 
mathematical reasoning: air at rest, and under a given state of 
temperature, is observed to maintain a certain relation between 
the pressure and the density ; hence the fundamental property 
of the fluid which is the subject of calculation is assumed to be 
the constancy of this relation, to the exclusion of all the circums 
stances which may cause it to vary. The fluids treated of in 
this kind of problems are rather hypothetical than real, yet not 
so different from real fluids but that the mathematical deduc- 
tions obtained respecting them admit of having the test of ex~ 
periment applied. I propose in this Report to confine myself 
entirely to problems of the second class,—those in the common 
theory of fluids. The reasons for making this limitation are, 
that both kinds together would afford too ample matter for one 
Report, and that those which I have selected are distinguished 
from the others by the different purpose in regard to science 
which correct solutions of them would answer: for the treat- 
ment of any hydrostatical or hydrodynamical questions which 
involve the consideration of molecular attraction and the repul: 
sion of heat, must proceed upon certain hypotheses respecting 
the mode of action of these forees, and the interior constitution 
of the fluid, as these are circumstances which from their nature 
cannot be data of observation; and hence, assuming the ma+ 
thematical reasoning founded on'the hypotheses to be correct, 
a satisfactory comparison of the theoretical deductions with 
facts must serve principally to establish the truth of the hypo- 
theses, and so to let us into secrets of nature which probably 
could never be known by any other process. But when the 
K2 


132 THIRD REPORT—1833. 


basis of calculation, as in the questions that will come before 
us, is some observed and acknowledged fact, solutions which 
satisfy experiments will first of all serve to confirm the truth of 
the mathematical reasoning, and then give us confidence in the 
theoretical results, which, as often happens, cannot readily re- 
ceive the test of experiment. Calculations of this kind do not 
add much to our conviction that the facts applied as the test of 
the theory are really consequences of those which are the basis 
of it. For instance, we feel satisfied, independently of any ma- 
thematical reasoning, that the motions of waves on the surface 
of water are consequences of the incompressibility of the fluid, 
and the law of equal pressure. But the purpose which these 
calculations answer of confirming methods of applying analysis 
is very important, particularly in regard to the higher class of 
physical questions, which M. Poisson has proposed to refer to 
a distinct department of science, under the title of Mathéma- 
tiqgue Physique, viz. those that require in their theoretical treat- 
ment some hypotheses respecting the interior constitution of 
‘bodies, and the laws of corpuscular action: for in questions of 
this nature, as well as in problems in the common theory of 
fluids, the mathematical reasoning conducts to partial differen- 
tial equations; and if the method of treating these, and of 
drawing inferences from their integrals, be established in one 
kind, it may be a guide to the method to be adopted in the 
other. It is plainly, then, desirable that the mathematical pro- 
cesses be first confirmed in the cases in which the basis of rea- 
soning is an observed fact, that the reasoning may proceed with 
certainty in those cases where it is based on an hypothesis, the 
truth of which it proposes to ascertain. 

_ The subjects of this Report may now be stated to be, the 
leading hydrostatical and hydrodynamical problems recently 
discussed, which proceed upon the supposition of an incom- 
pressible fluid, or of a fluid in which the quotient of the pres- 
sure divided by the density is a constant; and the end it has 
in view is, to ascertain to what extent, and with what success, 
analysis has been employed as an instrument of inquiry in these 
problems. Iam desirous it should be understood that I have 
not attempted to make a complete enumeration either of the 
questions that have been discussed in this department of science, 
or of the labours of mathematicians in those which have come 
under notice. It has rather been my endeavour to give some 
idea of the most approved methods of treating the leading 
problems, and the possible sources of error or defect in the 
solutions. In taking this course I hope I may be considered. to 
have acted sufficiently in accordance with the recommendation 


ae 


REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 133 


of the Committee for Mathematics, which was the occasion of 
my receiving the honour of a request to take this Report in 
hand. 

With the limitation above stated as to the subjects our Re- 
port is to embrace, we shall have scarcely anything to say on 
the analytical theory of hydrostatics. The problems of interest 
in this department were early*solved, and present no difficulty 
in principle, and little in the detail of calculation. The deter~ 
mination of the height of mountains by the barometer is a 
hydrostatical question, the difficulty of which does not consist 
in the analytical calculation, but only in ascertaining the law of 
the distribution of the atmospheric temperature. We shall not 
have to speak of the theories that have been invented to over- 
come this difficulty. Neither does it fall within the scope of this 
Report to notice the very valuable memoir of M. Poisson on 
the equilibrium of fluids*, which has for its object the deriva- 
tion of the general equations of equilibrium from a consideration 
of molecular attraction and the repulsion of caloric, and seems 
to have been composed in immediate reference to the theory 
of capillary attraction, which the author subsequently pub- 
lished. With regard to the problem of capillary attraction, we 
may remark, that it is not possible by any supposition respect- 
ing the forces which sustain or depress the fluid in the tube, 
to solve it as a question in the common theory of hydrostatics. 
M. Poisson has shown the insufficiency of Laplace’s theory, 
and by taking into account the molecular forces and the effect 
of heat, has proved that the explanation of the phenomenon is 
essentially dependent on a modification of the property which. 
is the basis of the common theory, viz. the incompressibility of 
the fluid. It does not fall within our province to say more on 
the celebrated theory of M. Poisson. 

One improvement I consider to have been recently made in 
the common theory of fluids. It has been usual to take the law 
of equal pressure as a datum of observation. Professor Airy, 
in his Lectures in the University of Cambridge, has shown that 
this property may be derived, by reasoning according to esta- 
blished mechanical principles, from another of a simpler kind, 
the notion of which may be gathered from observation, viz. 
that the division of a perfect fluid may be effected without the 
application of sensible force; from which it immediately follows 
that the state of equilibrium or motion of a fluid mass is not 
altered by mere separation of its parts by an indefinitely thin 
partition. A definition of fluids founded on this principle, and 


* Mémoires de U Académie des Sciences, Paris, tom, ix. 1830. 


134 THIRD REPORT—1833. 


a proof of the Jaw of equal pressure, are given at the beginning 
of the Elements of Hydrostatics and Hydrodynamics of Pro- 
fessor Miller*. Several advantages attend this mode of com- 
mencing the mathematical treatment of fluids. The principle 
is one which perfectly characterizes fluids, as distinguished in 
the internal arrangement of their particles from solids. It may 
be rendered familiar to the senses. It is, I think, necessary for 
the solutions of some hydrostatical and hydrodynamical pro- 
blems, particularly those of reflection}. Lastly, in reference 
to the department of science proposed to be called Physical 
Mathematics, the propositions of the common theory ought to 
be placed on the simplest possible basis, because the questions 
of most interest in that department are those which have in 
view the explanation of the phenomena that are the founda- 
tions of the reasoning in the other kind. ‘The solution of one 
such question is a great step in scientific generalization. It is 
plainly, therefore, of importance that the fact proposed for ex- 
planation should be the simplest that direct observation can 
come at. 

- The analytical theory of hydrodynamics is of a much more 
difficult nature than that of hydrostatics. The assumptions it is 
necessary to make to obtain even approximate solutions of the 
simplest problems of fluid motion betray the difficulty and im- 
perfection of this part of science. There are cases, however, of 
steady motion, that is, of motion which has arrived at a perma- 
nent state, so that the velocity is constantly the same in quantity 
and direction at the same point, which require a much more 
simple analysis than those which do not satisfy this condition. It 
does not appear that the equations applicable to this kind of mo- 
tion were obtained in any general manner till they were given in 
an Elementary Treatise on Hydrostatics and Hydrodynamics 
by Mr. Moseley t, who has derived them from a principle of so 
simple a nature, that, as it can be stated ina few words, it may 
be mentioned here. When the motion is steady, each particle 


in passing from one point to another, passes successively through 


the states of motion of all the particles which at any instant le 
on its path. 'This principle is valuable for its generality: it is 


equally applicable to all kinds of fluids, and will be true, whe- 


ther or not the effect of heat be taken into account, if only the 
condition of steadiness remains. The equations of motion are 
readily derived from it, because it enables us to consider the 


_* Cambridge 1831. 
+ Dr. Young employed an equivalent principle to determine the manner of 
the reflection of waves of water. See his Natural Philosophy, vol. ii. p. 64. 
} Cambridge 1830. 


ee eee 


REPORT ON HYDROSTATICS AND HYDRODYNAMIcS. 135 


motion of a single particle, in the place of the motion of an 
aggregate of particles. Though this mode of deriving them is 
the best possible on account of its simplicity, it was yet de- 
sirable to know how they may be obtained from the general 
equations of fluid motion. In a paper contained in the Trans- 
actions of the Philosophical Society of Cambridge*, the author 
of this Report has given a method of doing this, both for incom- 
pressible and elastic fluids, and has shown that a term in the 
general formule which gives rise to the complexity common to 
most hydrodynamical questions, disappears for this kind of 
motion. Euler had already done the same for incompressible 
fluids}. The instances in nature of fluid motion of the steady 
kind are far from uncommon; and it is probable that when the 
equations applicable to them are better known, and studied 
longer, they may be employed in very interesting researches. 
The motion of the atmosphere, as affected by the rotation of 
the earth, and a given distribution of the temperature due to 
solar heat, seems to be an instance of this kind. 

We will now proceed to consider in order the principal hydro- 
dynamical problems that have recently engaged the attention 
of mathematicians. For convenience we shall class them as 
follows :—I. Motion in pipes and vessels. II. The velocity of 
propagation in elastic fluids. III. Musical vibrations in tubes. 
IV. Waves at the surface of water. V. The resistance to the 
motion of a ball-pendulum. 

I. The motion of fluids in pipes and vessels has not been 
treated with any success, except in the cases in which the con- 
dition of steadiness is fulfilled. ‘The paper above alluded to, 
in the Transactions of the Philosophical Society of Cambridge, 
contains some applications of the equation of steady motion for 
incompressible fluids, to determine the velocity of water issuing 
from different kinds of adjutages in vessels of any shape: also 

-a theoretical explanation of a phenomenon which a short while 
ago excited some attention,—that of the attraction of a disc to 
an orifice through which a steady current either of water or air 
is issuing. 

In the Memoirs of the Paris Academy of Sciences there is 
an Essay by M. Navier on the motion of elastic fluids in ves- 
sels, and through different kinds of adjutages into the sur- 
rounding air, or from one vessel into another. For the sake of 
simplicity the author considers the fluid to be subject to a con- 
stant pressure, and consequently the motion to have arrived at 
a state of permanence. His calculations are founded upon the 


* Vol. iii. Part II]. + Mémoires de UV Académie de Berlin, 1755, p. 344. 
+ Tom. ix. 1830. 


136 AH THIRD REPORT—1833. 


hypothesis of parallel slices, which assumes the velocity to be 
the same, and in the same direction, and the density to be the 
same at all points of any section transverse to the axis of the 
vessel or pipe. This hypothesis is one of those that the theory 
of hydrodynamics has borrowed from experience to supply its 
defects. Lagrange has, however, shown theoretically * that it 
always furnishes a first approximation, the breadth of the ves- 
sel being considered a quantity of the first order, and the effect 
of the adhesion of the fluid to the sides of the vessel being neg- 
lected. It is right to observe, that in the problems M. Navier 
has considered, this hypothesis might have been in a great 
measure dispensed with: the expression he has given,—more 
correct than that commonly adopted for the velocity of issuing 
through a small aperture by which airs of different densities 
communicate,—might have been obtained by employing the equa- 
tion above mentioned of steady motion, as, in fact, Mr. Moseley 
has done}. This would be a preferable mode of treating such 
questions, because in every instance in which these auxiliary 
hypotheses are got rid of, something is gained on the side of 
theory. This memoir contains another hypothesis, which can- 
not be so readily dispensed with. Theory is at present quite 
inadequate to determine the retardation in the flow of fluids 
occasioned by sudden contractions or widenings in the bore of 
the pipe. It is found by experiments with water, that the re- 
tardation is sufficiently represented by taking account of the 
loss of vs viva which, on the hypothesis of parallel slices, will 
result from the sudden changes of velocity which must be sup- 
posed to take place at the abrupt changes in the bore of the 
pipe. M. Navier extends these considerations to elastic fluids. 
The theory manifests a sufficient agreement with the experi- 
ments it is compared with, and is valuable on account of the 
applications it may receive. 

II. The most interesting class of problems in hydrodynamics 
are perhaps those which relate to small oscillations. Newton 
was the first to submit the vibrations of the air to mathema- 
tical calculation. The propositions in the second book of the 
Principia, devoted to this subject, and to the determination of 
the velocity of sound, may be ranked among the highest pro- 
ductions of his genius. He has assumed that the vibratory 
motion of the particles follows the law of the motion of an oscil- 
lating pendulum. It was soon discovered that many other 
assumed laws of vibration would, by the same mode of reasoning, 


* Mécanique Analytique, Part II. § xi. art. 34. 
+ Elementary Treatise, p. 204. 


REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 137 


conduct to the same velocity of propagation. This, which was 
thought to be an objection to the reasoning, is an evidence of 
its correctness: for the plain consequence is, that the velocity 
of propagation is independent of the kind of vibration which 
we may arbitrarily impress on the fluid;—and so experience 
finds it to be. 

When the partial differential equation, which applies equally 
to the vibrations of the air and those of an elastic chord, had 
been formed and integrated, a celebrated discussion arose 
between Euler and D’Alembert as to the extent to which the 
integral could be applied; whether only to cases in which the 
motion was defined by a continuous curve, or also to motion 
defined by a broken and discontinuous line. It is well known 
that the question was set at rest by Lagrange, in two Disser- 
tations published in vol. i. and vol. ii. of the Miscellanea Tau- 
rinensia. The difficulty that arrested the attention of these 
eminent mathematicians was one of a novel kind, and peculiar 
to physical questions that require for their solution the integrals 
of partial differential equations. The difficulty of integration, 
which is the obstacle in most instances, had been overcome by 
D’Alembert. It remained to draw inferences from the inte- 
gral,—to interpret the language of analysis. When an aggre- 
gate of points, as a mass of fluid or an elastic chord, receives 
an arbitrary and irregular impulse, any point not immediately 
acted upon may have a correspondent irregular movement after 
the initial disturbance has ceased. This is a matter of experi- 
ence. Was it possible, then, that these irregular impulses, and 
the consequent motions, were embraced by the analytical calcu- 
lation? From Lagrange’s researches it follows that the func- 
tions introduced by integration are arbitrary to the same degree 
that the motion is so practically, and that they will therefore 
apply to discontinuous motions. (Of course we must except 
the practical disturbances which the limitations of the calcula- 
tion exclude,—those which are very abrupt, or very large.) 
This has been a great advance made in tke application of ana- 
lysis to physical questions. Had a different conclusion been 
arrived at, many facts of nature could never have come under 
the power of calculation.. The Researches of Lagrange, which 
will ever form an epoch.in the science of applied mathematics, 
establish two points principally: First, That the arbitrary func- 
tions, as we have been just saying, are not necessarily conti- 
nuous: Secondly, That (in the instance he considered) they are 
equivalent to an infinite series of terms having arbitrary con- 
stants for coefficients, and proceeding according to the sines of 
multiple arcs. This latter result, which appears to be true for 


1388 THIRD REPORT—1833. 


all linear partial differential equations of the second order, with 
constant coefficients, is valuable as presenting an analogy be- 
tween arbitrary constants and arbitrary functions. 

But the way in which Lagrange, after establishing these two 
points, proceeds to find the velocity of propagation, does not 
appear to me equally satisfactory with the rest of his reasoning. 
His method seems to be a departure from the principle which 
may be gathered from that of Newton. For, as was mentioned 
above, the reasoning of the Principia shows that the velocity of 
propagation is independent of all that is arbitrary. It seems 
important to the truth of the analytic reasoning, that it should 
not only obtain a constant velocity of propagation, but arrive at 
it by a process which is independent of the arbitrary nature of 
the functions; whereas the method which the name of La- 
grange has sanctioned, is essentially dependent on the discon- 
tinuity of the functions, that is, on their being arbitrary. With 
a view of calling attention to this difficulty, and as far as possi- 
ble removing it, the author of this Report read a paper before 
the Philosophical Society of Cambridge, which is published in 
Vol. iii. Part I. of their Transactions. Iam far from assert- 
ing that that Essay has been successful; but some service, I 
think, will be done to science if it should lead mathematicians to 
a reconsideration of the mode of mathematical reasoning to be 
employed in regard to the applications of arbitrary functions. 
if the. determination of the velocity of propagation in elastic 
fluids were the only problem affected by this treatment of arbi- 
trary functions, it would not be worth while to raise a question 
respecting the principle of the received method, as no doubt 
attaches to the result obtained by it; but there are other pro- 
blems, (one we shall have to consider,) the correct solutions of 
which mainly depend on the construction to be put upon these 
functions. The difficulty I am speaking of, which is one of a 
delicate and abstract nature, will perhaps be best understood 
by the following queries, which seem calculated to bring the 
point to an issue :—Can the arbitrary functions be immediately 
applied to any but the parts of the fluid immediately acted upon 
by the arbitrary disturbance, and to parts indefinitely near to 
these? To apply them to parts more remote, is it not necessary 
first to obtain the law of propagation? And do not the arbi- 
trary functions themselves, by the quantities they involve, fur- 
nish us with means of ascertaining the law of propagation, 
independently of any consideration of discontinuity ? 

Kuler and Lagrange determined the velocity of propagation 
in having regard tothe three dimensions of the fluid, on the li- 
mited: supposition that the initial disturbance is the same as’ to 


REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 139 


dénsity and velocity, at the same distance in every direction 
from a fixed point, which is the centre of it. Laplace first dis- 
pensed with this limitation in the case in which two dimensions 
only of the fluid are taken account of*. The principal cha- 
racter of his analysis is a new method of employing definite 
integrals. Finally, M. Poisson solved the same problem for 
three dimensions of the fluid}. This memoir deserves to be 
particularly mentioned for the interesting matter it contains. 
The object of the author is to demonstrate, in a more general 
manner than had been before done, some circumstances of the 
motions of elastic fluids which are independent of the particular 
motions of the fluid particles, such as propagation and reflection. 
The general problem of propagation just mentioned he solves 
by developing the integral of the partial differential equation of 
the second order in x, y, 2, and ¢, applicable to this case, in a 
series proceeding according to decreasing powers of the di- 
stance from the centre of disturbance, as it cannot be obtained. 
in finite terms, and then transforming the series into a definite 
integral,—a method which has of late been extensively em- 
ployed. The crossing of waves simultaneously produced by 
disturbances at several centres, is next considered, and this 
leads to a general solution of the problem of reflection at a plane 
surface. For the case in which the motions of the aérial parti- 
cles are not supposed small, the velocity of propagation along 
a line of air is shown to be the same as when they are small. 
This result is an inference drawn from the arbitrary disconti- 
nuity of the motion, on which it does not seem to depend. In 
a paper before alluded to}, the same result is obtained without 
reference to the principle of discontinuity. M. Poisson treats 
also of propagation in a mass of air of variable density, such as 
the earth’s atmosphere. His analysis is competent to prove, in 
accordance with experience, that the velocity of sound is the 
same as in a mass of uniform density, and that its intensity at 
any place depends, in addition to the distance from the point 
of agitation, only on the density of the air where the disturbancé 
is made. So that a bell rung in the upper regions of the air 
will not sound so loud as when rung by the same effort below, 
but will sound equally loud at all equal distances from the place 
where it is rung. 

In seeking for the general equations of the motion of fluids, 
(first obtained by Euler,).a quantity § is met with which, if it be 


* Mémoires de V Académie, An 1779. ; 

+ “Mémoire sur la Théorie du Son,”. Journal de l’Ecole Polytechnique, 
tom. vii. cah, xiv. 

} Transactions of the Philosophical Society of Cambridge, vol. iii. Part II. 

§ In M. Poisson’s writings this quantity is udz + vdy +udz. 


140 THIRD REPORT—1833. - 


an exact differential of a function of three variables, renders the 
subsequent analytical reasoning much simpler than it would be 
in the contrary case. This simplification has been proved by 
Lagrange to obtain in most of the problems of interest that are 
proposed for our solution*. Euler showed that the differential 
is inexact when the mass of fluid revolves round an axis so that 
the velocity is some function of the distance from the axis +. 
But no general method exists of distinguishing the instances in 
which the quantity in question is a complete differential, and 
when it is not. Nor is it known to what physical circumstance 
this peculiarity of the analysis refers. To clear up this point 
is a desideratum in the theory of hydrodynamics. M. Poisson 
has left nothing to be desired in the generality with which he 
has solved the problem of propagation of motion in elastic fluids ; 
for in the Memoirs of the Academy of Parist he has given 
a solution of the question, without supposing the initial disturb- 
ance to be such as to make the above-mentioned quantity an 
exact differential. His conclusions’ are, that the velocity of 
propagation is the same as when this supposition is made ; that 
the part of the motion which depends on the initial condensa- 
tions or dilations follows the same laws as in that case, but 
the part depending on the initial velocity does not return com- 
pletely to a state of repose after a determinate interval of time; 
that at great distances from the place of agitation there is no 
essential difference between the motion in the two cases. 

III. We turn now to the theory of musical vibrations of the 
air in cylindrical tubes of finite length. Little has been effected 
by analysis in regard to this interesting subject. The principal 
difficulty consists in determining the manner in which the mo- 
tion is affected by the extremities of the tube, whether open or 
closed, but particularly the openend. Those who first handled 
the question reasoned on the hypotheses, that at the open end 
the air is always of the same density as the external air to 
maintain an equilibrium with it, and at the closed end always 
stationary by reason of the stop. The latter supposition will 
be true only when the stop is perfectly rigid. It does not ma- 
terially affect the truth of the reasoning ; but if the other sup- 
position were strictly true, the sound from the vibrating column 
of air in the tube would not cease so suddenly as experience 
shows it does, when the disturbing cause is removed; neither 
on this hypothesis could the external air be acted on so as to 
receive alternate condensations and rarefactions, and transmit 


* Mécanique Analytique, Part II. § xi. art. 16. 
+ Mémoires de ! Académie de Berlin, 1755, p. 292. 
ft tom. x. 1831. : 


REPORT ON HYDROSTATICS AND uypropynamics. 141 


sonorous waves. ‘These objections to the old theory have been 
stated by M. Poisson, who proposes a new mode of considering 
the problem*. He reasons on an hypothesis which embraces 
both the case of an open and a closed end, viz. that the velo- 
city at each is in a constant ratio to the condensation. This 
ratio will be very large for the open end, and a very small frac- 
tion for the closed end. Its exact value in the latter case de- 
pends on the elasticity of the stop, and in the other on the mode 
of action of the vibrations on the external air,—to determine 
which is a problem of great difficulty, which M. Poisson has 
forborne to meddle with. His theory is not competent to assign 
a priori either the series of tones or the gravest that can be 
sounded by a tube of given length, but is more successful in 
determining the number of nodes and loops, and the intervals 
between them, when a given tone is sounded. To find the di- 
stances of the nodes and loops from the extremities of the tubes, 
he has recourse to the hypotheses of the old theory, which 
make the closed end the position of a node, and the open end 
the position of aloop. This, he says, will not be sensibly dif- 
ferent from the truth, if, in the one case, the stop be very un- 
yielding, and, in the other, the diameter of the tube be small. 
Recent researches on this subject, which we shall presently 
speak of, show that when the diameter is not very small the 
position of the loop is perceptibly distant from the open end. 

" The latter part of M. Poisson's memoir contains an applica- 
tion of the principles of the foregoing part to the vibrations of 
air in a tube composed of two or more cylinders of different dia- 
meters, and to the motion of two different fluids superimposed 
in the same tube. In the course of this latter inquiry, the au- 
thor determines the reflection which sound experiences at the 
junction of two fluids; and by an extension of like considerations 
to luminous undulations, obtains the same expressions for the 
relative intensities of light perpendicularly incident, and re- 
flected at a plane surface, as those given by Dr. Young in the 
Article Curomatics of the Supplement to the Encyclopedia 
Britannica. This subject was afterwards resumed by M. Pois- 
son at greater length in a very elaborate memoir ‘ On the Mo- 
tion of two Elastic Fluids superimposed +,” which is chiefly 
remarkable for the bearing which the results have upon the 
theory of light. 

_ At the last meeting, in May this year, of the Philosophical So- 
ciety of Cambridge, a paper was read by Mr. Hopkins, in which, 


* Mémoires de U Académie des Sciences, Paris, An 1817. 
+ Ibid. tom. x. p. 317. 


142 THIRD REPORT—1833. 


by combining analysis with a delicate set of experiments, te~ 
sults are obtained which are a valuable addition to this part of 
the theory of fluid motion. His experiments were made on a 
tube open at both ends, and the column of air within it was put 
in motion by the vibrations of a plate of glass applied close to 
one end. ‘The following are the principal results. The nodes 
are not points of quiescence, but of minimum vibration ;—the 
extremity of the tube most remote from the disturbance is not 
a place of maximum vibration, but the whole system of places 
of maximum and minimum vibration is shifted in a very sensible 
degree towards it ;—the distances of the places of maximum 
and minimum vibration from each other, and from that extre- 
mity, remain the same for the same disturbance, whatever be the 
length of the tube. This last fact Mr. Hopkins proves by his 
analysis must obtain. The shifting of the places of maximum 
and minimum vibration is not accounted for by the theory: nor 
is it probable that it can be, unless the consideration of the 
mode of action of the vibrations on the external air be entered 
upon,—an important inquiry, but, as I said before, one of great 
difficulty. I think also that the effect of the vibrations of the 
tube itself on the contained air ought to be taken into account. 
IV. The problem of waves at the surface of water is princi- 
pally interesting as furnishing an exercise of analysis. ‘The 
general differential equations of fluid motion assume a very sim- 
ple form for the case of oscillations of small velocity and extent, 
and seem to offer a favourable opportunity for the application 
of analytical reasoning. Yet mathematicians have not succeed- 
ed in giving a solution of the problem in any degree satisfactory, 
which does not involve calculations of a complex nature. We 
need not stay to inquire in what way Newton found the velo- 
city of the propagation of waves to vary as the square root of 
their breadths: he was himself aware of the imperfection of his 
theory. The question cannot be well entered upon without 
partial differential equations. Laplace was the first to apply to 
it a regular analysis. His essay is inserted at the end of a 
memoir on the oscillations of the sea and the atmosphere, in 
the volume of the Paris Academy of Sciences for the year 1776. 
The differential equations of the motion are there formed on 
the supposition that the velocities and oscillations are always so 
small that their products, and the powers superior to the first, 
may be neglected. The problem without this limitation be- 
comes so complicated that no one has dared to attempt it. La- 
place’s reasoning conducts to a linear partial differential equa- 
tion of the second order, consisting of two terms, which is 
readily integrated; but on account of the difficulty of obtaining a 


REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 143 


general solution from this integral, he makes a particular sup- 
position, which is equivalent to considering the fluid to be de- 
ranged from its state of equilibrium by causing the surface in 
its whole extent to take the form of a trochoid, ¢. e. a serpentine 
curve, of which the vertical ordinate varies as the cosine of the 
horizontal abscissa. The solution in question is of so limited 
a nature, that we may dispense with stating the results arrived 


at. 

In the volume of the Memoirs of the Academy of Berlin for 
the year 1786, Lagrange has given* a very simple way of 
proving, in the Newtonian method of reasoning, that the ve- 
locity of propagation of waves along a canal of small and con- 
stant depth and uniform width, is that acquired by a heavy 
body falling through half the depth. In the Mécanique Ana- 
lytique+ the same result is obtained analytically. ‘The princi- 

al feature of the analysis in this solution is, that the linear 
partial differential equation of the second order and of four va- 
riables, to which the reasoning conducts, is integrated approxi- 
mately in a series. Lagrange is of opinion, that on account of 
the tenacity and mutual adherence of the parts of the fluid, the 
motion extends only to a small distance vertically below the 
surface agitated by the waves, of whatever depth the fluid may 
be; and that his solution will consequently apply to a mass of 
fluid of any depth, and will serve to determine, from the ob- 
served velocity of propagation, the distance to which the motion 
extends downwards. 

The problem of waves was proposed by the French Institute 
for the prize subject of 1816. M. Poisson, whose labours are 
preeminent in every important question of Hydrodynamics, had 
already given this his attention. His essay, which was the first 
deposited in the bureau of the Institute, was read Oct. 2, 1815, 
just at the expiration of the period allowed for competition. It 
forms the first part of the memoir ‘‘ On the Theory of Waves,” 
published in the volume of the Academy for the year 1816, and 
contains the general formulz required for the complete solution 
of the problem, and the theory, derived from these formule, of 
waves propagated with a uniformly accelerated motion. In the 
month of December following, an additional paper was read by 
M. Poisson on the same subject, which forms the second part 
of the memoir just mentioned, and contains the theory of waves 
propagated with a constant velocity. These are much more 
sensible than the waves propagated with an accelerated motion, 
and are in fact those which are commonly seen to spread in 


* p. 192, + Part II. sect. xi. art. 36. 


144: THIRD REPORT—1833. 


circles round any disturbance made at the surface of water. 
No theory of waves which does not embrace these can be con- 
sidered complete. In the essay of M. Cauchy, which obtained 
the prize, and is printed in the Mémoires des Savans*, the 
theory of only the first kind of waves is given. This essay, 
however, claims to be more complete than the first part of 
M. Poisson’s memoir, because it leaves the function relative to 
the initial form of the fluid surface entirely arbitrary, and conse- 
quently allows of applying the analysis to any form of the body 
immersed to produce the initial disturbance. M. Poisson re- 
stricts his reasoning to a body, of the form of an elliptic para- 
boloid, immersed a little in the fluid, with its vertex downward 
and axis vertical; and as this form may have a contact of the 
second order, with any continuous surface, the reasoning may 
be legitimately extended to any bodies of a continuous form, 
but not to such as have summits or edges, like the cone, cy- 
linder and prism. This restriction having been objected to as a 
defect in the theory}, M. Poisson answers} that his analysis. 
is not at fault, but that one of the differential equations of the 
problem, which expresses the condition that the same particles 
of water remain at the surface during the whole time of motion, 
very much restricts the form which the immersed body may be 
supposed to have. When the initial motion is produced by the 
immersion of a body whose surface presents summits or edges, 
it is not possible, he thinks, to represent the velocities of the 
fluid particles by analytical formule, especially at the first in- 
stants of the agitation, when the motion must be very complicated, 
and the same points will not remain constantly at the surface. 

With the exception of the particular we have been mention- 
ing, the two essays do not present mathematical processes es- 
sentially different in principle. Attached to that of M. Cauchy, 
which was published subsequently to M. Poisson’s memoir, are 
valuable and copious additions, serving to clear up several 
points of analysis that occur in the course of the work, and re- 
ferring chiefly to integration by series and definite integrals, 
and to the treatment of arbitrary functions. Among these is a 
lengthened discussion of the theory of the waves uniformly 
propagated, the existence of which, as indicated by the analysis, 
had escaped the notice of both mathematicians in their first re- 
searches. In this discussion the velocities of propagation are 
determined of the two foremost waves produced by the immer- 

* vol. iii. 

+ Bulletin de la Société Philomatique, Septembre 1818, p. 129. 

+ “Note sur le Probléme des Ondes,” tom. viii. of Mémoires de lV Académie 
des Sciences, p. 571. : 


REPORT ON HYDROSTATICS AND HYDRODYNAMICS, 145 


sion and sudden elevation of bodies of the forms of a parabo- 
loid, a cylinder, a cone, and a solid, generated by the revolution 
of a parabola about a tangent at its vertex. To bodies of the 
last three forms, M. Poisson objects to extending the reasoning; 
and in the ‘‘ Note” above referred to, attempts to show that such 
an extension leads to results inconsistent with the principle of 
the coexistence of small vibrations. If we are not permitted to 
receive the analysis of M. Cauchy in all the generality it lays 
claim to, we must at least assent to the reasonableness of the 
following conclusion it pretends to arrive at, viz. that ‘the 
heights and velocities of the different waves produced by the 
immersion of a cylindrical or prismatic body depend not only 
on the width and height of the part immersed, but also on the 
form of the surface which bounds this part.”. There is also 
much appearance of probability in a remark made by the 
same mathematician, that the number of the waves produced 
may depend on the form of the immersed body and the depth 
of immersion. 

We proceed to say a few words on the contents of M. Pois- 
son’s memoir. He commences by showing, as well by @ priort 
reasoning as by an appeal to facts, that Lagrange’s solution 
cannot be extended to fluid of any depth. In his own solution 
he supposes the fluid to be of any uniform depth, but princi- 
pally has regard to the case which most commonly occurs of a 
very great depth: he neglects the square of the velocity of the 
oscillating particles, as all have done who have attempted this 
problem, and assumes, that a fluid particle which at any instant 
is at the surface, remains there during the whole time of the 
motion. This latter supposition seems necessary for the con- 
dition of the continuity of the fluid. With regard to the neg- 
lect of the square of the velocity, it does not seem that we can 
tell to what extent it may affect the calculations so well as in 
the case of the vibrations of elastic fluids, where the velocity of 
the vibrating particle is neglected in comparison of a known and 
constant velocity, that of propagation. M. Poisson treats first 
the case in which the motion takes place in a canal of uniform 
width, and, consequently, abstraction is made of one horizontal 
dimension of the fluid; and afterwards. the case in which the 
fluid is considered in its three dimensions. The former requires 
for its solution the integration of the same differential equation 
of two terms* as that occurring in Laplace’s theory. No use 
is made of the common integral of this equation, as, on account 
of the impossible quantities it involves, it would be difficult 


* In M. Poisson’s works this equation is PD + #® = 0. 


aux * dyr 
1833. L i 


146 THIRD REPORT—1833. 


to make it serve to determine the laws of propagation. It is 
remarkable that this integral is not necessary for solving the 
problem, although, as M. Poisson has shown in his first me- 
moir, ‘ On the Distribution of Heat in Solid Bodies,” and M. 
Cauchy in the Notes added to his ‘‘ Theory of Waves,” a solu- 
tion may be derived from it equivalent to that which they have 
given without its aid. We may be permitted to doubt whether 
its meaning is yet fully understood, and to hope that, by over- 
coming some difficulty in the interpretation of this integral, the 
problem of waves may receive a simpler solution than has hi- 
therto been given. Be this as it may, the process of integration 
adopted by M. Poisson leaves nothing to be wished for in regard 
to generality. It is easy to obtain an unlimited number of par- 
ticular equations not containing arbitrary functions, which will 
satisfy the differential equation in question, and to combine 
them all in an expression for the principal variable (¢), deve- 
loped in series of real or imaginary exponentials. This will be 
the most general integral the equation admits of, and (to use 
the words of M. Poisson,) ‘‘ there exist theorems, by means of 
which we may introduce into expressions of this nature, arbi- 
trary functions, which represent the initial state of the fluid: 
the difficulty of the question consists then in discussing the re- 
sulting formule, and discovering from them all the laws of the 
phenomenon. ‘The theory of waves furnishes at present the 
most complete example of a discussion of this sort.” 

In a Report like the present, it is not possible to give any 
very precise idea of the analysis which has been employed for 
solving the problem of waves. I have thought it proper to call 
attention to a process of reasoning which has been very exten- 
sively employed by the French mathematicians of the present 
day, and indeed may be considered to be the principal feature 
of their calculations in the more recent applications of mathe- 
matics to physical and mechanical questions. To understand 
fully the nature and power of the method, the works of Fourier, 
particularly The Analytical Theory of Heat, the Notes, before 
spoken of, to M. Cauchy’s “ ‘Theory of Waves,” and the two 
memoirs of M. Poisson ‘ On the Distribution of Heat in Solid 
Bodies,” must be studied. I will just refer to some parts of the 
writings of the last-mentioned geometer, where he has been 
careful to state in a concise manner the principle of the method 
in question. ‘There are some remarks on the generality of a 
main step in the process in the Bulletin de la Société Philoma- 
tique*. 'The note before spoken of in the eighth volume of the 


* An 1817, p. 180. 


REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 147 


Memoirs of the Academy concludes with a brief account of the 
history and principle of this way of expressing the complete 
integral by a series of particular integrals, and introducing the 
arbitrary function. But I would chiefly recommend the peru- 
sal of the remarks at the end of a memoir by this author “ On 
the Integration of some linear partial Differential Equations ; 
and particularly the general Equation of the Motion of Elastic 
Fluids.” To the memoir itself I beg to refer, by the way, as 
presenting a demonstration of the constancy of the velocity of 
propagation from an irregular disturbance in an elastic fluid, 
more simple and direct than that in the Journal de l’ Ecole Po- 
lytechnique. It contains also a general integral of the linear 
partial differential equation of three terms, which occurs in the 
problem of waves for the case in which the three dimensions of 
the fluid are taken account of; but the author does not consider 
this integral of much utility, because of the impossible quantities 
involved in it, and rather recommends the method of express- 
ing the principal variable by infinite series of exponentials. In 
fact, in the ‘Theory of Waves” this case is treated in a manner 
exactly analogous to that in which abstraction is made of one 
dimension of the fluid. 

It may be useful to state some of the principal results ob- 
tained by theory respecting the nature of waves, to give an idea 
of what the independent power of analysis has been able to ef- 
fect. 

With respect, first, to the canal of uniform width, the law of 
the velocity of propagation found by Lagrange is confirmed by 
M. Poisson’s theory when the depth is small, but not other- 
wise. 

When the canal is of unlimited depth, the following are the 
chief results : 

(1.) An impulse given to any point of the surface affects in- 
stantaneously the whole extent of the fluid mass. ‘The theory 
determines the magnitude and direction of the initial velocity of 
each particle resulting from a given impulse. 

_ (2.) “ The summit of each wave moves with a uniformly acce- 
lerated motion.” 

- This must be understood to refer to a series of very small 
waves, called by M. Poisson dents, which perform their move- 
ments as it were on the surface of the larger waves, which he 
calls “ Jes ondes dentelées.” Each wave of the series is found 
to have its proper velocity, independent of the primitive im- 
pulse. Waves of this kind have been actually observed: they 
are small from the first, and quickly disappear. 

(3.) At considerable distances from the place of disturbance, 

L2 


148 THIRD REPORT—1833. 


there are waves of much more sensible magnitude than the pre- 
ceding. Their summits are propagated with a uniform velocity, 
which varies as the square root of the breadth a@ fleur deau of 
the fluid originally disturbed. Yet the different waves which 
are formed in succession are propagated with different veloci- 
ties: the foremost travels swiftest. The amplitude of oscilla- 
tions of equal duration are reciprocally proportional to the 
square root of the distances from the point of disturbance. 

(4.) The vertical excursions of the particles situated directly 
below the primitive impulse, vary according to the inverse ratio 
of the depth below the surface. This law of decrease is not so 
rapid but that the motion will be very sensible at very consider- 
able depths: it will not be the true law, as the theory proves, 
when the original disturbance extends over the whole surface 
of the water, for the decrease of motion in this case will be 
much more rapid. 

The results of the theory, when the three dimensions of the 
fluid are considered, are analogous to the preceding, (1), (2), (8), 
(4), and may be stated in the same terms, excepting that the am- 
plitudes of the oscillations are inversely as the distances from 
the origin of disturbance, and the vertical excursions of the par- 
ticles situated directly below the disturbance vary inversely as 
the square of the depth. 

There is a good analysis of M. Poisson’s theory, and a com- 
parison of many of the results with experiments, in a Treatise 
by M. Weber, entitled Wellenlehre auf Experimente gegriin- 
det*. The experiments of M. Weber were made in a manner 
not sufficiently agreeing with the conditions supposed in the 
theory to be a correct test of it. They, however, manifest a 
general accordance with it, and confirm the existence of the 
small accelerated waves near the place of disturbance, and of a 
sensible motion of the fluid particles at considerable depths 
below the surface. In one particular, in which the theory ad- 
mits of easy comparison with experiment, it is not found to 
agree. When the body employed to cause the initial agitation 
of the water is an elliptic paraboloid, with its vertex downwards 
and axis vertical, and consequently the section in the plane of 
the surface of the water an ellipse, theory determines the velo- 
city of propagation to be greater in the direction of the major 
axis than in that of the minor in the proportion of the square 
root of the one to the square root of the other. This result, 
which it must be confessed has not an appearance of probabi- 
lity, is not borne out by experience. 


* Leipzig, 1825, 


REPORT ON HYDROSTATICS AND HYDRODYNAMICS, 149 


The theory has been also put to the test of*experiment by 
M. bidone, who succeeded in overcoming in great measure an 
obstacle in the way of making the experiments according to the 
conditions supposed in the theory, arising from the adhesion of 
the water to the immersed body*. His observations confirm 
the existence and laws of motion of the accelerated waves. - 

V. Scarcely anything worth mentioning has been effected 
by theory in regard to the resistance of fluids to bodies moving 
in them. The defect of every attempt hitherto made has 
arisen from its proceeding upon some hypothesis respecting 
the law of the resistance; for instance, that it varies as the ve- 
locity, or as the square of the velocity: whereas the law, which 
cannot be known @ priori, ought to be a result of the calcula- 
tion, which should embrace not only the motion of the body, 
but that of every particle of the fluid which moves simulta- 
neously with it. The only problem that has been attempted 
to be solved on this principle, is one of very considerable in- 
terest, relating to the correction to be applied to the pendulum 
to effect the reduction to a vacuum. ‘The memoir of M. Pois- 
son, ‘‘ On the Simultaneous Motions of a Pendulum and of the 
surrounding Air,” was read before the Royal Academy of Paris 
in August 1831, and is inserted in vol. xi. of their Memoires. 
He takes the case of a spherical ball suspended by a very slen- 
der thread, the effect of which is neglected in the calculations; 
the ball is supposed to perform oscillations of very small ampli- 
tude, so that the air in contact with its surface is sensibly the 
same during the motion. A simpler problem of resistance can- 
not be conceived. M. Poisson considers the effect which the 
friction of the particles of air against the surface of the ball 
may have on its motion, and comes to the conclusion that'the 
time of the oscillations is not affected by it, but only their ex- 
tent. The most important result of the theoretical calculation 
is, that the correction which has been usually applied for the 
reduction to a vacuum, and calculated without considering the 
motion of the air, must be increased by one half. This he finds 
to agree sufficiently with some experiments of Captain Sabine. 
He also adduces forty-four experiments of Dubuat, made fifty 
years ago, upon oscillations in water, and three upon oscilla- 
tions in air. These give nearly the same numerical result, and 
agreeing nearly with the value 14. ‘The experiments, however, 
of M. Bessel give results which coincide with Dubuat’s for os- 
' cillations in water, but determine the correction in air for re- 
duction to a vacuum to be very nearly double that hitherto 


* See vol. xxv. of the Memoirs of the Royal Academy of Turin. 


150 THIRD REPORT—1833. 


applied, instead of once and a half. M. Poisson thinks that 
the calculations of M. Bessel leave some room for doubt, and 
objects to the discordance of the values obtained for air and 
water, which, according to his own theory, ought to agree. 
More recent experiments of Mr. Baily *, which, from their num- 
ber and variety, and the care taken in performing them, are 
entitled to the utmost confidence, give the value 1°864 for 
spheres of different materials one inch and a half in diameter, 
and 1°748 for spheres two inches in diameter, the latter bein 
nearly the size of those for which M. Bessel obtained 1:946. 
The theory of M. Poisson does not recognise any difference in 
the value of the coefficient for spheres of different diameters. 
The discrepancies that thus appear between theory and expe- 
riment, and between the experiments themselves, show that 
there is much that requires clearing up in this important sub- 
ject. As far as theory is concerned, it is easily conceivable that 
much must depend upon the way in which the law of trans- 
mission of the motion from the parts of the fluid immediately 
acted on by the sphere to the parts more remote is to be deter- 
mined: and, as it is the province of this Report to point out 
any possible source of error in theory, I will venture again 
to express my doubts of the correctness of the principle em- 
‘ployed in the solution of this problem, of making the deter- 
mination of the law of transmission depend on the arbitrary 
discontinuity of the functions introduced by integration, the law 
itself not being arbitrary +. 

A singular fact, relating to the resistance to the motion of 
bodies partly immersed in water, has been recently established 
by experiments on canal navigation, by which it appears that a 
boat, drawn with a velocity of more than four or five miles an 
hour, rises perceptibly out of the water, so that the water-line 
is not so distant from the keel as in a state of rest, and the re- 
sistance is less than it would be if no such effect took place. 
Theory, although it has never predicted anything of this na- 
ture, now that the fact is proposed for explanation, will proba- 
bly soon be able to account for it on known mechanical prin- 
ciples. 


The foregoing review of the theory of fluid motion, incom- 


* Philosophical Transactions for 1832, p. 399. 

+ In an attempt at this problem made by myself, and published subsequently 
to the Meeting of the Association, the value of the coefficient is found to be 2, 
without accounting for any difference for spheres of different diameters. See 
ra London and Edinburgh Philosophical Magazine and Journal for Septem- 

er 1833. 


REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 151 


plete as it is, may suffice to show that this department of science 
is in an extremely imperfect state. Possibly it may on that ac- 
count be the more likely to receive improvements; and I am 
disposed to think that such will be the case. But these im- 
provements, I expect, will be available not so much in practical 
applications, as in reference to the great physical questions of 
light, heat and electricity, which have been so long the subjects 
of experiment, and the theories of which require to be perfected. 
For this purpose a more complete knowledge of the analytical 
calculation proper for the treatment of fluids in motion may be 
of great utility. 


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[ 153 ] 


Report on the Progress and Present State of our Knowledge 
of Hydraulics as a Branch of Engineering. By GEorGE 
Rennit, Esq., F.R.S., §e. Sc. 


Parr I. 


THE paper now communicated to the British Association for 
the Advancement of Science comprises a Report on the pro- 
gress and present state of our knowledge of Hydraulics as a 
branch of Engineering, with reference to the principles already 
established on that subject. 

Technically speaking, the term hydraulics signifies that 
branch of the science of hydrodynamics which treats of the 
motion of fluids issuing from orifices and tubes in reservoirs, 
or moving in pipes, canals or rivers, oscillating in waves, or 
opposing a resistance to the progress of solid bodies at rest. 

We can readily imagine that if a hole of given dimensions be 
pierced in the sides or bottom of a vessel kept constantly full, 
the expenditure ought to be measured by the amplitude of the 
opening, and the height of the liquid column. 

If we isolate the column above the orifice by a tube, it ap- 
pears evident that the fluid will fall freely, and follow the laws 
of gravity. But experiment proves that this is not exactly the 
case, on account of the resistances and forces which act in a 
contrary direction, and destroy part of, or the whole, effect. 
The development of these forces is so extremely complicated 
that it becomes necessary to adopt some auxiliary hypothesis 
or abbreviation in order to obtain approximate results. Hence 
the science of hydrodynamics is entirely indebted to experi- 
ment. The fundamental problem of it is to determine the efflux 
of a vein of water or any other fluid issuing from an aperture 
made in the sides or bottom of a vessel kept constantly full, or 
allowed to empty itself. Torricelli had demonstrated that, 
abstracting the resistances, the velocities of fluids issuing from 
very small orifices followed the subduplicate ratio of the pres- 
sures. This law had been, in a measure, confused by sub- 
sequent writers, in consequence of the discrepancies which 
appeared to exist between the theory and experiment; until 
_Varignon remarked, that when water escaped from a small 
opening made in’ the bottom of a cylindrical vessel, there ap- 
peared to be very little, or scarcely any, sensible motion in the 


‘ 


154 THIRD REPORT—1835. 


particles of the water; from which he concluded that the law 
of acceleration existed, and that the particles which escaped at 
every instant of time received their motion simply from the 
pressure produced by the weight of the fluid column above the 
orifice, and that the weight of this column of fluid ought to 
represent the pressure on the particles which continually escape 
from the orifice; and that the quantity of motion or expenditure 
is in the ratio of the breadth of the orifice, multiplied by the 
square of the velocity, or, in other words, that the height of 
the water in the vessel is proportional to the square of the ve- 
locity with which it escapes; which is precisely the theorem of 
Torricelli. This mode of reasoning is in some degree vague, 
because it supposes that the small mass which escapes from 
the vessel at each instant of time acquires its velocity from the 
pressure of the column immediately above the orifice. But 
supposing, as is natural, that the weight of the column acts on 
the particle during the time it takes to issue from the vessel, it 
is clear that this particle will receive an accelerated motion, 
whose quantity in a given time will be proportional to the 
pressure multiplied by the time: hence the product of the 
weight of the column by the time of its issuing from the orifice, 
will be equal to the product of the mass of this particle by the 
velocity it will have acquired; and as the mass is the product 
of the opening of the orifice, by the small space which the 
particle describes in issuing from the orifice, it follows that the 
height of the column will be as the square of the velocity ac- 
quired. This theory is the more correct the more the fluid 
approaches to a perfect state of repose, and the more the 
dimensions of the vessel exceed the dimensions of the orifice. 
By a contrary mode of reasoning this theory became insufficient 
to determine the motions of fluids through pipes of small dia- 
meters. It is necessary, therefore, to consider all the motions 
of the particies of fluids, and examine how they are changed 
and altered by the figure of the conduit. But experiment teaches 
us that when a pipe has a different direction from the vertical 
one, the different horizontal sections of the fluid preserve their 
parallelism, the sections following taking the place of the pre- 
ceding ones, and so on; from which it follows (on account of 
the incompressibility of the fluid) that the velocity of each 
horizontal section or plate, taken vertically, ought to be in 
the. inverse ratio of the diameter of the section. It suffices, 
therefore, to determine the motion of a single section, and the 
problem then becomes analogous to the vibration of a com- 
pound pendulum, by which, according to the theory of James 
Bernoulli, the motions acquired and lost at each instant of time 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 155 


form an equilibrium, as may be supposed to take place with 
the different sections of a fluid in a pipe, each section being 
animated with velocities acquired and lost at every instant of 
time. 

The theory of Bernoulli had not been proposed by him until 
long after the discovery of the indirect principle of vs viva by 
Huygens. The same was the case with the problem of the mo- 
tions of fluids issuing from vessels, and it is surprising that no 
advantage had been taken of it earlier. Michelotti, in his experi 
mental researches de Separatione Fluidorum in Corpore Ani- 
mali, in rejecting the theory of the Newtonian cataract, (which 
had been advanced in Newton’s Mathematical Principles, in 
the year 1687, but afterwards corrected in the year 1714,) sup- 
poses the water to escape from an orifice in the bottom of a 
vessel kept constantly full, with a velocity produced by the 
height of the superior surface ; and that if, immediately above 
the lowest plate of water escaping from the orifice, the column 
of water be frozen, the weight of the column will have no effect 
on the velocity of the water issuing from the orifice; and that 
if this solid column be at once changed to its liquid state, the 
effect will remain the same. The Marquis Poleni, in his work 
De Castellis per que derivantur Fluviorum Aque, published at 
Padua in the year 1718, shows, from many experiments, that 
if A be the orifice, and H the height of the column above it, 
the quantity of water which issues in a given time is represented 
by 2AH x baad, whereas if it spouted out from the orifice 
with a velocity acquired by falling from the height H, it ought 
to be exactly 2 A H, so that experiment only gives a little more 
than half the quantity promised by the theory; hence, if we 
were to calculate from these experiments the velocity that the 
water ought to have to furnish the necessary quantity, we 
should find that it would hardly make it reascend 4rd of its 
height. These experiments would have been quite contrary 
to expectation, had not Sir Isaac Newton observed that water 
issuing from an orifice ths of an inch in diameter, was contracted 
$3ths of the diameter of the orifice, so that the cylinder of water 
which actually issued was less than it ought to have been, 
according to the theory, in the ratio of 441 to 625; and aug- 
menting it in this proportion, the opening should have been 
2AH oan or #ths of the quantity which ought to have issued 
on the supposition that the velocity was in the ratio of the 
square root of the height; from which it was inferred that the 
theory was correct, but that the discrepancy was owing to cer- 


156 THIRD REPORT—1833. 


tain resistances, which experiment could alone determine. The 
accuracy of the general conclusion was affected by several 
assumptions, namely, the perfect fluidity and sensibility of the 
mass, which was neither affected by friction nor cohesion, and 
an infinitely small thickness in the edge of the aperture. : 

Daniel Bernoulli, in his great work, Hydrodynamica, seu de 
Viribus et Motibus Fluidorum Commentaria, published at Stras- 
burgh in the year 1738, in considering the efflux of water from 
an orifice in the bottom of a vessel, conceives the fluid to be 
divided into an infinite number of horizontal strata, on the fol- 
lowing suppositions, namely, that the upper surface of the fluid 
always preserves its horizontality; that the fluid forms a con- 
tinuous mass; that the velocities vary by insensible gradations, 
like those of heavy bodies; and that every point of the same 
stratum descends vertically with the same velocity, which is 
inversely proportional to the area of the base of the stratum ; 
that all sections thus retaining their parallelism are contiguous, 
and change their velocities imperceptibly ; and that there is 
always an equality between the vertical descent and ascent, or 
vis viva: hence he arrives, by a very simple and elegant pro- 
cess, to the equations of the problem, and applies its general 
formulz to several cases of practical utility. When the figure 
of the vessel is not subject to the law of continuity, or when 
sudden and finite changes take place in the velocities of the 
sections, there is a loss of vis viva, and the equations require 
to be modified. John Bernoulli and Maclaurin arrived at the 
same conclusions by different steps, somewhat analogous to the 
cataract of Newton. The investigations of D’Alembert had 
been directed principally to the dynamics of solid bodies, until 
it occurred to him to apply them to fluids ; but in following the 
steps of Bernoulli he discovered a formula applicable to the 
motions of fluid, and reducible to the ordinary laws of hydro- 
statics. The application of his theory to elastic and non-elastic 
bodies, and the determination of the motions of fluids in flexible 
pipes, together with his investigations relative to the resistance 
of pipes, place him high in the ranks of those who have contri- 
buted to the perfection of the science. 

The celebrated Euler, to whom every branch of science owes 
such deep obligations, seems to have paid particular attention 
to the subject of hydrodynamics ; and in attempting to reduce 
the whole of it to uniform and general formule, he exhibited a 
beautiful example of the application of analytical investigation 
to the solution of a great variety of problems for which he was 
so famous. The Memoirs of the Academy of Berlin, from the 
year 1768 to 1771, contain numerous papers relative to fluids 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 157 


flowing from orifices in vessels, and through pipes of constant 
or variable diameters. “ But it is greatly to be regretted,” 
says M. Prony, “that Euler had not treated of friction and 
cohesion, as his theory of the linear motion of air would have 
applied to the motions of fluids through pipes and conduits, 
had he not always reasoned on the hypotheses of mathematical 
fluidity, independently of the resistances which modify it.” 

In the year 1765 a very complete work was published at 
Milan by Pau] Lecchi, a celebrated Milanese engineer, entitled 
Idrostatica esaminata ne’ suoi Principi e Stabilite nelle suot 
Regole della Mensura della Acque correnti, containing a com- 
plete examination of all the different theories which had been 
proposed to explain the phenomena of effluent water, and the 
doctrine of the resistance of fluids. The author treats of the 
velocity and quantity of water, whether absolutely or relatively, 
which issues from orifices in vessels and reservoirs, according 
to their different altitudes, and inquires how far the law applies 
to masses of water flowing in canals and rivers, the velocities 
and quantities of which he gives the methods of measuring. 
The extensive and successful practice of Lecchi as an engineer 
added much to the reputation of his work *. 

In the year 1764 Professor Michelotti of Turin undertook, 
at the expense of the King of Sardinia, a very extensive series 
of experiments on running water issuing through orifices and 
additional tubes placed at different heights in a tower of the 
finest masonry, twenty feet in height and three feet square 
inside. The water was supplied by a channel two feet in width, 
and under pressures of from five to twenty-two feet. ‘The 
effluent waters were conveyed into a reservoir of ample area, 
by canals of brick-work lined with stucco, and having various 
forms and declivities; and the experiments, particularly on the 
efflux of water through differently shaped orifices, and addi- 
tional tubes of different lengths, were most numerous and 
accurate, and Michelotti was the first who gave representations 
of the changes which take place in the figure of the fluid vein, 
after it has issued from the orifice. His experiments on the 
velocities of rivers, by means of the bent tube of Pitot, and by 
an, instrument resembling a water-wheel, called the stadera 
idraulica, are numerous and interesting; but, unfortunately, 
their reduction is complicated with such various circumstances 
that it is difficult to derive from them any satisfactory conclu- 
sions. But Michelotti is justly entitled to the merit of having 
made the greatest revolution in the science by experimental 


* See also Memorie Idrostatico-storiche, 17738. 


158 THIRD REPORT—1833. 


investigation*. ‘The example of Michelotti gave a fresh sti- 
mulus to the exertions of the French philosophers, to whom, 
after the Italians, the science owes the greatest obligations. 
Accordingly, the Abbé Bossut, a most zealous and enlight- 
ened cultivator of hydrodynamics, undertook, at the expense 
of the French Government, a most extensive and accurate se- 
ries of experiments, which he published in the year 1771, 
and a more enlarged edition, in two volumes, in the year 
1786, entitled Traité Théorique et Expérimental d’Hydro- 
namique. ‘The first volume treats of the general principles of 
hydrostatics and hydraulics, including the pressure and equili- 
brium of non-elastic and elastic fluids against inflexible and 
flexible vessels; the thickness of pipes to resist the pressure 
of stagnant fluids; the rise of water in barometers and pumps, 
and the pressure and equilibrium of floating bodies; the ge- 
neral principles of the motions of fluids through orifices of dif- 
ferent shapes, and their friction and resistance against the 
orifices; the oscillations of water in siphons; the percussion 
and resistance of fluids against solids; and machines moved by 
the action and reaction of water. ‘The second volume gives a 
great variety of experiments on the motions of water through 
orifices and pipes and fountains; their resistances in rectan- 
gular or curvilinear channels, and against solids moving through 
them; and lastly, of the fire- or steam-engine. In the course 
of these experiments he found that when the water flowed 
through an orifice in a thin plate, the contraction of the fluid 
vein diminished the discharge in the ratio of 16 to 10; and when 
the fluid was discharged through an additional tube, two or 
three inches in length, the theoretical discharge was diminished 
only in the ratio of 16 to 13. In examining the effects of fric- 
tion, Bossut found that small orifices discharged less water in 
proportion than large ones, on account of friction, and that, as 
the height of the reservoir augmented, the fluid vein contracted 
likewise; and by combining these two circumstances together, 
he has furnished the means of measuring with precision the 
quantity of water discharged either from simple orifices or 
additional tubes, whether the vessels be constantly full, or be 
allowed to empty themselves. He endeavoured to point out 
the law by which the diminution of expenditure takes place, 
according to the increase in the length of the pipe or the num- 
ber of its bends; he examined the effect of friction in dimi- 
nishing the velocity of a stream in rectangular and curvilinear 
channels; and showed that in an open canal, with the same 


* Sperimenti Idraulici, 1767 and 1771. 


ON HYDRAULICS AS A BRANCH OF EGINEERING. 159 


height of reservoir, the same quantity of water is always dis- 
charged, whatever be the declivity and length; that the ve- 
locities of the waters in the canal are not as the square roots 
of the declivities, and that in equal declivities and depth of the 
canal’the velocities are not exactly as the quantities of water 
discharged ; and he considers the variations which take place 
in the velocity and level of the waters when two rivers unite, 
and the manner in which they establish their beds. 

His experiments, in conjunction with D’Alembert and Con- 
dorcet, on the resistance of fluids, in the year 1777, and his 
subsequent application of them to all kinds of surfaces, in- 
cluding the shock and resistance of water-wheels, have justly 
entitled him to the gratitude of posterity. The Abbé Bossut 
had opened out a new career of experiments ; but the most dif- 
ficult and important problem remaining to be solved related to 
rivers. It was easy to perform experiments with water running 
through pipes and conduits on a small scale, under given and 
determined circumstances: but when the mass of fluid rolled 
in channels of unequal capacities, and which were composed of 
every kind of material, from the rocks amongst which it accu- 
mulated to the gravel and sand through which it forced a pass- 
age,—at first a rapid and impetuous torrent, but latterly hold- 
ing a calm and majestic course,—sometimes forming sand-banks 
andislands, at other times destroying them, at all times capri- 
cious, and subject to variation in its force and direction by 
the slightest obstacles,—it appeared impossible to submit them 
to any general law. 

Unappalled, however, by these difficulties, the Chevalier 
Buat, after perusing attentively M. Bossut’s work, undertook 
to solve them by means of a theorem which appeared to him 
to be the key of the whole science of hydraulics. He consi- 
dered that if water was in a perfect state of fluidity, and ran in 
a bed from which it experienced no resistance whatever, its 
motion would be constantly accelerated, like the motion of a 
heavy body descending an inclined plane; but as the velocity 
of a river is not accelerated ad infinitum, but arrives at a state 
of uniformity, it follows that there exists some obstacle which 
destroys the accelerating force, and prevents it from impressing 
‘upon the water a new degree of velocity. This obstacle must 
therefore be owing either to the viscidity of the water, or to 
the resistance it experiences against the bed of the river; from 
which Dubuat derives the following principle:—That when 
water runs uniformly in any channel, the accelerating force 
which obliges it to run is equal to the sum of all the resistances 
which it experiences, whether arising from the viscidity of the 
water or the friction of its bed. Encouraged by this discovery, 


160. THIRD REPORT—1833. 


and by the application of its principles to the solution of a great 
many cases in practice, Dubuat*was convinced that the motion 
of water in a conduit pipe was analogous to the uniform motion 
of a river, since in both cases gravity was the cause of motion, 
and the resistance of the channel or perimeter of the pipes the 
modifiers. He then availed himself of the experiments of Bossut 
on conduit pipes and artificial channels to explain his theory: 
the results of which investigations were published in the year 
1779. M. Dubuat was, however, sensible that a theory of so 
much novelty, and at variance with the then received theory, 
required to be supported by experiments more numerous and 
direct than those formerly undertaken, as he was constrained 
to suppose that the friction of the water did not depend upon 
the pressure, but on the surface and square of the velocity. 
Accordingly, he devoted three years to making fresh experi- 
ments, and, with ample funds and assistance provided by the 
French Government, was enabled to publish his great work, 
entitled Principes d Hydraulique vérifiés par un grand nombre 
d Expériences, faites par Ordre du Gouvernement, 2 vols. 1786, 
(a third volume, entitled Principes d Hydraulique et Hydro- 
namique, appeared in 1816);—in the first instance, by repeating 
and enlarging the scale of Bossut’s experiments on pipes (with 
water running in them) of different inclinations or angles, of 
from 90° to gguth part of a right angle, and in channels of from 
1} line in diameter to 7 and 8 square toises of surface, and sub- 
sequently to water running in open channels, in which he ex- 
perienced great difficulties in rendering the motion uniform: 
but he was amply recompensed by the results he obtained on 
the diminution of the velocity of the different parts of a uniform 
current, and of the relation of the velocities at the surface and 
bottom, by which the water works its own channel, and by the 
knowledge of the resistances which different kinds of beds pro- 
duce, such as clay, sand and gravel; and varying the experiments 
on the effect of sluices, and the piers of bridges, &c., he was ena- 
bled to obtain a formula applicable to most cases in practice*. 
Thus, let V = mean velocity per second, in inches. 

d = hydraulic mean depth, or quotient which arises 
from dividing the area or section of the canal, 
in square inches, by the perimeter of the part 
in contact with the water, in linear inches. 

s = the slope or declivity of the pipe, or the sur- 

face of the water. = 
16:087, the velocity in inches which a body 
acquires in falling one second of time. 


ve) 
II 


* Edinburgh Encyclopedia, Art. Hypropynamics, by Brewster. 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 161 


nm = an abstract number, which was found by ex- 
periment to be equal to 243-7. 


/ng (Vd — 01) 
Ws — log. Ws +16 


Such are some of the objects of M. Dubuat’s work. But his 
hypotheses are unfortunately founded upon assumptions which 
render the applications of his theory of little use. It is evident 
that the supposition of a constant and uniform velocity in rivers 
cannot hold: nevertheless he has rendered great services to 
the science by the solution of many important questions relating 
to it; and although he has left on some points a vast field open 
to research, he is justly entitled to the merit of originality and 
accuracy. 

Contemporary with Dubuat was M. Chezy, one of the most 
skilful engmeers of his time: he was director of the Ecole des 
Ponts et Chaussées, and reported, conjointly with M. Perronet, 
on the Canal Yvette. He endeavoured to assign, by experiment, 
the relation existing between the inclination, length, trans- 
versal section, and velocity of a canal. In the course of this 
investigation he obtained a very simple expression of the velo- 
city, involving three different variable quantities, and capable, 
by means of a wii experiment, of being applied to all cur- 
rents whatever. He assimilates the resistance of the sides and 
bottom of the canal to known resistances, which follow the law 
of the square of the velocity, and he obtains the following sim- 
ple formula : 


v= Ved 
28S 


then v 


ll 


— 0:3 (/d — 0'1). 


, where g is = 16-087 feet, the velocity acquired 


by a heavy body after falling one second. 

d = hydraulic mean depth, equal to the area of the section 
divided by the perimeter of the part of the canal in 
contact with the water. 

$ = the slope or declivity of the pipe. 

2 = an abstract number, to be determined by experiment. 

In the year 1784, M. Lespinasse published in the Memoirs 

of the Academy of Sciences at Toulouse two papers, contain- 
ing some interesting observations on the expenditure of water 
through large orifices, and on the junction and separation of 
rivers. The author had performed the experiments contained 
in his last paper on the rivers Fresquel and Aude, and on that 
part of the canal of Languedoc below the Fresquel lock, towards 
its junction with that river. 


As we before stated, M. Dubuat had classified with much 
1833. M 


162 THIRD REPORT—1833. 


sagacity his observations on the different kinds of resistance 
experienced in the motion of fluids, and which might have led 
him to express the sum of the resistances by a rational function 
of the velocity composed of two or three terms only. Yet the 
merit of this determination was reserved to M. Coulomb, who, 
in a beautiful paper, entitled ‘‘ Expériences destinées a déter- 
miner la Cohérence des Fluides et les Lois de leurs Résistances 
dans les Mouvemens trés lents,” proves, by reasoning and facts, 

Ist, That in extremely slow motions the part of the resist- 
ance is proportional to the square of the velocity. 

2ndly, ‘That the resistance is not sensibly increased by in- 
creasing the height of the fluid above the resisting body. 

ordly, That the resistance arises solely from the mutual co- 
hesion of the fluid particles, and not from their adhesion to the 
body upon which they act. 

4thly, That the resistance in clarified oil, at the temperature 
of 69° Fahrenheit, is to that of water as 17-5:1; a proportion 
which expresses the ratio of the mutual cohesion of the par- 
ticles of oil to the mutual cohesion of the particles of water. 

M. Coulomb concludes his experiments by ascertaining the 
resistance experienced by cylinders that move very slowly and 
perpendicularly to their axes, &c. 

This eminent philosopher, who had applied the doctrine of 
torsion with such distinguished success in investigating the 
phenomena of electricity and magnetism, entertained the idea 
of examining in a similar manner the resistance of fluids, con- 
trary to the doctrines of resistance previously laid down. 
M. Coulomb proved, that in the resistance of fluids against 
solids, there was no constant quantity of sufficient magnitude 
to be detected; and that the pressure sustained by a moving 
body is represented by two terms, one which varies as the 
simple velocity, and the other with its square. 

The apparatus with which these results were obtained con- 
sisted of discs of various sizes, which were fixed to the lower 
extremity of a brass wire, and were made to oscillate under a 
fluid by the force of torsion of the wire. By observing the 
successive diminution of the oscillations, the law of resistance 
was easily found. The oscillations which were best suited to 
these experiments continued for twenty or thirty seconds, and 
the amplitude of the oscillation (that gave the most regular re- 
sults) was between 480 the entire division ofthe disc, and 8 or 
10 divisions from zero. 

The first who had the happy idea of applying the law of 
Coulomb to the case of the velocities of water running in na- 
tural or artificial channels was M. Girard, Ingénieur en chef 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 163 


des. Ponts et Chaussées, and Director of the Works of the 
Canal !Ourcgq at Paris *. 

He is the author of several papers on the theory of running 
waters, and of a valuable series of experiments on the motions 
of fluids in capillary tubes. 

M. Coulomb had given a common coefficient to the two terme 
of his formula representing the resistance of a fluid,—one pro- 
portional to the simple velocity, the other to the square of the 
velocity. M. Girard found that this identity of the coefficients 
was applicable only to particular fluids under certain circum- 
stances ; and his conclusions were confirmed by the researches 
of M. Prony, derived from a great many experiments, which 
make the coefficients not only different, but very inferior to the 
value of the motion of the filaments of the water contiguous to 
the side of the pipe. 

The object of M. Girard’s experiments was to determine 
this velocity; and this he has effected in a very satisfactory 
manner, by means of twelve hundred experiments, performed 
with a series of copper tubes, from 1°83 to 2:96 millimetres in 
diameter, and from 20 to 222 centimetres in length; from which 
it appeared, that when the velocity was expressed by 10, and 
the temperature was 0, centigrade, the velocity was increased 
four times when the temperature amounted to 85°. When the 
length of the capillary tube was below that limit, a variation of 
temperature exercised very little influence upon the velocity 
of the issuing fluid, &c. 

It was in this state of the science that M. Prony (then having 
under his direction different projects for canals,) undertook to 


' reduce the solutions of many important problems on running 


water to the most strict and rigorous principles, at the same 
time capable of being applied with facility to practice. 

For this purpose he selected fifty-one experiments which 
corresponded best on conduit pipes, and thirty-one on open 
conduits. Proceeding, therefore, on M. Girard’s theory of the 
analogy between fluids and a system of corpuscular solids or 
material bodies, gravitating in a curvilinear channel of indefinite 
length, and occupying and abandoning successively the dif- 
ferent parts of the length of channel, he was enabled to express 
the velocity of the water, whether it flows in pipes or in open 
conduits, by a simple formula, free of logarithms, and requiring 
merely the extraction of the square root f. 


* Essai sur le Mouvement des Eaux courantes: Paris 1804. Recherches 
sur les Eaux publiques, §c. Devis général du Canal l’Ourcg, &c. 
t Mémoires des Savans Etrangers, §c. 1815. 


M2 


164 THIRD REPORT—1833. 


Thus v = — 0:0469754 + /%0:0022065 + 3041°47 x G, 
which gives the velocity in metres: or, in English feet, 
v = — 0°1541131 + /0:023751 + 328066 x G. 
When this formula is applied to pipes, we must take G=iDK, 
which is deduced from the equation K = PE aa When 
it is applied to canals, we must take G = RI, which is deduced 
from the equation I = a R being equal to the mean radius of 


Dubuat on the hydraulic mean depth, and I equal to the sine 
of inclination in the pipe or canal. M. Prony has drawn up ex- 
tensive Tables, in which he has compared the observed velo- 
cities with those which are calculated from the preceding for- 
mulz, and from those of Dubuat and Girard. In both cases 
the coincidence of the observed results with the formule are 
very remarkable, but particularly with the formule of M. Prony. 
But the great work of M. Prony is his Nouvelle Architecture 
Aydraulique, published in the year 1790. This able produc- 
tion is divided into five sections, viz. Statics, Dynamics, Hydro- 
statics, Hydrodynamics, and on the physical circumstances 
that influence the motions of Machines. The chapter on hydro- 
dynamics is particularly copious and explanatory of the motions 
of compressible and incompressible fluids in pipes and vessels, 
on the principle of the parallelism of the fluid filaments, and 
the efflux of water through different kinds of orifices made in 
vessels kept constantly full, or permitted to empty themselves ; 
he details the theory of the clepsydra, and the curves described 
by spouting fluids; and having noticed the different phenomena 
of the contraction of the fluid vein, and given an account of the ex- 
periments of Bossut, M. Prony deduces formule by which the re- 
sults may be expressed with all the accuracy required in practice. 

In treating of the impulse and resistance of fluids, M. Prony 
explains the theory of Don George Juan, which he finds con- 
formable to the experiments of Smeaton, but to differ very ma- 
terially from the previously received law of the product of the 
surfaces by the squares of the velocities, as established by the 
joint experiments of D’Alembert, Condorcet and Bossut, in the 
year 1775. The concluding part of the fourth section is de- 
voted to an examination of the theory of the equilibrium and 
motion of fluids according to Euler and D’Alembert; and by a 
rigorous investigation of the nature of the questions to be de- 
termined, the whole theory is reduced to two equations only, in 
narrow pipes, according to the theory of Euler, showing its 
approximation to the hypothesis of the parallelism of filaments. 


| 
| 


i 
| 


| 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 165 


The fifth and last section investigates the different circum- 
stances (such as friction, adhesion and rigidity,) which influence 
the motions of machines. 

A second volume, published in the year 1796, is devoted to 
the theory and practice of the steam-engine. Previously to the 
memoir of M. Prony, Sur le Jaugeage des Eaux courantes, in 
the year 1802, no attempt had been made to establish with cer- 
tainty the correction to be applied to the theoretical expendi- 
tures of fluids through orifices and additional tubes. The phe- 
nomenon had been long noticed by Sir Isaac Newton, and illus- 
trated by Michelotti by a magnificent series of experiments, 
which, although involving some intricacies, have certainly 
formed the groundwork of all the subsequent experiments upon 
this particular subject. _ 

By the method of interpolation, M. Prony has succeeded in 
discovering a series of formulz applicable to the expenditures 
of currents out of vertical and horizontal orifices, and to the con- 
traction of the fluid vein; and in a subsequent work, entitled 
Recherches sur le Mouvemens des Eaux courantes, he establishes 
the following formule for the mean velocities of rivers. 

When V = velocity at the surface, 

and U = mean velocity, 
U = 0°816458 V, 
which is about + V. 

These velocities are determined by two methods. Ist, By a 
small water-wheel for the velocity at the surface, and the im- 
proved tube of Pitot for the velocities at different depths below 
the surface. 

If h = the height of the water in the vertical tube above the 
level of the current, the velocity due to this height will be deter- 


eo ee metres 

mined by the formula V = /2¢h = \/ 19-606 h = 4°429 Wh. 

When water runs in channels, the inclination usually given 
amounts to between z3,th and ,3,th part of the length, which 
will give a velocity of nearly 14 mile per hour, sufficient to 
allow the water to run freely in earth. We have seen the incli- 
nation very conveniently applied in cases of drainage at 745th 
and 55th, and some rivers are said to have g,55th only. 
__M. Prony gives the following formule, from a great number 
~ of observations : 

If U = mean velocity of the water in the canal, 

I = the inclination of the canal per metre, 
R = the relation of the area to the profile of its perimeter, 

we shall have 


U = — 0:07 + 0-005 + 3233. R.1; 


166 THIRD REPORT—1833. 


and for conduit pipes, 


calling U = the mean velocity, 
Z = the head of water in the inferior orifice of the pipe, 
L = the length of the pipe in metres, 
D = the diameter of the pipe, 


we shall have 


U = — 0:0248829 + /0:000619159 + 717-857 DZ 
ap 


or, where the velocity is small, 

U = 26°79 / DZ; 

L 

that is, the mean velocities approximate to a direct ratio com- 
pounded of the squares of the diameters and heads of water, 
and inversely as the square root of the length of the pipes: 
and by experiments made with great care, M. Prony has found 
that the formula 


U = — 0:0248829 + /0-000619159 + 717-857 D Z 
L 
scarcely differs more or less from experiments than J, or 24. 
The preceding formule suppose that the horizontal sections, 
both of the reservoir and the recipient, are great in relation 
to the transverse section of the pipe, and that the pipe is kept 
constantly full *. 

In comparing the formule given for open and close canals, 
M. Prony has remarked that these formule are not only similar, 
but the constants which enter into their composition are nearly 
the same; so that either of them may represent the two series 
of phenomena with sufficient exactness. 

The following formula applies equally to open or close canals: 

U = — 00469734 + +/(0-0022065 + 3041-47 4). 
But the most useful of the numerous formule given by M. Prony 
for open canals is the following : 


* According to Mr. Jardine’s experiments on the quantity of water delivered 
by the Coniston Main from Coniston to Edinburgh, the following is a compa- 
rison : Scots Pints. 


Actual delivery of Coniston Main.........sceesceeeeeeees 189-4 
Ditto by Eytelwein’s formula .........see+e++e+e0s 189:77 
Ditto by Girard’s formula ..........-.eseeseeeeees 188-26 
Ditto by Dubuat’s formula ..............ceseeeeees 188-13 


Ditto by Prony’s simple formula......... es 
Ditto by Prony’s tables......... ne ache ei ctube sees 180°7 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 167 


Let g = the velocity of a body falling in one second, 
w = the area of the transverse section, 
. p = the perimeter of that section, 
I = the inclination of the canal, 
Q = the constant volume of water through the section, 
U = the mean velocity of the water, 
R = the relation of the area to the perimeter of the section; 


then Ist, 0-000436 U + 0-003034 U? = g IR = g ta 
10 ais Oke Q, 


Ww 


ordly, Rw? — 0:0000444499 . w . — 0:000309314: 


This last equation, containing the quantities 
~~ QI wand R = = 


shows how to determine one of them, and, knowing the three 
others, we shall have the following equations: ; 


Iw? 


Q? _ 
Aielae es 


thly, p= TO00ISEO w + ONSET 
s . 2 
sey, 1 = 200000159 @ w+ 7000300814 @") 
* 2 * 
Ginly, w = 0-000196 4 0000486)" + 4(0-008084) gR 1] Q 


2¢eRI1 


_ These formulz are, however, modified in rivers by circum- 
stances, such as weeds, vessels and other obstacles in the 
rivers; in which case M. Girard has conceived it necessary to 
introduce into the formule the coefficient of correction = 1°7 
as a multiplier of the perimeter, by which the equations will be, 
-* p — 1°7(0:000436 U + 0-:003034 U?) = gIw. 
The preceding are among the principal researches of this 
distinguished philosopher *. 
' In the year 1798, Professor Venturi of Modena published a 
very interesting memoir, entitled Sur la Communication laté- 
rale du Mouvement des Fluides. Sir Isaac Newton was well 
acquainted with this communication, having deduced from it 
the propagation of rotary motion from the interior to the exte- 
vior of a whirlpool; and had affirmed that when motion is pro- 
_ pagated in a fluid, and has passed beyond the aperture, the 


* Recherches Physico-mathématiques sur la Théorie des Eaux courantes, 
par M. Prony. 


168 THIRD REPORT—1833. 


motion diverges from that opening, as from a centre, and is 
propagated in right lines towards the lateral parts. The sim- 
ple and immediate application of this theorem cannot be made 
to a jet or aperture at the surface of still water. Circumstances 
enter into this case which transform the results of the principle 
into particular motions. It is nevertheless true that the jet 
communicates its motion to the lateral parts without the orifice, 
but does not repel it in a radial divergency. M. Venturi illus- 
trates his theory by experiments on the form and expenditure 
of fluid veins issuing from orifices, and shows how the velocity 
and expenditure are increased by the application of additional 
tubes; and that in descending cylindrical tubes, the upper ends 
of which possess the form of the contracted vein, the expense 
is such as corresponds with the height of the fluid above the 
inferior extremity of the tube. The ancients remarked that a 
descending tube applied to a reservoir increased the expendi- 
ture*. D’Alembert, Euler and Bernoulli attributed it to the 
pressure of the atmosphere. Gravesend, Guglielmini and others 
sought for the cause of this augmentation in the weight of the 
atmosphere, and determined the velocity at the bottom of the 
tube to be the same as would arise from the whole height of 
the column, including the height of the reservoir. Guglielmini 
supposed that the pressure at the orifice below is the same for 
a state of motion as for that of rest, which is not true. In the 
experiments he made for that purpose, he paid no regard either 
to the diminution of expenditure produced by the irregularity 
of the inner surface of the tubes, or the augmentation occa- 
sioned by the form of the tubes themselves. But Venturi esta- 
blished the proposition upon the principle of vertical ascension 
combined with the pressure of the atmosphere, as follows : 

Ist, That in additional conical tubes the pressure of the at- 
mosphere increases the expenditure in the proportion of the 
exterior section of the tube to the section of the contracted 
vein, whatever be the position of the tube. 

2ndly, That in cylindrical pipes the expenditure is less than 
through conical pipes, which diverge from the contracted vein, 
and have the same exterior diameter. This is illustrated by 
experiments with differently formed tubes, as compared with a 
plate orifice and a cylindrical tube, by which the ratios in point 
of time were found to be 41", 31" and 27", showing the advan- 
tage of the conical tube. 

ordly, That the expenditure may be still further increased, 


* “Calix devexus amplits rapit.’”—Jrontinus de -Aqueductibus. See also 
Pneumatics of Hero. 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 169 


in the ratio of 24 to 10, by a certain form of tube,—a circum- 
stance of which he supposes the Romans were well aware, as 
appears from their restricting the length of the pipes of con- 
veyance from the public reservoirs to fifty feet; but it was not 
perceived that the law might be equally evaded by applying a 
conical frustrum to the extremity of the tube. 

M. Venturi then examines the causes of eddies in rivers; 
whence he deduces from his experiments on tubes with en- 
larged parts, that every eddy destroys part of the moving force 
of the current of the river, of which the course is permanent 
and the sections of the bed unequal, the water continues more 
elevated than it would have done if the whole river had been 
equally contracted to the dimensions of its smallest section,—a 
consequence extremely important in the theory of rivers, as the 
retardation experienced by the water in rivers is not only due 
to the friction over the beds, but to eddies produced from the 
irregularities in the bed, and the flexures or windings of its 
course: a part of the current is thus employed to restore an 
equilibrium of motion, which the current itself continually de- 
ranges. As respects the contracted vein, it had been pretended 
by the Marquis de Lorgna* that the contracted vein was. no- 
thing else but a continuation of the Newtonian cataract; and 
that the celerity of the fluid issuing from an orifice in a thin 
plate is much less than that of a body which falls from the 
height of the charge. But Venturi proved that the contraction 
of the vein is incomparably greater than can be produced by 
the acceleration of gravity, even in descending streams, the 
contraction of the stream being 0°64, and the velocity nearly 
the same as that of a heavy body which may have fallen through 
the height of the charge. ‘These experimental principles, which 
are in accordance with the results of Bossut, Michelotti and 
Poleni, are strictly true in all cases where the orifice is small in 
proportion to the section of the reservoir, and when that orifice 
is made in a thin plate, and the internal afflux of the filaments 
is made in an uniform manner round the orifice itself. Venturi 
then shows the form and contraction of the fluid vein by in- 
creased charges. His experiments with the cone are curious ; 
and it would have been greatly to be regretted that he had 
stopped short in his investigations, but for the more extensive 
researches of Bidone and Lesbros. M. Hachette, in opposition 
to the theory of Venturi, assigns, as a cause of the increase by 
additional tubes, the adhesion of the fluid to the sides of the. 
tubes arising from capillary attraction. 


* Memorie della Societa Italiana, vol. iv. 


170 ; THIRD REPORT—1833. 


. In the year 1801, M. Eytelwein, a gentleman well known to 
the public by his translation of M. Dubuat’s work into German, 
(with important additions of his own,) published a valuable 
compendium of hydraulics, entitled Handbuch der Mechanik 
und der Hydraulik, in which he lays down the following rules. 

1. That when water flows from a notch made in the side of 

a dam, its velocity is as the square of the height of 
the head of the water; that is, that the pressure and 
consequent height are as the square of the velocity, the 
proportional velocities being nearly the same as those 
of Bossut. 

. That the contraction of the fluid vein from a simple orifice 
in a thin plate is reduced to 0°64. 

. For additional pipes the coefficient is 0°65. 

For a conical tube similar to the curve of contraction 0°98. 

. For the whole velocity due to the height, the coefficient 
by its square must be multiplied by 8-0458. 

. For an orifice the coefficient must be multiplied by 7°8. 

. For wide openings in bridges, sluices, &c., by 6°9. 

- For short pipes 6°6. 

For openings in sluices without side walls 5:1. 

Of the twenty-four chapters into which M. Eytelwein’s * work 
is divided, the seventh is the most important. The late Dr, 
Thomas Young, in commenting upon this chapter, says: 

. The simple theorem by which the velocity of a river is de- 
termined, appears to be the most valuable of M. Eytelwein’s 
improvements, although the reasoning from which it is deduced 
is somewhat exceptionable. ‘The friction is nearly as the square 
of the velocity, not because a number of particles proportional 
to the velocity is torn asunder in a time proportionally short,— 
for, according to the analogy of solid bodies, no more falcon is 
destroyed by friction when the motion is rapid than when slow, 
—but because when a body is moving in lines of a given curva- 
ture, the deflecting forces are as the squares of the velocities ; 
and the particles of water in contact with the sides and bottom 
must be deflected, in consequence of the minute irregularities 
of the surfaces on which they slide, nearly in the same curvi- 
linear path, whatever their velocity may be. At any rate (he 
continues) we may safely set out with this hypothesis, that the 
principal part of the friction is as the square of the velocity, 
and the friction is nearly the same at all depths}; for Professor 
. Robison found that the time of oscillation of the fluid ina bent 


es) 


Or 09 


Nolo hs eer) 


* See Nicholson’s translation of Eytelwein’s work. z 
+ See my “Experiments on the Friction and Resistance of Fluids,” Philo- 
sophical Transactions for 1831. 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 171 


tube was not increased. by increasing the pressure against the 
sides, being nearly the same when the principal part was si- 
tuated horizontally, as when vertically. The friction will, how- 
ever, vary, according to the surface of the fluid which is in 
contact with the solid, in proportion to the whole quantity of 
fluid; that is, the friction for any given quantity of water will 
be as the surface of the bottom and sides of a river directly, 
and as the whole quantity in the river inversely ; or, supposing 
the whole quantity of water to be spread on a horizontal sur- 
face equal to the bottom and sides, the friction is inversely as 
the height at which the river would then stand, which is called 
the hydraulic mean depth*.” It is, therefore, calculated that 
the velocities will be a mean proportional between the hydraulic 
mean depth and the fall, or +4ths of the velocity per second. 
Professor Robison informs us, that by the experiments of 
Mr. Watt on a canal eighteen feet wide at the top, seven feet 
at the bottom, and four feet deep, having a fall of four inches 
per mile, the velocities were seventeen inches per second at the 
surface, fourteen inches per second in the middle, and ten inches 
per second at the bottom, making a mean velocity of fourteen 
inches per second; then finding the hydraulic mean depth, and 


dividing the area of the section by the perimeter, we have a 
_ or 29°13 inches ; and the fall in two miles being eight inches, 
we have 4/(8 x 29°13) = 15-26 for the mean proportional of 
+aths, or 13-9 inches, which agrees very nearly with Mr. Watt's 
velocity. 

The Professor has, however, deduced from Dubuat’s elabo= 
rate theories 12568 inches. But this simple theorem applies 
only to the straight and equable channels of a river. Ina 
curved channel the theorem becomes more complicated ; and, 
from observations made in the Po, Arno, Rhine, and other 
rivers, there appears to be no general rule for the decrease of 
velocity going downwards. M. Eytelwein directs us to deduct 
from the superficial velocity ;;3 for every foot of the whole 
depth. Dr. Young thinks ;3,ths of the superficial velocity suf- 
ficient. According to Major Rennell, the windings of the river 
Ganges in a length of sixty miles are so numerous as to reduce 
the declivity of the bed to four inches per mile, the medium 
rate of motion being about three miles per hour, so that a mean 
hydraulic depth of thirty feet, as stated to be 2rds of the 
velocity per second, will be 4°47 feet, or three miles per hour. 
Again, the river when full has thrice the volume of water in it, 
and its motion is also accelerated in the proportion of 5 to 3; 


* See Nicholson’s Journal for 1802, vol. iii. p. 31. 


172 THIRD REPORT—1835. ‘ 


and, assuming the hydraulicmean depth to be doubled at the 
time of the inundation, the velocity will be increased in the 
ratio of 7 to 5; but the inclination of the surface is probably 
increased also, and consequently produces a further velocity of 
from 1:4 to 1:7. M. Eytelwein agrees with Genneté*, that a 
river may absorb the whole of the water of another river equal 
in magnitude to itself, without producing any sensible elevation 
in its surface. This apparent paradox Genneté pretends to 
prove by experiments, from observing that the Danube absorbs 
the Inn, and the Rhine the Mayne rivers; but the author evi- 
dently has not attended to the fact, as may be witnessed in the 
junction of rivers in marshes and fenny countries,—the various 
rivers which run through the Pontine and other marshes in 
Italy, and in Cambridgeshire and Lincolnshire in this country : 
hence the familiar expression of the waters being overridden is 
founded in facts continually observed in these districts. We 
have also the experiments of Brunings in the Architecture Hy- 
draulique Générale de Wiebeking, Wattmann’s Mémoires sur 
U Art de construire les Canaux, and Funk Sur l Architecture 
Hydraulique générale, which are sufficient to determine the 
coefficients under different circumstances, from velocities of 
2ths to 74 feet, and of transverse sections from 1 to 19135 
square feet. The experiments of Dubuat were made on the 
canal of Jard and the river Hayne; those of Brunings in the 
Rhine, the Waal and Ifrel; and those of Wattmann in the 
drains near Cuxhaven. 

M. Eytelwein’s paper contains formule for the contraction 
of fluid veins through orifices +, and the resistances of fluids 
passing through pipes and beds of canals and rivers, according 
to the experiments of Couplet, Michelotti, Bossut, Venturi, 
Dubuat, Wattmann, Brunings, Funk and Bidone. 

In the ninth chapter of the Handbuch, the author has en- 
deavoured to simplify, nearly in the same manner as the motion 
of rivers, the theory of the motion of water in pipes, observing 
that the head of water may be divided into two parts, one to 
produce velocity, the other to overcome the friction; and that 
the height must be as the length and circumference of the sec- 
tion of the pipe directly, or as the diameter,—and inversely as 
the area of the section, or as the square of the diameter. 


* Expériences sur le Cours des Fleuves, ou Lettre & un Magistrat Hollandais, 
par M. Genneté. Paris 1760. 

+ “Recherches sur le Mouvement de |’Eau, en ayant egard 4 la Contraction 
qui a lieu au Passage par divers Orifices, et 4 la Resistance qui retard le Mouve- 
ment, le long des Parois des Vases ; par M. Eytelwein,”"—Mémoires de ' Aca- 
démie de Berlin, 1814 and 1815. 


amon 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 173 


In the allowance for flexure, the product of its square, multi- 
plied by the sum of the sines of the several angles of inflection, 
and then by ‘0038, will give the degree of pressure employed 
in overcoming the resistance occasioned by the angles; and de- 
ducting this height from the height corresponding to the velo- 
city, will give the corrected velocity *. 

M. Eytelwein investigates, both theoretically and experi- 
mentally, the discharge of water by compound pipes,—the mo- 
tions of jets, and their impulses against plane and oblique sur- 
faces, as in water-wheels, in which it is shown that the hydraulic 
pressure must be twice the weight of the generating column, as 
deduced from the experiments of Bossut and Langsdorft; and 
in the case of oblique surfaces, the effect is stated to vary as the 
square of the sine of the angle of incidence; but for motions 
in open water about 2ths of the difference of the sine from the 
radius must be added to this square. 

The author is evidently wrong in calculating upon impulse 
as forming part of the motion of overshot wheels; but his 
theory, that the perimeter of a water-wheel should move with 
half the velocity of a given stream to produce a maximum effect, 
agrees perfectly with the experiments of Smeaton and others +. 

The author concludes his highly interesting work by exa- 
mining the effects of air as far as they relate to hydraulic ma- 
chines, including its impulse against plane surfaces on siphons 


* Hence, if f denote the height due to the friction, 
d = the diameter of the pipe, 
rf @ = a constant quantity, 
we shall have, firs we and V? =e 


But the height employed in overcoming the friction corresponds to the differ- 
ence between the actual velocity and the actual height, that is, f= h — a 
where b is the coefficient for finding the velocity from the height. 

al ieh _bBdh—dVv? VP dah 

PSMA TOT ee aoe 
Now Dubuat found 0 to be 6°6, and a4? was found to be 0-0211, particularly 
when the velocity is between six and twenty-four inches per second. Hence 


_ Hence we have, vy? 


, 436 dh dh 
ales = 45:5 ai al a 
> a = ponte Y= 3 , 
2) Vv (i 
or more accurately, V=50 i+ 50a)" 


__ + The author of this paper has made a great many experiments on the max- 
imum effect of water-wheels ; but the recent experiments of the Franklin Insti- 
tution, made on a more magnificent scale, and now in the course of trial, eclipse 
everything that has yet been effected on this subject. See also Poncelet, Mé- 
moire sur les Roues Hydrauliques, and Aubes Courbes par dessous, §c. 1827. 


174 THIRD REPORT—1833. 


and pumps of different descriptions, horizontal and inclined 
helices, bucket-wheels, throwing-wheels, and lastly, on instru- 
ments for measuring the velocity of streams of water. A very 
detailed account of the work was given in the Journal of the 
Royal Institution, by the late Dr. Young. But it is due to 
MM. Dubuat and Prony to state, that M. Eytelwein has 
exactly followed the steps of these gentlemen in his Theory of 
the Motion of Water in open Channels. 

In the year 1809 a valuable series of experiments upon the 
motions of waters through pipes, was made by MM. Mallet 
and Vici at Rome, and afterwards by M. Prony*. 

It had been proved, by experiments made with great care, 
that the diminution of velocity, and consequent expenditure in 
pipes, was not in the ratio of the capacity of the pipes, as Fron- 
tinus had supposed in his valuation of the product of the an- 
cient module or calice ; and as it was desirable to ascertain the 
actual product of the three fountains now used at Rome, a se- 
ries of experiments was undertaken by these gentlemen; the 
principal result of which was, that a pipe, of which the gauge 
was five onces}, furnished }th more water than five pipes of 
one once, on account of the diminution of the velocity by 
friction in the ratio of the perimeters of the orifices as com- 
pared with their areas. 

M. Mallet also made a great many researches relative to the 
distribution of water in the different cities and towns of En- 

land and France, with a view to their application at Paris; of 
all of which he has published an account. 

The researches that had been made hitherto on the expendi- 
ture of water through orifices, had for their object the deter- 
mination of the velocity and magnitude of the section, by which 
it is necessary to multiply the velocity to obtain the expense. 
But although these be the'first elements for consideration, they 
are not sufficient; for the fluid vein presents other phenomena 
equally important, both in the theory and its application, 
namely, the form and direction of the vein after it has issued 
from the orifice. The former phenomena, as we before stated, 
had been long noticed by Michelotti and others, but nothing 
precise had been established on the forms and remarkable phe- 
nomena of the fluid vein itself. Venturi had given three ex- 
amples. 

M. Hachette, in two memoirs presented to the Académie 
Royale des Sciences in 1815 and 1816, also considered the. 


* Notices Historiques, par M. Mallet. Paris 1830. 
+ French measure, or 0°03059 French kilolitres. 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 175 


form of veins; and in his Traité des Machines, he states that 
he had already given a description of veins issuing from circu- 
lar, elliptical, triangular and square orifices, without having 
entered into any detail respecting them, so that that part of 
the subject was in a great measure involved in doubt. In 1829 
a paper, entitled ‘‘ Expériences sur Ja Forme et sur la Direction 
desVeines et des Courans d’ Eau, lancés par diverses Ouvertures,” 
was read to the Academy of Sciences at Turin by M. Bidone,. 

iving an account of a series of experiments made in the years 
1826 and 1827, in the Hydraulic Establishment of the Royal 
University. The results of these experiments are divided into: 
five articles. The first. gives, a description of the apparatus 
and mode of proceeding, and the. figures obtained from veins 
expended from rectilinear and curvilinear orifices, with salient 
angles pierced. in. vertical plates, and whose perimeters are 
formed by straight and curved lines, varying upwards of fifty 
different ways, with variable and invariable changes, from zero 
to twenty-two French, feet: the area.of water was equal to one 
square inch. The sections of the veins were taken at different 
distances from the aperture. .The results are extremely curi- 
ous, as illustrating the influence of pressure and divergence on 
part of a fluid mass not in equilibrio, and may be assimilated to 
the phenomena presented by the undulation of streams of 
light. The author contents himself with stating the results, 
which are further illustrated by diagrams. 

In a second paper, read to the Accademia delle Scienze in 
April following of the same year (1829), M. Bidone enters into 
a theoretical consideration of his experiments, in which he re- 
presents the greatest contraction of the fluid vein to take place 
at a distance not exceeding the greatest diameter of the orifice, 
whatever be the shape; from which it results that the expres- 
sion for the expense of the orifice is equal to the sum of the 
product of each superficial element multiplied by the velocity. 
of the fluid vein; and as it was determined by experiment 
that the area of the vena contracta was from 0°60 to 0°62 of 
the area of the orifice, it follows that this coefficient of con-; 
traction, multiplied by the velocity due to the charge, repre- 
sents the expenditure. 

’ M. Bidone considers the case of a fluid vein reduced to a 


of the component filaments, but solely on their direction, a re- 
sult conformable to experiment. 


176 THIRD REPORT—1833. 


He then determines, from the results of M. Venturoli*, the 
absolute magnitude of the contracted section of the vein (issuing 
from a circular orifice) to be exactly 2rds of the orifice, the 
correction due to the contraction depending upon the ad- 
hesion and friction of the fluid against the perimeter of the ori- 
fice, and the ratio of the area of the vein to the area of the 
orifice: the same for all orifices. Hitherto the magnitude of 
fluid veins, as determined by direct measurements, had given 
greater coefficients than the effective expenditure allowed. 

Michelotti, with a pressure of twenty feet, with orifices of one 
and two inches in diameter, found the coefficient 0-649 

Bassuth: Macraitqagoels (00s eice 214 OnE 
petdackiadeepawal..0 iabais ogls peipag OES 
Mientras aid yeu. pegs ise da9 ti Obae 
Biytelborgimn ds: thay hed: nde LOO W aide AGA 
Hachette... 2h cian naalbes iyo du soe 
Mewaomis ia sualsise seis ariel! ocat Abit tO 
Heebshatuuas atsiite 10.0 seigtihhe 1:2 volhinge be, GEO 
Brindley and Smeaton . . . «. . . 0°631 
Banke na wie Alagus Ap iid Obs eae 
Rennie $yics. s1sej sag, ery eageuder, ideas] a 

In several experiments the ratio rarely exceeded 0°620; so 
that the discrepancy must have arisen from inaccuracies in the 
measurement of the fluid vein and orifice. 

In the year 1827, it having been considered desirable to re- 
peat the experiments of Bossut and Dubuat, application was 
made to the French Government by General Sabatier, Com- 
mander-in-chief of the Military School at Metz, for permission 
to undertake a series of experiments on a scale of magnitude 
sufficient to establish the principles laid down by those authors, 
and serve as valuable practical rules for future calculations. 
The apparatus consisted, Ist, of an immense basin, having 
an area of 25,000 square metres; 2nd, of a smaller reservoir, 
having a superficial area of 1500 square metres, and a depth 
of 3°70 metres, so contrived, by means’of sluices, as to have a 
complete command of the level of the water during the experi- 
ment; 3rd, of a basin directly communicating with the second 
basin, 3°68 metres in length, and 3 metres in width, to receive 
the product of the orifices; 4th, a basin or gauge capable of 
containing 24,000 litres. 


* Elementi di Meccanica e d’Idraulica: Milano 1818. Recherche Geome- 
triche fatte nella Scuola degli Ingegneri pontifici d’ Acque e Strade, Tanno 
1821. Milano. 


+ “On the Friction and Resistance of Fluids,” Philosophical Transactions of 


1831. 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. Vit 


The time was constantly noticed by an excellent stop-watch, 
made by Breguet; and the opening of the orifices, the charges 
of the fluid in the reservoir, as well as the level of the water in 
the gauge basin relative to each expense of fluid, were always 
measured to the tenth of a millimetre, so that, even under the 
most unfavourable circumstances, the approximation was at least 
to z4,dth part of the total result. The total disposable fall or 
height, counting from the ordinary surface of the Moselle river, 
was four metres, from which two metres were deducted for the 
gauge basin, leaving only a fall of two metres under the most 
favourable circumstances; and in the subsequent experiments 
of 1828 the height never exceeded 1°60 metre, sufficiently 
high for all practical purposes. An apparatus was provided 
for regulating the height of the orifice and the surface of the 
water in the reservoirs, and for tracing with the greatest accu- 
racy the forms and sections of the fluid veins before and after 
issuing from the orifices, and the depressions experienced by 
the surface of the water previously to its issuing from an open- 
ing of twenty centimetres square, the upper side of which was 
on a level with the surface of the water in the reservoir. These 
depressions are recorded in the Tables, 

Ist, On the expenditure of water through rectangular verti- 
cal orifices, twenty centimetres square, and varying in height 
from one to twenty centimetres, under charges of from ‘0174 
of a metre to 16901 metre: 

2ndly, On the expenditures of water from the similar-sized. 
orifices, open at the top, but under charges of from two to 
twenty-two centimetres. 

The whole is comprised in eleven Tables of 241 experiments, 
to which is added a twelfth Table, showing the value of the co- 
efficients of contraction for complete orifices, from twenty cen- 
timetres square to one centimetre, calculated according to the 
following formula: 

D for the height of the orifices, where* 


D=loV2gh=I1(h—-h) V2¢ — being the_theo- 


retical expense relative to the velocity ; 

D’=21 /2¢ (hE —W3)= 81 (hV 2gh—-WV 2h) 
or the theoretical expense, having regard to the influence of 
the orifice. 


* That is, where / = 0-20 metre, being the horizontal breadth of all the orifices; 
h = the charge of the fluid on the lower part of the orifice ; 
h'= the charge in the upper or variable side of the orifice ; 
o = h —h' the thickness of the vein of water. 


1833, = 


178 THIRD REPORT—1833. 


The conclusions to be derived from these Tables are, 

Ist, That for complete orifices of twenty centimetres square 
and high charges, the coefficient is 0-600; with the charge 
equal to four or five times the opening of the orifice, the co- 
efficient augments to 0°605; but beyond that charge the co- 
efficient diminishes to 0°593. : 

2ndly, That the same law maintains for orifices of ten and 
five centimetres in height, the coefficients being for ten centi- 
metres 0°611, 0-618, 0°611 respectively, and for five centi- 
metres in height 0°618, 0°631, 0°623. 

Lastly, That with orifices of three, two and one centimetres 
in height, the law changes very rapidly, and the coefficients 
increase as the opening of the orifice becomes less, being for 
one centimetre 0°698, the smallest height of the orifice, to 0°640 
for three centimetres. 

These remarkable discrepancies from the results of Bidone 
and others are attributed by MM. Lesbros and Poncelet to 
differences in the construction of the apparatus or in the mode 
of measurement adopted by the latter gentlemen; but in gene- 
ral the coincidences are sufficiently satisfactory, and they are 
the more accurately confirmed by the subsequent investigations 
‘of MM. D’Aubuisson and Castel at Toulouse *. As respects 
water issuing from the openings or notches made in the sides 
of dams, or what we should term incomplete orifices, it appears 
that the coefficient obtained by the ordinary formula of Dubuat, 
or Lhi/ 2 gh, augments from the total charge of twenty-two cen- 
timetres when it is from 0°389 to two centimetres when it be- 
comes 0:415 ; hence we may safely adopt M. Bidone’s coefficient 
of 0:405, or, according to MM. Poncelet and Lesbros’ theory 
0-400, for calculating expenditures through notches in dams. 
From these and other experiments the authors are led to con- 
clude, that the law of continuity maintains for indefinite heights 
both with complete and incomplete orifices, and that the same 
coefficient can be obtained by adopting in both cases the same 
formula. The authors observe that the area of the section of 
the greatest contraction of the vein, considered as a true 
square, is exactly two thirds of the area of the orifice; a fact 
which goes to prove that there is no certain comparison be- 
tween the mean theoretical or calculated velocities, by means 
of the formula now used, and the mean effective velocities de- 
rived from the expenditure. P 

The authors conclude their memoir by recommending. their 
experiments for adoption in all cases of plate orifices situated 


* Annales de Chimie et de Physique for 1880, tom.*xliv. p. 225. 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 179 


at a distance from the sides and bottom of the reservoir, pro- 
mising to investigate with similaraccuracy in a future memoir 
the cases which may occur to the contrary. 

A note is appended to the memoir by M. Lesbros, contain- 
ing formule for calculating the effective expenditure of com- 
plete orifices; and also a 'l'able of constants, which gives the 
effective expenditure of each orifice as compared with experi- 
ment. We have been thus particular in detailing the results of 
MM. Lesbros and Poncelet’s work, because they have com- 
prehended all the cases upon which there remained any doubts, 
and with very few exceptions are in accordance with the expe- 
riments of Brunacci, Navier, Christian, Gueymard, D’Aubuis- 
son, and by the author of this paper*. So that in point of 
accuracy and laborious investigation, the authors of these va- 
luable accessions to our knowledge, not only merit our grati- 
tude, but have very amply replied to the liberality of the French 
Government. 

_ Having thus endeavoured to elucidate the labours of the 
foreign philosophers who have contributed so greatly to the 
progress of hydraulics, it only remains for us to notice the 
scanty contributions of our countrymen to the science. While 
France and Germany were rapidly advancing upon the traces 
of Italy, England remained an inactive spectator of their pro- 


‘gress, contented with the splendour of her own Newton, to 


receive from foreigners whatever was original or valuable in 
the science. The Philosophical Transactions, rich as they 
are in other respects, scarcely contain a single paper on this 
subject founded on any experimental investigations. Some 
erroneous and inconclusive inferences from Newton, by Dr. 
Jurin; a paper on the Measure of Force, by Mr. Eames; a 
paper on Wiers, by Mr. Roberts; another on the Motion and 
Resistance of Fluids, by Dr. Vince; and a summary of Bossut 
and Dubuat’s Experiments on the Motions of Fluids through 
‘Tubes, by Dr. Thomas Young, comprise nearly the whole of 
the papers on hydraulics in the Philosophical Transactions. 
The various treatises on the subject published by Maclaurin, 
Emerson, Dr. Matthew Young, Desaguliers, Clare and Switzer, 
with the exception of the theoretical investigations, are compiled 
principally from the works of foreigners; and it was not until the 
subject was taken up by Brindley, Smeaton, Robison, Banks 
and Dr. Thomas Young, that we were at all aware of our defi- 
ciency. Practical men were either necessitated to follow the un- 
certain rules derived from their predecessors, or their own expe- 
rience and sagacity, for the little knowledge they possessed. 

' _ * Philosophical Transactions for 1831, . 


N 2 


180 THIRD REPORT—1833. 


On the subject of hydrometry we were equally ignorant; and 
although the Italian collection had been published several 
years previously, and was well known on the Continent, it was 
not until Mr. Mann published an abstract of that collection 
that we were at all aware of the state of the science abroad. 

Under these circumstances the author of this paper was in- 
duced, in the year 1830, to undertake a series of experiments to 
ascertain, Ist, The friction of water against the surface of a 
cylinder, and discs revolving in it, at different depths and ve- 
locities: from which it appeared, that with slow velocities the 
friction approximated the ratio of the surfaces, but that an in- 
crease of surface did not materially affect it with increased velo- 
cities; and that with equal surfaces the resistances approxi- 
mated to the squares of the velocities. 

2ndly, To ascertain the direct resistances against globes 
and dises revolving in air and water alternately: from which it 
resulted, that the resistances in both cases were as the squares 
of the velocities; and that the mean resistances of circular discs, 
square plates, and globes of equal area, in atmospherical air, 
were as under: 


Circular discs . . 25°180 ...... 1:18 
Square plates . . 22°010 im air, . . 1°36 in water. 
Round globes. . 10°627..... . O75 


3rdly, That with circular orifices made in brass plates of 
eth of an inch in thickness, and having apertures of 4, 3, 2,2 
of an inch respectively, under pressures varying from one to 
four feet, the average coefficients of contraction were, 


for altitudes of 1 foot. ...... 0-619 
yO ss! | SP ra a 0°621 
For additional tubes of glass the coefficient was, 
TOR Bet eee rT aN 0°817 
AREER VIE Oe 0°806-. 


4thly, That the expenditures through orifices, additional 
tubes, and pipes of different lengths, of equal areas and under 
the same altitude as compared with the expenditure through a 
pipe of 30 feet in length, are as 


1:3 for orifices, 

1:4 for additional tubes, 

1 : 3-7 for a pipe 1 foot in length, 
irs BG 8 feet ————-, 
1:20 ——4 , 
1: 1-4 2 ———_———_.. 


5thly, That with bent rectangular pipes } an inch in diameter, 
and 15 feet in length, the expenditures were diminished with 
fourteen bends two thirds, as compared with a straight pipe, 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 181 


and with twenty-four right angles, one third ; but did not seem 
to observe any decided law. 

In several experiments tried on a great scale, the results 
gave from one fifth to one sixth of the altitude for the fric- 
tion. In the case of the Coniston main, which conducts the 
water from the reservoir at Coniston to the castle of Edin- 
burgh, the diameter of which is 44 inches, the length 14,930 
feet, and the altitude 51 feet, it was proved by Mr. Jardine 
that the formule of Dubuat and Eytelwein approximated to 
the real results very nearly; and in some experiments made on 
a great scale by the author of this paper, these formulze were 
found equally applicable. In several experiments made in the 
year 1828, on the water-works at Grenoble, by M. Gueymard, 
it was found that pipes of six and eight French inches in dia- 
meter furnished only two thirds of the water indicated by the 
formulz of M. Prony; but when of nine inches diameter, the 
formula approximated very nearly. In M. Gueymard’s expe- 
riment the altitude of the reservoir above the point of delivery 
was 8°453 metres, or 27°73 English feet. The height to which 
the water was required to be elevated was 5°514 metres, or 
18 feet; the volume of water required was 954 litres, or 33°6 
cubic feet; the length of the pipe was 3200 metres, or 10498 
feet. ‘There were eight gentle curves in the system, but en- 
larged beyond the average diameter of the parts of the pipe; 
from which it resulted that the height to which, the water was 
delivered was only two thirds of the height of the reservoir *. 

In the preceding short but imperfect history of the science 
of hydraulics we have confined our attention to the experi- 
mental researches that have been made on spouting fluids only. 
In a future communication I hope to examine the state of 
our knowledge of the natural phenomena of rivers, and the 
causes by which they are influenced; at present it is extremely 
limited, and although we have many works upon the subject, 
very little seems to be known either of their properties or of 
the laws by which they are governed. , 


* According to M. Prony’s theory, the height raised would only have been 
5°514 metres instead of 5°671 metres. The difficulty, however, of making ex- 
periments on a great scale will always prove an obstacle to the right solution 
of the question, in as much as it exacts that the pipe be of the same dia- 
meter throughout, that is, perfectly straight, and free from bends, and the 
charge of water invariable. For this purpose M. Prony has calculated Tables 
showing the relation subsisting between the expenditure, diameter, length, the 
total inclination of the pipes, and the difference of pressure at its extremities. 


182 THIRD REPORT—1833; te 


APPENDIX. 


Since the foregoing Report was read to the British Associa- 
tion a paper, entitled ‘“‘ Mémoire sur la Constitution des Veines 
Liquides lancées par des Orifices Circulaires en mince paroi,” 
has been communicated to the Academy of Sciences at Paris, 
by M. Félix Savart, 26 Aott 1833. The author, after detailing 
very minutely the different phanomena presented by liquid 
veins issuing from circular orifices perforated in thin plates, 
attached to the bottom and sides of vessels, illustrates his po- 
sitions by a series of curious experiments on the vibrations and 
sounds of the drops which issue from the annular rings or pipes 
formed by the troubled part of the liquid. The results of these 
experiments are best given in his own words. 

«1°, Toute veine liquide lancée verticalement de haut en bas 
par un orifice circulaire pratiqué dans une paroi plane et hori- 
zontale est toujours composée de deux parties bien distinctes 
par l’aspect et la constitution. La partie qui touche a l orifice est 
un solide de révolution dont toutes les sections horizontales 
vont en décroissant graduellement de diameétre. Cette premiére 
partie de la veine est calme et transparente, et ressemble 4 un 
tige de cristal. La seconde partie, au contraire, est toujours 
agitée, et parait dénuée de transparence, quoiqu elle soit ce- 
pendant d’une forme assez réguliére pour qu'on puisse facile- 
ment voir quelle est divisée en un certain nombre de ren- 
flemens allongés dont le diamétre maximum est toujours plus 
grand que celui de lorifice. 


* 2°, Cette seconde partie de la veine est composée de gouttes 


bien distinctes les unes des autres, qui subissent pendant leur 
chute, des changemens périodiques de forme, auxquels sont dues 
les apparences de ventres ou renflemens régulitrement espacés 
que l’inspection directe fait reconnaitre dans cette partie de la 
veine, dont la continuité apparente dépend de ce que les gouttes 


se suecédent a des intervalles de temps qui sont moindres que. 


la durée de la sensation produite sur la rétine par chaque goutte 
en particulier. 

“30, Les gouttes qui forment la partie trouble de la veine 
sont produites par des renflemens annulaires qui prennent 
naissance trés pres de lorifice, et qui se propagent a des inter- 


valles de temps égaux, le long de la partie limpide de la veine,. 


en augmentant de volume A mesure quiils descendent, et qui 


enfin se séparent de l'extrémité inférieure de la partie limpide: 


et continue 4 des intervalles de temps égaux 4 ceux de leur 
production et de leur propagation. 


ON HYDRAULICS AS A BRANCH OF ENGINEERING. 183. 


«40, Ces renflemens annulaires sont engendrés par une suc- 
eession périodique de pulsations qui ont lieu 4 l’orifice méme ; 
de sorte que la vitesse de I’écoulement, au lieu d’étre uniforme, 
est périodiquement variable. 

_ “5°, Le nombre de ces pulsations, méme pour des charges 
foibles, est toujours assez grand, dans un temps donné, pour 
qu’elles soient de l’ordre de celles qui, par la fréquence de leur 
retour, peuvent donner lieu a des sons perceptibles et compa- 
rables. Ce nombre ne dépend que de la vitesse de l’écoule- 
ment, a laquelle il est directement proportionnel, et du diamétre 
des orifices, auquel il est inversement proportionnel. [I ne pa- 
rait altéré ni par la nature du liquide, ni par la température. 

«6°, L’amplitude de ces pulsations peut étre considérable- 
ment augmentée par des vibrations de méme période commu- 
niquées A la masse entiére du liquide et aux parois du réservoir 
qui le contient. Sous cette influence étrangére, les dimensions 
et état de la veine peuvent subir des changemens remarqua- 
bles: la longueur de la partie limpide et continue peut se 
réduire presqu’a rien, tandis que les ventres de la partie trouble 
acquiérent une régularité de forme et une transparence qu’ils 
ne possédent pas ordinairement. Lorsque le nombre des vibra- 
tions communiquées est différent de celui des pulsations qui 
ont lieu a Yorifice, leur influence peut méme aller jusqu’a 
changer le nombre de ces pulsations, mais seulement entre de 
certaines limites. 

«7°, La dépense ne parait pas altérée par l’amplitude des 
pulsations, ni méme par leur nombre. 

** 8°. La resistance de lair n’influe pas sensiblement sur la 
forme et les dimensions des veines, non plus que sur le nombre 
des pulsations. 

«9°, La constitution des veines lancées horizontalement ou 
méme obliquement de bas en haut ne différe pas essentiellement 
de celle des veines lancées verticalement de haut en bas ; seule- 
ment le nombre des pulsations a l’orifice parait devenir d’autant 
moindre que le jet approche plus d’étre lancé verticalement de 
bas en haut. 

** 10°. Quelle que soit la direction de la veine, son diamétre 
décroit toujours trés rapidement jusqu’a une petite distance de 
Yorifice ; mais quand la veine tombe verticalement, le décroisse- 
ment continue jusqu’a ce que la partie limpide se perde dans 
la partie trouble: il en est encore de méme quand la veine est 
lancée horizontalement, quoiqu’alors le décroissement suive une 
loi moins rapide. Lorsque le jet est lancé obliquement de 
bas en haut, et qu'il forme avec lhorizon un angle de 25° 4 45°, 
toutes les sections normales a la courbe qu'il décrit deviennent 


184 THIRD REPORT—1833. 


sensiblement égales entre elles, 4 partir de la partie contractée 
que touche a l’orifice. Enfin, pour des angles plus grands que 
45°, le diamétre de la veine va en augmentant depuis la partie 
contractée jusqu’a la naissance de la portion trouble; de sorte 
que c’est seulement alors qu'il existe une section qu’on peut 
4 juste titre appeler section contractée.” 


185 


Report on the Recent Progress and Present State of certain 
Branches of Analysis. By Grorcs Peacock, M.A., F.R.S., 
E.G.S., F.Z.S., £.R.A.S., F.C.P.S., Fellow and Tutor of 
Trinity College, Cambridge. 


Tux present Report was intended in the first instance to have 
comprehended some notice of the recent progress and present 
state of analytical science in general, including algebra, the 
application of algebra to geometry, the differential and integral 
calculus, and the theory of series: a very little progress, how- 
ever, in the inquiries which were required for the execution of 
this undertaking convinced me of the necessity of confining 
them within much narrower limits, unless I should have ven- 
tured to occupy a much larger space in the annual publication 
of the Proceedings and Reports of the British Association than 
could be properly or conveniently allotted to one department 
of science, when so many others were required to be noticed. 
It is for these reasons that I shall restrict my observations, 
in the following Report, to Algebra, Trigonometry, and the 
Arithmetic of Sines; at the same time I venture to indulge a 
hope of being allowed, upon some future occasion, to bring 
before the Members of the Association some notice of those 
higher branches of analysis which at present I feel myself 
compelled, though reluctantly, to omit. . 

Algebra.—The science of algebra may be considered under 
two points of view, the one having reference to its principles, 
and the other to its applications: the first regards its complete- 
ness as an independent science; the second its usefulness and 
power as an instrument of investigation and discovery, whether 
as respects the merely symbolical results which are deducible 
from the systematic developement of its principles, or the ap- 
plication of those results, by interpretation, to the physical 
sciences. 

Algebra, considered with reference to its principles, has re- 
ceived very little attention, and consequently very little im- 
provement, during the last century; whilst its applications, 
using that term in its largest sense, have been in a state of 
continued advancement. Many causes have contributed to this 
comparative neglect of the accurate and logical examination of 
the first principles of algebra: in the first place, the proper 


186 THIRD REPORT—1833. 


assumption and establishment of those principles involve meta- 
physical difficulties of a very serious kind, which present them- 
selves to a learner at a period of his studies when his mind has 
not been subjected to such a system of mathematical discipline 
as may enable it to cope with them: in the second place, we are 
commonly taught to approach those difficulties under the cover 
of a much more simple and much less general science, by steps 
which are studiously smoothed down, in order to render the 
transition from one science to the other as gentle and as little 
startling as possible; and lastly, from the peculiar relation 
which the first principles of algebra, in common with those of 
other scierices of strict demonstration, bear to the great mass 
of facts and reasonings of which those sciences are composed. | 

It is this last circumstance which constitutes a marked distine- 
tion between those sciences which, like algebra and geometry, 
are founded upon assumed principles and definitions, and the 
physical sciences: in one case we consider those principles and 
definitions as ultimate facts, from which our investigations pro- 
ceed in one direction only, giving rise to a series of conclusions 
which have reference to those facts alone, and whose correct- 
ness or truth involves no other condition than the existence of 
a necessary connexion between them, in whatever manner the 
evidence of that existence may be made manifest ; whilst in the 
physical sciences there are no such ultimate facts which can be 
considered as the natural or the assumable limits of our inves- 
tigations. It is true, indeed, that in the application of algebra 
or geometry to such sciences, we assume certain facts or prin- 
ciples as possessing a necessary existence or truth, investing 
them, as it were, with a strictly mathematical character, and 
making them the foundation of a system of propositions, whose 
connexion involves the same species of evidence with that of the 
succession of propositions in the abstract sciences; but in as- 
signing to such propositions their proper interpretation in the 
physical world, our conclusions are only true to an extent which 
is commensurate with the truth and universality of application 
of our fundamental assumptions, and of the various conditions 
by which the investigation of those propositions has been sup- 
posed to be limited; in other words, such conclusions can be 
considered as approximations only to physical truth; for such 
assumed first principles, however vast may be the superstrue> 
ture which is raised upon them, form only one or more links in 
the great chain of propositions, the termination and foundation 
of which must be for ever veiled in the mystery of the first 
cause. : ‘ 

It is not my intention to enter upon the examination of the 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 187 
general relations which exist between the speculative and physi- 
cal sciences, but merely to point out the distinction between the 
ultimate objects of our reasonings in the one class and in the 
other: in the first, we merely regard the results of the science 
itself, and the logical accuracy of the reasoning by which they 
are deduced from assumed first principles; and all our conclu- 
sions possess a necessary existence, without seeking either for 
their strict or for their approximate interpretation in the nature 
of things: in the second, we found our reasonings equally upon 
assumed first principles, and we equally seek for logical accu- 
racy in the deduction of our conclusions from them ; but both 
in the principles themselves and in the conclusions from them, 
we look to the external world as furnishing by interpretation 
corresponding principles and corresponding conclusions ; and 
the physical sciences become more or less adapted to the ap- 
plication of mathematics, in proportion to the extent to which 
our assumed first principles can be made to approach to the 
most simple and general facts or principles which are discover- 
able in those sciences by observation or experiment, when di- 
vested of all incidental and foreign causes of variation; and 
still more so, when the causes of such variation can be di- 
stinctly pointed out, and when their extent and influence are 
reducible to approximate at least, if not to accurate estimation. 

The first principles, therefore, which form the foundation 
of our mathematical reasonings in the physical sciences being 
neither arbitrary assumptions nor necessary truths, but really 
forming part of the series of propositions of which those sci- 
enees are composed, can never cease to be more or less the 
subject of examination and inquiry at any point of our re- 
searches: they form the basis of those interpretations which 
are perpetually required to connect our mathematical with the 
corresponding physical conclusions; and even supposing the 
immediate appeal to them to be superseded, as will frequently 
be the case, by other propositions which are deducible from 
them, they still continue to claim our attention as the proposi- 
tions which terminate those physical and logical inquiries at 
which our mathematical reasonings begin. But in the abstract 
sciences of geometry and algebra, those principles which are 
the foundation of those sciences are also the proper limits of 
our inquiries; for if they are in any way connected with the phy- 
sical sciences, the connexion is arbitrary, and in no respect af- 
fects the truth of our conclusions, which respects the evidence 
of their connexion with the first principles only, and does not 
require, though it may allow, the aid of physical interpretation. 
- It is true that there exists a connexion between physical and 


188 THIRD REPORT—1833. 


- 


speculative geometry, as well as between physical and specula- 
tive mechanics; and if in speculative geometry we regarded 
the actual construction and mensuration of the figures and solids 
in physical geometry alone, the transition from one science to 
the other being made by interpretation, then speculative geo- 
metry and speculative mechanics must be regarded as sciences 
which were similar in their character, though different in 
their objects: but we cultivate speculative geometry without 
any such exclusive reference to physical geometry, as an in- 
strument of investigation more or less applicable, by means 
of interpretation, to all sciences which are reducible to mea- 
sure, and whose abstract conclusions, in whatever manner 
suggested or derived, possess a great practical value altogether 
apart from their applications to practical geometry; whilst the 
conclusions in speculative mechanics are valuable from their 
applications to physical mechanics only, and are not other- 
wise separable from the conclusions of those abstract sciences 
which are employed as instruments in their investigation. 

This separation of speculative and physical geometry was 
perfectly understood by the ancients, though their views of its 
application to the physical sciences were extremely limited ; 
and it is to the complete abstraction of the principles of specu- 
lative geometry that we must in a great measure attribute the 
vast discoveries which were made by its aid in the hands of 
Newton and his predecessors, when a more enlarged and phi- 
losophical knowledge of the laws of nature supplied those phy- 
sical axioms or truths which were required as the medium of 
its applications ; and though it was destined to be superseded, 
at least in a great degree, by another abstract science of much 
greater extent and applicability, yet it was enabled to maintain 
its ground for a considerable time against its more powerful 
rival, in consequence of the superior precision of its prin- 
ciples and the superior evidence of its conclusions, when con- 
sidered with reference to the form under which the principles 
and conclusions of algebra were known or exhibited at that 
period. 

Algebra was denominated in the time of Newton specious or 
universal arithmetic, and the view of its principles which gave 
rise to this synonym (if such a term may be used) has more or 
less prevailed in almost every treatise upon this subject which 
has appeared since his time. In a similar sense, algebra has 
been said to be a science which arises from that generalization 
of the processes of arithmetic which results from the use of 
symbolical language: but though in the exposition of the prin- 
ciples of algebra, arithmetic has always been taken for its foun- 


REPORT ON CERTAIN BRANCHES OF ANALYSIS, 189 


dation, and the names of the fundamental operations in one 
science have been transferred to the other without any imme- 
diate change of their meaning, yet it has generally been found 
necessary subsequently to enlarge this very narrow basis of so 
very general a science, though the reason of the necessity of 
doing so, and the precise point at which, or the extent to which, 
it was done, has usually been passed over without notice. The 
science which was thus formed was perfectly abstract, in what- 
ever manner we arrived at its fundamental conclusions; and 
those conclusions were the same whatever view was taken of 
their origin, or in whatever manner they were deduced; but a 
serious error was committed in considering it as a science which 
admitted of strict and rigorous demonstration, when it certainly 
possessed no adequate principles of its own, whether assumed 
or demonstrated, which could properly justify the character 
which was thus given to it. | 

There are, in fact, two distinct sciences, arithmetical and 
symbolical algebra, which are closely connected with each 
other, though the existence of one does not necessarily deter- 
mine the existence of the other. The first of these sciences 
would be, properly speaking, wniversal arithmetic: its general 
symbols would represent numbers only; its fundamental ope- 
rations, and the signs used to denote them, would have the same 
meaning as in common arithmetic; it would reject the inde- 
pendent use of the signs + and —, though it would recognise the 
common rules for their incorporation, when they were preceded 
by other quantities or symbols: the operation of subtraction 
would -be impossible when the subtrahend was greater than 
the quantity from which it was required to be taken, and there- 
fore the proper zmpossible quantities of such a scienee-would 
be the negative quantities of symbolical algebra; it would re- 
ject also the consideration of the multiple values of simple 
roots, as well as of the negative and impossible roots of equa- 
tions of the second and higher degree: it is this species of al- 
gebra which alone can be legitimately founded upon arithmetic 
as its basis. 

Mr. Frend *, Baron Maseres, and others, about the latter 
end of the last century, attempted to introduce arithmetical 


* The Principles of Algebra, by William Frend, 1796; and The true The- 
ory of Equations, established on Mathematical Demonstration, 1799. The fol- 
Towing extracts from his prefaces to these works will explain the nature of his 
‘views : 

“ The ideas of number are the clearest and most distinct of the human mind: 
the acts of the mind upon them are equally simple and clear. There cannot 
be confusion in them, unless numbers too great for the comprehension of the 


190 THIRD REPORT— 1833. 


to the exclusion of symbolical algebra, .as the only form of it 
which was capable of strict demonstration, and which alone, 
therefore, was entitled to be considered as a science of strict and 
logical reasoning. ‘The arguments which they made use of 
“were unanswerable, when advanced against the form under 
which the principles of algebra were exhibited in the elemen- 
tary and all other works of that period, and which they have 
continued to retain ever since, with very trifling and unimpor- 
tant alterations; and the system of algebra which was formed 
by the first of these authors was perfectly logical and complete, 
the corinexion of all its parts being capable of strict demon- 
stration; but there were a great multitude of algebraical re- 
sults and propositions, of unquestionable value and of unques- 
tionable consistency with each other, which were irreconcila- 
ble with such a system, or, at all events, not deducible from it; 
and amongst them, the theory of the composition of equations, 
which Harriot had left in so complete a form, and which made 
it necessary to consider negative and even impossible quan- 


learner are employed, or some arts are used which are not justifiable. The 
first error in teaching the first principles of algebra is obvious on perusing a few 
pages only of the first part of Maclaurin’s Algebra. Numbers are there divided 
Into two sorts, positive and negative: and an attempt is made to explain the 
nature of negative numbers, by allusions to book debts and other arts. Now 
when a person cannot explain the principles of a science, without reference to 
a metaphor, the probability is, that he has never thought accurately upon the 
subject. A number may be greater or less than another number: it may be 
added to, taken from, multiplied into, or divided by, another number ; but in 
other respects it is very intractable; though the whole world should be destroyed, 
one will be one, and three will be three, and no art whatever can change their 
nature. You may put a mark before one, which it will obey; it submits to be 
taken away from another number greater than itself, but to attempt to take it 
away from a number less than itself is ridiculous. Yet this is attempted by 
algebraists, who talk of a number less than nothing, of multiplying a negative 
number into a negative number, and thus producing a positive number, of a 
number being imaginary. Hence they talk of two roots to every equation of 
the second order, and the learner is to try which will succeed in a given equa- 
tion: they talk of solving an equation which requires two impossible roots to 
make it soluble: they can find out some impossible numbers, which being 
multiplied together produce unity. This is all jargon, at which common sense 
recoils; but from its having been once adopted, like many other figments, it 
finds the most strenuous supporters among those who love to take things upon 
trust and hate the colour of a serious thought.” 

“ From the age of Vieta, the father, to this of Maseres, the restorer of alge- 
bra, many men of the greatest abilities have employed themselves in the pursuit 
of an idle hypothesis, and have laid down rules not founded in truth, nor of any 
sort of use in a science admitting in every step of the plainest principles of 
reasoning. If the name of Sir Isaac Newton appears in this list, the number 
of advocates for errour must be considerable. It is, however, to be recollected, 
that for a much longer period, men scarcely inferiour to Newton in genius, and 
his equals, probably, in industry, maintained a variety of positions in philoso- 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 191 


tities as having a real existence in algebra, however vain might 
be the attempt to interpret their meaning. 

Both Mr. Frend and Baron Maseres were sensible of the con- 
sequences of admitting the truth of this theory of the compo- 
sition of equations as far as their system was concerned, and it 
must be allowed that they have struggled against it with con- 
siderable ingenuity: they admitted the possibility of multiple 
real, that is, positive roots, and which are all equally congruous 
to the problem whose solution was required through the medium 
of the equation, indicating an imdetermination in the problem 
proposed: but it would be easy to propose problems leading to 
equations whose roots were real and positive, and yet not con- 
gruous to the problem proposed, whose existence must be ad- 
mitted upon their own principles; and if so, why not admit the 
‘existence of other roots, whether negative or impossible, to 
which the algebraical solution of the problem might lead, though 
they might admit of no very direct interpretation, in conformity 
with the expressed conditions of the problem*? 


phy, which were overthrown by a more accurate investigation of nature; and 
if the name Ptolemy can no longer support his epicycles, nor that of Des 
Cartes his vortices, Newton’s dereliction of the principles of reasoning cannot 
establish the fallacious notion, that every equation has as many roots.as it has 
dimensions.” 
This notion of Newton and others is founded on precipitation. Instead of 
a patient examination of the subject, an hypothesis which accounts for many 
appearances is formed; where it fails, unintelligible terms are used; in those 
terms indolence acquiesces: much time is wasted on a jargon which has the 
_ ‘appearance of science, and real knowledge is retarded. Thus volumes upon 
volumes have been written on the stupid dreams of Athanasius, and on the im- 
possible roots of an equation of n dimensions.” 

This work of Mr. Frend, though containing many assertions which show 
great distrust of the results of algebraical science which were in existence at 
the time it was written, presents a very clear and logical view of the principles 
of arithmetical algebra. 

The voluminous labours of Baron Maseres are contained in his Scriptores 
Logarithmici, and in a thick volume of Tracts on the Resolution of Cubic and 
Biquadratic Equations. He seems generally to have forgotten that any change 

_ had taken place in the science of algebra between the age of Ferrari, Cardar, 
Des Cartes, and Harriot, and the end of the 18th century; and by considering 
all algebraical formule as essentially arithmetical, he is speedily overwhelmed 
by the same multiplicity of eases (which are all included in the same really al- 
gebraical formula) which embarrassed and confounded the first authors of the 
science. 

' * Thus, in the solution of the following problem: ‘Sold a horse for 24/., 
and by so doing lost as much per cent. as the horse cost me: required the 
prime cost of the horse?”’ we arrive at the equation 

; 100 # — #? = 2400; 

if we subtract both sides of this equation from 2500, we get 

2500 — 100 2 + a? = 100, 

dedi or a? — 1007 + 2500 = 100, 
inasmuch as the quantities upon each side of the sign = are in both cases 


192 THIRD REPORT—1833. 


If the authors of this attempt at algebraical reform had been 
better acquainted with the more modern results of the science, 
they would have felt the total inadequacy of the very limited 
science of arithmetical algebra to replace it; and they would 
probably have directed their attention to discover whether any 
principles were necessary to be assumed, which were not neces- 
sarily deducible as propositions from arithmetic or arithmetical 
algebra, though they might be suggested by them. As it was, 
however, these speculations did not receive the consideration 
which they really merited; and it is very possible that the 
attempt which was made by one of their authors to connect the 
errors in reasoning, which he attacked, as forming part only of 
a much more extensive class to which the human mind is liable 
from the influence of prejudice or fashion, had a tendency to 
divert men of an enlarged acquaintance with the results of 
algebra from such a cautious and sustained examination of them 
as was required for their refutation, or rather for such a correc- 
tion of them as was really necessary to establish the science of 
algebra upon its proper basis. 

I know that it is the opinion of many persons, even amongst 
the masters * of algebraical science, that arithmetic does supply 


identical with each other: if we extract the square root on both sides, re- 
jecting the negative value of the square root, we’get in the first case 

50 — x = 10, 
and in the second, 

az — 50= 10. 
The first of these simple equations gives us x =40, and the second « = 60, both 
of which satisfy the conditions of the problem proposed: the two roots which 
are thus obtained, strictly by means of arithmetical algebra, show that the pro- 
blem proposed is to a certain. extent indeterminate, Mr. Frend and Baron 
Maseres contended that multiple real roots, which are always the indication 
of a similar indetermination in the problems which lead to such equations, 
might be obtained by arithmetical algebra alone, and that all other roots were 
useless fictions, which could lead to no practical conclusions. But it is very easy 
to show, that incongruous and real, as well as negative and impossible roots, 
may equally indicate the impossibility of the problem proposed: thus, if it 
was proposed “to find a number the double of whose square exceeds three 
times the number itself by 5,”’ we shall find } and — 1 for the roots of the 
resulting equation, both of which equally indicate the impossibility of the pro- 
blem proposed, if by number be meant a whole positive number. 

* Cauchy, who has enriched analysis with many important discoveries, 
and who is justly celebrated for his almost unequalled command over its lan- 
guage, has made it the principal object of his admirable work, entitled Cours 
d@’ Analyse de l’ Ecole Royale Polytechnique, to meet the difficulties which pre- 
sent themselves in the transition from arithmetical to symbolical algebra : and 
though he admits to the fullest extent the essential distinction between them, 
in the ultimate form which the latter science assumes, yet he considers the 
principles of one as deducible from those of the other, and presents the rules 
for the concurrence and incorporation of signs ; for the inverse relation of the 
operations called addition and subtraction, multiplication and division; for 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 193 


a sufficient basis for symbolical algebra. considered under its 
most. general form; that symbols, considered as representing 
numbers, may represent every kind of concrete magnitude ; 


the indifference of the order of succession of different algebraical operations, 
as so many theorems founded upon the ordinary principles and reasonings of 
arithmetic. In order to show, however, the extraordinary vagueness of the 
reasoning which is employed to establish these theorems, we will notice some 
of them in detail: On représente, says he, les grandeurs qui doivent servir d’ac- 
croissements, par des nombres précédés du signe +, et les grandeurs qui doivent 
servir de diminutions par des nombres précédés du signe —. Cela posé, les signes 
+ et — placés devant les nombres peuvent étre compares, suivant la remarque 
quien a été faite}, & des adjectifs placés aupres de leurs substantifs. It is unques- 
tionable, however, that in the most common cases of the interpretation of 
specific magnitudes affected with the signs + and —, there is no direct refer- 
ence either to increase or diminution, to addition or to subtraction. Hesub- 
sequently gives those signs a conventional interpretation, as denoting quan- 
tities which are opposed to each other; and assuming the existence of quan- 
tities affected by independent signs, and denoting + A by a, and — A by 8, 
he says that 


+a=+A +6=-—A 
—a=—A —b=+A4; 
and therefore, 
+(fA)=4+A +(-AN =A 
—(+A)=—A —(-A=++A4; 


which he considers as a sufficient proof of the rule of the concurrence of 
signs in whatever operations they may occur ; though it requires a very slight 
examination of this process of reasoning to show that it involves several ar- 
bitrary assumptions and interpretations which may or may not be consistent 
with each other. In the proofs which he has given of the other fundamental 
theorems which we have mentioned above, we shall find many other instances 
of similar confusion both in language and in reasoning: thus, ‘‘ subtraction 
is the inverse of addition in arithmetic; then therefore, also, subtraction is 
the inverse of addition in algebra, even when applied to quantities affected 
with the signs + and —, and whatever those quantities may be.” But is 
this a conclusion or an assumption? or in what manner can we explain in 
words the process which the mind follows in effecting such a deduction? 
“If a and b be whole numbers, it may be proved that a 0 is identical with 
ba: therefore, a b is identical with 5 a, whatever a and b may denote, and 
whatever may be the interpretation of the operation which connects them.” 
But any attempt to establish this conclusion, without a previous definition 
of the meaning of the operation of multiplication when applied to such quanti- 
ties, will show it to be altogether impracticable. The system which he has fol- 
lowed, not merely in the establishment of the fundamental operations, but 
likewise in the interpretation of what he terms symbolical expressions and 
symbolical equations, requires the introduction of new conventions, which are not 
the less arbitrary because they are rendered necessary for the purpose of 
making the results of the science consistent with each other: some of those 
conventions I believe to be necessary, and others not; but in almost every in- 
stance I should consider them introduced at the wrong place, and more or 
pee eenieeatly with the professed grounds upon which the science is 
ounded. 


. 1} By Buée in the Philosophical Transactions, 1806. 
1833. oO 


194 THIRD REPORT—1833. 


that the operations of addition, subtraction, multiplication and 
division are used in one science and in the other in no sense 
which the mind may not comprehend by a practicable, though 
it may not be by a very simple, process of generalization ; that 
we may be enabled by similar means to conceive both the use 
and the meaning of the signs + and —, when used independ- 
ently ; and that though we may be startled and somewhat em- 
barrassed by the occurrence of impossible quantities, yet that 
investigations in which they present themselves may generally 
be conducted by other means, and those difficulties may be 
evaded which it may not be very easy or very prudent to en- 
counter directly and openly. 

In reply, however, to such opinions, it ought to be remarked 
that arithmetic and algebra, under no view of their relation to 
each other, can be considered as one science, whatever may be 
the nature of their connexion with each other; that there is 
nothing in the nature of the symbols of algebra which can es- 
sentially confine or limit their signification or value; that it is 
an abuse of the term generalization* to apply it to designate 
the process of mind by which we pass from the meaning of a—b, 
when a is greater than 8, to its meaning when a is less than 0, 
or from that of the product a b, when a and 6 are abstract num- 
bers, to its meaning when a and 6 are concrete numbers of the 
same or of a different kind; and similarly in every case where 
a result is either to be obtained or explained, where no pre- 
vious definition or explanation can be given of the operation 
upon which it depends: and even if we should grant the legiti- 
macy of such generalizations, we do necessarily arrive at a new 
science much more general than arithmetic, whose principles, 
however derived, may be considered as the immediate, though 
not the ultimate foundation of that system of combinations of 
symbols which constitutes the science of algebra. It is more 
natural and philosophical, therefore, to assume such principles 
as independent and ultimate, as far as the science itself is con- 
cerned, in whatever manner they may have been suggested, so 
that it may thus become essentially a science of symbols and 
their combinations, constructed upon its own rules, which may 


* The operations in arithmetical algebra can be previously defined, whilst 
those in symbolical algebra, though bearing the same name, cannot: their 
meaning, however, when the nature of the symbols is known, can be generally, 
but by no means necessarily, interpreted. The process, therefore, by which we 
pass from one science to the other is not an ascent from particulars to generals, 
which is properly called generalization, but one which is essentially arbitrary, 
though restricted with a specific view to its operations and their results admit- 
ting of such interpretations as may make its applications most generally useful. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 195 


be applied to’ arithmetic and to all other sciences by interpreta- 
tion: by this means, interpretation will follow, and not precede, 
the operations of algebra and their results; an order of suc- 
cession which a very slight examination of their necessary 
changes of meaning, corresponding to, the changes in the spe- 
cific values and applications of the symbols involved, will very 
speedily make manifest. 

But though the science of arithmetic, or of arithmetical al- 
gebra, does not furnish an adequate foundation for the science 
of symbolical algebra, it necessarily suggests its principles, or 
rather its laws of combination; for in as much as symbolical al- 
gebra, though arbitrary in the authority of its principles, is not 
arbitrary in their application, being required to include arith- 
metical algebra as well as other sciences, it is evident that their 
rules must be identical with each other, as far as those sciences 
proceed together in common: the real distinction between them 
will arise from the supposition or assumption that the symbols 
in symbolical algebra are perfectly general and unlimited both 
in value and representation, and that the operations to which 
they are subject are equally general likewise. Let us now 
consider some of the consequences of such an assumption. 

A system of symbolical algebra will require the assumption 
of the independent use of the signs + and —. 

For the general rule in arithmetical algebra* informs us, 
that the result of the subtraction of 6 + ¢ from a is denoted 
by a — b —c, or that a — (b +c) = a — b —¢, its application 
being limited by the necessity of supposing that b + ¢ is less 
than a. The general hypothesis made in symbolical algebra, 
namely, that symbols are unlimited in value, and that operations 
are equally applicable in all cases, would necessarily lead us 
to the conclusion that a — (6 +c) =a—6-—ce for all values 
of the symbols, and therefore, also, when 6 = a, in which case 
we have 

a—(@+c)=a-—-a-c=—Cc. 
In a similar manner, also, we find 
a—(a—ec)=a-—a+c=+ec=C¢}. 

We are thus necessarily led to the assumption of. the exist- 
ence of such quantities as — ¢ and + ce, or of symbols preceded 


.* Whatever general symbolical conclusions are true in arithmetical algebra 
must be true likewise in symbolical, algebra, otherwise one science could not 
include the other, This is a most important principle, and will be the subject 
of particular notice. hereafter. , 

¢ For it appears from arithmetical algebra that a —a= 0, and thata —a@ 


+b=5 
02 


196 THIRD REPORT—1833. 


by the independent signs * + and —, which no longer denote 
operations, though they may denote affections of quantity. It 
appears likewise that + ¢ is identical with c, but that — cis a 
quantity of a different nature from ¢: the interpretation of its 
meaning must depend upon the joint consideration of the spe- 
cific nature of the magnitude denoted by a, and of the symbolical 
conditions which the sign —, thus used, is required to satisfy}. 

In a similar manner, the result of the operation, or rather 
the operation itself, of extracting the square root of such a 
quantity as a — b is impossible, unless a is greater than b. To 
remove the limitation in such cases, (an essential condition in 
symbolical algebra,) we assume the existence of a sign such 
as “ —1; so that if we should suppose b = a+ c, we should 
get /(a —b) = V {a—(a+c)} = V(a—a—c) = V(—o) 
= 7% —I1c{. Ina similar manner, in order to make the ope- 
ration universally applicable, when the nx root of a — 6 is 


required, we assume the existence of a sign / —1, for which, 
as will afterwards appear, equivalent symbolical forms ‘can al- 


ways be found, involving “ —1 and numerical quantities. 

By assuming, therefore, the independent existence of the 
signs +, —, V1, and 7 —1, (1), and (—1)*§, we shall obtain 
a symbolical result in all those operations, which we call addi- 
tion, subtraction, multiplication, division, extraction of roots, 
and raising of powers, though their meaning may or may not be 
identical with that which they possess in arithmetic. Let us 
now inquire a little further into the assumptions which deter- 
mine the symbolical character and relation of these funda- 
mental operations. 

The operations called addition and subtraction are denoted 
by the signs + and —. 

They are the inverse of each other. 


* That is, not preceded by other symbols as in the expressions a — c and 
a+e. 

+ Amongst these conditions, the principal is, that if —c be subjected to 
the operation denoted by the sign —, it will become identical with + ec: thus, 
a—(—c)=a-+c. It does not follow, however, that the sign — thus used, 
must necessarily admit of interpretation. 

+ The symbolical form, however, of this and of similar signs is not arbi- 
trary, but dependent upon the general laws of symbolical combination. 

I do not assert the necessity of considering such signs as 4/—1, (1)*, 
(—1)", as forming essentially a part of the earliest and most fundamental as- 
sumptions of algebra: the necessity for their introduction will arise when 
those operations with which they are connected are first required to be con- 
sidered, and will in all cases be governed by the general principle above men- 
tioned. F 


REPORT ON. CERTAIN BRANCHES OF ANALYSIS. 197 


‘In the concurrence of the signs + and —, in whatever man- 
ner used, if two like signs come together, whether + and +, or 
— and —, they are replaced by the single sign + ; and when 
two unlike signs come together, whether + and —, or — and +, 
they are replaced by the single sign —. 

‘When different operations are performed or indicated, it is 
indifferent in what order they succeed each other. 

The operations called multiplication and division are de- 
noted by the signs x and ~, or more frequently by a conven- 
tional position of the quantities or symbols with respect to 
each other: thus, the product of a and 6 is denoted by a x 8, 
a.b, orab; the quotient of a divided by 6 is denoted by 


~ b, or by ©. 
a ‘ate Ys 


The operations of multiplication and division are the inverse 
of each other. 

In the concurrence of the signs + and — in multiplication or 
division, if two like signs come together, whether + and +, or 
— and —, they are replaced by the single sign +; and if two un- 
like signs come together, whether + and —, or — and +, they 
are replaced by the single sign —. 

When different operations succeed each other, it is not indif- 
rent in what order they are taken. 

We arrive at all these rules, when the operations are defined 
and when the symbols are numbers, by deductions, not from 
each other, but from the definitions themselves: in other words, 
these conclusions are not dependent upon each other, but upon 

‘the definitions only. In the absence, therefore, of such defini- 
tions of the meaning of the operations which these signs or 
forms of notation indicate, they become assumptions, which are 
‘independent of each other, and which serve to define, or rather 
to interpret* the operations, when the specific nature of the 
symbols is known; and which also identify the results of those 
operations with the corresponding results in arithmetical alge- 
bra, when the symbols are numbers and when the operations are 
arithmetical operations. 

The rules of symbolical combination which are thus assumed 


* To define, is to assign beforehand the meaning or conditions ofa term or 
operation ; to interpret, is to determine the meaning of a term or operation 
conformably to definitions or to conditions previously given or assigned. It is 
for this reason, that we define operations in arithmetic and arithmetical alge- 
bra conformably to their popular meaning, and we interpret them in symboli- 
a ai conformably to the symbolical conditions to which they are sub- 
ject. 


198 THIRD REPORT—1883. 


have been suggested only by the corresponding rules in arith- 
metical algebra. They cannot be said to be founded upon them, 
for they are not deducible from them; for though the opera- 
tions of addition and subtraction, in their arithmetical sense, 
are applicable to all quantities of the same kind, yet they ne- 
cessarily require a different meaning when applied to quanti- 
ties which are different in their nature, whether that difference 
consists in the kind of quantity expressed by the unaffected 
symbols, or in the different signs of affection of symbols de- 
noting the same quantity; neither does it necessarily follow 
that in such cases there exists any interpretation which can be 
given of the operations, which is competent to satisfy the re- 
quired symbolical conditions. It is for such reasons that the 
investigation of such interpretations, when they are discover- 
able, becomes one of the most important and most essential of 
the deductive processes which are required in algebra and its 
applications. 

Supposing that all the operations which are required to be 
performed in algebra are capable of being symbolically de- 
noted, the results of those operations will constitute what are 
called equivalent forms, the discovery and determination of 
which form the principal business of algebra. The greatest 
part of such equivalent forms result from the direct applica- 
tion of the rules for the fundamental operations of algebra, 
when these rules regard symbolical combinations only: but 
in other cases, the operations which produce them being nei- 
ther previously defined nor reduced to symbolical rules, unless 
for some specific values of the symbols, we are compelled to 
resort, as we have already done in the discovery and assump- 
tion of the fundamental rules of algebra themselves, to the re- 
sults obtained for such specific values, for the purpose of dis- 
covering the rules which determine the symbolical nature of 
the operation for ald values of the symbols. As this principle, 
which may be termed the principle of the permanence of equi- 
valent forms, constitutes the real foundation of all the rules of 
symbolical algebra, when viewed in connexion with arithmeti- 
cal algebra considered as a science of suggestion, it may be 
proper to express it in its most general form, so that its autho- 
rity may be distinctly appealed to, and some of the most im- 
portant of its consequences may be pointed out. 

Direct proposition: 

Whatever form is algebraically equivalent to another when 
expressed in general symbols, must continue to be equivalent, 
whatever those symbols denote. 

Converse proposition : 


REPORT ON. CERTAIN BRANCHES OF ANALYSIS. 199 


Whatever equivalent form is discoverable in arithmetical 
algebra considered as the science of suggestion, when the sym- 
bols are general in their form, though specific in their value, 
will continue to be an equivalent form when the symbols are 
general in their nature as well as in their form *. 

The direct proposition must be true, since the laws of com- 
bination of symbols by which such equivalent forms are de- 
duced, have no reference to the specific values of the symbols. 

‘The converse proposition must be true, for the following 
reasons : 

If there be an equivalent form when the symbols are general 
in their nature as well as in their form, it must coincide with 
the form discovered and proved in arithmetical algebra, where 
the symbols are general in their form but specific in their na- 
ture ; for in passing from the first to the second, no change in 
its form can take place by the first proposition. 

Secondly, we may assume the existence of such an equivalent 
form in symbols which are general both in their form and in 
their nature, since it will satisfy the only condition to which 
all such forms are subject, which is, that of perfect coincidence 
with the results of arithmetical algebra, as far as such results. 
are common to both sciences. 

Equivalent forms may be said to have a necessary existence 
when the operation which produces them admits of being de- 
fined, or the rules for performing it of being expressly laid 
down: in all other cases their existence may be said to be 
conventional or assumed. Such conventional results, however, 
are as much real results as those which have a necessary ex- 
istence, in as much as they satisfy the only condition of their 
existence, which the principle of the permanence of equivalent 
forms imposes upon them: thus, the series for (1 + x)" has a 
necessary existence whenever the nature of the operation upon 
1 + 2 which it indicates can be defined; that is, when» is a 
whole or a fractional, a positive or negative, number +; but for 
all other values of x, where no previous definition or interpre- 
tation of the nature of the operation which connects it with its 
equivalent series can be given, then its existence is conventional 
only, though, sythBolically speaking, it is equally entitled to be 
considered as an equivalent form in one case as in the other. 

It is evident that a system of symbolical algebra might be 


_ * Peacock’s Algebra, Art. 132. 1 

_ + The meaning of (1 + x)" cannot properly be said to be defined when n 
is a fractional number, whether positive or negative, or a negative whole num- 
ber, but to be ascertained by interpretation conformably to the principle of 
‘the permanence of qquivalent forms. ' 


200 THIRD REPORT—1833. 


formed, in which the symbols and the conventional operations 
to which they were required to be subjected would be perfectly 
general both in value and application. If, however, in the con- 
struction of such a system, we looked to the assumption of such 
rules of operation or of combination only, as would be sufficient, 
and not more than suflicient, for deducing equivalent forms, 
without any reference to any subordinate science, we should be 
altogether without any means of interpreting either our opera- 
tions or their results, and the science thus formed would be 
one of symbols only, admitting of no applications whatever. It 
is for this reason that we adopt a subordinate science as a sci- 
ence of suggestion, and we frame our assumptions so that our 
results shall be the same as those of that science, when the 
symbols and the operations upon them become identical like- 
wise : and in as much as arithmetic is the science of calculation, 
comprehending all sciences which are reducible to measure and 
to number ; and in as much as arithmetical algebra is the imme- 
diate form which arithmetic takes when its digits are replaced 
by symbols and when the fundamental operations of arithmetic 
are applied to them, those symbols being general in form, 
though specific in value, it is most convenient to assume it as - 
the subordinate science, which our system of symbolical algebra 
must be required to comprehend in all its parts. 'The principle 
of the permanence of equivalent forms is the most general ex- 
pression of this law, in as much as its truth is absolutely neces- 
sary to the identity of the results of the two sciences, when the 
symbols in both denote the same things and are subject to the 
same conditions. It was with reference to this principle that 
the fundamental assumptions respecting the operations of ad- 
dition, subtraction, multiplication and division were said to be 
suggested by the ascertained rules of the operations bearing 
the same names in arithmetical algebra. The independent use 
of the signs + and —, and of other signs of affection, was an as- 
sumption requisite to satisfy the still more general principle of 
symbolical algebra, that its symbols should be unlimited in value 
and representation, and the operations to which they are sub- 
ject unlimited in their application. 

In arithmetical algebra, the definitions of the operations de- 
termine the rules; in symbolical algebra, the rules determine 
the meaning of the operations, or more properly speaking, they 
furnish the means of interpreting them: but the rules of the 
former science are invariably the same as those of the latter, 
in as much as the rules of the latter are assumed with this view, 
and merely differ from the former in the universality of their 
applications: and in order to secure this universality of their 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 201 


applications, such additional signs * are assumed, and of such a 
symbolical form, as those applications may render necessary. 
We call those rules, or their equivalent symbolical consequences, 
assumptions,in as much as they are not deducible as conclusions 
from any previous knowledge of those operations which have 
corresponding names: and we might call them arbitrary as- 
sumptions, in as much as they are arbitrarily imposed upon a 
science of symbols and their combinations, which might ‘be 
adapted to any other assumed system of consistent rules. In 
the assumption, therefore, of a system of rules such as will make 
its symbolical conclusions necessarily coincident with those of 
arithmetical algebra, as far as they can exist in common, we in 
no respect derogate from the authority or completeness of sym- 
bolical algebra, considered with reference to its own conclu- 
sions_and to their connexion with each other, at the same time 
that we give to them a meaning and an application which they 
would not otherwise possess. 

It follows from this view of the relation of arithmetical and 
symbolical algebra, that all the results of arithmetical algebra 
which are general in form are true likewise in symbolical. 
algebra, whatever the symbols may denote. This conclusion 
may be said to be true in virtue of the principle of the perma- 
nence of equivalent forms, or rather it may be said to be the 
proper expression of that principle. Its consequences are most 
important, as far as the investigation of the fundamental pro- 
positions of the science are concerned, in as much as it enables 
us to investigate them in the most simple cases, when the 
operations which produce them are perfectly defined and un- 
derstood, and when the general symbols denote positive whole 
numbets. If the conclusions thus obtained do not involve in 
their expression any condition which is essentially connected 
with the specific values of the symbols, they may be at once 
transferred to symbolical algebra, and considered as true for 
all values of the symbols whatsoever +. 

Thus, coefficients in arithmetical algebra, such as m in m a, 
which are general in form, lead to the interpretation of such 


* There is no necessary limit to the multiplication of such signs : the signs 
++, —, (1)” and (—1)? and their equivalents (for the symbolical form of such 
signs is not arbitrary), comprehend all those signs of affection which are re- 
quired by those operations with which we are at present acquainted. 

- F Some formule are essentially arithmetical: of this kindis1.2.3...7, 
in which r must be a whole number. The formula spite 
is symbolical with respect to m, but arithmetical with respect to 7. Such cases, 


and their extension to general values of r, will be more particularly considered 
hereafter. 


202 THIRD REPORT—1833. 


>. 


expressions as ma in symbolical algebra, when m is a number 
whole or fractional, and a any symbol whatsoever. When m, n 
and a are whole numbers, it very readily appears that ma + na 
= (m + n) a, and that ma — na = (m — n)a: the same con- 
clusions are true likewise for all values of m, x and a. In 
arithmetical algebra we assume a’, a®, a‘, &c., to represent aa, 
aaa,aaaa, &c., and we readily arrive at the conclusion that 
a”™ x a®* = a”™*", when m and m are whole numbers: the same 
conclusion must be true also when m and z are any quantities 
whatsoever. In a similar manner we pass from the result 
(a”)" = a™”, when z is a whole number, to the same conclusion 
for all values of the symbols *. 

The preceding conclusions are extremely simple and element- 
ary, but they are not obtainable for all values of the symbols 
by the aid of any other principle than that of the permanence 
of equivalent forms: they are assumptions which are made in 
conformity with that principle, or rather for the purpose of 
rendering that principle universal; and it will of course follow 
that all interpretations of those expressions where m and n are 
not whole numbers must be subordinate to such assumptions. 


1 1 
Thus, + oe 3 = (G ta a) a@ =a, and therefore 5 must 


mean one half of a, whatever a may be; a xad=at? 


1 
=a =, and therefore a® must mean the square root of a, 
whatever a@ may be, whenever such an operation admits of 


é e anny a Z 
interpretation. In a similar manner 3 must mean one third 


3 


part, and a* the cube root of a, whatever a may be, and simi- 
larly in other cases: it follows, therefore, that the interpreta- 


tion of the meaning of a, a’, &c., is determined by the general 


* The genera] theorems ma + na= (m+n) aandma—na=(m—n)a, 
1 m 
™m _ = 
a™ x a® = a™t" and “= a™—™, (a™)" = a™" and (a) * =a”? which 
a 


are deduced by the principle of the permanence of equivalent forms, and which 
are supplementary to the fundamental rules of algebra, are of the most essen- 
tial importance in the simplification and abridgement of the results of those 
operations, though not necessary for the formation of the equivalent results 
themselves. It also appears from the four last of the above-mentioned theorems 
that the operations of multiplication and division, involution and evolution, are 
performed by the addition and subtraction, multiplication and division, of the 
indices, when adapted to the same symbol or base. If such indices or logarithms 
be calculated and registered with reference to a scale of their corresponding 
numbers, they will enable us to reduce the order of arithmetical operations by 
two unities, if their orders be regulated by the following scale ; addition (1), sub- 
traction (2), multiplication (3), division (4), involution (5), and evolution (6). 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 203 


principle of indices, and also that we ought not to say that we 


assume a? to denote Wa, and a’ to denote 4/a, as is commonly 
done*, in as much as such phrases would seem to indicate that 
such assumptions are independent, and not subject to the same 
common principle in all cases. 

In all cases of indices which involve or designate the inverse 
processes of evolution, we must have regard likewise to the 
other great principle of symbolical algebra, which authorizes 
the existence of signs of affection. The square root of a may 


be either affected with the sign + or with the sign —; for + a* 


x + a’, and — a* x — a’, will equally have for their result 
+ aora, by the general rule for the concurrence of similar 
signs and the general principle of indices : in a similar manner 
r 


a> may be affected with the multiple sign of affection (1), if 


x 

there are any symbolical values of (1)? different from + 1 (equi- 
valent to the sign +), which will satisfy the requisite symbo- 
lical conditions. It is the possible existence of such signs of 
affection, which is consequent upon the universality of alge- 
braical operations, which makes it expedient to distinguish be- 
tween the results which are not affected by such signs, and 
the same results when affected by them. The first class of 
results or values are such as are alone considered in arithmeti- 
cal algebra, and we shall therefore term them arithmetical va- 
lues, though the quantities themselves may not be arithmetical : 
the second class may be termed algebraical values, in as much 
as they are altogether, as far as they are different from the 
arithmetical values, the results of the generality of the opera- 
tions of symbolical algebra. 

This distinction may generally be most conveniently ex- 
pressed by considering such a sign as a factor, or a symbolical 
quantity multiplied according to the rule for that operation 
into the arithmetical value: in this sense + 1 and — 1 may be 
considered as factors which are equivalent to the signs + and 
—, that is, equivalent to affecting the quantities into which 
they are multiplied with the signs + and —, according to the 


* Wood’s Algebra, Derinirions. 

+ That is, if there is any symbolical expression different from 1, such as 
= tw and wae the cubes of which are identical with 1. 
Tn a similar manner we may consider the existence of multiple values of 1" or 


(— 1)", and, therefore, of multiple signs of affection corresponding to them, as 
consequent upon the general laws of combination of symbolical algebra, and as 
results to be determined from those laws, and whose existence, also, is de- 
pendent upon them. 


204: THIRD REPORT—1839. 


general rule for the concurrence of signs. In a similar manner 
we may consider (1)? (a)? as equivalent to (a)? ; (1) (a)? as 
equivalent to (a)*; (1)” a” as equivalent to a"; (— 1)" (a)" as 
equivalent to (— a)", and similarly in other cases: in all such 
cases the algebraical quantity into which the equivalent sign 
or its equivalent factor is multiplied, is supposed to possess its 
arithmetical value only *. 

The series for (1 + x)", when x is a whole number, may be 
exhibited under a general form, which is independent of the 
specific value of the index; for such a series may be continued 
indefinitely in form, though all its terms after the (m + 1)th 
must become equal to zero. Thus, the series 


(yaya (yp (1 tae Sie 
n(n—1)..(n—r-+41) 
ade SEE aa 


indefinitely continued, in which » is particular in value (a whole 
number) though general in form, must be true also, in virtue 
of the principle of the permanence of equivalent forms, when 
nm is general in value as well as in form7. 

This theorem, which, singly considered, is, of all others, the 
most important in analysis, has been the subject of an almost 
unlimited variety of demonstrations. Like all other theorems 
whose consequences present themselves very extensively in 
algebraical results, it is more or less easy to pass from some 
one of those consequences to the theorem itself: but all the 
demonstrations which have been given of it, with the excep- 
tion of the principle of one given by Euler{, have been con- 
fined to such values of the index, namely, whole or fractional 
numbers, whether positive or negative, as made not only 
the development depend upon definable operations, but like- 
wise assumed the existence of the series itself, leaving the form 
of its coefficients alone undetermined. It is evident, however, 
that if there existed a general form of this series, its form could 


+ a” + &e.) 


* This separation of the symbolical sign of affection from its arithmetical 
subject, or rather the expression of the signs of affection explicitly, and not im- 
plicitly, is frequently important, and affords the only'means of explaining many 
‘paradoxes (such as the question of the existence of real logarithms of negative 
numbers), by which the greatest analysts have been more or less embarrassed. 

+ If such a series should, for any assigned value of m, have more symbolical 
values than one, one of them will be the arithmetical value, inasmuch as one 


symbolical value of 1” is always 1. 
+ In the Nov. Comm. Petropol. for 1774. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 205 


be detected for any value of the index whatever, which was 
general in form, and therefore, also, when that index was a 
whole number ; a case in which the interpretation of the opera- 
tion designated by the index, as well as the performance of the 
operation itself, was the most simple and immediate. 

That such a series, likewise, would satisfy the only sym- 
bolical conditions which the general principles of indices im- 
poses upon the binomial, might be very easily shown ; for if 
m and n be whole numbers, then if the two series 

m(m—1) 9 


(1+ a2)" = 1" (1 + nx + ae a - &e.) 


be multiplied together, according to the rule for that purpose, 
we must obtain 


(Pears e+" ( + (m+ n)x + it eh &e.) 


a series in which m + has replaced m or n in its component 
factors: and in as much as we must obtain the same symbolical 
result of this multiplication, whatever be the specific values of 
mand 2, it follows, that if the same form of these series repre- 
sents the development of (1 + x)" and (1 + x)", whatever m and 
n may be, then, likewise, the series for the product of (1 + a)” 
and (1 + a)", or (1 + x)"*", would be that which arose from 
putting m + n in the place of m or m in each of the component 
factors. If, therefore, we assumed S (m) and S (nm) to represent 
the series for (1 + 2)” and (1 + 2)", when m and n are any 
quantities whatsoever, then (1 + xy x (1 + x) = (1+ x)"*" 
=S(m +n) =S(m) x S(m); or, in other words, the series 
will possess precisely the same symbolical properties with the 
binomial to which they are required to be equivalent. 

It is the equation a” x a” = a”*”, for all values of m and n, 
which determines the interpretation of a” or a", when such an 
interpretation is possible ; in other words, such quantities pos- 
sess no properties which are independent of that equation. ‘The 
same remark of course extends to (1 + 2)", for all values of n, 
and similarly, likewise, to those series which are equivalent to 
it. That all such series must possess the same form would be 
evident from considering that the symbolical properties of 
(1 + x)" undergo no change for a change in the value of n, and 
that no series could be permanently equivalent to it whose form 


206 THIRD REPORT—1833. 


was not equally permanent likewise. In assuming, therefore, the 
existence of such a permanent series, our symbolical conclu- 
sions are necessarily consistent with each other, and it is the 
interpretation of the operations which produce them, which 
must be made in conformity with them. It is true that we can 
extract the square or the cube root of 1 + a, and we can also 
determine the corresponding series by the processes of arith- 


metical algebra ; and we likewise interpret (1 + x)? and (1+ at 
to mean the square and the cube root of 1 + a, in conformity 
with the general principle of indices. The coincidence of the 


series for (1 + 2)® and (1 + a), whether produced by the 
processes of arithmetical algebra, or deduced by the principle © 
of the permanence of equivalent forms from the series for 
(1 + 2x)", would be a proof of the correctness of our interpreta- 
tion, not a condition of the truth of the general principle itself. 

In order to distinguish more accurately the precise limits of 
hypothesis and of proof in the establishment of the fundamental 
propositions of symbolical algebra, it may be expedient to re- 
state, at this point in the progress of our inquiry, the order in 
which the hypotheses and the demonstrations succeed each 
other. 

We are supposed to be in possession of a science of arith- 
metical algebra whose symbols denote numbers or arithmetical 
quantities only, and whose laws of combination are capable of 
strict demonstration, without the aid of any principle which is 
not furnished by our knowledge of common arithmetic. 

The symbols in arithmetical algebra, though general in form, 
are not general in value, being subject to limitations, which are 
necessary in many cases, in order to secure the practicability 
or possibility of the operations to be performed. In order to 
effect the transition from arithmetical to symbolical algebra, we 
now make the following hypotheses : 

(1.) The symbols are unlimited, both in value and in repre- 
sentation. 

(2.) The operations upon them, whatever they may be, are 
possible in all cases. 

(3.) The laws of combination of the symbols are of such a 
kind as to coincide wniversally with those in arithmetical algebra 
when the symbols are arithmetical quantities, and when the 
operations to which they are subject are called by the same 
names as in arithmetical algebra. 

The most general expression of this last condition, and of its 
connexion with the first hypothesis, is the Jaw of the perma- 


-~ 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 207 


nence of equivalent forms, which is our proper guide in the 
establishment of the fundamental propositions of symbolical 
algebra, in the invention of the requisite signs, and in the de- 
termination of their symbolical form: but in the absence of the 
complete enunciation of that law, we may proceed with the in- 
vestigation of the fundamental rules for addition, subtraction, 
multiplication and division, and of the theorems for the collec- 
tion of multiples, and for the multiplication and involution of 
powers of the same symbol, which will, in fact, form a series of 
assumptions which are not arbitrary, but subordinate to the 
conditions which are imposed by our hypotheses : but if we 
suppose those conditions to be incorporated into one general 
law, whose truth and universality are admitted, then those as- 
sumptions become necessary consequences of this law, and 
must be considered in the same light with other propositions 
which follow, directly or indirectly, from the first principles of 
a demonstrative science. In the same manner, if we assume the 
existence of such signs as are requisite to secure the universality 
of the operations, the symbolical form of those signs, and the 
laws which regulate their use, will be determined by the same 
principles upon which the ordinary results of symbolical al- 
gebra are founded. 

The natural and necessary dependence of these two methods 
of proceeding upon each other being once established, we may 
adopt either one or the other, as may best suit the form of the 
investigation which is under consideration : the great and im- 
portant conclusion to which we arrive in both cases being, the 
transfer of all the conclusions of arithmetical algebra which are 
general in form (that is, which do not involve in their expres- 
sion some restriction which limits the symbols to discontinuous 
values,) to symbolical algebra, accompanied by the invention or 
use of such signs (with determinate symbolical forms) as may 
be necessary to satisfy so general an hypothesis. 

There are many expressions which involve symbols which 
are necessarily discontinuous in their value, either from the 
form in which they present themselves in such expressions or 
from some very obvious conventions in their use: thus, when 
we say that 
cos « = cos (2Qra+ 2), 


and — cos x = cos { (2r+1)7+ x} 


propositions which are only true when 7 is a whole number, 
the limitation is conveyed (though imperfectly) by the con- 
ventional use of 2r and 2 r + 1 to express even and odd num- 
bers; for otherwise there would be no sufficient reason for not 


208 THIRD REPORT—1833. 


using the simple symbol r both in one case and the other. In 
a similar manner, in the expression of Demoivre’s theorem 


(cos 6+ /—I1 sin 6)" 
= cos (2raxr+né6)+ /—l1sin (Qrnx + n), 


we may suppose 7 to be any quantity whatsoever*, but 7 is ne- 
cessarily a whole number. : 

In some cases, however, the construction of the formula it- 
self will sufficiently express the necessary restriction of the 
values of one or more of its symbols, without the necessity of 
resorting to any convention connected with their introduction: 
thus, the formula 1 x 2 x 3......7, commencing from 1, is 
essentially arithmetical, and limited by its form to whole and 
positive values of r. The same is the case with the formula 
r(r —1)....3.2.1, where some of the successive and strictly 
arithmetical values of the terms of the series r, r — 1, &c., are 
put down; but the formula r (r — 1) (ry — 2).... is subject to 
no such restriction, in as much as any number of such factors 
may be formed and multiplied together, whatever be the value 
of r. In a similar manner, the formula 


n(n —1)... (n—r + Vy 
eae t 


7a see 7 


which is so extensively used in analysis, is unlimited with re- 
spect to the symbol , and essentially limited with respect to 
the symbol r: it is under such circumstances that it presents 
itself in the development of (1 + «)”. 

In the differential calculus we readily find 


CE ig n—?r 
Sa ly <b) cate al ala ; 
and in a similar manner also 
d* 
dar +, 271'4+C, 2°? +..C,) =n (n—1)..(n—r+ Dae: 


in both these cases the value of » is unlimited, whilst the value 
of r is essentially a positive whole number; in other words, 


* The investigation of this formula (like the equivalent series for (1 + zx)” 
when v is a general symbol,) requires the aid of the principle of the perma- 
nence of equivalent forms, in common with all other theorems connected with 
the general theory of indices. The formula above given involves also impli- 
citly any sign of affection which the general value of x may introduce: for 

(cos 6+ 4/—1 sin 6)” = (1)" (cos n 6+ /—1 sin n 9) 
= (cos 2raa+ a/—1 sin 2rn7z) (cosnd +./—l1 sin n 8) 
=cos(Qrnax+né) + Y—l1sin Qrner +n) 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 209 


the-principle of equivalent forms might-be extended to this for- 
mula (supposing it to be investigated for integral values of n 
and r,) as far as the symbol 7 is concerned only. This arithme- 
tical coefficient of differentiation (if such a term may be ap- 
plied to it,) will present itself in the expression of the rth dif- 
ferential coefficient (r being a whole positive number,) of all al- 
gebraical functions *; and it is for this reason that we are appa- 
rently debarred from considering fractional or general indices 
of differentiation when applied to such functions, and that we 
are consequently prevented from treating the differential and 
integral calculus as the same branch of analysis whose general 
laws of derivation are expressed by common formule. 

But is it not possible to exhibit the coefficient of differentia- 
tion under some equivalent form which may include general 
values of the index of differentiation? It is well known that 
the definite integral ig: 


n 2 Se 
(adopting Fourier’s notation,) is equal to 1.2....n, when zis 


a whole number; and that consequently, under the same cir- 
cumstances, the coefficient of differentiation or 


1 1 n 
A= —F ED) 1 ee st 


Pe ax (Io a be 
0 8 


and in as much as the form of this equivalent expression is not 
restricted to integral and positive values of r, we may assume 


) 1 déu  (—1)".2.3... (74-1) at 
* Thus if w= TGs) we have F735 = Gf arti ) 
sedi utiegetee TI ee ao &e.} 
2.3 2 ch ia al ee ES oo 


BEE ann 1 h dru 1 ee ee 
and if u = Va 4 ay Fe ave Tor = Tl —aryrte 


1 r(@—l) 1 r(r—1) (r —2) (”®— 3) : 
Ba At Se iy ag Oe 

+ This definite integral, the second of that class of transcendents to which 
Legendre has given the name of Integrales Euleriennes, was first considered 
by Euler in the fifth volume of the Commentarii Petropolitani, in a memoir 
on the interpolation of the terms of the series : 

1+1x24+1x2x34+1x2x3x4+4 &e., 
which is full of remarkable views upon the generalization of formule and 
their mterpretation. The same memoir contains the first solution of a pro- 
blem involving fractional indices of differentiation, 
. P 


210 THIRD REPORT—1833. 


‘it to be permanent, so long as we do not at the same time as- 
sume the necessary existence and interpretation of equivalent 
results. If, however, such results can be found, either gene- 
rally, or for particular values (not integral and positive) of r, 
apart from the sign of integration, the consideration of the values 
of the corresponding differential coefficients will involve no 
other theoretical difficulty than that which attends the transition 
from integral to fractional and other values of common indices. 

Euler, in his Differential Calculus*, has given the name of 
inexplicable functions to those functions which are apparently 
restricted by their form to integral and positive values of one or 
more of the general symbols which they involve: of this kind 
are the functions 


LOS WB ita: HO An, 


x, 
ci et se 
gener dS ay 
res hora ae aces 


and innumerable others which present themselves in the theory 
of series. The attempts which he has made to interpolate the 
series of which such functions form the general terms, are 
properly founded upon the hypothesis of the existence of per- 
manent equivalent forms, though it may not be possible to ex- 
hibit the explicit forms themselves by means of the existing 
signs and symbols of algebra. In the cases which we have 
hitherto considered, the forms which were assumed to be per- 
manent had a real previous existence, which necessarily re- 
sulted from operations which were capable of being defined. 
In the case of inexplicable functions, the corresponding perma- 
nent forms which hypothetically include them, may be consi- 
dered as having an hypothetical existence only, whose form 
degenerates into that of the inexplicable function in the case of 
integral and positive values of the independent variable or va- 
riables. It is for the expression of such cases that definite 
integrals find their most indispensable usage. 


* Institutiones Calculi Differentialis, Capp. xvi. et xvii. See also an admira- 
ble posthumous memoir of the same author amongst the additions to the 
Edition of that work printed at Pavia in 1787. He had been preceded in such 
‘researches by Stirling, an author of great genius and originality, whose la- 
bours upon the interpolation of series and other subjects have not received 
the attention to which they are justly entitled. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. ait 


It is easy to construct formule which may exhibit the possi- 
bility of their thus degenerating into others of a much more 
simple form, when one or more of the independent variables be- 
come whole numbers: of this kind is the formula 


a+ PsnQre+6)+ysn(2rnx +0) + &e. 


a+ Psiné + ysiné +, &e. x O(r), 


which is, or is not, identical with ¢$ (r), according as r is a whole 
or a fractional number: such functions are termed wndulating 
functions by Legendre *. We can conceive also the possible 
existence of many other transcendents amongst the unknown 
and undiscovered results of algebra, which may possess a simi- 
lar property. 

The transcendent 


Sf, éx (to oy 
0 8a 


mentioned above, possesses many properties which give it an 
uncommon importance in analysis, and most of all from its fur- 
nishing the connecting link in the transition from integral 
and positive to general indices of differentiation in algebraical 
functions. If we designate, as Legendre has done, 


1 1 r 
or dx (log +) by P(1 +7), 


we shall readily derive the fundamental equation 
Bein steel (ep hd wees @) 
which is in a form which admits of all values ofr. It appears’ 


* Traité des Fonctions Elliptiques, tom. xi. p. 476. 
+ In as much as 
dv-r eh oat, T (1 +) 


aan ern Cae 


and 


>————, gr® = > 

dan-r +1 I (r) 
it follows that r A = B, and therefore also that 

: r 1 
Ttit+n = T(r) or rT (r) = Tr(1+7), 
which is the equation (1): and it is obvious that the transition from 
qdn-r qdn—r+1 
d an-r am to dan—-rt+l an 
(which is equivalent to the simple differentiation of A 2’, when A is a 
constant coefficient), will lead to the same relation between [' (1 + 7) and 
r (r), whatever be the value of r, whether positive or negative, whole or frac- 
tional. Legendre has apparently limited this equation to positive values of r, 
P2 " 


nm—rtl 
a Fe OE co a ference 


212 THIRD REPORT—1833. 


also from this equation that if the values of the transcendent 
I (r) can be determined for all values of 7 which are included 


(Fonctions Elliptiques, tom. ii. p. 415,) a restriction which is obviously unne- 
cessary. 

There are two cases in which the coefficient of x"—7 in the equation - 

aa PA +2) 
dat ~ V(1+n2—r) 

requires to be particularly considered: the first is that in which this coeffi- 
cient becomes infinite ; the second, that in which it becomes equal to zero. 

The numerator I' (1 +n) will be infinite when n is any negative whole 
number; the denominator I’ (1 + ™—r) will become infinite when » — r is 
any negative whole number, and in no other case: if n be a negative whole 
number, and if r be a whole number, either positive or negative, such that n — r 


g“n-Tr 


uf 
is negative, then the coefficient ep) becomes finite, in as much as 
8 ra+n2—n 
T'(0 
IT (—2z) (if ¢ be a whole number) = mos De? and T' (0) disap- 


Be Side 4 ab 1 5 e 
pears, therefore, by division: thus all the coefficients of Baer etn infinite, 


unless 7 be a negative whole number, such as — m, in which case it becomes 
1.2..m.(—1)™, a result which is easily verified. In a similar manner it 


dr 1 ; 
would appear that the coefficients of = dcr ew infinite, when 7 is a posi- 


tive number, unless r be a negative whole number equal to, or greater than, 2. 


The coefficient ear ot m) 7) will become equalto zero, when 1+2—r 
is, and when 1 + 2 is nof, equal to zero or to any negative whole number ; for, 
under such circumstances, the denominator is infinite and the numerator is 
Sinite. 

As the most important consequences will be found to result from these 
critical values of the coefficient of differentiation, we shall proceed to examine 
them somewhat in detail. 

(1.) The simple differentials or differential costicients of constant quantities - 
are equal to zero, whilst the differentials or differential coefficients to general 
indices (positive whole numbers being excepted,) are variable. 


Thus 


RE ec 5 A A SEH Per a al ada 
dat da TG) ° Jae” qa * 
a(Tl) 2 /Sx d—a TQ) 
= + es = 
T@ ° ors = ve da-| — TQ) a xt! = az; 
and similarly in other cases. ~ 


(2.) The differentials of zero to general indices (positive whole numbers 
being excepted) are not necessarily equal to zero. 
Thus, if we suppose 
aingk fa C 
nape dg T (0) Sir Snare iat, aaalk 0) rd—s—r)" 
if x be a positive whole number, I (1— ) = », and this expression is finite un- 
less T (1 —n— 1) = , in which case it is zero: if 7 be also a positive whole 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 213 


between any two successive whole numbers, they can. be de- 
termined for all other values of r. Euler * first assigned the 


number, it is always xero: if r= — 1, itis finite whenn=1: ifr = —2, 
it is finite when n= 1 or n= 2: ifr = — 3, it is finite whenn = 1, orn = 2, 
orn=3: and generally if r be any negative whole number, there will be 
finite values corresponding to every value of n from 1 to — 7; we thus get 


2 = Cxe+C) 
—_ = ae + Cia+ Co 
= = Seo + peer + Cn—2.@ + Cy-1. 


This is the true theory of the introduction of complementary arbitrary func- 
tions in the ordinary processes of integration. ny af 
More generally, if 7 be not a whole number, 


PO Cc Hunk (1 —n) an, 
da? ro rda—nz—nr) 

which will be finite when n is a positive whole number and when 1 —2 —r is. 

not a negative whole number : thus if m be any number in the series 1, 2, 3..+, 


and if r = 3, then 


d? 0 Cc ing Cc F iO Higey 
se ie OOF ——_— «&r 
aoe CD rey "TCH 


and so on for ever : consequently, 


Chet tee 
72 aes <4 =! + &c. in infinitum. 
ct we ee x? a? 


In a similar manner, we shall find 


-F f 1 Ce C3 
Be bank Ct yp ty + &e. i infinitum, 


a 
dx? 
The knowledge of these complementary arbitrary functions will be found of 
great importance for the purpose of explaining some results of the general 
differentiation of the same function under different forms which would other- 
wise be irreconcileable with each other. 
~ (3.) The differential coefficient will be zero, when n is not, and whenn —r 
is, a negative whole number. 


Thus, 
d2 x __ Baz a? ae d® at a? “Fayre 
i Peg, ahi Same nie 
d x* d x? dx * 


and similarly in other cases. 
(4.) The differentials of & are not necessarily equal to ©, but may be finite. 


If we represent » by C I'(0), we shall find 


* Commentarii Petrop., vol. y. 1731. 


214 ' THIRD REPORT—1833. 


value of I (5) = 7, by the aid of the very remarkable ex- 


pression for 7, which Wallis derived from his theory of inter- 


eee car 


rs = CI(0) eae Ih wLigae(ns barry 
whenever + is a positive whole oa 
Conversely eke : ia a 
Shelia 0 (0 
ie ti ae) Saabs 
a ie = I (0) a 
d x-2 (2) , 
Gr Bret PMO) ing 
d x-3 T (3) if 
dr. a} — T (0) gr-) 
CaO E Ye) Aonk* 


the arbitrary complementary functions being omitted. 

(5.) The occurrence of infinite values of the coefficient of differentiation will 
generally be the indication of some essential change of form in the transition 
from the primitive function to its corresponding differential coefficients. 

Thus, 

d-} 1 T (0) 


peed EL VP ahi tices aes MOO bh ; 
dx-1 x Tae * ERAS 
this last result or value of y ro: x being obtained by the ordinary process 
of integration: and generally, 
ad". 4-1 Las T Tr (0) gr-) ar-l C, a7r—2 & 
Br aa ay ae ee 


the first term of which is infinite, in all ‘cases in which 7 is not a negative 
whole number, in which case it becomes equal to (—1)-"1.2...(— 7) a7—}, 
the complementary arbitrary functions also disappearing. If we suppose, 


0 
however, 7 to be a positive whole number, and if we replace r oy - «9 by 
its transcendental value chat determined, we shal] get 

d-ra-l.. be gr-2 
dar 35 G eee ta ait faead 
which may be replaced by 
af a-) _ gr-1 a ar- 
STG {loge + (—1)r T(r). ra Arh Ob ihe Sw 
which is in a form which is true for all values of tng 9h a aud which 
coincides, for integral values of r, with the form determined by the ordinary 


process of integration, 
More generally, 


+ &c., 


aah er) TO ¢) Le OPN hee Gyan 
aban 7 are ee Se a ee se I (r—n) mite 
which is finite, whilst r is less than x; and when r and x are whole numbers, 


T (”—vr) 
~ Ta 


becomes = (— 1)”. ar—", omitting complementary functions. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 215: 


polations; and subsequently, by a much more direct process, 
which lead to the equation, 


roy)rd n= a 


Tv 
~@ (when r at 


If, under the same circumstances, 7 be greater than , the coefficient of dif- 
ferentiation becomes infinite, and its value, determined as above, becomes 


xar-n C; at="-! 


=Tr@rer—nt+) {loge +C} + rT @7—n + &e. 
arn 
=Toprer w+) {log 2 + (—1)r TP @—a41) TP (n—r) + Cc} 
’ ie: C, ar—n-1 a 
' T@—n) ‘ 


which is in a form adapted to all values of r. 

The cases which we have considered above are the only ones in which the 
coefficient of differentiation will become infinite, in consequence of the intro- 
duction of log z in the expression of its value. We shall have occasion here- 
after to notice more particularly the meaning of infinite values of coefficients 
as indications of a change in the constitution of the function into which they 
are multiplied. 

(6.) If w = (a + 6)”, then 

du I (1+ 2) a Car-l 
det, TAL at OPS A eye + 


: d d3 
Forifv=az+b, then 5~ =a and — = 0; and therefore 


du T(1 +2) dv\t Car! 

gait alee atl =) tir¢oyy a 

Thus if u = (x + 1), we get 
fe FO +n + hes 
dat ee r(—y)e" r(— pat 


 @HI)® 4+ 4 Sy — + &e. 


tae 
If we replace (z + 1)? by 22 + 2 +2 1, we shall get 
ds FG) 2 ena at) Ey 
af a), eee ¢ Te ae 
C on 


fh Seep te bh ee) 
Tiy)e  r(—a)e* 


8 3 4 1 2 C 
a ae, ees Tes ge 


It thus appears that the two results may be made to coincide with each 


other, when (x + 1)? in the first of them is developed, by the aid of the proper 
arbitrary functions. 

The necessity of this introduction of arbitrary functions to restore the re- 
quired identity of the expressions deduced for the same differential coefficients, 
presents itself also in the ordinary processes of the integral colcaalncts thus, 
if u= (x + 1)*, we find 


216 Cane, THIRD REPORT—1833. 


Legendre, following closely in the footsteps of this illustrious - 
analyst, has succeeded in the investigation of methods by which 
the values of this transcendent I (r) may be calculated to any 
required degree of accuracy for all positive values of r, and has 


(d-24u (x + 1)4 
snhaMeea = Cogagct Oiectale 


d-1y-"  @ +138 
3 
at a3 a? 2 1 
ag ct ig ih eg ok fp Pee Ge. 
If we replace (x + 1)? by a? + 2” + 1, we find 
d-2u at x3 = 
da”, eeEst 3p at CA t Oe 
d-2 : : 
It is obvious that these two values of — cannot be made identical, 
without the aid of the proper arbitrary functions. 


dru 
(7.) Let « = v" where v = f(x): and let it be required to find 7-5 - 


dr ; 
The general expression for —< , when 7 is a whole number, is generally 


extremely complicated, though the law of formation of its terms can always 
be assigned. If the inexplicable expressions in the resulting series be re~ 
placed by their proper transcendents, the expression may be generalized for 
any value of r. 


d2 dr 
If a4 = p and if 8 =c, aconstant quantity, then —= n (n—1) 
eee (n—7+1) v7 pr 


r (r — 1) cv r(r—1)(r—2) (r—3) cw 
Kae eT gf tie Gore Gare pt ef 


20 AY (e a) 4 r(r—1)" | ev } 
—~ T(l+n2—r) aie {1+ l.(n—r+1)° pP ut 
C. 4 Cy +5 
he rye Oe eT ee ey + &c. 
which is in a form adapted to any value of r. 
I ar 
ie= Ji pa md" = -p> we shall find 
1 — 
a ——— az esa 1 Le: ais 5 Biss Py / te 

ad Jima) = Ws {2 ae gags ~ QM go we. } 

d x? 

pp erty Oy ae 
a x? 


Rational functions of « may be resolved into a series of fractions, whose 
denominators are of the form (x + a)”, and whose numerators are constant 
quantities, whose rth differential coefficients may be found by the methods 
givenabove. Irrational functions must he treated by general methods similar 
to that followed in the example just given, which will be more or less com- 
plicated according to the greater or less number of successive simple differ- 
entials of the function beneath the radical sign, which are not equal to zero. 


REPORT ON. CERTAIN BRANCHES OF ANALYSIS. QF 


given tables of its logarithmic values to twelve places of decimals, 
with columns of three orders of differences for 1000 equal in- 
tervals between 1 and 2*; and similar tables have been given 
by Bessel and by others. We may therefore consider ourselves 
to be in possession of its numerical values under all circum- 
stances, though we should not be justified in concluding from 
thence that their explicit general symbolical forms are either 
discoverable or that they are of such a nature as to be ex- 
pressible by the existing language and signs of algebra. 
The equation 
T(r) =(r—1) @ —2).... YQ —m T(r —m), 
P(r) 

(ry = 1) @—2).. @ — my 
where m is a whole number, will explain the mode of passing 
from the fundamental transcendents, when included between 
yr = 0 and 1, or between r = 1 and 2, to all the other derived 
transcendents of their respective classes}. The most simple of 
such classes of transcendents, are those which correspond to 


r(}) =v 


which alone require for their determination the aid of no higher 
transcendents than circular arcs and logarithms. In all cases, 
also, if we consider I(r) as expressing the arithmetical value of 
the corresponding transcendent, its general form would require 
the introduction of the factor 1”, considered as the recipient 
of the multiple signs of affection which are proper for each dif- 
ferential coefficient, if we use that term in its most general 
sense. 

In the note, p. 211, we have noticed the principal properties 
of these fractional and general differential coefficients, partly 
for the purpose of establishing upon general principles the 


basis of a new and very interesting branch of analysist, and 


or C(r —m) = 


* Fonctions Elliptiques, tom. ii. p. 490. 
1 3 1 ‘ 5 3al 
Th ECL jaws pat (2 Neds ; (a = 
+ Thus, F (= vnt( ~vmr(l)=43 vn, 


a mae" pe ARN bic 28 Hi pit ot siBe 
ee i v=, 0 (=) each aah audeaace 
V x, &e. 

{ The consideration of fractional and general indices of differentiation was 
first suggested by Leibnitz, in many passages of his Commercium Epistolicum 
with John Bernouilli, and elsewhere; but the first definite notice of their 
theory was given by Euler in the Petersburgh Commentaries for 1731: they 
have also been considered by Laplace and other writers, and particularly by 
Fourier, inhis great work, La Théorie de la Propagation de la* Chaleur. The 
last of these illustrious authors has considered the general differential coeffi- 


218 THIRD REPORT—1833. 


partly for the purpose of illustrating the principle of the per- 
manence of equivalent forms in one of the most remarkable 
examples of its application. The investigations which we have 
given have been confined to the case of algebraical functions, 


cients of algebraical functions, through the medium of their conversion into 
transcendental functions by means of the very remarkable formula, 


2 + © + © 
gent fl O(a)da f Q (a) dqcos g (« — @), 
-@ -o 


which immediately gives us, 


ox 2 + +O dr 
=f O(a)de f Q (a) aq gar cosg (&—%)5 
re) 


dT 
dat 
and the requisite definite integrations effected. If, indeed, we grant the prac- 
ticability of such a conversion of @ (#) in all cases, and if we suppose the 
difficulties attending the consideration of the resulting series, which arise 
from the peculiar signs, whether of discontinuity or otherwise, which they 
may implicitly involve, to be removed, then we shall experience no embarrass- 
ment or difficulty whatever in the transition from integral to general indices 
of differentiation. 

In the thirteenth volume of the Journal de I’ Ecole Polytechnique for 1832, 
there are three memoirs by M. Joseph Liouville, all relating to general in- 
dices of differentiation, and one of them expressly devoted to the discussion 
of their algebraical theory. The author defines the differential coefficient of 
the order « of the exponential funetion e”” to be m” &”*, and consequently 
the jth differential coefficient of a series of such functions denoted by 2 A,,¢”” 


must be represented by = Aj, m“ e"*. If it be granted that we can properly 
define a general differential coefficient, antecedently to the exposition of any 
general principles upon which its existence depends, then such a definition 
ought to coincide with the necessary conclusions deduced by those principles 
in their ordinary applications: but the question will at once present itself, 
whether such a definition is dependent or not upon the definition of the simple 
differential coefficient in this and in all other cases. In the first case it will be 
a proposition, and not a definition, merely requiring the aid of the principle 
of the permanence of equivalent forms for the purpose of giving at least an 


which can be determined, therefore, if 


cos g (« — «) can be determined, 


: : apes E 
hypothetical existence to — for general, as well as for integral values 


of %. M. Liouville then supposes that all rational functions of # are ex- 
pressible by means of series of exponentials, and that they are consequently 
reducible to the form = A,, e”, and are thus brought under the operation of 
his definition. Thus, if x be positive, we have, 


1 lo 9) 
Fs if S da, 


es ee) 
i =f e #7 (_ aw )hda, 
d xk 0 


and therefore, 


REPORT ON.CERTAIN BRANCHES OF ANALYSIS. 219 


and have been chiefly directed to meet the difficulties connected 
with the estimation of the values of the coefficient of differen- 
tiation in the case of fractional and general indices. If we 
should extend those investigations to certain classes of tran- 


which is easily reducible to the form, 


dep 
ae _ (Erte), 
a gite 
an expression which we have analysed in the note on p. 211. This part of 


M. Liouville’s theory is evidently more or less included in M. Fourier’s views, 
which we have noticed above. The difficulties which attend the complete 


("TA +e) 
gite 
principle of equivalent forms alone can reconcile, will best show how little 


developement of the formula for all values of «, which the 


progress has been made when the «th differential coefficient of _ is reduced 
to such a form. e 

M. Liouville adopts an opinion, which has been unfortunately sanctioned by 
the authority of the great uames.of Poisson and Cauchy, that diverging series 
should be banished altogether from analysis, as generally leading to false 
results ; and he is consequently compelled to modify his formule with refer- 
ence to those values of the symbols involved, upon which the divergency or 
convergency of the series resulting from his operations depend. In one sense, 
as we shall hereafter endeavour to show, such a practice may be justified ; but 
if we adopt the principle of the permanence of equivalent forms, we may 
safely conclude that the limitations of the formule will be sufficiently ex- 
pressed by means of those critical values which will at once suggest and re- 
quire examination. The extreme multiplication of cases, which so remark- 
ably characterizes M. Liouville’s researches, and many of the errors which he 
has committed, may be principally attributed to his neglect of this important 
principle. 

It is easily shown, if 6 be an indefinitely smal] quantity, that 


4 br e—hr . emBx __ e—NBz 
"2B mane” 
and that consequently any integral function La A, e+... Ap xP, involving _ 
integral and positive powers of x only, may be expressed by = Am ¢””, where 


m is indefinitely small; and conversely, also, = A,, e”* may, under the same 


circumstances, be always expressed by a similar integral function of «. M. Liou- 
ville, by assuming a particular form, 


2B yy sr 
where C is arbitrary, and A indefinitely small, to represent zero, and differen- 
tiating, according to his definition, gets 


Gin @ yet Medes! gp Gau/=D, 


sayy 2/8 2 


but it is evident that by altering the form of this expression for zero we might 


d® are 
show that was equal either to zero or to infinity; and that in the latter 


dx 


220 | (THIRD REPORT—1833. | 


scendental functions, such as e”*, sinm a, and cosma, we 
shall encounter no such difficulties, in as much as the differen- 
tials of those functions corresponding to indices which are ge- 
neral in form, though denoting integral numbers, are in a form 


case the critical value infinity might be merely the indication of the existence 
1 


a 0. which were 
dat 
not expressible by any rational function of e** under a finite form and. in- 
volving indefinitely small indices only. And such, in fact, would be the re- 
sult of any attempt to differentiate this exponential expression for 2 or its 
powers, with respect to fractional or negative indices. It has resulted from 
this very rash generalization of M. Liouville that he has assigned as the ge- 
neral form of complementary arbitrary functions, 

C+ C, «+ Ce 22 + C3 23 + &e., 
which is only true when the index of differentiation is a negative whole 
number. 

Most of the rules which M. Liouville has given for the differentiation of 
algebraical functions are erroneous, partly in consequence of his fundamental 
error in the theory of complementary arbitrary functions, and partly in 
consequence of his imperfect knowledge of the constitution of the formula 


LTA)” ‘thus, after deducing the formula 


of negative or fractional powers of x in the expression for 


Td+u—r)' 
1 
df o=4=— (—1)".a’. T(a+r) 
EY iil ole Ga + ont . 
aur 


which is only true when ” is a whole number, he says that no difficulty pre- 
sents itself in its treatment, whilst 2 + ris >0, but that T(n +r) be- 
comes infinite, when x + r<{0, in which case he says that it must be 
transformed into an expression containing finite quantities only, by the aid 
of complementary functions; whilst, in reality, IT (~ +r) is only infinite 
when n+ 71 is zero or a negative whole number, and the forms of the com- 
plementary functions, such as he has assigned to them, are not competent to 
effect the conversion required. In consequence of this and other mistakes, 
1 : ’ 
i dr, ———_— P 
in connexion with the important case (ax + 6)", nearly all his conclu- 
d xr 

sions with respect to the general differentials of rational functions, by means 
of their resolution into partial fractions, are nearly or altogether erroneous. 

The general differential coefficients of sines and cosines follow immediately 
from those of exponentials, and present few difficulties upon any view of their 
theory. In looking over, however, M. Liouville’s researches upon this sub- 
ject, 1 observe one remarkable example of the abuse of the first principles of 

1 
d? Mie 
reasoning in algebra. There are two values of ere *, one positive and 
da? 

the other negative, considered apart from the sign of m, whether positive or 


: 1 3 
negative: but if we put cosma = Zz cos me + gicos m=, We get 


1 4 
d? cosmax 3. d? cosmux 


1 
2 , if 
Geos wae} CSRS 4 -oon, 
aa? d x? d x" 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 228 


which is adapted to the immediate application of the general 
principle in question. 
Thus; if w= e””, we get 
du cy 1 Us Oh a) Loe d” u 
hae ah i pease oa Te eT fe 
‘when ris a whole number, and therefore, also, when 7 is any 
quantity whatsoever. 


— gm & 
Syne 5 


if : du Ph sat te ad? u fi 
—! — =m Si a a | nee SI 
uU sim mx, ne PD} ’ dx 


d” u CEN te d 
os + mx), 00's eae m sin (S + mz) when 7 is a whole 


number, and therefore generally. In a similar manner if 


u = cos ma, or rather u = cos m(1)* a, (introducing 1” as 
a factor in order to express the double sign of m x, if de- 
termined from the value of its cosine,) then we shall find 


ve = (mV 1)" cos {75 + (m / 1) zl, whatever be the 
x 


value of r. Ifu = e"*cosmax, we get, by very obvious re- 
eee ‘ ee ke Oy ie 
ductions, making p = Gas and # = cos 5 
d™u 
d a 
It is not necessary to mention the process to be followed in ob- 


= pre"® cos (mx + n4). 


de eT se te ne oe ee cee ee NET EAT TT ETE TERT oe LERE ROTELLA) 


and if we combine arbitrarily the double values of the two parts of the second 


member of this equation, we shall get four values of vee instead of 

ie ; and, in a similar manner, if we should resolve cos pst any number 
ciate ee 

of parts, we should get double the number of values of ree eS, If this 

principle of arbitrary combinations of algebraical values iid from a.com- 


mon operation was admitted, we must consider. as having two values, 


a 
1—«z 
and its equivalent series 

: sin a+at tat + &e. : 

as having an infinite number. But it is quite obvious that those expressions 
which involve implicitly or explicitly a multiple sign must continue to be 
estimated with respect to the same value of this sign, however often the reci- 
pient of the multiple sign may be repeated in any derived series or expression. 
The case is different in those cases where the several terms exist indepen- 
dently of any explicit or implied process of derivation. 


222 THIRD REPORT—1833. 


taining the general differential coefficients of other expressions, 
such as (cos x)", cos max x cos nx, &c., which present no kind 
of difficulty. In all such cases the complementary arbitrary 
functions will be supplied precisely in the same manner as for 
the corresponding differential coefficients of algebraical func- 
tions. 

The transition from the consideration of integral to that of 
fractional and general indices of differentiation is somewhat 
startling when first presented to our view, in consequence of 
our losing sight altogether of the principles which have been 
employed in the derivation of differential coefficients whose in- 
dices are whole numbers: but a similar difficulty will attend 
the transition, in every case, from arithmetical to general values 
of symbols, through the medium of the principle of the perma- 
nence of equivalent forms, though habit and in some cases im- 
perfect views of its theory, may have made it familiar to the 
mind. We can form distinct conceptions of m.m, m.m.m, 
m.m.m....(r), where m is a whole number repeated twice, 
thrice, or 7 times, when r is also a whole number; and we 
can readily pass from such expressions to their defined or as- 
sumed equivalents m*, m°,...m”: ina similar manner we can rea- 
dily pass from the factorials* 1.2,1.2.3,...1.2...7r, to 
their assumed equivalents (3), (4), ... (1 +7), aslongas r 
is a whole number. The transition from m” and I (1 + r) when 
r is awhole number, to m’ and r'(1 + 7) when r is a general 
symbol, is made by the principle of the equivalent forms; but by 
no effort of mind can we conneet the first conclusion in each case 
with the last, without the aid of the intermediate formula, involv- 
ing symbols which are general in form though specific in value ; 
and in no instance can we interpret the ultimate form, for 
values of the symbols which are not included in the first, by 
the aid of the definitions or assumptions which are employed 
in the establishment of the primary form. In all such cases 
the interpretation of the ultimate form, when such an interpre- 
tation is discoverable, must be governed and determined by a 
reference to those general properties of it which are inde- 
pendent of the specific values of the symbols. 


* Legendre has named the function T(1 +7) =1.2...r, the function 
gamma. Kramp, who has written largely upon its properties, gave it, in his 
Anulyse des Refractions Astronomiques, the name of faculté numérique; but in his 
subsequent memoirs upon it in the earlier volumes of the Annales des Ma- 
thématiques of Gergonne he has adopted the name of factorial function, which 
Arbagost proposed, and which I think it expedient to retain, as recalling to 
mind the continued product which suggests this creature of algebraical lan- 
guage. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 293 


The law of derivation of the terms in Taylor’s series, 
du au =ih? Cu he 
Ds ned Ba AN Maa net yee hh een trlenen 
Birth pp rihink 7g Tigi th Tas PT aT 

is the same as in the more general series 
du! _ a u d™u h dtu he 
dat dat! dat" 1 * dart?’ 1.2 
u 


and if we possess the law of derivation of As and of wed we 
dx d x 


can find all the terms of both these series, whatever be the 
value of r. The first of these terms must be determined through 
the ordinary definitions of the differential calculus ; the second 
must be determined in form by the same principles, and gene- 
ralized through the medium of the principle of equivalent 
forms. Both these processes are indispensably necessary for 


Tr 
the determination of ots : but it is the second of them which 


; - dt du 
altogether separates the interpretation of os from that of ae 


d’u 
d a” 
ticular cases in which the symbols in both are identical in 
value. 

There are two distinct processes in algebra, the direct and 
the inverse, presenting generally very different degrees of dif- 
ficulty. In the first case, we proceed from defined operations, 
and by various processes of demonstrative reasoning we arrive 
at results which are general in form though particular in value, 
and which are subsequently generalized in value likewise: in 
the second, we commence from the general result, and we are 
either required to discover from its form and composition some 
equivalent result, or, if defined operations have produced it, to 
discover the primitive quantity from which those operations 
have commenced. Of all these processes we have already given 
examples, and nearly the whole business of analysis will consist 
in their discussion and developement, under the infinitely varied 
forms in which they will present themselves. 

The disappearance of undulating and of determinate func- 
tions with arbitrary constants, upon the introduction of inte- 
gral or other specific values of certain symbols involved, is one 
of the chief sources * of error in effecting transitions to equiva- 


or rather of 


when r is a whole number, unless in the par- 


* The theory of discontinuous functions and of the signs of discontinuity 
will show many others. 


224. THIRD REPORT—1833. 


lent forms, whethe'the process followed be “direct or inverse. 
Many examples of the first kind may be found in the researches 
of Poinsot respecting certain trigonometrical series, which 
will be noticed hereafter, and which had been hastily gene- 
ralized by Euler and Lagrange; and a remarkable example of 
the latter has already been pointed out, in the disappearance 
of the functions with arbitrary constants in the transition from 
d” u 
da’ 
ral discussion of such cases, however, would lead me to an 
examination of the theory of the introduction of determinate 
and arbitrary functions in the most difficult processes of the 
integral calculus and of the calculus of functions, which would 
carry me far beyond the proper limits and object of this Re- 
port. I have merely thought it necessary to notice them in 
this place for the purpose of showing the extreme caution 
which must be used in the generalization of equivalent results 
by means of the application of the principle of the permanence 
of equivalent forms*. 

The preceding view of the principles of algebra would not 
only make the use and form of derivative signs, of whatever 
kind they may be, to be the necessary results of the same ge- 
neral principle, but would also show that the interpretation of 
their meaning would not precede but follow the examination of 
the circumstances attending their introduction. I consider it 
to be extremely important to attend to this order of succession 
between results and their interpretation, when those results 
belong to symbolical and not to arithmetical algebra, in as much 
as the neglect of it has been the occasion of much of the con- 
fusion and inconsistency which prevail in the various theories 
which have been given of algebraical signs. I speak of dera- 


u to , when r becomes a whole positive number. The gene- 


* Euler, in the Petersburgh Acts for 1774, has denied the universality of 
this principle, and has adduced as an example of its failure the very remark- 
able series 


1—a™ (1—a™) (l—a™-!) | d@—a™) (i — a™—1) (1 — a™—2) 
te * 1—a i 1—@ + &e., 


which is equal to m, when m is a whole number, but which is apparently not 
equal to m, for other values of m, unless at the same time a= 1: the occur- 
rence, however, of zero as a factor of the (m + 1)th and following terms in 


: Os 
the first case, and the reduction of every term to the form > in the second, 


would form the proper indications of a change in the constitution of the equi- 
valent function corresponding to these values of m and a, of which many ex- 
amples will be given in the text. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 225. 


vative signs as distinguished from those pilnitive signs of ope- 
ration which are used in arithmetical algebra; but such signs, 
though accurately defined and limited in their use in one sci- 
ence, will cease to be so in the other, their meaning being de- 
pendent in symbolical algebra, in common with all other signs 
which are used in it, upon the symbolical conditions which they 
are required to satisfy. 

I will consider, in the first place, signs of affection, which are 
those symbolical quantities which do not affect the magnitudes, 
though they do affect the specific nature, of the quantities into 
which they are incorporated. 

Of this kind are the signs + and —, when used independ- 
ently; or their equivalents + 1 and — 1, when considered as 
symbolical factors ; the signs (+ 1)" and (— 1)", or their sym- 
bolical equivalents 

cos 2rna + /—] sn2rnz and cos (2r+1)n27+ 
¥—l sin (2r +1) a2; 
dy errnaVv ai aad e@rt 1) na/—1 

The affections symbolized by the signs + 1 and — 1 admit 
of very general interpretation consistently with the symbolical 
conditions which they are required to satisfy, and particularly 
so in geometry: and it has been usual, in consequence of 
the great facility of such interpretations, to consider all quan- 
tities affected by them (which are not abstract) as possible, 
that is, as quantities possessing in all cases relations of exist- 
ence which are expressible by those signs. It should be kept 
in mind, however, that such interpretations are in no respect 
distinguished from those of other algebraical signs, except in 
the extent and clearness with which their conditions are sym- 
bolized in the nature of things. 

The other signs of affection, different from + 1 and — 1, 
which are included in (1)" and (— 1)", are expressible generally 
by cos§ + “—1sin 6, or by a + B VW —1,where « and 6 may have 
any values between 1 and — 1, zero included, and where «? + 6? 
= 1. To all quantities, whether abstract or concrete, expressed 
by symbols affected by such signs, the common term impossible 
has been applied, in contradistinction to those possible magni- 
tudes which are affected by the signs + and — only. 

Tf, indeed, the affections symbolized by the signs included. 
under the form cos § + “ —1 sin 4, admitted in no case of an in- 
terpretation which was consistent with their symbolical condi- 
tions, then the term impossible would be correctly applied to 
a Tyeee affected by them: but in as much as the signs + and 

. Q 


226 THIRD REPORT—1833. 


—, when used independently, and the sign cos 6+ “—I sin6, 
when taken in its most enlarged sense, equally originate in the 
generalization of the operations of algebra, and are equally in- 
dependent of any previous definitions of the meaning and extent 
of such operations, they are also equally the object of inter- 
pretation, and are in this respect no otherwise distinguished 
from each other than by the greater or less facility with which 
it can be applied to them. 

Many examples* of their consistent interpretation may be 
pointed out in geometry as well as in other sciences: thus, if 
+ aand — a denote two equal lines whose directions are op- 
posite to each other, then (cos § + “—1 sin 4) a may denote 
an equal line, making an angle @ with the line denoted by + a; 
and consequently a “ —1 will denote a line which is perpen- 
dicular to + a. This interpretation admits of very extensive 
application, and is the foundation of many important conse- 
quences in the application of algebra to geometry. 

The signs of operation + and — may be immediately inter- 
preted by the terms addition and subtraction, when applied to 
unaffected symbols denoting magnitudes of the same kind: if 
they are applied to symbols affected with the sign —, these 
signs, and the terms used to interpret them, become convertible. 
Thus a + (— 6) =a — 6, and a — (— 6b) = a + J; or the al- 

ebraical swm and difference of a and — 5, is equivalent to the 
algebraical difference and sum of a and b: but if they are applied 
to lines denoted by symbols affected by the signs cos 6 + ” —1. 
sin 6, and cos 6 + ” —1 sin @, the results will no longer de- 
note the arithmetical (or geometrical) swm and difference of the 
lines in question, but the magnitude and position of the dia- 
gonals of the parallelogram constructed upon them, or upon 
lines which are equal and parallelto = 
them. Thus, if we denote the line 
AB by a, and the line A C at right 
angles to it by 6 / — 1, and if we 
complete the parallelograms ABDC 
and ABCE, thena + 6 VW — | will 
denote the diagonal A D, and a — 6  —1 will denote the 
other diagonal B C, or the equal and parallel line A E. 
It is easily shown that a+b /W—1= V(@ + 5) (cos 4 
ae cos-la - 
+ 7% —1 sin $), (where 6 = Vien RB anda— b6 ¥—1 
= /(a? + 5) {cosé — / —1 sin 4} ; it follows, therefore, that 
' * Peacock’s Algebra, chap. xii. Art. 437, 447, 448, 449. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 227 


a+b/%—1 anda—bW—1 may be considered as repre- 
senting respectively a single line, equal in magnitude to 
(a? + b?)*, and affected by the sign cos# + /—1sin$@ in 
one case, and by the sign cos § — “ — 1 sin@ in the other; or 
as denoting the same lines through the medium of the opera- 
tions denoted in the one case by +, and in the other by —, 
upon the two lines at right angles to each other, which are de- 
noted by a and 6b W—1. 

We have spoken of the signs of operation + and —, as di- 
stinguished from the same signs when used as signs of affection, 
and we have also denominated a + 6 W—1, anda —b VW —1, 
the sum and difference of a and 6 “ —1, though they can no 
longer be considered to be so in the arithmetical or geometrical 
sense of those terms; but it is convenient to explain the mean- 
ing of the same sign by the same term, though they may be 
used in a sense which is not only very remote from, but even 
totally opposed to}, their primitive signification; and such a 
licence in the use both of signs and of phrases is a necessary 
consequence of making their interpretation dependent, not upon 
previous and rigorous definitions as is the case in arithmetical 
algebra, but upon a combined consideration of their symbolical 
conditions, and the specific nature of the quantities represented 
by the symbols. It is this necessity of considering all the re- 
sults of symbolical algebra as admitting of interpretation sub- 
sequently to their formation, and not in consequence of any 
previous definitions, which places all those results in the same 
relation to the whole, as being equally the creations of the 
same general principle: and it is this circumstance which jus- 


* The arithmetical quantity ,/(a?+ 6?) has been called the modulus of 
a + b,/—1 by Cauchy, in his Cours d’ Analyse, and elsewhere. It is the single 
unaffected magnitude which is included in the affected magnitude a+ 6 4/—1: 
conversely the affected magnitude (cos 6+ 4/—1 sin 6) 4/a? + 0? is reducible 


. 3 Seer 3 a 
to the equivalent quantity a + 6 ,/—1, if cos é= VeLe and therefore 
sin 6 Vere 
+ The sum of a and —4, or a + (— d), is identical with the difference of a 
and 6, or with a —b. The term operation, also, which is applied generally to 
the fact of the transition from the component members of an expression to the 
final symbolical result, will only admit of interpretation when the nature of the 
process which it designates can be described and conceived. In all other cases 
we must regard the final result alone. Thus, if a and 5 denote lines, we can 
readily conceive the process by which we form the results a + 6 and a — 5, at 
least when a is greater than b. But when we interpret a + b ,/—1 to mean a 
determinate single line with a determinate position, we are incapable of con- 
celying any process or operation through the medium of which it is obtained. 
Q2 


228. THIRD REPORT—1833. 


tifies the assertion, which we have made above, that quantities 
or their symbols affected by the signs +, —,or cos6+ “%—I. 
sin 6, are only distinguished from each other by the greater or 
less facility of their interpretation. 

The geometrical interpretation of the sign / — 1, when 
applied to symbols denoting lines, though more than once 
suggested by other authors, was first formally maintained by 
M. Buée in a paper in the Philosophical Transactions for 1806*, 
which contains many original, though very imperfectly deve- 
loped views upon the meaning and application of algebraical 
signs. In the course of the same year a small pamphlet was pub- 
lished at Paris by M. Argand, entitled Essai sur une Maniére 
de représenter les Quantités Imaginaires, dans les Construc- 
tions Géométriques, written apparently without any knowledge 
of M. Buée’s paper. In this memoir M. Argand arrives at this 
proposition, That the algebraical sum { of two lines {, estimated 
both according to magnitude and direction, would be the dia- 
gonal of the parallelogram which might be constructed upon 
them, considered both with respect to direction and magnitude, 
which is, in fact, the capital conclusion of this theory. This 
memoir of M. Argand seems, however, to have excited very 
little attention ; and his views, which were chiefly founded upon 
analogy, were too little connected with, or rather dependent 
upon, the great fundamental principles of algebra, to entitle 
his conclusions to be received at once into the great class 
of admitted or demonstrated truths. It would appear that 
M. Argand had consulted Legendre upon the subject of his me- 
moir, and that a favourable mention of its contents was made 
by that great analyst in a letter which he wrote to the brother 
of M. J. F. Frangais, a mathematician of no inconsiderable 
eminence. It was the inspection of this letter, upon the death 
of his brother, which induced M. Frangais to consider this 
subject, and he published, in the fourth volume of Gergonne’s 
Annales des Mathématiques for 1815, a very curious memoir 
upon it, containing views more extensive, and more completely 
developed than those of M. Argand, though generally agreeing 
with them in their character, and in the conclusions deduced 
from them. This publication led to a second memoir upon the 
same theory from M. Argand, and to several observations 
upon it, in the same Journal, from MM. Servois, Frangais, 
and Gergonne, in which some of the most prominent objections 
to it were proposed, and partly, though very imperfectly, an- 


* This paper was read in 1805. + La somme dirigée. 
t Lignes dirigées. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 229 


swered. No further notice appears to have been taken of these 
researches before the year 1828, when Mr. Warren’s treatise on 
the geometrical representation of the square roots of negative 
quantities * was published. In this work Mr. Warren proposes 
to give a geometrical representation to every species of quan- 
tity; and after premising definitions of addition, subtraction, 
multiplication and division, involution and evolution, which are 
conformable to the more enlarged sense which interpretation 
would assign to those operations when applied to lines repre- 
sented in position as well as in magnitude; and after showing 
in great detail the coincidence of the symbolical results obtained 
from such definitions with the ordinary results of arithmetical 
and symbolical algebra, he proceeds to determine the meaning 
of the different symbolical roots of 1 and — 1, when applied 
to symbols denoting lines, under almost every possible circum- 
stance. The course which Mr. Warren has followed leads 
almost necessarily to very embarrassing details, and perhaps, 
also, to the neglect of such comprehensive propositions as can 
only derive their authority from principles which make all the 
results of algebra which are general in form independent of the 
specific values and representation of the symbols: but at the 
same time it must be allowed that his conclusions, when viewed 
in connexion with his definitions, were demonstrably true; a. 
character which could not be given to similar conclusions when 
they were attempted to be derived by the mere aid of the arith- 
metical definitions of the fundamental operations of algebra. 
This objection to the course pursued by Mr. Warren will 
more or less apply to all attempts which are made to make the 
previous interpretations of algebra govern the symbolical con- 
clusions; for though it is always possible to assign a meaning 
to algebraical operations, and to pursue the consequences of 
that meaning to their necessary conclusions, yet if the laws of 
combination which lead to such conclusions are expressed 
through the medium of general signs and symbols, they will 
cease, when once formed, to convey the necessary limitations 
of meaning which the definitions impose upon them. It is for 
this reason that we must in all cases consider the laws of com- 
bination of general symbols as being arbitrary and independent 
in whatever manner suggested, and that we must make our in- 
terpretations of the results obtained conformable to those laws, 
and not the laws to the interpretations: it is for the same reason, 
likewise, that our interpretations will not be necessary, though 


: ‘s A Treatise on the Geometrical Representation of the Square Roots of Ne- 
gative Quantities, by the Rev.-John-Warren, M.A., Fellow and Tutor of Jesus 
College Cambridge. 1828. - 


230 THIRD REPORT—1833. 


governed by necessary laws, except so far as those interpreta- 
tions are dependent upon each other. Thus, if a be taken to 
represent a line in magnitude, it is not necessary that (cos 4 
+ 4/—1 sin6)ashould represent a line equal in length to the one 
represented by a, and also making ,an angle 6 with the line re- 
presented by a; but if (cos § + 4/—1 sin 6) a, may, consistently 
with the symbolical conditions, represent such a line, without 
any restriction in the value of 6, then, if it does represent such 
a line for one value of 9, it must represent such a line for every 
value of 4 included in the formula. It is only in such a sense 
that interpretations can be said in any case to have a necessary 
and inevitable existence. 

It is this confusion of necessary and contingent truth which 
has occasioned much of the difficulty which has attended the 
theories of the interpretation of algebraical signs. It has been 
supposed that a meaning could be transmitted through a suc- 
cession of merely symbolical operations, and that there would 
exist at the conclusion an equally necessary connexion between 
the primitive definition and the ultimate interpretation, as be- 
tween the final symbolical result and the laws which govern it. 
So long as the definitions both of the meaning of the symbols 
and of the operations to which they are required to be subject 
are sufficient to deduce the results, those results will have a 
necessary interpretation which will be dependent upon a joint 
consideration of all those conditions; but whenever an operation 
is required to be performed under circumstances which do not 
allow it to be strictly defined or interpreted, the chain of con- 
nexion is broken, and the interpretation of the result will be 
no longer traceable through its successive steps. This must 
take place whenever negative or other affected quantities are 
introduced, and whenever operations are to be performed, 
either with them, or upon them, even though such quantities 
and signs should altogether disappear from the final result. 

This principle of interpretation being once established, we 
must equally consider —1, /—1, cos# + —1 sin 9, as signs 
of impossibility, in those cases in which no consistent meaning 
can be assigned to the quantities which are affected by them, 
and in those cases only: and it must be kept in mind that the 
impossibility which may or may not be thus indicated, has re- 
ference to the interpretation only, and not to the symbolical 
result, considered as an equivalent form: for all symbolical 
results must be considered as equally possible which the signs 
and symbols of algebra, whether admitting of interpretation or 
not, are competent to express. But there will be found to be 
many species of impossibility which will present themselves in 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 231 


considering the relations of formule with a view to their equi- 
valence, and also under other circumstances, which will be in- 
dicated by such means as will destroy all traces of the equiva- 
lence which would otherwise exist. 

The capacity, therefore, possessed by the signs of affection 


involving / —1 of admitting geometrical or other interpreta- 
tions under certain circumstances, though it adds greatly to 
our power of bringing geometry and other sciences under the 
dominion of algebra, does not in any respect affect the general 
theory of their introduction or of their relation to other signs: 
for, in the first place, it is not an essential or necessary pro- 
perty of such signs; and in the second place, it in no respect 
affects the form or equivalence of symbolical results, though it 
does affect both the extent and mode of their application. It 
would be a serious mistake, therefore, to suppose that such inci- 
dental properties of quantities affected by such signs constituted 
their real essence, though such a mistake has been generally 
made by those who have proposed this theory of interpretation, 
and has been made the foundation of a charge against them by 
others, who have criticised and disputed its correctness*. 


* This charge is made by Mr. Dayies Gilbert in a very ingenious paper in 
the Philosophical Transactions for 1831, “‘ On the Nature of Negative and Im- 
possible Quantities.” He says that those mathematicians take an incorrect 
view of ideal quantities,—mistaking, in fact, incidental properties for those 
which constitute their real essence,—who suppose them to be principles of 
perpendicularity, because they may in some cases indicate extension at right 
angles to the directions indicated by the correlative signs + and —; for with 
an equal degree of propriety might the actually existing square root of a quan- 
tity be taken as the principle of obliquity, in as much as in certain cases it 
indicates the hypothenuse of a right-angied triangle. In reply to this last 
observation, it may be observed, that I am not aware that in any case the 
sign 4/— 1 has had such an interpretation given to it. 

It is quite impossible for me to give an abridged, and at the same time a fair 
view of Mr. Davies Gilbert’s theory, within a compass much smaller than the 
contents of his memoir. But I might venture to say that his proof of the rule 
of signs rests upon some properties of ratios or proportions which no arith- 
metical or geometrical view of their theory would enable us to deduce. In con- 
sidering, also, imaginary quantities as creations of an arbitrary definition, en- 
dowed with properties at the pleasure of him who defines them, he ascribes to 
them the same character as to all other symbols and operations of algebra; 


but in saying “that quantities affected by the sign ,/— 1 possess a potential 
existence only, but that they are ready to start into energy whenever that sign 
is removed,” he appears to me to assert nothing more than that symbols are 


impossible or not, according as they are affected by the sign ,/— 1 or not. 
Again, in examining the relation of the terms of the equation 


e+ f/®—-1l=(Y4+ fP—1*=y+ny" 1 vP—1 


pi fe Aarts tet ; 
eee Dp? ge a1) 4 EY. ah => yt 3 ye + &e., 


232 THIRD REPORT—1833. 


Signs of transition are those signs which indicate a change 
in the nature or form of a function, when considered in the 
whole course of its passage through its different states of ex- 
istence. Such signs, if they may be so designated, are gene- 
rally zero and tnfinity. 

Zero and infinity are negative terms, and if applied to desig- 


he denies the correctness of the reasoning by which it is inferred that the 
second term of the first, and the even terms of the second members of this 
equation are equal to one another (when « is less than 1), because they are 
the only terms which are homogeneous to each other, in as much as we thus 
ascribe real properties to ideal quantities; and he endeavours to make this 
equality depend upon an assumed arbitrary relation between x and y, though 
it is obvious that if y = cos 6, we shall find « = cos n 6, and that, therefore, 
this relation is determinate, and not arbitrary. A little further examination 
of this conclusion would show that it did not depend upon any assumed 
homogeneity of the parts of the members of this equation to each other, but 
upon the double sign of the radical quantity which is involved upon both 
sides. 

In arithmetical algebra, where no signs of affection are employed or recog- 
nised, both negative and imaginary quantities become the limits of operations ; 
and when this science is modified by the introduction of the independent 
signs + and — and the rule for their incorporation, the occurrence of the 
square roots of negative quantities, by presenting an apparent violation of the 
rule of the signs, becomes a new limit to the application of this new form of 
the science. The same algebraists who have acquiesced in the propriety of 
making the first transition in consequence of the facility of assigning a meaning 
to negative quantities, at the same time that they retained the definitions and 
principles of the first science, were startled and embarrassed when they came 
to the second; for it was very clear that no attempt could be made to recon- 
cile the existence and use of such quantities, consistently with the main- 
tenance of that demonstrative character in our reasonings which exists in 
geometry and arithmetic, where the mind readily comprehends the nature of 
the quantities employed, and of the operations performed upon them. The 
proper conclusion in such a case would be that the operations performed, as 
well as the quantities employed, were symbolical, and that the results, though 
they might be suggested by the primitive definitions, were not dependent 
upon them. If no real conclusions had been obtained by the aid of such 
merely symbolical quantities, they would probably have continued to be re- 
garded as algebraical monsters, whose reduction under the laws of a regular 
system was not merely unnecessary, but altogether impracticable. But it was 
soon found that many useful theories were dependent upon them; that any 
attempt to guard against their introduction in the course of the progress of 
our operations with symbols would not merely produce the most embarrassing 
limitations, when such limitations were discoverable, but that they would 
present themselves in the expression of real quantities, and would furnish at 
the same time the only means by which such quantities could be expressed. 
A memorable example of their occurrence under such circumstances presents 
itself in what has been called the irreducible case of cubic equations. 

In the Philosophical Transactions for 1778 there is a paper by Mr. Playfair 
on the arithmetic of impossible quantities, in which the definable nature of 
algebraical operations is asserted in the most express terms, and in which 
the truth of conclusions deduced by the aid of imaginary symbols is made to 
depend upon the analogy which exists between certain geometrical properties 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 283 


nate states of quantity, are equally inconceivable. We are ac- 
customed, however, to speak of quantities as infinitely great 
and infinitely small, as distinguished from jfinéte quantities, 
whether great or small, and to represent them by the symbols 
co and 0. It is this practice of designating such inconceivable 
states of quantity by symbols, which brings them, in some de- 


of the circle and the rectangular hyperbola. It is well known that the circle 
and rectangular hyperbola are included in the same equation y= »/(1 — 2”), 
if we suppose 2 to have 

any value between + © 

and — o: let a circle be 

described with centre C P I; 

and radius CA =1, and 

upon the production of 

this radius, let a rectan- cM [A 

gular hyperbola be de- 

scribed whose semiaxis is 

1, in a plane at right an- 

gles to that of the circle: 

if # denote the angle A C P, 

then the circular cosine and sine (C M and P M) are expressed by 

A, aT se Pb V/A a lie OWS 1 
—_____——___ an — 
2 2 /—1 

respectively; whilst the hyperbolic cosine and sine (to adopt the terms pro- 
posed by Lambert) corresponding to the angle 6 ,/—1 (in a plane at right 
angles to the former) are expressed by 

efpte—é fy pale b+e-o of _--8 

and /—1 a , or by 5 and 7 

if they be considered as determined by the following conditions; namely, 
that (hyp. cosine)? — (hyp. sine)’ = 1, and that hyp. cos 6 = hyp. cos — 4, 
and hyp. sine 6 = — hyp. sine — 6. A comparison of these processes in the 
circle and hyperbola would show, says Mr. Playfair, that investigations which 
are conducted by real symbols, and therefore by real operations, in the hy- 
perbola, would present analogous imaginary symbols, and therefore analogous 
imaginary operations in the circle, and conversely ; and that the same species 
of analogy which connects the geometrical properties of the circle and hyper- 
bola, connects the conclusions, of the same symbolical forms, when conducted 
by real and imaginary symbols. 

This attempt to convert an extremely limited into a very general analogy, 
and to make the conclusions of symbolical algebra dependent upon an insu- 
lated case of geometrical interpretation, would certainly not justify us in 
drawing any genera! conclusions from processes involving imaginary symbols, 
unless they could be confirmed by other considerations. The late Professor- 
Woodhouse, who was a very acute and able scrutinizer of the logic of ana- 
lysis, has criticised this principle of Mr. Playfair with just severity, in a paper 
in the Philosophical Transactions for 1802, ‘On the necessary truth of certain 
conclusions obtained by means of imaginary expressions.’? The view which he 
has taken of algebraical equivalence, in cases where the connexion between 
the expressions which were treated as equivalent could not be shown to be 
the result of a defined operation, makes a very near approach to the principle 


234 THIRD REPORT—1833. 


gree, under the ordinary rules of algebra, and which compels us 
to consider different orders both of ¢nfinities and of zeros, though 
when they are considered without reference to their symbo- 
lical connexion, they are necessarily denoted by the same sim- 
ple symbols oo and 0; thus there is a necessary symbolical di- 


stinction between (00 )3, co and (00 yi and between (0)2, 0 and 
(0); though when considered absolutely as denoting infinity 
in one case and zero in the other, they are equally designated 
by the simple symbols « and 0 respectively. 

Though the fundamental properties of 0 and ©, considered 
as the representatives of zero and infinity, are suggested by the 
ordinary interpretation of those terms, yet their complete in- 
terpretation, like that of other signs, must be founded upon the 


of the permanence of equivalent forms: thus, supposing, when 2 is a real 
quantity, we can show that 
a a3 


ew =1l+e¢+ 75+ pag t+ kes 


but that we cannot show in a similar or any other manner that 


VT ayaa . feet 
vA TIM! ae Tact are: aes 
then the equivalence in the latter case is assumed, by considering 7 dea 
as the abridged symbol for the series of terms 
— a a3 /— 1 
Re Day WD 


in other words, the form which is proved to be true for values of the symbols 
which are general in form, though particular in value, is assumed to be true 
in all other cases. 

It is true that such a generalization could not be considered as legitimate, 
without much preparatory theory and without considerable modifications of 
our views respecting nearly all the fundamental operations and signs of arith- 
metical algebra; but I refer with pleasure to this incidental testimony to the 
truth and universality of this important law, from an author whose careful 
and bold examination of the first principles of analytical calculation entitle 
his opinion to the greatest consideration. 

Mr. Gompertz published, in 1817 and 1818, two tracts on the Principles and 
Application of Imaginary Quantities, containing many ingenious and novel 
views both upon the correctness of the conclusions obtained by means of ima- 
ginary quantities and also upon their geometrical interpretation. The first 
of these tracts is principally devoted to the establishment of the following 
position: ‘‘ That wherever the operation by imaginary expressions can be 
used, the propriety may be explained from the capability of one arbitrary 
quantity or more being introduced into the expressions which are imaginary 
previously to the said arbitrary quantity or quantities being introduced, so 
as to render them real, without altering the truth they are meant to express ; 
and that, in consequence, the operation will proceed on real quantity, the 
introduced arbitrary quantity or quantities necessary to render the first steps 
of the reasoning arguments on real quantity, vanishing at the conclusion; 


+ &c.; 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 235 


consideration of all the circumstances under which they pre- 
sent themselves in symbolical results. In order, therefore, to 
determine some of the principles upon which those interpreta- 
tions must be made, it will be proper to examine some of the 
more remarkable of their symbolical properties. 


and from whence it will follow that the non-introduction of such can pro- 
duce nothing wrong.” Thus, 2? + az + b, which is equal to 


{VG+4y+VG-9} 
«{VG+5Y-VG-)} 


is also equal to 


(rari VE 18} 
x {Vergy te-VG ~o+e)h 


whatever be the value of the quantity 6; a conclusion which enables us to 

reason upon real quantities and to make 6= 0, when the primitive factors 

= =T —% =T 

are required. Similarly, if instead of ie ae 

%/B=1 —2,/p=1 y/1 —2%,/=1 

ee vee =y— R’, and if instead of AAT AMG 
2 2V/7—1 

e/ p-1 ft os V1 po 
2/p—1 

éVe-! = y—R!' + /B—1 (z— R), a result which degenerates into 


the well known theorem e VAT — y+ V—12,if6=0. Many other ex- 


amples are given of this mode of porismatizing expressions, (a term derived by 
Mr.Gompertz from the definition of porisms in geometry,) by which operations 
are performed upon real quantities which would be otherwise imaginary : 
and if it was required to satisfy a scrupulous mind respecting the correctness 

_of the real conclusions which are derived by the use of imaginary expressions, 
there are few methods which appear to me better calculated for this purpose 
than the adoption of this most refined and beautiful expedient. 

The second tract of Mr. Gompertz appears ta have been suggested by 
M. Buée’s paper in the Philosophical Transactions, to which reference has been 
made in the text : it is devoted to the algebraical representation of lines both 
in position and in magnitude, as a part of a theory of what he terms func- 
tional projections, and embraces the most important of the conclusions obtained 
by Argand and Frangais, with whose researches, however, he does not appear 
to have been acquainted. I should by no means consider the process of rea- 
soning which he has followed for obtaining these results to be such as would 
naturally or necessarily follow from the fundamental assumptions of algebra : 
but it would be unjust to Mr. Gompertz not to express my admiration of the 
skill and ingenuity which he has shown in the treatment of a very novel 
subject and in the application of his principles to the solution of many curious 
and difficult geometrical problems. 


= y, Wwe suppose 


=) 4, 


we suppose 


= « — R, we shall find, whatever 6 may be, 


236 THIRD REPORT—1833. 


If we assume a to denote a finite quantity, then 
(1.) a+O=a,andatwn=to. 


o 


Consequently 0 does not affect a quantity with which it is 
connected by the sign + or —, whilst «, similarly connected 
with such a quantity, altogether absorbs it. 

a a 

2.) ax0=0,ax@ =; 7 =~ and = 0. 

It is this reciprocal relation between zero and infinity which 
is the foundation of the great analogy which exists between 
their analytical properties. ; 

(3) If these symbols be considered absolutely by themselves, 
without any reference to their symbolical origin, then we must 
consider 9 = 1 and =1. 

0 00 

But if those symbols be considered as the representatives 

equally of all orders of zeros and infinities respectively, then 


0 


ve : - 
— and > may represent either | or a or 0 or ©, its final 


0 
form and value being determined, when capable of determina- 
tion, by an examination of the particular circumstances under 
which those symbols originated. ‘The whole theory of vanish- 
ing fractions will depend upon such considerations. 

Having ascertained the principal symbolical conditions which 
0 and © are required to satisfy, we shall be prepared to con- 
sider likewise the principle of their interpretation. The exami- 
nation of a few cases of their occurrence may serve to throw 
some light upon this inquiry. 

Let us consider, in the first place, the interpretation of the 


critical values 0, © and +. in the formule which express the 


values of « and y in the simultaneous equations, 
ax+by=e 
ae Oy =e 

In this case we find 


8 
ll 
) 
=} 
a 

S 
ll 


5 febt b Aaftedinny .0Ge 

Biol sept Wep Bilin GAs 
Goria yar ty ae sg NON Sect ete 
TT Power ;, and therefore pis 7, then @ = 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 237 


In this case a! = ma, b! = ma,and c = mc, and the second 
equation is deducible from the first, and does not furnish, there- 
fore, a new condition: under such circumstances, therefore, 
the values of x and y are really indeterminate, and the occur- 
0 
0 
sign, or rather the indication of that indetermination. . 


! 
If be not equal to a 


and y = ©. In this case we have a’ = ma, b/ = mb, but ¢’ is 
not equal to mc; and the conditions furnished are inconsistent, 
or more properly speaking tmpossible. In this case, the occur- 
rence of the sign o in the expressions for 2 and y is the sign 
or indication of this inconsistency or impossibility: and it should 
be observed that no infinite values of x and y, if the infinities 
thus introduced were considered as real existences and ¢denti- 
cal in both equations, would satisfy the two equations any more 
than any two finite values of x and y which would satisfy one 
of them. We may properly interpret 0 in this case by the 
term émpossible. 


! 
If = = ms but if BS be not equal to 4 then wx is zero and y 


rence of — in the values of the expressions for x and y is the 


! 
but if be equal to aa then 2 = o 


is finite, and therefore possible. It is in this sense that we 
should include zero amongst the possible values of x or y, a 
use or rather an abuse of language to which we are somewhat 
familiarized, from speaking of the zero of quantity as an exist- 
ing state of it in the transition from one affection of quantity to 
another. 

If we should take the equations of two ellipses, whose semi- 
axes are a and J, a’ and Bb’ respectively, which are 
2 


xv 4 
atepa)h 
x 2 
et peat 


and consider them as simultaneous when expressing the co- 
ordinates of their points of intersection, then we should find 


V{B — 8} V{a@ — a?) 


r= BF G2 and y= a? Ce ee 
ea? Be Bb? 
b 


If we suppose any ed or the ellipses to be similar, and at the 


same time 6 not equal to 0’, then x = © and y = ©, which 


238 THIRD REPORT—1833. 


would properly be interpreted to mean that under no cir- 
cumstances whatever, whether in the plane of 2 y or in the 
plane at right angles to it, in which the hyperbolic portions* of 
curves expressed by those equations are included, would a point 
of intersection or a simultaneous value of # and y exist: or in 
other words, the sign or symbol o would in this case mean 
that such intersection was zmpossible. If we supposed s = 4 
and also 6 = b’,or the ellipses to be coincident in all their parts, 
then we should find « = a and y = 2, indicating that their 
values were indeterminate, in as much as every part in the iden- 
tical curves would be also a point of intersection, and would fur- 
nish therefore simultaneous values. If we should suppose 3 
a a 
6 we? 


greater than 0’, a greater than a’, and then 


we should find 


not equal to 


xw«=aandy = 6 /—I, 
or x =a/—1 andy = 8, 
according as . is less or greater than > In this case, one 
ellipse entirely includes the other, but the hyperbolic portions 
at right angles to their planes, which are in the direction of 
the major axis in one case and in that of the minor axis in the 
other, will intersect each other at points whose coordinates are 
the values of 2 and y above given: it would appear, therefore, 
that the tmpossible intersection of the curves would be indi- 
cated by the sign or symbol o alone, and not by /—1. 

The preceding example is full of instruction with respect to 
the interpretation of the signs of algebra, when viewed in con- 
nexion with the specific values and representations of the sym- 
bols; and there are few problems in the application of algebra 
to the theory of curve lines which would not furnish the mate- 
rials for similar conclusions respecting them: but it is chiefly 
with reference to the connexion of those signs with changes in 
the nature of quantities, and in the form and constitution of ex- 
pressions, that their interpretations will require the most care- 
ful study and examination. We shall proceed to notice a few of 
such cases. 


2 2 
* If in the equation <5 “IB a = 1, we suppose y replaced by y W—1, 


and the line which it represents when not affected by “—1 to be moved 
through 90° at right angles to the plane of x y, we shall find an hyperbola 
included in the equation of the ellipse. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 239 


- The second member of the equation 


1 1 b b? 
As a a ak 


preserves the same form, whatever be the relation of the values 
of a and 5, and the operation, which produces it, is equally prac- 
ticable in all cases. As long asa is greater than b, a—b is po- 
sitive, and there exists, or may be conceived to exist, a perfect 
arithmetical equality between the two members of the equa- 


tion. If, however, a = b, we have 7 upon one side and the 


sum of an infinite series of units multiplied into | Upon the 


other, and both the members are correctly represented by 0; 
but if a be less than b, we have a negative and a finite value 
upon one side of the equation, and an infinite series of perpe- 
tually increasing terms upon the other, forming one of those 
quantities to which the older algebraists would have applied 
the term plus quam infinitum, and which we shall represent by 
the sign or symbol «. It remains to interpret the occurrence 
of such a sign under such circumstances. 


The first member of this equation z ‘ 3 is said to pass 


through infinity when its sign changes from + to —, or con- 
versely : its equivalent algebraical form presents itself in a se- 
ries which is incapable of indicating the peculiar change in the 
nature of the quantity designated by Peay 2 which accompa- 
nies its change of sign. The infinite values, therefore, of the 
equivalent series (for in its general algebraical form, where no 
regard is paid to the specific values of the symbols, it is still 
an equivalent form,) is the indication of the impossibility of ex- 


hibiting the value of z ul A in a series of such a form under 


such circumstances. 
Let us, in the second place, consider the more general series 
for (a — 6)", or . 
D0 (tame De ae 


MD psi see? Ue 


(abe =ardi—n.% ca 


n(n—1)(n—2) B 
ge de 


The inverse ratio of the successive coefficients of this series 


240 ’ THIRD REPORT—1833. 


approximates continually to — 1 as a limit, and the terms be- 
come all positive or all negative, according as the first negative 


coefficient is that of an odd or of an even power of es It follows, 


therefore, that if a be greater than 8, the series will be conver- 
gent and finite in all cases; if a be equal to 8, it will be 0, |, 
or ©, according as x is positive, 0, or negative; and if a be 
less than 4, it will be énjinite. 

The occurrence of the last of these signs or values is an in- 
dication generally that some change has taken place in the na- 
ture of the quantity expressed by (a — 6)", in the transition 
from a > bto a< b, which is of such a kind that the correspond- 
ing series is not competent to express it: thus, if m = 4, then 
(a — b)" is affected with the sign 4/—1 when a is less than 6, 
whilst no such sign is introduced nor introducible into the equi- 
valent series corresponding to such relative values of a and b: 
and a similar change will take place, whenever a transition 
through zero or infinity takes place. 

In this last case (a — 6)" would appear to attain to zero or in- 
jfinity, but not to pass through it, and no change would appa- 
rently take place in its affection corresponding to the change of 
affection of a — 6; but the corresponding series will under the 
same circumstances change from being finite to infinite, a cir- 
cumstance which we shall afterwards have occasion to notice, 
and which we shall endeavour to explain in the course of our 
observations upon the subject of diverging and converging 
series. 

In the preceding examples the sign or symbol «© has not 
presented itself immediately, but has replaced an infinite series 
of terms, whose sum exceeded any finite magnitude ; and it may 
be considered as indicating the zncompetence of such a series 
to express the altered state or conditions of the quantity or 
fraction to which it was required to be altogether, as well as 
algebraically, equivalent. In the examples which follow, it will 
present itself immediately and will be found to be the indica- 
tion of a change in the algebraical form of the term or terms in 
' which it appears, or rather that no terms of the form assigned. 
can present themselves in the required equivalent series or 
expression. 


n+ 
The integral /a"dax = = it C is said to fail when 
n = — 1, in as much as it appears that under such circum- 


a—1 


stances trey becomes ©, which is an indication that the va- 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. P41 


riable part of ford xis no longer expressible by a function 
grtt 


n+1 

by independent considerations. A knowledge, however, of the 

nature of its form in this particular case has enabled algebraists 

to bring it under a general form, by which the sign of failure 

or impossibility is replaced by the sign of indetermination 
; antl grt — gntl 

for if we put — pp C= 


under the form » but by one which must be determined 


o 5 + C, (borrowing 


a the | 
a wrt . . 
Paro from the arbitrary constant,) we shall get an expression 
which becomes iG when x = — 1, and whose value, determined 


0 
according to the rules which are founded upon the analytical 
properties of 0, will be log x + C. 

A more general example of the same kind, including the one 
which we have just considered, is given in the note to page 
211, where it is required to determine the general form of 

_ she EI ih 
dat ge NO Gar 
values of r: a formula is there constructed, from our knowledge 
of the form in the excepted case, which is capable of correctly 
expressing its value in all cases whatever. 

The cases in which the series of Taylor is said to fail are of 


a similar nature. Thus, if u =¢(«)=2+ Vx —a, then 
aah? Beh he 


Pes = pi ical igs Sl Mg alec : 
Re het =u + ook to eit duida2 aes 


P = (where z is a positive number) for all 


and if we suppose x = 4, all the differential coefficients ae 

F] 
= &c., become infinite, which is an indication that no terms 
of such a form exist in its developement, which becomes, under 
such circumstances, a + /h. The reasons of this failure in 
such cases have been very completely explained by Lagrange 
and other writers; but it is possible, by presenting the deve- 
lopement which constitutes T'aylor’s series under a somewhat 
different and a somewhat more general form, that the series may 
be so constructed as to include all the excepted cases. 

There are two modes in which the developement of ¢ (a + h) 
according to powers of h may be supposed to be effected. In 
the first and common mode we begin by excluding all those 


Be x the developement whose existence would be incon= 
3 R 


242 THIRD REPORT—1833. 


sistent with general values of the symbols: in the second we 
should assume the existence of all the terms which may cor- 
respond to values of the symbols, whether general or specific, 
and then prescribe the form which they must possess, con- 
sistently with the conditions which they are required to satisfy. 
If we adopt this second course, and assuming u = ¢ (x) and 
u’ = 9(« + h), if we make 


w=utAh?+Bh+ Ch + &c., 


the inquiry will then be, if there be such a term as A h*, where 
A is a function of x or a constant quantity, and a is any quantity 
whatsoever, what are the properties of A by which it may be 
determined? For this purpose we shall proceed as follows. 
It is very easy to show, from general considerations, that 
if «’ be considered successively as a function of x and of h, 
dul , 

— = wes for all values of 7, whether whole or fractional, 
positive or negative: it will follow, therefore, (adopting the 
principles of differentiation to general indices which have been 
laid down in the note, p. 211,) that 


dtu F(1 + @) een r(1 + bd) 
aa TET, Se r(il+6—a) 
omitting the arbitrary complementary functions, which will in- 
volve powers of k. Ina similar manner we shall get 
ia a a ne Coe 
dat dat t dat ‘sian dx* 
If these results be identical with each other, we shall find 


‘BAb-* + @e; 


h® + &e. 


1 dea. A 
and, therefore, A = rato)’ ae since I'(1) = 1, It is easy 


to extend the same principle to the determination of the other - 
coefficients, and we shall thus find 


y dtu, he au : he 
u=u+ 7e¢ Pada) da Td+d) 
or, in other words, it follows that the coefficient of any power 
of h whose index is r will be 
1 du 
ri +r) da" 


+ &e.; (1.) 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 243 


The next step is to adapt the series (1) to the different cases 
which an examination of the constitution of the function w will 
present to us. 

If we suppose x to possess a general value, then w! and w 
will possess the same number of values, and no fractional 
power of / can present itself in the developement. In this case 
r(1+a)=1.2... a, and it may be readily proved that 
the successive indices a, b, c, &c., are the successive numbers 
1, 2, 3, &c., and that consequently, 

a du du Bu he 
da? 1.2" dw 1.2.3 
It will also follow that the series for uw’ can involye no negative 
and integral power of 4; for in that case the factorial I'(1 + a), 
which appears in its denominator, would become o, and the 
term would disappear. If it should appear, also, that for spe- 
cific values of 2 any differential coefficient and its successive 
values should become infinite, they must be rejected from the 
developement, in as much as in that case the equation 
d*u 

ri+aA= dat 
would no longer exist, which is the only condition of the intro- 
duction of the corresponding terms. In other words, those 
terms in the developement of u' must be equally obliterated, 
which, under such circumstances, become either 0 or . 

If the general differential coefficient of u could be assigned, 
its examination would, generally speaking, enable us to point 
out its finite values wherever they exist, for those specific va- 
lues of the symbols which make the integral differential coeffi- 
cients zero or infinity. For all such values there will be a cor- 
responding term in the developement of w’ under those cireum- 
stances. Thus, if we suppose wu = « a — x, we shall find 


re ce) [re-9-G-yeh 


if we make x = a, this expression will be neither zero nor 


+ &c. 


infinity in two cases only, which are when r = * and when | 


a: ae 
‘=>: in the first case we get, 


244 THIRD REPORT—1833. 
and in the second we get, 
— 38 
1 3 = » oe s 
r(5) 4-3 
2) dx 21 x0x(@—a) 
since '(1) = 1 = 0 F'(O), and the symbol 0 in the denominator 
3 


= —1; 


aa aeag is a stmple zero. The corresponding developement 


of u’ under such circumstances is 
A eh? tla B*, 
a result which is very easily verified. 

If we pay a proper regard to the hypotheses which deter- 
mine the existence of terms in the series for w for specific 
values of the independent variable, we shall be enabled without 
difficulty to select the indices of the differential coefficients 
which can present themselves amongst the coefficients of the 
different powers of / in the developement. For, in the first 


place, 4», and the differential coefficient whose index is = will 
possess the same number of values, and the same signs of affec- 


™m 
tion. If there be a term in w which = P (x — a)", where P 
neither becomes zero nor infinity, when x = a, and where the 
multiple values of P, if any, are independent of those contained 


d= .P. (« — aje 
d an 
which is independent of (a — ayn is P. a and that 
dun 


in (« — ar, then it will appear that the term of 


m™ 

dn*u 

m 
dxn 
x =a, or, if finite, introducing, through the medium of the 
factorial function by which they are multiplied, multiple values 
which are greater in number than those contained in w’, must 
be rejected, as forming no part of the developement. It will of 
- course follow, that the function P will become, under such cir- 
cumstances, a function of h, and if we represent it by P’, and 
denote its values, and those of its successive differential coeffi- 
cients, when h = 0, by p, p’, p", pl”, &c., we shall find 


P= 7 1 h* mw he 
=ptph+ pap" 7-9-3 + ke. 


all the other terms of ,» being either zero or infinity when 


REPORT ON CERTAIN BRANCHES OF ANALYSIS, 245 


none of which become zero or infinity, in as much as P does not 
vanish when x = a. 
If there exist other terms in zu of a similar kind, such as 
, mm" 


m 
Q (x — 5)", R (w — c)””, &c., the same observations will apply 
to them. Such terms will correspond to values of x, which 
make radical expressions of any kind zero or infinity, and the 
form of the function « must be modified when necessary, so 
that such radicals may present themselves in single terms of 


‘ ™ ; 
the form P (« — a)". The same observations will apply to ne- 
gative as well as positive values of Fe unless we suppose a 


negative whole number. The principle of the exception in this 
last case may be readily inferred from the remarks in the note, 


p- 211, on the subject of the values of ae a = when 7 isa 
whole number. If we suppose, therefore, « to involve terms 
m m! 


such as P (x — a)", Q (n — 5)”, &c., the most general form 
under which its developement can be put, supposing all terms 
which become zero or infinity for specific values of x to be 
rejected, will be as follows: 


fu hh d? u h3 


Bi wick ~du La ie tra 
Bar Mitigate rene ue 10a. i 
eae meee 
Wee r(i+” 
n 
pale ane ae - 
+ .Q). — — ¢ &e. : 
za«—b oy ate ‘agent ; 


or, 
du d?u_ h? Bu .h 

‘a pers 6 Sent Ape aaa ae es 

int ga iaet 1 oae Te a ree 


ig ee 2 


a ' h m 
+ “(pt phtp. 5+ Sic.) 


tI— 


b—b h? feed 
+ Fos (gt gata’ + Se. ) he 


+ &e. 


-s s ! u i 4 a= 
We have introduced the discontinuous signs or factors . 


cm & 


246 THIRD REPORT—1833. 


b6—b 
zx~b’ 
but which are zero for all other values of 2, to show that the 
terms into which they are multiplied disappear from the deve- 
lopement in all cases except for such specific values of a. 

The existence of the terms of the series for w’ is hypothetical 
only, and the equation which must be satisfied, as the essential 
condition of the existence of any assigned hypothetical term, at 
once directs us to reject those terms which would lead to infi- 
nite values of the differential coefficients, as well as those which 
possess multiple values which are incompatible with those con- 
tained in w’. It is quite obvious that upon no other principle 
could we either reject such infinite values, or justify the con- 
nexion of a series of terms with the general form of a’, which 
have no existence except for specific values of 2 The con- 
clusion obtained is of considerable importance, in as much as it 
shows that the series of Taylor, if considered and investigated 
as having a contingent, and not a necessary existence, may be 
so exhibited as to comprehend all those cases in which it is 
commonly said to fail: and it will thus enable us to bring under 
the dominion of the differential calculus many peculiar cases in 
its different applications which have hitherto required to be 
treated by independent methods. 

Thus, if it was required to determine the value of the fraction 


( pe all a’)? 


&c., which become equal to 1 when 2 = a or a = 8, &e., 


, when x = a, we should find it to be, 


x? (a — a)? 
dt 
oo cae Lae a’)? 
d x? 
di Ms 
++ 2? (a2 — a)? 
"0 ( ) 
or, 
dz 
(# + a)?- (« — a)? 
+ a 
a = @a? 4/20; 
di? a? 


a conclusion which would be justified by the developement of 
the numerator and denominator of this fraction by the complete 
form of Taylor’s series, when 2 = a. 

Many delicate and rather obscure questions in the theory of 
maxima and minima, particularly those which Euler has deno- 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. Q47 


minated maxima and minima of the second species, and others 
also relating to the singular or critical points of curve lines, 
must depend for their dilucidation upon this more general view 
of Taylor’s series, as connected with the consideration of ge- 
neral differential coefficients *. 


* Euler has devoted an entire chapter of his Calculus Differentialis to the 
examination of what he terms the differentials of funciions in certain peculiar 
cases. It is well known that he adopted Leibnitz’s original view of the prin- 
ciples of the differential calculus, and considered differentials of the first and 
higher orders as infinitesimal values of differences of the first and higher orders. 

uch a principle necessarily excludes the consideration of differential coefficients 
as essentially connected with determinate powers of the increment of the inde- 
pendent variable, which may be said to constitute the essence of Taylor's 
theorem, and which must be the foundation of all theories of the differential 
calculus, which make its results depend upon the relation of forms, and not 
upon the relation of values. As long, however, as the independent variable 
continues indeterminate, the symbolical values of the differentials are the same 
upon both hypotheses. But when we come to the consideration of specific va- 
lues of the independent variable which make differential coefficients above or 
below a certain order, infinite or zero, then such a view of the nature of dif- 
ferentials necessarily confounds those of different orders with each other. Thus, 


if y= a? + (« — a), Euler makes, when r=a, dy=(d x)=, instead of 
dy ay ‘ 

(f/9\= 8 , = (a)?. fy=2axr— 2 + a (a — 2%), he makes, 

MG) a” 


when « = a, dy = a ./— 2a. da, instead of 


a? (d a) {* /2 L. =1\ 

pe a +av 

These examples are quite sufficient to make manifest the inadequacy of 
merely arithmetical views of the principles of the differential calculus to ex- 
hibit the correct relation which exists between different orders of differentials, 
and, @ fortiori, therefore, between different orders of differential coefficients. 

M. Cauchy, in his Legons sur le Calcul Infinitesimal (published in 1823), has 
attempted to conciliate the direct consideration of infinitesimals with the purely 
algebraical views of the principles of this calculus, which Lagrange first securely 
established; and it may be very easily conceded that no attempt of this able 
analyst, however much at variance with ordinary notions or ordinary practice, 
would fail from want of a sufficient command over all the resources of analysis. 
He considers ail infinite series as fallacious which are not convergent, and that, 
consequently, the series of Taylor, when it takes the form of an indefinite series, 
is not generally true. It is for this reason that he has transferred it from the 
differential to the integral calculus, and exhibits it as a series with a finite 
number of terms completed by a definite integral. It is very true that M. Cauchy 
has perfectly succeeded in dispensing with the consideration of infinite series in 
the establishment of most of the great principles of the differential and integral 
calculus; but I should by no means feel disposed to consider his success in over- 
coming difficulties which such a course presents as a decisive proof of the expe- 
diency of following in his footsteps. The fact is, that if the operations of algebra 
be general, we must necessarily obtain indefinite series, and if the symbols we 
employ are general likewise, it will be impossible to determine, in most cases, 


dty= 


248 THIRD REPORT—1833. 


Signs of discontinuity are those signs which, in conformity 
with the general laws of algebra, are equal to 1 between given 
limits of one or more of the symbols involved, and are equal to 
zero for all their other values. If merely conventional signs 
were required, we might assume arbitrary symbols for this 
purpose, attaching to them far greater clearness as diventical 
marks, the limits of the symbol or symbols between which the 
sign of discontinuity was supposed to be applied. Thus, we 
might suppose 7D,” to denote 1, when x was taken between 
0 and a, to denote zero for all other values; 7D, ,%, to denote 1, 
when x was taken between a and a + 3, and zero for all other 
values ; and similarly in other cases. 

Thus, if y= a+ Band y = a’ x + 6! were the equations 
of two lines, and if we supposed that the generating point whose 
coordinates are x and y was taken in the first line between the 
limits 0 and a, and in the second line between the limits a and 8, 
then we should have generally, 


y= "D,; (ax + B) ph °T),” (a! x + B’) (1.) 


the convergency or divergency of the series which result. It is only, therefore, 
when we come to specific values that a question will arise generally respecting 
the character of the series: and it is only when we are compelled to deduce the 
function which generates the series from the application of the theory of limits 
to the aggregate of a finite number of its terms, that its convergency or diver- 
gency becomes important as affecting the practicability of the inquiry : in short, 
it must be an erroneous view of the principles of algebra which makes the result 
of any general operation dependent upon the fundamental laws of algebra to be 
fallacious. The deficiency should in all such cases be charged upon our power 
of interpretation of such results, and not upon the results themselves, or upon the 
certainty and generality of the operations which produce them: in short, the 
rejection of diverging series from analysis, or of such series as may become 
divergent, is altogether inconsistent with the spirit and principles of symbolical 
algebra, and would necessarily bring us back again to that tedious multipli- 
cation of cases which characterized the infancy of the science. A very instruc- 
tive example of the consequences of adopting such a system may be seen in the 
researches of M. Liouville, which have been noticed in the note at p. 217. 

Lagrange in his Théorie des Fonctions Analytiques, and in his Calcul des 
Fonctions, has given theorems for determining the limits between which the 
remainder of Taylor’s series, after a finite number of terms, is situated: and 
the same subject has been very fully discussed in a memoir by Ampére, in the 
sixth volume of the Journal de Ecole Polytechnique. Such theorems are ex- 
tremely important in the practical applications of this series, but they in no 
respect affect either the existence or the derivation of the series itself. Itisa 
very common error to confound the order in which the conclusions of algebra 
present themselves, and to connect difficulties in the interpretation and appli- 
cation of results with the existence of the results themselves: and it is the in- 
fluence of this prejudice which has induced some of the greatest modern ana- 
lysts, not merely to deny the use, but to dispute the correctness of diverging 
series. 

Messrs. Swinburne and Tylecote, the joint authors of a Treatise on the true 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 249 


_ Thus, if in the triangle A C B, we 
draw C D, a perpendicular from the 


ms a 
vertex to the base, and if we suppose et 
AD=a, AB= 4, A the origin of a 
the coordinates, A B the axis of 2, yan beni 

o 1 ‘ 
4 ’ 


y = a x the equation of the line AC, Ke 
and y = a! « + 6! the equation of the 4 B 
line B C, then we should find that the value of y represented 
by the equation 
y= “D,°.ax2 + °D;" (a! a + B')* (2.) 

would be confined to the two sides A C and B C of the triangle 
A BC, excepting only the point C, which corresponds to the 
common limit of the discontinuous signs, For if we suppose 
*D,” and *D,’ to be true up to their limits, we shall find, when 
& = a, that 7D,’ + 7D," = 2. If we replace, however, 


2 £ a 2T)y.2 avT).@ b—5 
D, by ..— aa and D; by D; aienet 


a= 
ta 


Developement of the Binomial Theorem, which was) published in 1827, have 
contended vigorously for the restriction of the meaning of the sign = to simple 
arithmetical equality, and would reject its use when placed between a function 
and its developement, unless its complete remainder, after a finite number of 
terms, should replace the remaining terms of the series; or unless, when the 
indefinite series was supposed to be retained, the value or the generating func- 
tion of this remainder could be assigned. In conformity with this principle 
they have assigned the remainder in the series for (a + x)", which they exhibit 
under the following form : 
“(a+ a)*=a%®+na"-le+.... nie LL chelate cute Wan +) gn-r ge 
1 (r + 1) a 

1 Om cas Sens, wala Pil ey Aenea EP doy 

Fat ©) @int ja) aaa 1 ‘(@+part2 


(r+) (7% +2)...(2—1) a®-7-1) | 
the remainder being (a + 2)" 27 +1 multiplied into x — r terms of the deve- 


1 1 

{@+a)— ape t 8 ear 

The method which they have employed for this purpose, which is extremely 
ingenious, succeeds for integral values of m, whether positive or negative, but 
fails to assign the law when the index is fractional. But my own views of the 
principles of symbolical algebra would, of course, induce me to attach very little 
value to results which were exhibited in such a form as to be incapable of being 
generalized, a defect under which the formula given above evidently labours. 


_ * The conventional sign 7D,7 might be replaced, though not with perfect 
sre 4 1 
propriety, by the definite integral -—> S, "de. 


lopement of 


250 THIRD REPORT—1833. 


and if we make, therefore, 
4 i ot gray ‘, 
y=}ps— Sars }p, —Jaih@e +e) (3.) 


the equation will be true for the ordinate of every point of the 
sides A C and CB of the triangle A BC. 

More generally, if we suppose y = 9, 7, ¥ = 22, ¥ = $34, 
y = 9, x, &c., to be the equations of a series of curves, then the 
equation of a polylateral curve composed of the several portions 
of the separate curves corresponding to values of «, included 
between the limits a and b, b and c, ¢ and d, &c., would be, 


a b—b r oe 
y= \7D: — 2S) ee + (*DP-2=*) ge 


r 


a. (De a g = ‘) Q,2 + &e.; (4.) 
the value of the ordinate at each successive limit being replaced 
by that of the succeeding curve. In this manner, if we should 
grant the existence of the sign of discontinuity, we should be 
enabled to represent the equations of polygons, and of poly- 
lateral curves of every description. 

It remains to consider the nature of the expressions which 
are competent to express 7D;". 

The expressions which have been generally proposed for this 
purpose are either infinite series, or their equivalent definite 
integrals. Le Comte de Libri, however, a Florentine analyst 
of distinguished genius, has proposed * a finite exponential ex- 
pression which will answer this purpose. The examination of 
the expression 


ellog 0) log 0) (v — a) 


would readily show that its value is 1 when z is greater than 
a, and that it is 0 when 2 is equal to or less than a. It will 
therefore follow that the product 


e (log 0) 2098 (#~ a) ye log 0) 2080) 0-2) 


is equal to 1 between the limits @ and 4, and is equal to 0 at 
those limits, and for all other values. And, in as much as 


* Mémoires de Mathématigue et de Physique, p. 44. Florence 1829. The 
author has since been naturalized in France, and has been chosen to succeed 
Legendre as a member of the Institute: he has made most important additions 
to the mathematical theory of numbers. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 251 
208°) = 0, we may replace the preceding product by the equi- 
valent expression 

(v= a) (6 = 2) 

QS eravhaxo8 avout 
a—a b—b 
t—a «—b’ 
has been applied by Libri to the expression of many important 
theorems in the theory of numbers*. 


G0 
The definite integral BA nae sin r has been shown by 


This expression, which is equivalent to 7D,” — 


Euler} and many other writers, to be equal to = when ~ is 


when z is negative. It 


— 7 


positive, to 0 when 2 is 0, and to 3 


follows, therefore, that 


2 fdr. (b—a) f G+9 | 
=f a ee oe rT COS fev lniagial Tr 


) = 1 war. 1 Qdr. 
| =~, Taine — art — J = sin (@ — b)r 


is equal to 1, when z is between the limits a and 8, to = when ats 
at those limits, and to zero, for all other values. If we denote the 
eae 2 f®dr . (b—a) { (a + b) 
definite integral oy os sin —3— r cos4 a = est ? 
by C;”, we shall get, 
nae ey a—@ Saye 

ihe to vides toe CL -amt) Bws EBLE 
and consequently the equation of a polylateral curve, suchas 
that which is expressed by equation (4), will be, 

y=Cr. 92+ C.. Ge+Ci. a2 4+&e., 
in as much as at the limits we have $, (b) = 9, (6), @2 (c) = 95 (¢), 


and consequently for such limits Cy” 9, (2) + C,? 9, (b) = 9, (6 
= 9, (b), and not 2 ¢, (5). a ie 

All definite integrals which have determinate values within 
given limits of a variable not involved in the integral sign, may 
be converted into formule which will be equal to { within those 


* Creile’s Journal for 1830, p. 67. 

+ Inst. Calc. Integ., tom. iv.; Fourier, Théorie de la Chaleur, p.442.; Frul- 
loni, Méemorie della Societé Italiana, tom. xx. p. 448, ; Libri, Mémoires de Ma= 
thématique et de Physique, p. 40. HB OO y 


252 THIRD REPORT—1833. 


limits and also including the limits, and to zero for all other 
values *. But the expressions which thence arise, though fur- 
nishing their results in strict conformity with the laws of sym- 
bolical combinations, possess no advantage in the business of 
calculation beyond the conventional and arbitrary signs of dis- 
continuity which we first adopted for this purpose: but though 
it is frequently useful and necessary to express such signs ea- 
plicitly, and to construct formule which may answer any as- 
signed conditions of discontinuity, yet such conditions will be 
also very commonly involved implicitly, and their existence and 
character must be ascertained from an examination of the pro- 
perties of the discontinuous formule themselves. We shall now 
proceed to notice some examples of such formule. 
The well known series + 


© é {Vs ee bey 
seth een Pa srentyhl ot 
ret>= sing 5g singe + 3 sind a q sin 4a + &e. (1.) 


is limited to integral values of r, whether positive or negative, 


and to such values of ra + = as are included between *. and 


— = the value of r, therefore, is not arbitrary but condi- 


* If a definite integral (C) has determinate values @,, a, .. - én, within 
the limits of the variable a and b, and no others, the values at those limits 
being included, and if C be equal to zero for all values beyond those limits, 
then we shall find 


apy tie Cra Ee a): oc Se 2g 


a, X tg XK. an 


thus in the case considered in the text, we get 
*p,? = —2(C—1) (c— +) pl = aCe ee 


+ The principle of the introduction of r x in equation (1.) by which it is ge- 
neralized, will be sufficiently obvious from the following mode of deducing it: 


2 Vai * we 
log {tere fome hee} ta 4 ee 
e 
Oe reat et oem /Leae senda aa 
—s{eev ingoaenoth $i pe Vay seat in ad 


and, therefore, dividing by 2 ./—1, and replacing the exponential expressions 
by their equivalent values, we get 


e : a. In. ‘a 
vr. — =sinz — — sin 22+ — sin3«—— sin42+ &c., 
’ a 2 2 a 3 4 + 


where a upon the second side of the equation may have any value between 
+ © and —o, 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 258 


tional. If we successively replace, therefore, x by e + x and 


Fi x, we shall get 


- 
* . Zz Th. 1 
Ae Sy rs gt ostt se sin2v — 3 cos 3 # 
— sin da + &e. 


“~ iy 1 
ran+——-—=cosr— > sn2r — > cos3r 


412 2 3 
-. = sin 4a + &c. 
Adding these two series together and dividing by 2, we get 


CF) + Zs cose— | cos 324 pe ean (2.) 


4 3 3 

If x be included between + and — = then r = 0 and x’ = 0, 
and we get 

og 1 1 

q = cose — gcosda + — cos 5a — &e, (3.) 
If z be included between + and oe) thenr = — 1 andr’ = 0, 
and we get 

sew J cos 82 + + cos 52 —& 4. 

BP gh ES ap ene e+ — cos Sax Cc. (4.) 

tan 37 5a Of it —7 
If the limits of z be a and Ta and ae ree and 
=i a — _ and — a we shall obtain values of the series 
(2.), which are alternately *- and — A, f 
~ Again, if in equation (1.), or ra + s = sing — a sin2 x 
+ ania PF toa fs sin 4.2 + &c., we replace « by r — 2, we 


shall get 
Ee | OL) ae 
ee + 5 = sing + g sin2a + 3g sind.2 + qsindae + &e, 


Adding these equations together and dividing by 2, we get 


254 THIRD REPORT—1833. 


(x + 7’) a Pe Te Te We 
a t+ 7 = sine + 3 sind2 + = sind x + &e. (5.) 
which may be easily shown to be equal to = and — = alter- 


4 4 
nately, in the passage of x from 0 to 7, from 7 to 27, from 27 
to 3 7, &c., or from 0 to— 7, from — ¢ to —27, &e.: its 
values at those limits are zero. 

The series (2.) and (5.) have been investigated by Fourier, in 
his Théorie de la Chaleur *, by a very elaborate analysis, which 
fails, however, in showing the dependence of these series upon 
each other and upon the principles involved in the deduction 
of the fundamental series: and they present, as we shall now 
proceed to show, very curious and instructive examples of dis- 
continuous functions. 

x 
2 
Q A P, making an angle with the axis of x, whose tangent is 


The equation y = — is that of an indefinite straight line, 


io 


1 Phy. ; 
@? and which passes through the origin of the coordinates : 
whilst the equation 


y = sine — 3 sind zt + + sine = os sin 42 + &c. 
is that of a series of terminated straight lines, d'c, dC, DC, 
&e., passing through points a, A, A’, &c., whieh are distant 
27 from each other: the portion d C alone coincides with the 


primitive line, whose equation is y = 3 


- Again, the line whose equation is y = is parallel to the 


| 


| 
> 
a 


ea 
es 
ar 
Ls) 


* From page 167 to 190; also 267 and 346. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 255 


axis of x at the distance 7 above it: the line whose equation is 


eg a is also parallel to the axis of x, at the distance * below 


4 
it: the line whose equation is 


1 4 1 
y= cost — 3 cos 3a + — cos 5a — &e. 


consists of discontinuous portions of the first and second of 
those lines, whose lengths are severally equal to. The values 
as 
2 
are equal to zero, since the equidistant points D and C, ¢ and d, 
are common to both equations at those points. 

* It would appear, therefore, in the cases just examined, that 
the conversion of one member of the equation of a line into a 
series of sines and cosines would change the character of that 
equation from being continuous to discontinuous, the coinci- 
dence of the two equations only existing throughout the ex- 
tent of one complete period of circulation of the trigonometrieal 
series: and more generally, if, in any other case, we could ef- 
fect this conversion of one member of the equation of a curve 
into a series of sines or cosines, it is obvious that the second 
equation must be discontinuous, and that the coincidence 
would take place only throughout one period of circulation, 
whether from 0 to z or from — z to > It remains therefore 


of y at the points B and 8, corresponding to z = — and — oy 


to consider whether such a conversion is generally practicable. 
Let us take equidistant points in the axis of the curve 

whose equation is y = ¢ 2, between the limits 0 and, those 

limits being excluded: if we denominate the correspondin 

values of the ordinate by y;, Ya, +++» Yn and if it be ee 

to express the values of these ordinates by means of a series 

of sines (of z terms) such as 

a, sinz+a,sin2r4+a;,sn3a+....+a,snuez, 


then we shall get the following z equations to determine the 
coefficients @,, Ao, 3. +--+» An 


T SOME . oT Sli he 
ioe apes rg Pile yo ae neha ere noe telat 
5 res . Ag bh ,.2n 
ola ne aa Pe a ar st Oe aren 

eit S pki: 5 aaa EN . Ont. 
Y¥3 >= a, <a +1 + G2 ea 1 + a3 LL ears | rhe On SUD Sr oe |? 


256 THIRD REPORT—1833. 


. . . . . . . . . . . . . . . . . . . . 


. ne - 2nn . Ont - er 
Yn i Vea aT + Gy SIN > ro | ae eau IEE | +.» Oy sine a, 
If any assigned coefficient a, be required to be determined 
from this system of equations, we must multiply * them seve- 
rally by 


Qma . dm 
——_ in ——.,. 
n+l 


when all the coefficients except a,, will disappear from the sum 
of the resulting equations: and we shall thus find 


nme 
n+’ 


_ mn : 
2sin 74 -. 2sin 


Be pad { eam sin 2m _ nme 
OF Apa Qa ab iF meds 5a Bice | - 


It would thus appear that it is always possible to determine a 
series of sines of m terms with finite and determinate coeffi- 
cients, which shall be the equation of a curve which shall have 
m points in common with the curve whose equation is y = ¢ 2, 
within the limits corresponding to values of x between 0 and 7; 
and it is obvious that the greater the number of those points, 
the more intimate would be the contact of these two curves 
throughout the finite space corresponding to those limits. If 
we should further suppose the number of those points to be- 
come infinitely great, then the number of terms of the trigono- 
metrical series would be infinite likewise, and the coincidence of 
the curve which it expresses with the curve whose equation is 
y = > x, would be complete within those limits only, producing 
a species of contact to which the term finite osculation has been 
applied by Fourier +. Beyond those limits the curves would 
have no necessary relation to each other. 

It would follow, also, from the preceding view of the theory of 
finite osculations, that the curve expressed by y = $ x might be 
perfectly arbitrary, continuous, or discontinuous. Thus, it might 
express the sides of a triangle, or of a polygon, or of a multi- 
lateral curve, or of any succession of points connected by any 
conceivable law ; for in all cases when the corresponding or- 
dinates of equidistant points are finite, we shall be enabled to 
determine values of the coefficients a,, which are finite or zero 
by the process which has been pointed out above. 


* This is the process proposed by Lagrange in his ‘‘ Théorie du Son,” in 
the third volume of the Turin Memoirs, as stated by Poisson in his memoir 
on Periodic Series, &c., in the 19th cahier of the Journal de l’ Ecole Polytech- 
nique. 

t Théorie dela Chaleur, page 250. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 257 


__ The hypothesis of » being infinite would convert the series 
for a, into the definite integral * 
4 


er eae te 
=f, gxsinmxrdx, 
a 


Mm 7 ; : . « 
= 2 and =: d x: or otherwise if we 


n+ 1 n+l 
assume the existence of the series 


if we make 


Px =a,snz+az,sin2xrx+...admsinmn + &e., 


it may be readily shown, by multiplying both sides of the equa- 
tion by sin m «x d x, that 


2 
= — x“sinm«xdx: 
An JT () 


and in a similar manner, if we should assume 


$x =a,cosOxr + a,cosxr +...d,cosmax + &e., 
that 


2 0 ; 
On = =f; gxcosmaxdxt. 
Thus, if we should suppose ¢z = cos x, we should find 


sin 6 a + &e.| 


bs 


6 
1.3 3.5 5.7 


a very singular result, which is of course only true between the 
limits 0 and z, excluding those limits {. 


cose = {7 gsin 2 + cs sin 4a -+ 


If we should suppose ¢ z = a constant quantity = between 


the limits 0 and a, and that it is equal to sero between « and 7, 
we should find 


| (= Cosa). 50> (1 —'cos 22) «. © cos 3 2) 

oe a oe a ee + ag ae 
sn32r+ &e, 

excluding the limiting value «, when the value of the series is 


only = §. 


If we should suppose ¢ 2 = *D,’. ax + *D,”. (ex + 6), 
which is the equation of the sides of a triangle (excluding the 


-* Poisson, Journal de I’ Ecole Polytechnique, cahier xix. p. 447. 

_*+ Fourier, Théorie de la Chaleur, pp.235 & 240. 
} Ibid., p. 238 ; Poisson, Journal de ’ Ecole Polytechnique, cahier xix. p. 418. 
§ Fourier, Théorie de /a Chaleur, p. 244. 

1833. s 


258 THIRD REPORT—1833. 


limit 2 = 4), whose base is represented by 7, then we shall 


find * ; 
bi r ; sn22  sn3z _ 
gu = — {ax + (a—a!) d} {sine — behing se. } 


rs kd Cony) — ‘i win ace n aos 9 we. |, 


The trigonometrical series, in this last case, would represent a 
series of triangles placed alternately in an inverse position with 
respect to each other; and a similar observation would apply 
to the discontinuous curves which are represented by any series 
of sines and cosines. Thus, if y = ¢ x be the equation of the 
curve PC C’Q, and if we suppose 


y=?ex=a,snze + agsin22+a,sn32+ &e., 
between the limits 0 and 7; and if we make AB =a, A A’ 
= 27, AB'=3rn, &c., we shall get a discontinuous curve, 
consisting of a series of similar arcs, C D, aC", C’ D', &c., 
placed successively in an inverse relation with respect to each 
other upon each side of the axis of x, of which one are C D 
alone coincides with the primitive curve. 

If we should suppose the same curve to be expressed be- 
tween the limits 0 and a bya series of cosines or 


y= Gx =a + a, cosx + a,cos22 + &e., 


and if we make A B=7, Ab = —7, AA’ = 27, AB =3z, 
&c., then the trigonometrical equation will represent a discon- 
tinuous curve dC D C’ D’, of which the portions C D and Cd, 


* Fourier has given a particular case of this series, p. 246. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 259 


C! D! and C' D will be symmetrical by pairs; but one portion 
only, C D, will necessarily coincide with the primitive curve. 
The theory of discontinuous functions has recently received 
considerable additions from a young analyst of the highest pro- 
mise, Mr. Murphy, of Caius College, Cambridge. In an admi- 
rable memoir on the Inverse Method of Definite Integrals *, he 
has given general methods for representing discontinuous func- 
tions, of one or a greater number of breaks, by means which are 
more directly applicable to the circumstances under which they 
present themselves in physical problems than those which have 
been proposed by Fourier, Poisson, and Libri. Mr. Murphy 
had already, in a previous memoir }, given a most remarkable 
extension to the theory of the application of Lagrange’s theo- 
rem to the expression of the least root of an equation, which 
we shall have occasion to notice hereafter; and he has shown 
that if g (x) be an integral function of z then the coefficient of 


= in the developement of — log = will represent the least 


root of the equation ¢2 = 0. We thus find that the least 
of the two quantities « and will be represented by the coeffi- 


cient of = in the series for log saat te | which is 


(« 8) (« 6)? 
JPat ed : (7 u By) a ~igae ‘ (058) + &e. (I.) 


and if we replace « and B by es and $ the feast of the two 


sila: ty 1 : as 
quantities — and ae the greatest of the two quantities # and 


B, will be represented by 
2 
a ee ee 
axptaa (2+* * ke ct ae (ctf) + Be (2.)¢ 


Zz 


& 
~ 


* Transactions of the Philosophical Society of Cambridge, vol. iv. p. 374. 
t+ Ibid. p. 125. 
t+ If we represent the series (2.) by S, we shall get 

dr-1§s 1 


oe Sor Oy 
(—1)"-1T(n) da™-! ~ an os 


dea 
traction within and without a spherical shell, which is 0 or Li where « is 


oe 
the distance from the centre. 


= would represent the at- 


according as @ is greater or less than 8: thus 


s2 


260 THIRD REPORT—1833. 


Thus, ify — «7 — 6B = Oand y — @’ x — 6’ = 0 be the equa- 


AB P oD 


tions of two lines B C and DC, forming a triangle with a por- 
tion B D of the axis of x, then the system of lines which they 
form will be expressed by the product 

(y—ax—B) (y—avax—f)=0. (3.) 
Now it is obvious that if common ordinates P M, PM’ be 
drawn to the two lines, the /east of them will belong to the sides 
of the triangle B C D; if we denote, therefore, P M and P M! 
by y, and yo, the equation 


(Yi Yo)” Og) 
oo Shas: a ; in: aaa ee ok 
ete eee) +0 14.6 (TES mao 
will become the equation of the sides of the triangle BC D, 
when y, and y are replaced by their values; for y will denote 
P M for one side and p ™ for the other. 

In order to express a discontinuous function ¢, which as- 
sumes the successive forms ¢,, $5, $3, &c., for different values 
of a variable which it involves between the limits a and 6, 6 
and y, y and 8, &c., Mr. Murphy assumes 8 (4% 2), 8 (6, 2), 


S (y, 2), &c., to denote the coefficient of 3 in the several series 


for 


ere Ae OE a8), es 


& 
and supposes 


Se Mo” 


If « be less than x or 2 greater than a, then S (a, x) = a, 
a8 2) = 1: if p be less than 2, ‘hen oe ds (B, 9) 


wane Bake 
dS (y, 2) 
dy 


dS (8, s 


neg 6 Saeeru errs rae 


and Peet rore 


= 1: if y be less thane, then = |, and so on; con- 


- 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 261 


‘sequently, in the first case we have ¢ = f, = ¢,: 
in the second, 


@=f, + fo = %, and therefore f, = ¢, — ¢,: 
in the third, 
9=f, + fo + fs = 3, and therefore f; = $3 — $2. 
It appears therefore that } 


dS (a,z LSB; 8 dS (y, 2 
9= 9, te) + 0) e+ @s— 9) + &e 


is a formula which is competent to express all the required 
conditions of discontinuity *. 

Equivalent forms may. be considered as permanent within 
the limits of continuity, and no further, unless the requisite 
signs of discontinuity, whether implicit or explicit, exist upon 
both sides of the sign =: thus, the equation 


*D,cosx = “ire Zat+ zinta + son sin6a + we. 
is permanent within the limits indicated by the sign 7D and 
no further, and similarly in most of the cases which have been 
considered above. The imprudent extension of such equivalent 
forms, which has arisen from the omission of the necessary 
signs of discontinuity, has frequently led to very erroneous 
conclusions; thus, the equation 

es _ 12-1 1 Lin ee a — 1 Ae par 
ee ere eet 2° (@@+12" 8° (@41) te. 
which is true for all values of 2 between 0 and «, has been 
extended to all values of « between — 0 and + «, and has 
thus been made the foundation of an argument for the identity 
of the logarithms of the same number, both when positive and 
negative. 

There are two species of discontinuity which we have consi- 
dered above, one of which may be called instantaneous and the 
other finite: the first generally accompanies such changes of 
form as are consequent upon the introduction of critical values 


* These formule would require generally a correction at their limits, in 
order to render them symbolically general. The nature of these corrections 
may in most cases be easily applied from the observations which we have 
made above. E 

_ + This series is given by M. Bouvier in tke 14th volume of Gergonne’s 
Annales des Mathématiques. The conclusion referred to in the text assumes 
the identity of the logarithms of 2? and of (— 2)*, which is in fact the whole 
_guestion in dispute. vii 


262 THIRD KEPORT—1833. 


of the variables, when the corresponding equivalent form no 
longer exists, or when the conditions which determined its exist- 
ence no longer apply ; the second restricts the existence of the 
equivalent form to limits of the variable which have a finite dif- 
ference from each other. In neither case, if we suppose the con- 
ditions of the discontinuity to be implicitly involved, or if we 
suppose the explicit signs of discontinuity to be assumed con- 
formably to the general laws of algebra, can we consider the 
law of the permanence of equivalent forms to be violated. It is 
only when a continuous formula is assumed to be equivalent to 
a discontinuous formula, without the introduction of the requi- 
site sign of discontinuity to limit the extent of the continuous 
formula, that we can suppose this fundamental law to be vio- 
lated or the asserted equation between such expressions to be 
false. Many important errors have been introduced into ana- 
lysis from the neglect of those conditions. 

The identity of the values of powers of 1, whose indices 
are general whole numbers, and also of the sines and cosines 
of angles which differ from each other by integral multiples of 
360°, is a frequent source of error in the generalization of equi- 
valent forms, when the symbols which express those indices or 
multiples are no longer whole numbers. <A very remarkable 
example of both these sources of error has occurred in the for- 
mula 

(2 cos x)” = cosm (2r a + x) + mcos(m — 2) (2rm + 2) 

a a cos (m — 4) (2ra +42) + &e. 


+ / —] {sinm (2ra + x) + m sin (m — 2) (2r7 4+ 2) 
m (m —) sin (m — 4) @re + 2) + &e. (1.) 


If we suppose m to be a whole number, this equation degene- 
rates into 


2cos 2)” = cosmx +mcos(m — 2)a oe ae 
cos (m — 4) x + &e. (2.) 


the series first discovered by Euler, and which he assumed to - 
be true for all values of m. If, however, we should suppose m 


to be a fraction of the form P, we should have q values of 
the first member of the equation (2.), and only one of the second. 
And if we should confine our attention to the arithmetical 
value (7) of the first, it would not be equal to the second, un- 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 263 


less m was a whole number ; for if we should denote the series 
of cosines 
cos (m 2ra + x) + m cos (m — 2) (2Qra + x) + &e., by Cr, 
and the series of sines 
sin m (2r7m + 2) + msin (m — 2) (Qra+ 2) + &c., by Sr, 
we should find, when cos ~ is positive, 
CG, ; S; 


~cos2mrar sin2mrn’ 


P 


and when cos is negative, 
ak Cr Sr 
P ~ cos m Qr+1)9r = ‘sin m (2r + 1)7° 
It will follow, therefore, that when r is not a whole number, p 
will be expressible indifferently by a series of cosines or of 
sines, unless cos 2mra7 = 0 or sin 2mra = 0, when cos z is 
positive, or cos m (2r +1) r=0, or sinm (2r + 1)r= 0, 
when cos « is negative. 
In a similar manner, assuming 
n? (n? — 2?) 
1.2.3.4 
n® (n® — 2°) (n? — 4?) 
oy tee 26 


2 
n 
MS bs —_...cos* 7 + cos? x 


cos*° x + &c. 


and 
2 2 


X’=n { cos “= = cos? x 


+ oe cos? x — ae. } 
we shall find 
cosn (2Qr 7 + x) = cosn (27 4 5) ».¢ 
+ cos (n — 1) (27 + = wm. X’, 
If we suppose 7 to be equal to zero, this equation will become 
cos ne = cos >. X + con X!, 


which is the form which has been erroneously assigned by La- 
grange * and Lacroix} as generally true for all values of x. 
Many other examples of similar wndulating functions, ex- 


* Calcul des Fonctions, chap. xi. 
+ Traité du Calcul Diff. et Intég., tom. i. p. 264. 


264 ‘ THIRD REPORT—1833. 


pressing the various relations between the cosines and sines of 
multiple arcs and the powers of simple ares, whether ascending 
or descending, have been given by Lagrange * and other writers 
as general, which are either degenerate forms of the correct 
and more comprehensive equations, or altogether erroneous. 
Poisson had pointed out some of the inconsistencies to which 
some of these imperfect equations lead, and had slightly hinted 
at their cause and their explanation ; and the discussion of such 
cases became soon.afterwards a favourite subject of speculation 
with many writers in the Mathematical Journals of France + and 
Germany {; but the complete theory and correction of these 
expressions was first given by M. Poinsot in an admirable me- 
moir which was read to the Academy of Sciences of Paris in 
1823, and published in 1825. They form a most remarkable 
example of expressions extremely simple and elementary in 
their nature, which have escaped from the review and analysis 
of the greatest of modern analysts, in forms which were not 
merely imperfect, but in some cases absolutely erroneous. 

The difficulties which have presented themselves in the 
theory of the logarithms of negative numbers, as compared with 
those of the same numbers with a positive sign, have had a 
very similar origin. If we consider the signs ‘of quantities as © 
factors of their arithmetical values, and if we trace them through- 
out the whole course of the changes which they undergo, we 
shall find many examples of results which are identical when 
considered in their final equivalent forms, but which are not 
in every respect identical when considered with respect to their 
derivation: thus (+ a)? is identical with (— @)*, when consi- 
dered in their common result + a*, but not when considered 
with respect to their derivation. Let us now consider their se- 
veral logarithms, the common arithmetical value of the logarithm 
of a being denoted by p: 


log (+ a)? = log (12 a2 =4r0 /—1 + 2p (1.) 
log (— a)? = log (— 1? a2 = (27 +1) 24 YW —1 4+ 2p~ (2.) 
log a =log1.a®@ =2r%rV7—1+4 2p (3.) 


It thus appears that the values of log (+ a)? and log (— a)? are 
included amongst those of log a”, but not conver sely; and also 
that the values of log (+ a) and log(— a)’, the arithmetical 
value being excepted, are not included in each other. 


> Correspondence sur ? Ecole Polytechnique, tom. ii. p. 212. 
+ In Gergonne’s-dnnales des Mathématiques, tom. Xiv. XV. XVI. XVii. 
{ In Crelle’s Journal fiir die reine und angewandte Mathematik, Berlin. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 265. 


Again, if we consider — a” as originating from (— 1) (+ a)”, 
we shall get 
log — a” = (2r + Qmr +1)rV—l + mp*: 


if we suppose m = a y = 0 andr! = — 1, we shall get 
log — Vaate = Flog a = log ¥ a3 


or the logarithm of a negative quantity will be identical with 
the logarithm of the same quantity with a positive sign. Ina 


similar manner, if we suppose ™ = me where p is prime to 7, 
pi ~nandr =25+, then 2r +2 me’ + 1 = 0, and the 


corresponding logarithm of — a” will coincide with the arith- 
metical logarithm of a”. We should thus obtain possible loga- 
rithms of negative numbers in those cases in which we should 
be prepared to expect them from the ordinary definition} of 
logarithms. 

In the absence of all knowledge of the specific process of de- 
rivation of quantities, such as a” and — a”, we should consider 
their logarithms as ‘dentical with those of 1”. A and (—1) 1”. 4 
where A is the arithmetical value of a”: and in considering 
the different orders of logarithms which correspond to the same 
value of a” or of — a”, they will be found to differ from each 
other by the logarithms of 1” and (— 1) 1” only, which are 
Omrt f—land 2r+2mr' + lyrv—l respectively. The 
logarithms in question are Napierian logarithms whose base is é. 
If we should suppose the logarithms to be calculated to any 
other base, we should replace the Napierian logarithms of 1” 
and (— 1) 1” by the logarithms of those quantities (or signs) 
multiplied by the modulus M: the same remarks will apply to 
such logarithms which have been made with respect to Na- 

-pierian logarithms. 

The question of the identity of the logarithms of the same 
number, whether positive or negative, was agitated between 
Leibnitz and Bernoulli, between Euler and D’Alembert, and 
has been frequently resumed in later times. The arguments in 


~ * Peacock’s Algebra, p. 569. 
+ The logarithm being defined to be the index of the power of a given base 
which is equal to a given number, it would follow, since at = ++ n, that— 


is equally the logarithm of + and — n. The same remark ay plies to all in- 
dices or logarithms which are rational fractions with éven dencminatars. 


266 THIRD REPORT—1833. 


favour of the affirmative of this proposition, which were for the 
most part founded upon the analytical interpretation of the pro- 
perties of the hyperbola and logarithmic curve, were not en- 
titled to much consideration, inas much as they were not drawn 
from an analysis of the course followed in the derivation of the 
symbolical expressions themselves and from the principles of 
interpretation which those laws of derivation authorized. A 
very slight examination of those principles, combined with a re- 
ference to those upon which algebraical signs of affection are in- 
troduced, will readily show the whole of the very limited num- 
ber of cases in which such a proposition can be considered to 
be true *. 


_ * In the 15th volume of the Annales des Muthématiques of Gergonne, there 
is an ingenious paper by M. Vincent on the construction of the logarithmic 
and other congenerous transcendental curves. Thus, if y = e” there willbe in 
the plane of x y a continuous branch such as is commonly considered, and a 
discontinuous branch corresponding to those negative values of y which arise 
from values of x, which are expressible by rational fractions with even deno- 
minators: thus, if we suppose the line between « = 0 and xz = 1 to be di- 
vided into an even number 2p of parts, (where p is an odd number,) the 
values of x will form a series of fractions, 


yet. Bin 2p—1 


2p p 2p p 2p 
which have alternately odd and even denominators, and which correspond 
therefore to values of y which are alternately single and double. If we may 
suppose, therefore, a curve to be composed of the successive apposition of 
points, the complete logarithmic curve will consist of two symmetrical 
branches, one above and the other below the axis of x, one of which, in cor- 
responding parts of the curve, will have double the number of points with the 
other. The inferior curve, therefore, may in this sense be considered as dis- 
continuous, being composed of an infinite number of conjugate points, forming, 
in the language of M. Vincent, une branche pointillée. ‘lhe same remark ap- 
plies to other exponential curves, such as the catenary, &c. 

It was objected to this theory of M. Vincent by M. Stein, another writer 
in the same journal, that every fractional index in this interval might be con- 
verted into an equivalent fraction with an even denominator, which would 
give a double possible value of the ordinate, which would be different from 
that given by the fractional index in its lowest terms; and that consequently 
there would necessarily be a double ordinate for every point of the axis, and 
therefore also a double number, cne positive and the other negative, corre- 
sponding to every logarithm. In reply to this objection, it is merely neces- 

we 2.2 oe rae 
sary to observe that the values of a” and a”? or of 1” and 1”? are in every 
respect identical with each other, the m p values in the second case consisting 
merely of p periodical repetitions of those in the first. 

In a paper in the Philosophical Transactions for 1829, Mr. Graves has given 
a very elaborate analysis of logarithmic formule, and has arrived at some 
conclusions of great generality which it is difficult to reconcile with those 
which have been commonly received. Amongst some others may be men- 
tioned the formula which he has given for the Napierian logarithms of 1, 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 267 


Convergency and Divergency of Series.—The subject of di- 
vergent series, their origin, their interpretation and their use 
in analysis, is one of great importance and great difficulty, and 
has been and continues to be the occasion of much controversy 
and doubt. I shall feel it necessary, for such reasons, to notice 
it somewhat in detail. 

If the operations of algebra be considered as general, and 
the symbols which are subject to them as unlimited in value, 
it will be impossible to avoid the formation of divergent as 
well as of convergent series: and if such series be consi- 
dered as the results of operations which are definable, apart 
from the series themselves, then it will not be very important 
to enter into such an examination of the relation of the arith- 
metical values of the successive terms as may be necessary to 
ascertain their convergency or divergency; for, under such 
circumstances, they must be considered as equivalent forms 
representing their generating function, and as possessing, for 
the purposes of such operations, equivalent properties. ‘Thus, 
if they result from the division of the numerator of an alge- 
braical fraction by its denominator, then they will produce the 
numerator when multiplied into the denominator or divisor: if 
they result from the extraction of the square or cube root of 
an algebraical expression, then their square or cube will pro- 
duce that expression; and similarly in other cases, no regard 


2rea 
which is not 2r* V. —1, but i Pip which, though it includes the 


former, is not included by it. It appears to me, however, that there exists a 
fundamental error in the attempt which has been made by Mr. Graves to 
generalize the ordinary logarithmic formule upon the same principles which 
have been applied by Poinsot to the generalization of the trigonometrical series 
which have been noticed in the text. He assumes f (4) = cos + V—1 sin@ 
— ef aoe and makes the series for f (4) and f —1 (), combined with the equa- 
tion f (« 4) = a value of f (6)”, and therefore f—' f 6 = 2r x + 6, the foun- 


dation of his logarithmic developements: in other words, he makes e afeat 


periodic quantity the base of his system of logarithms, an assumption which 
is essential to the truth of the formula f~' f 6= 27 x + @ and to the gene- 
ralization of the series for f—~' @ by means of it; an hypothesis which is al- 
together at variance with our notions of logarithms as ascertained by the ordi- 
nary definition. The logarithms of + 1 and of (+ 1)” alone, for very obvious 
reasons, can be considered as possessing such a character. 

Though I have felt myself called upon to state my objections to the fun- 
damental principle assumed in this memoir of Mr. Graves, and consequently 
to many of the conclusions which are founded upon it, yet I think it right at 
the same time to observe that it displays great skill and ingenuity in the con- 
duct of the investigations, and is accompanied by many valuable and ori- 
ginal observations upon the general principles of analysis. 


268 THIRD REPORT—1833. 


being paid in such cases to terms which are at an infinite di- 
stance from the origin. 

It is this last condition, which, though quite indispensable, 
is rather calculated to offend our popular notions of the values 
of series as exhibited in their sums. We speak of series as 
having swms when the arithmetical values of their terms are 
considered, and when the actual expression for the sum of n 
terms does not become infinite when z is infinite, or when, in 
the absence of such an explicit expression, we can show from 
other considerations that its value is finite. In all other cases 
the series, arithmetically speaking, may be considered as di- 
vergent, and therefore as having no sum *, if the word sum be 
used in an arithmetical sense only, as distinguished from gene- 
rating function. 

We are in the habit of considering quantities which are én- 
finitely great and infinitely little as very differently circum- 
stanced with respect to their relation to finite magnitude. We 
at once identify the latter with zero, of which we are accus- 
tomed to speak as if it had a real existence ; but if we subject 
our ideas of zero and infinity to a more accurate analysis, we 
shall find that it is equally impossible for us to conceive either 
one or the other as a real state of existence to which a mag- 
nitude can attain or through which it can pass. But it is the 
relation which magnitudes in their finite and conceivable state 
still bear to other magnitudes in their course of continued in- 
crease or continued diminution, which enables us to consider 
their symbolical relations when they cease to fe finite ; and 
whilst quantities infinitely little are neglected as being absorbed 
in a finite magnitude, so likewise finite magnitudes are consi- 
dered as being absorbed in infinity, and therefore neglected 
when considered with relation to it. The principle, therefore, 
of neglecting terms beyond a finite distance from the origin, in 
converging series, is both safe and intelligible, whilst the case is 
very different with respect to the neglect of similar terms in a 
diverging series. Of such series it is said that they have no 
arithmetical sum; but it may be said in the same sense of all 
algebraical series involving general symbols that they have no 
sum. But it is not the business of symbolical algebra to deal 
with arithmetical values, but with symbolical results only ; and 
such series must be considered with reference to the functions 
which generate them, and the laws of the operations employed 
for that purpose. The neglect, therefore, of terms beyond a 


* This would appear Cauchy’s view of the subject: see the 6th chapter 
of his Cours d’ Analyse, 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 269 


a ; 
finite distance from the origin would be perfectly safe as far as 
it does not influence the determination of the series from the 
generating function, or the generating function from the series ; 
and it is upon this principle that the practice is both founded 
and justified. A few examples may make this reasoning more 
plain. 

Let it be required to determine the function which generates 
the series 

ataxr+ax*?+ ax+ &e. * (L.) 
Let s be taken to represent this function, and therefore 
s=atax+ax?+ax*+ &e. 
=atue{atar+ax?+ax+ &e.}. 

a+2£s: consequently 

a 
dui ee. 
If the arithmetical values of the terms of this series be con- 


sidered, and if z be less than 1, then 


oS 


is the sum of the 


l—wz 
series: in all other cases it is its generating function. 
We may consider, however, s (whether it expresses a swm or 
a generating function) as identical with s,, s,,s3, &c., in the 
seyeral expressions 
§&=@425, ; 
S=at ax wx 5, 
s=a+art+azr+ xs, 
s=atart+az*4+...a2" 4 x5,,: 


for if the number of terms of the series s be expressed by x 
and if » be infinite, we must consider 5), Sg, 53, @... Sm as abso- 
lutely zdentical expressions ; for otherwise we must consider an 
infinite as possessing the properties of an absolute number, and 
must cease to regard énfinities with finite differences as iden 
tical quantities when compared with each other. It is for this 
reason that we assume it as a principle that no regard must be 
paid to terms at an infinite distance from the origin, whatever 
their arithmetical values may be. 
The sum of the series 
a—-ata—a-+t &e. 
was assigned by Leibnitz, upon very singular metaphysical 
; : a of eee 
considerations, to be Zz? the principle just stated would allow 


us to put 


270 THIRD REPORT—1833. 
s=a—(a—ata—a+ &c.) 


a 
= a —s; and therefore s = Pat 


* The same principle would show that the equation 


z=at+f(atsf(atsfaat...)) 


is identical with the equation 


ex=at+f(a); 
and that 
e=af(af(as(..))) 
is identical with 
ie = aefa(m). 


The example in the text is the most simple case of a class of periodic series, 
the determination of whose sums to infinity has been the occasion of much 
controversy and of many curious researches. The general property of such 
series is the perpetual recurrence of the same group of terms whose sum is 
equal to zero: thus, if there should be p terms in each group, and if the num- 
ber of terms n = m p + i, their sum would be identical with that of the 7 first 
terms of the series; and if we should denote those terms by a, do,... a,, 
and if we should take the successive values of this sum for all the values of ¢ 
between 1 and p inclusive, their aggregate value would be represented by 


pa + (p—1) a+ (p—2) as+--- ay 
of which the average (A) or mean would be represented by 


P 

If this periodic series was continued to infinity, it was contended by Daniel 
Bernoulli, in memoirs in the 17th and 18th volumes of Novi Commentarii 
Petropolitani, for 1772 and 1773, that its sum would be correctly represented 
by the average (A), in as much as it was equally probable that any one of 
the p values would be the true one. Upon this principle it would follow, 
that of the apparently identical series 

1—1+1—1+4+1—&c.... 

DENS Say 6 a ed oe eee ne 

14+0+0—1+4+1+4+0+4+0—1+4 &c. 


the first would be equal to a the second to = and the third to 


=. In the 
same manner we should find 

1+1—1—1+1+4+1—1—14, &. 
equal to 1, and 


1+1+0—1—14+1+4+1+4+0—1—141+4+1+4&c. 
equal to = The same observations would apply to the series 


1+ cosz+ cos2x+ cos3x+ cos4z + &e. 
and 
1+ cosz +0+4 cos2a2+ cos3x%+0+4cos4x+ &c. 
where x is commensurable with 2 x. 
These conclusions, however, though curious and probable, rested upon no 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 271 


If we consider this principle of the identity of series, whose 
terms within a finite distance from the origin are identical, as 
established, we shall experience no difficulty in admitting the 
perfect algebraical equivalence of such series, and their gene- 


secure basis founded upon the general principles of analysis, and their truth 
was not, therefore, generally admitted amongst mathematicians. In the year 
1798, Callet, the author of the logarithmic tables which go by his name, pre- 
sented a memoir to the Institute for the purpose of showing that the sums 
of such periodic series were really indeterminate: thus, if we divide 1 by 
1+ wand subsequently make x= 1, we get 


P— 7 i WE ie, (1.) 
the value of which is = In a similar manner, if we divide 1 + x by 
1+ 2+ 22, we get for the quotient 

1— a+ 23 — 25 + x — a8 4+ &e., 
which becomes the same series (1.), though the value of the generating func- 


tion under the same circumstances becomes os The same remark applies 


to the result of the division of 1 + «+ a? + ..2” byl +2+a2?4+..2", 
which produces the same series (1.) when « = 1, though under such circum- 


stances its generating function becomes fy 


This memoir of Callet gave occasion to a most elegant Report upon this 
delicate point of analysis by Lagrange, who justified upon very simple prin- 
ciples the conclusion of Daniel Bernoulli. The series which results from the 
division of 1 + 2 by 1+ + 2%, if the deficient terms be replaced, becomes 


14+0.¢2—24 040. at— ao + of + 0.07 — 2 4 &e., 
which degenerates, when « = 1, into the series 
Gh A de Orme ied obs Bee 


and not into the series (1.). The same remark applies to the series which 
arises from the divisionof 1+ 2+ ..a”byl+ae+....2°..n7m; 
which becomes, when x = 1, 


14+0+4+0+4+0+&. —1+0+0+4&. +1404 &e, 


- which is equal, by Bernoulli’s rule, to =. 


But it is not necessary to resort to this expedient for the purpose of deter- 
mining the sums of such series ; for the series 


Q, + dg2 + ay a? + 2G 2P! ea, oP + &e. 
is a recurring series resulting from the developement of 
a, + do2+a,2274+ .. Ce ne 
we EE eas ae 
1— a? 
which becomes > when «= 1. If we replace « by ~, this fraction will 


become 


272 THIRD REPORT—1833. 


rating functions. For the same principle would justify us in 
rejecting remainders after an infinite number of terms, whatever 
their arithmetical values may be; for such remainders can in- 
fluence no terms at a finite distance from the origin, and there- 
fore can in no respect affect any reverse operation, by which 
it may be required to pass from the series to any expression 
dependent upon the generating function. Thus, if 


& ynie 24% 1s 
BE ata Si Ce MMs Pe sy ets og ae ee 


we shall get 

Bi i 12). 8: =, 08, 
if we reject remainders after an infinite number of terms; and 
similarly in other cases. It would thus appear that algebraical 
equivalence is not necessarily dependent upon the arithmetical 
equality of the series and its generating function. 

It is, however, an inquiry of the utmost importance to be able 
to ascertain when this arithmetical equality exists; or, in other 
words, to ascertain under what circumstances we can determine 
the sum of the series, either from our knowledge of the law of 
formation of its successive terms, or approximate, to any required 


nye oe age? Ths, a, z 


ti ; 
which becomes by the application of the ordinary rule of the differential cal- 
culus, when s = 1 or2z = 1, 
pyA+t p—VDat.. 4, 
Pp > 
which is the average or mean value determined by Bernoulli’s rule. 

The discussion of the values of these periodic series has-been resumed by 
Poisson in the twelfth volume of the Journal de l’ Ecole Polytechnique. He 
considers them as the limits of these series when considered as converging 
series, a view of their origin and meaning which is almost entirely coincident 
with that of Lagrange. Thus, the sum of the series 


sing + psin (x + q) + p’ sin (2a + q) + &e. 


is equal to 
sin g + p sin (2 — q) 
1—2pcosa +p?’ 
when p is less than 1, an expression which degenerates, when p = 1, into 


1 gen il fis 
i sin g + cy cos g cot >» 
which may be considered, therefore, as the limit of the sum of the series 


sin g + sin (a + g) + sin (2% + ¢) + &e, in infin. 


# 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 273 


degree of accuracy; to its value by the aggregation of a finite 
number of those terms. Many tests of the summability of series 
(considered as different from the determination of their gene- 
rating functions,) have been proposed, possessing very different 
degrees of certainty and applicability. The geometrical series 
which we have just been considering is convergent or divergent, 
that is, semmable or not, according as z is greater or less than 1; 
and it is convenient, for this and for other reasons, to assume it 
as the measure of the convergency or divergency of other series. 
If it can be shown that a converging geometrical series can be 
formed whose terms within a finite and assignable distance from 
the origin become severally greater than those corresponding 
to them of the assigned series, then that series is convergent. 
And if it can be shown that a divergent geometric series can be 
formed whose terms within a finite and assignable distance from 
the origin are severally less than those corresponding to them 
of the assigned series, then that series is divergent*. Such 
tests are certain, as far as they are applicable; but there may 
be many cases, both of divergent and convergent series, which 
they are not sufficiently delicate to comprehend. 

It would appear from the preceding observations that. di- 
verging series have no arithmetical sums, and consequently 


* Peacock’s Algebra, Art. 824, and following. Cauchy, Cours d’ Analyse 
Algébrique, chap. vi. This last work contains the most complete examination 
of the tests of convergency with which I am acquainted. 

The measure of convergency mentioned in the text, which was first sug- 
gested and applied by D’Alembert, will immediately lead to the following : 

“If w, represent the n‘* term of a series, it is convergent (or will become so) 

1 
if the superior limit of (w,,)” be less than 1, when x is infinite; divergent in 
the contrary case.” 

“Tf the limit of the ratio w, , , to u, be less than 1, the series is convergent, 
and divergent in the contrary case.” 

Many other consequences of these and other tests are mentioned by Cauchy 
in the work above referred to. 

M. Louis Olivier, in the second volume of Crelle’s Journal, has proposed th 
following test of convergency. ‘If the limit of the value of the product nu, 
be finite or zero when 7 is infinite, then the series is divergent in the first case, 
and convergent in the second.” This principle, however, though apparently 
very simple and elementary, has been shown by Abel, in the same Journal, to 
be not universally true. Thus, the series 


1 1 af 1 
2 log 2 + 3 log 3 G 4 log 4 mr preci 9 n log n ar 
may be shown to be infinite, though the product x 7, is equal to xero when n is 


infinite. The same acute and original analyst has shown that there is no func- 
tion of n whatever which multiplied into w, will produce a result which is zero 


or finite when n is infinite, according as the series is convergent or divergent. - 


1833. : T 


Q74 THIRD REPORT—1833. 


admit of no arithmetical interpretation. And it will be after- 
wards made to appear that such series do not include in their 
expression, at least in many cases, all the algebraical conditions 
of their generating functions. Before we proceed, however, to 
draw any inferences from this fact, it may be expedient in the 
first instance to give a short analysis of some of the cireum- 
stances in which such series originate. 
The series 
1 1 aia 
ene Hy a + af + &c. 

is convergent or divergent according as a is greater or less 
than 6. As this series is incapable, from its form, of receiving 
a change of sign corresponding to a change in the relation of 
a and b to each other, it would evidently be erroneous in the 
latter case if it admitted of any arithmetical value, in as much 
as it would then be equivalent to a quantity which is no longer 
arithmetical. In this case, therefore, the series may be replaced 
by the symbol ©, which is the proper sign of transition, (see 
page 237,) which indicates a change in the constitution of the 
generating function, of such a kind as to be incapable of being 
expressed by the series which is otherwise equivalent to it. 
The same observations apply to the equation 


Bi eid 6b  n(n—1) 
(a — bp sardine + a Tale DR 
n(n —1)(n—2) B 
zs 1 ' 2 3 + beh 


as we have already stated in our remarks upon signs of transition, 
in page 237. It will be extremely important, however, to examine, 
both in this and in other cases, the circumstances which attend 
the transition from generating functions to their equivalent 
series, in as much as they will serve to explain some difficulties 
which have caused considerable embarrassment. 
The two series 
1 1 2b 3b 468 
=e{it+ yt Jorn + &e.} 


(a — b)? a? 


and 
1 1 2a 8ar 4a 
(b Tu ay = 2 41 a5 Bh 55 ob) a5 ee at &e. } 


will be divergent in one case, and convergent in the other, 
whatever be the relation of a and 8, though they both equally 


REPORT ON- CERTAIN BRANCHES OF ANALYSIS. 275 


vigil 1 1 

Represent Tse ab + geand &-—2ab+a 
braically, as well as arithmetically, equivalent to each other. It 
might be contended, therefore, that in this instance the signo, 
which replaces one of the two series, is no indication of a change 
in the constitution of the generating function which is conse- 
quent upon a change of the sign of a— bor b—a. But 
though a? — 2a 6 + 6b? is equal to (a — b)?, and 6? — 2ab + a? 
to (6 — a)?; and though a’ — 2a 6 + 8? is identical in value 
and signification with 6°? — 2a 6 + a® when they are considered 
without reference to their origin, yet we should not, on that 
account, be justified in considering (a — 6)? and (6 — a)? as 
algebraically identical with each other. The first is equal to 
(+ 1)? (a — 6)*, and the second to (— 1)? (a — 6); or the first 
to (— 1)? (6 — a)’, and the second to (+ 1)? (6 — a)?. But the 
signs (+ 1)? and (— 1)? are not algebraically identical with each 
other, though identical when considered in their common result, 
in as much as their square and other roots and logarithms are 
different from each other*. It follows, therefore, that there is 


, which are alge- 


a symbolical change in the quantity denoted by sartroiig in its 


passage through infinity, which is indicated by the infinite value 
of the equivalent series, in as much as it is not competent to ex- 
press, in its developed form, the algebraical change which its 
generating function has undergone. The same remarks will 
apply to the series for (a — 6)" and (6 — a)", in all cases in 
which x is a negative even number. When 2 is a negative odd 
number, the change of constitution of the generating function 
is manifest, and requires no explanation. 
The two series 


1 1 b Fn Baiy 
eran etei stake} 


1 a  @ a. at 
— = — 1 | ati 75>. 0C~C*”«C—SS= oF . 
b+a at i ii: ay we. } 
correspond to the same generating function, though one of 
them is divergent, and the other convergent. But the divergent 
series, whose terms are alternately positive and negative, cannot 
be replaced by the symbol oo , in as much as it does not indicate 


., " Thus, if @ denote a line, (+ a)? and (— a)? can only be considered as 
identical in their common result a2. When (-+ a)? and (— a)? are considered 
with reference to each other, they are not identical quantities, though equal te 
each other. 

T2 


276 THIRD REPORT—1833. 


any change in the constitution of the generating function. They 
may both of them, therefore, be considered as representing the 
value of this function, though in one case only can we approxi- 
mate to its arithmetical value by the aggregation of any number 
of its terms *. 

Similar observations would apply to the series 


aw”, nb n(n—1) 6B 

(a+b =a He 2 a ost — ee + &e.\ 
when z is not a positive whole number. In all such cases, the 
developement will sooner or later become a series, whose terms 
are alternately negative and positive, and which will be di- 
vergent or convergent, according to the relation of a and b to 
each other. More generally we might assume it as a general 
proposition, “ that divergent series which correspond to no 
change in the constitution of the generating function, will have 
their terms or groups of terms alternately positive and nega- 
tive :” and conversely, ‘that divergent series which correspond 
to a change in the constitution of the generating function, will 
have all their terms or groups of terms affected with the same 
sign, whether + or —, and the whole series may be replaced 
by the symbol «.” 

In both these propositions the change of which we speak is 
that which corresponds to those values of the symbols which 
convert the equivalent series from convergency to divergency, 
and conversely. 

I am not aware of any proof of the truth of these important 
propositions which is more general than that which is derived 
from an induction founded upon an examination of particular 
cases. But such or similar conclusions might be naturally ex- 
pected to follow from the fundamental principles and assump- 
tions of symbolical algebra. If the rules of algebra be perfectly 
general, all symbolical conclusions which follow from them must 
be equally true: and those rules have been so assumed, that 
when the symbols of algebra represent arithmetical quantities, 
the operations with the same names represent arithmetical 
operations, and become symbolical only when the correspond- 
ing arithmetical operations are no longer possible. It will be 
essential, therefore, to the perfection of algebraical language 
that it should be competent to express fully its own limitations. 


1 bs 1 as *, : 
i =— — =) — —— ] 
* The equations s = Z - and s 7 5 will equally give us 
t= :: b in one case, and s = an in the other, whatever be Lae relation 


of and b. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. O77 


Such limitations will be conveyed by the introduction of signs 
of affection, of signs of transition, or of signs of discontinuity, 
which may be involved either implicitly or explicitly. It is for 
such reasons that all those signs must be considered in the 
interpretation of algebraical formule, and their occurrence will 
at once suggest the necessity of such an examination of the 
circumstances of their introduction as may be required for their 
correct explanation*. 

We thus recognise two classes of diverging series, which are 
distinct in their origin and in their representation. The first 
may be considered as involving the symbol or sign oo implicitly, 
and as capable, therefore, of the same interpretation as we give 
to the sign when it presents itself explicitly. The second re- 
presents finite magnitudes, which in their existing form are 
incapable of calculation by the aggregation of any number of 
their terms. Such series are in many cases capable of trans- 
formations of form, which convert them into equivalent con- 
verging series; and in some cases, where such a transformation 
is not practicable, or is not effected, the approximate values of 
the generating functions may be determined, from indirect con- 
siderations, supplied by very various expedients. 

The well known transformation of the series 


az—b2#4+cx#—dat+ex—f2'+ &e., 


which Euler has given}, into the equivalent series 


r 2 


a as FY 
Tta”"” (+a"" 
would be competent to convert a great number of divergent 


series of the second class into equivalent convergent series, or 
into such as would become so. In this manner the Leibnitzian 


series 


2 ae Se Aa pee ee a & 


1—1+4+1—1+4 &c. 


may be shown to be equal to = The series 


1—3+6—10+4+15—21 + &e. 


* The essential character of arithmetical division is that the quotient should 
approximate continually to its true value, and that the terms of the quotient 
which are introduced by each successive operation should be less and less con- 


. . : 1 1 
tinually. In the formation, therefore, of the quotient of = ma and Pai 


the analogy between the arithmetical and algebraical operation would cease to 
exist, unless @ was greater than 8, or unless the several terms in the quotient 
went on diminishing continually. oer a 

+ Institutiones Calculi Differentialis, Pars posterior, cap. i. 


278 _— REPORT—1833. 


of triangular numbers to as The series 
1—44+9-—16 +4 25 — &e. 
of square numbers to 0. The series of tabular logarithms 
log 2 — log 3 + log 4 — log 5 + &c., 


would be found to be equal to ‘0980601 nearly. If we should 
suppose x negative and greater than 1, the original and the 
transformed series would become divergent series of the first 
class. 

The series 

aS evils ap ers (a—1 (a—1) 

log a = (a — 1) Tat oe car eh ge + &c. 
is divergent when a is greater than 2, and convertible by Euler’s 
formula into the convergent series 


@-1) , 1 @-1F, 1 @-1, 1@-H 


a 2 a? 3 a 4 a‘ 


+ &c.; 


or by the method of Lagrange into the series 
n (Wa —1)— 5 (Wa —1P + 5 (= 1) — &e., 


which may be made to possess any required degree of con- 
vergency. But it is not necessary to produce further examples 
of such transformations, which embrace a very great part of the 
most refined artifices which have been employed in analysis. 

One of the most remarkable of these artifices presents itself 
in a series to which Legendre has given the name of demicon- 
vergent*, The factorial function (1 + «) is expressed by the 
continuous expression 


(=) (29 a)? R, 
e 
where R is a quantity whose Napierian logarithm is expressed 


by 
A B c 


122 Weeds ao oat cine 


where A, B, C, &c., are the numbers of Bernoulli. The law of 
formation of these numbers, as is well known, is extremely 


* Fonctions Elliptiques, tom. il. chap. ix. p. 425. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 279. 


irregular, and after the third term they increase with great 
rapidity. The series under consideration, therefore, even for 
considerable values of x, becomes divergent after a certain 
number of terms. But an approximate value of the series will 
be obtained from the aggregation of the convergent terms only: 
and it has been proved by a German analyst* that the error 
which is thus made in the value of the generating function will 
in this case be less than the last of the convergent, or the first 
of the divergent, terms. 

It has been usual amongst some later mathematicians of the 
highest rank to denominate diverging series, without any di- 
stinction of their class, as false, not merely when arithmetical 
values are considered, but also when employed as equivalent 
forms, in purely symbolical processes. ‘The view of their ori- 
gin and nature which we have taken above would explain the 
sense in which they might be so considered in relation both to 
arithmetical processes and to the calculation of arithmetical 
values. It seems, however, an abuse of terms to apply the term 
false to any results which necessarily follow from the laws of 
algebra. M. Poisson, perhaps the most illustrious of living 
analysts, has referred, in confirmation of this opinion, to some 
examples of erroneous conclusions produced through the me- 
dium of divergent series}; and as the question is one of great 
importance and of great difficulty, I shall venture to notice them 
in detail. 

Let it be required to express the value of 


5 ioe dx 
y" =f. {(l1— 2azx + a*) (1—2b62 + B*)}# 
by means of series. 

Assuming K = (1 — 2a + a®)-*and K’ = (1 — 2b + 0?)-4, 
let us suppose K and K’ developed according to ascending and 
descending powers of a and 6 respectively ; or, 

e =1l+aX,+a°X,+ aX,+ &e. 


K’=1+6X,+ #X,+ &X, + &e. 


1 1 1 1 
K= 7+ exit pret airst &c. 
; 1 1 1 1 
Kazt pit pXet pXst &e. 


* Erchinger in Schrader’s Commentatio de Summatione Seriei, &c. Weimar 
1818. 4 
+ Journal de l Ecole Polytechnique, tom. xii. 


280. THIRD REPORT—1833. 


The coefficients X,, X,, X3, &c., are reciprocal* functions, pos- 
1 
sessing the following remarkable property, that /” : Xm Xn de 
=k 


= 0, in all cases, unless » = m, in which case ue X,X,d x 
1 -1 
~On+1 
The knowledge of this property will readily enable us to de- 
termine the following four different values of =: 


ab ab? @&@b 


mpsieeey Oe OE, we. 
1 a a a 

ha Div ok BUT Botot es: 
1 b b? b? 

: ogee tget+zatrat ke. 
1 1 1 1 


“= Got see + sae t Tat t 
Whatever be the relation of @ and 6 to each other and to 1, 
two of these four series are convergent, and two of them di- 
vergent. But it appears from the examination of the finite in- _ 


+1 : 
tegral /_ K K! da, that one only of these two convergent 
-1 


series gives the correct value of z, being that which arises from 
the combination of the two convergent developements of K and 
K’, whilst the incorrect value arises from the combination of a 
convergent developement of K with a divergent developement 
of K’, or conversely. The conclusion which is drawn from this 
fact is, that the introduction of the divergent developement of 
K or of K’ vitiates the corresponding value of z, even though 
that value is expressed by a convergent series. Let us now 


+1 
examine how far the definite integral of f K K’ dz will jus- 
tify such an inference. e 


If we denote K K’ by - we shall easily find, 


* Functions which possess this property have been denominated reciprocal 
functions by Mr. Murphy, in a second memoir on the Inverse Method of Defi- 
nite Integrals, in the fifth volume of the 7ransactions of the Philosophical Society 
of Cambridge, in which general methods are given for discovering all species of 
such functions, and where one very remarkable form of them is assigned. The 
functions referred to in the text were first noticed by Legendre, in his first me- 
moir on the Attraction of Ellipsoids, and subsequently, at great length, in the 
Fifth Part of his Exercices du Calcul Intégral. Cauchy has used the term recipro- 
eal function in a different sense; see Exercices des Mathématiques, tom. ii. p. 141. 


REPORT ON ‘CERTAIN BRANCHES OF ANALYSIS. 281 


ere (eee bg g a) + const. ; 
ie, P 35 log zi ¥ ae 


and if we denote by r and 7’ the extreme values of p, when x 
= —1 and x = +1, we shall find, 


‘ ee ed et Srvuh tab ale see 

J-1 p 4V7ab 2 lar Jab —4ab— (a+ b)(1+ab)8? 
inasmuch as POP is 4a 6 — (a + b) (1 + @b) in one case, and 
—4ab—(at+ + 8) (1 + wb) inthe other. It will appear like- 
wise that r and 7’ will have the same sign, whether + or —, 
in as much as p will preserve the same sign ‘thr oughout the whole 
course of the integration. If, therefore, Hh a + a)(1+ a), 
then r = + (1 — a) (1 — 8); and if r’ = — (1 + a) (1 4+ 3B), 
then r = — (1 — a) (1 — 6). It thus appears that (1 — a) (1— 6) 
must have the same sign with (1 + a) (1 + 0), and consequently 
ifa 7 1, and b 7 1, we shall have, 


_? (a—1) (6-1) Vab +406 — (a+b) (1+ad) 
4 “ie ‘(a+1) (641) fab — 4ab6 — (a +b) (1 +45) 


; 1 _(4ab +19 
Aad ~ (Wab—1) 


(striking out the common divisor 


| be Meme e 
BFE BOe Wa beet ee 
If a Z1 and 6 21, we shall find r = (1 — a) (1 — 6), and 


1+ Vab 
svat” \L— Vab 
If a Z1 and 6 71, we shall find r = (1 — a) (1 — 5), and 


1 vb + xii) 
laa iY | SS = + ° 
2Vab °\Wb— Va)? 
_Ifa@ 71 and 6 21, we shall find r = (a — 1) (1 — 8), and 
1 a Vat Wvb 
== eaiiie Wat 
Lien api ave + ibh cata 
It would thus appear that the definite integral would furnish 
erroneous values of = if no attention was paid to those values 
of the factors of r and r’, which the circumstances of the inte- 
gration require: and it may be very easily shown that an atten- 
tion to the developements of K and K’ will, with equal certainty, 
enable us s to select the proper developement forz. T hus, ifa 7 1 


Bi 8 sa b= 


z= 


= + 28. 


s: => 


282 THIRD REPORT—1833. 

1 
f(a — 2b — IE 
and the value of x (z,) is determined by the combination of the 


two last developements. In a similar manner, if a Z1 and bZ1, 
 (z,) will be formed by the combination of the two first. If 


1 

a Z\and 6 71, thenr = (1—a)(b—1) (=a? (bby 
and the value of z (z,) is formed by the combination of the first 
and third developement. And if a 71 and 6 2 1, then the value 
of = (z3) will be formed by the combination of the second and 
third developements: in other words, the selection of the de- 
velopements is not arbitrary, in as much as {(1 — a)?}7* and 
{(a@ — 1)?}-* ought not to be considered, as we have already 
shown, as identical quantities, 

These combinations of the convergent and divergent series 
form all the four values of z, of which it appears that one value 
alone is correct for any assigned relation of a and 6 to 1, being 
that which arises from the combination of the convergent series 
for K and K’ only. The considerations, however, which deter- 
mine the selection of the correct developement of z are as de- 
finite and certain when the general series are employed as when 
that value is determined directly from the definite integral 
which expresses the value of z. It would appear to me, there- 
fore, that not only was the employment of divergent series 
necessary for the determination of a// the values of z, but that 
when the theory of their origin is perfectly understood they 
are perfectly competent to express all the limitations which are 
essential to their usage. The attempt to exclude the use of 
divergent series in symbolical operations would necessarily im- 
pose a limit upon the universality of algebraical formule and 
operations which is altogether contrary to the spirit of the 
science, considered as a science of symbols and their combina- 
tions. It would necessarily lead to a great and embarrassing 
multiplication of cases; it would deprive almost all algebraical 
operations of much of their certainty and simplicity; and it 
would altogether change the order of the investigation of results 
when obtained, and of their interpretation, to which I have so fre- 
quently referred in former parts of this Report, and upon which 
so many important conclusions have been made to depend. 

Elementary Works on Algebra.—There are few tasks the 
execution of which is so difficult as the composition of an ele- 
mentary work ; and very few in which, considering the immense 
number of such works, complete success is so rare. They re- 
quire, indeed, a union of qualities which the class of writers 
who usually undertake such works are not often competent to 


and 6 7 1, we have r = (a — 1) (6 — 1) 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 283 


furnish. Great simplicity in the exposition and exemplification 
of first principles, a perfect knowledge of the consequences to 
which they lead, and great forbearance in not making them an 
occasion for the display of the peculiar opinions or original re- 
searches of their authors. 

There is, in fact, only one elementary work which is entitled 
to be considered as having made a very near approach to per- 
fection. ‘The Elements of Euclid have been the text-book of 
geometers for two thousand years; and though they labour 
under some defects, which may or may not admit of remedy, 
without injury to the body of the work, yet they have not re- 
ceived any fundamental change, either in the propositions them- 
selves, or in their order of succession, or in the principles of 
their demonstrations, in the propriety of which geometers of 
any age or country have been found to acquiesce. It is true that 
both the objects and limits of the science of geometry are per- 
fectly defined and understood, and that systems of geometry 
must, more or less, necessarily approach to a common arrange- 
ment, in the order of their propositions, and to common prin- 
ciples as the bases of their demonstrations. But even if we 
should make every allowance for the superior simplicity of the 
truths to be demonstrated, and for the superior definiteness of 
the objects of the science to be taught, and also for the superior 
sanction and authority which time and the respect and accept- 
ance of all ages have assigned to this remarkable work, we may 
well despair of ever seeing any elementary exposition of the prin- 
ciples of algebra, or of any other science, which will be entitled to 
claim an equal authority, or which will equally become a model to 
which all other systems must, more or less, nearly approximate. 

There are great difficulties in the elementary exposition of 
the principles of algebra. As long as we confine our attention 
to the principles of arithmetical algebra, we have to deal with 
a science all whose objects are distinctly defined and clearly un- 
derstood, and all whose processes may be justified by demon- 
strative evidence. If we pass, however, beyond the limits which 
the principles of arithmetical algebra impose, both upon the re- 
presentation of the symbols, and upon the extent of the opera- 
tions to which they are subject, we are obliged to abandon the 
aid which is afforded by an immediate reference to the sensible 
objects of our reasoning. In the preceding parts of this Report 
we have endeavoured to explain the true connexion between 
arithmetical and symbolical algebra, and also the course which 
must be followed in order to give to the principles of the latter 
in their most general form such a character as may be adequate 
to justify all its conclusions. But the necessity which is thus 


284 THIRD REPORT—1833. 


imposed upon us of dealing with abstractions of a nature so 
complete and comprehensive, renders it extremely difficult to 
give to the principles of this science such a form as may bring 
them perfectly within the reach of a student of ordinary powers, 
and which have not hitherto been invigorated by the severe dis- 
cipline of a course of mathematical study. 

The range of the science of algebra is so vast, and its appli- 
cations are so various, both in their objects and in their degrees 
of difficulty, that it is quite impossible to fix absolutely the 
proper proportion of space which should be assigned to the 
developement of its different departments. If a system of al- 
gebra could be confined to the statement of fundamental prin- 
ciples, and to the establishment of fundamental propositions 
only, it might be possible to approximate to a fixed standard, 
which should possess the requisite union of simplicity and of 
sufficient generality. But it is a science which cannot be taught 
by an exposition of principles and their general consequences 
only, but requires a more or less lengthened institution of ex- 
amples of many of its different applications, in order to produce 
in the student mechanical habits of dealing with symbols and 
their combinations. The extent also to which such develope- 
ments are necessary will vary greatly with the capacities of dif- 
ferent students, and it would be quite impossible to determine 
any just mean between diffuseness and compression which shall 
be best adapted to the wants of the general average of students, 
or to the systems of instruction followed by the general average 
of teachers. 

In the early part of the last century the Algebra of Maclaurin 
was almost exclusively used in the public education of this 
country. It is unduly compressed in many of its most essential 
elementary parts, and is also unduly expanded in others which 
have reference to his own discoveries. It was written, however, 
in a simple and pure taste, and derived no small part of its 
authority as a text-book from the great and well-merited repu- 
tation of its author. It was subsequently, ina great measure, 
superseded, in the English Universities at least, by the large 
work of Sanderson, which was composed by this celebrated 
teacher to meet the wants of his numerous pupils. It was, in 
consequence, swelled out to a very unwieldy size by a vast 
number of examples worked out at great length; and it laboured 
under the very serious defect of teaching almost ‘exclusively 
arithmetical algebra, being far behind the work of Maclaurin 
in the exposition of general views of the science. At the latter 
end of the last century Dr. Wood, the present learned and 
venerable master of St. John’s College Cambridge, in conjunc- 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 285 


tion with the late Professor Vince, undertook the publication 
of a series of elementary works on analysis, and on the appli- 
cation of mathematics to different branches of natural philo- 
sophy, principally with a view to the benefit of students at the 
Universities. The works of the latter of these two writers have 
already fallen into very general neglect, in consequence partly 
of their want of elegance, and partly in consequence of their 
total unfitness to teach the more modern and improved forms 
of those different branches of science. But the works of his 
colleague in this undertaking have continued to increase in 
circulation, and are likely to exercise for many years a consi- 
derable influence upon our national system of education; for 
they possess in a very eminent degree the great requisites of 
simplicity and elegance, both in their composition and in their 
design. ‘The propositions are clearly stated and demonstrated, 
and are not incumbered with unnecessary explanations and 
illustrations. There is no attempt to bring prominently forward 
the peculiar views and researches of the author, and the dif- 
ferent parts of the subjects discussed are made to bear a proper 
subordination to each other. It is the union of all these qua- 
lities which has given to his works, and particularly to his 
Algebra, so great a degree of popularity, and which has se- 
cured, and is likely to continue to secure, their adoption as 
text-books for lectures and instruction, notwithstanding the 
absence of very profound and philosophical views of the first 
principles, and their want of adaptation, in many important 
particulars, to the methods which have been followed by the 
great continental writers. 

In later times a great number of elementary works on algebra, 
possessing various degrees of merit, have been published. 
Those, however, which have been written for purposes of in- 
struction only, without any reference to the advancement of 
new views, either of the principles of the science, or to the ex- 
tension of its applications, have generally failed in those great 
and essential requisites of simplicity, and of adequate, but not 
excessive, illustration, for which the work of Dr. Wood is so 
remarkably distinguished ; whilst other works, which have pos- 
sessed a more ambitious character, have been generally devoted 
too exclusively to the developement of some peculiar views of 
their authors, and have consequently not been entitled to be 
generally adopted as text-books in a system of academical or 
national education. There are, however, many private reasons 
which should prevent the author of this Report from enlarging 
upon this part of his subject, who is too conscious that there 
are few defects which he could presume to charge upon the | 


286 THIRD REPORT—1833. 


works of other authors from which he could venture to exempt 
his own. 

The elementary works on algebra and on all other branches 
of analytical and physical science which have been published 
in France since the period of the Revolution, have been very 
extensively used, not merely in this country, but in almost 
every part of the continent of Europe where the French lan- 
guage is known and understood. The great number of illus- 
trious men who took part in the lectures at the Normal and 
Polytechnic Schools at the time of their first institution, and 
the enlarged views which were consequently taken of the prin- 
ciples of elementary instruction and of their adaptation to the 
highest developement of the several sciences to which they 
lead, combined with the powerful stimulus given to the human 
mind in all ranks of life, in consequence of the stirring events 
which were taking place around them, at once placed the scien- 
tific education of France immensely in advance of that of the 
rest of Europe. The works of Lagrange, particularly his Caleul 
des Fonctions and his Théorie des Fonctions Analytiques, which 
formed the substance of lectures given at the Ecole Polytech- 
nique, exhibited the principles of the differential and integral 
calculus in a new light, and contributed, in connexion with his 
numerous other works and memoirs, which are unrivalled for 
their general elegance and fine philosophical views, to fami- 
liarize the French student with the most perfect forms and 
with the most correct and at the same time most general prin- 
ciples of analytical science. The labours of Monge also, upon 
the application of algebra to geometry, succeeded in bringing 
all the relations of space, with which every department of na- 
tural philosophy is concerned, completely under the dominion 
of analysis *, and thus enabled their elementary and other 
writers to exhibit the mathematical principles of every branch 
of natural philosophy under analytical and symmetrical forms. 
Laplace himself gave lectures on the principles of arithmetic 
and of algebra, which appear in the Séances de l’Ecole Nor- 
male and in the Journal de l’ Ecole Polytechnique; and there are 
very few of the illustrious men of science, of that or of a subse- 
quent period, who have done so much honour to France, who 
have not been more or less intimately associated with carrying 


* The developement of the details of this most important branch of analy- 
tical science, which has been so extensively and successfully cultivated in 
France, is greatly indebted to Monge’s pupils in the Polytechnic School, 
many of whom have subsequently attained to great scientific eminence: their 
results are chiefly contained in the three volumes of Correspondance sur 
U Ecole Polytechnique. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 287 


on the business of national education in its highest departments. 
The influence of such men has been felt not merely in the very 
general diffusion of scientific knowledge in that great nation, 
but also in the form and character of their elementary books, 
which are generally remarkable for their precision and clear- 
ness of statement, for their symmetry of form, and for their 
adaptation to the most extensive developement of the several 
sciences upon which they treat. ; 

The elementary works of M. Lacroix upon almost every de- 
partment of analytical science have been deservedly celebrated : 
they possess nearly all the excellences above enumerated as 
characteristic of French elementary writers, and they are also 
remarkable for the purity and simplicity of the style in which 
they are written*. The Cours des Mathématiques Pures of 
M. Francceur possesses merits of a similar kind, being too 
much compressed, however, for the purposes of self-instruction, 
though well adapted to form a basis for the lectures of a teacher. 
The works of M. Garnier are chiefly valuable for their careful 
illustration of, and judicious selection from, the writings of 
Lagrange, and are well calculated to make the general views 
and principles of that great analyst and philosopher familiar 
to the mind of a student. The Arithmetic, Algebra}, and Appli- 
cation of Algebra to Geometry, of M. Bourdon are works of 
more than ordinary merit, and present a very clear and fully 
developed view of the elements of those sciences. Many other 
works have been published of the same kind and with similar 
views by Reynaud, Boucharlat and other writers. 

I am too little acquainted with the elementary works which 
are used in the different Universities of Germany to be able to 
express any opinion of their character. Those which I have seen 
have been wanting in that precise and symmetrical form which 
constitutes the distinguishing merit of the French elementary 
writers ; but they are generally copious, even to excess, in their 
examples and illustrations. The immense developement which 
public instruction, in all its departments, has received in that 
country would lead us to conclude that they possess elementary 
mathematical works, which are at least not inferior to those which 


* Before the Revolution, the Cours des Mathématiques Pures et Appliquées 
of Bezout, in six volumes, was generally used in public education in France : 
it is a work much superior to any other publication of that period of a simi- 
lar kind which was to be found in any European language. 

_+ A part of the Algebra of Bourdon has been translated and highly com- 
mended by Mr. De Morgan, a gentleman whose philosophical work on Arith- 
“metic and whose various publications on the elementary and higher parts of 
mathematics, and particularly those which have reference to mathematical 
education, entitle his opinion to the greatest consideration. 


288 THIRD REPORT—1833. 


exist in other languages : and the labours of Gauss, Bessel, and 
Jacobi, and the numerous and important memoirs which appear 
in their public Journals and Transactions upon the most difficult 
questions of analysis and the physical sciences, sufficiently show 
that the mathematical literature of this most learned nation is 
not less diligently and successfully cultivated than that which 
belongs to every other department of human knowledge. 

The combinatorial analysis, which Hindenburg first intro- 
duced, has been cultivated in Germany with a singular and 
perfectly national predilection *; and it must be allowed that it 
is well calculated to compress into the smallest possible space 
the greatest possible quantity of meaning. In the doctrine of 
series it is also frequently of great use, and enables us to ex- 
hibit and to perceive relations which would not otherwise be 
easily discoverable. Without denying, however, the advantages 
which may attend either the study or the use of the notation of 
the combinatorial analysis, it may be very reasonably doubted 
whether those advantages form a sufficient compensation for 
the labour of acquiring an habitual command over the use and 
interpretation of a conventional symbolical language, which is 
necessarily more or less at variance with the ordinary usage and 
meaning of the symbols employed and of the laws of their com- 
binations. These objections would apply, if such a conven- 
tional use of symbolical language was universally adopted and 
understood; but they acquire a double force and authority, 
when it appears that they are only partially used in the only 
country | in which the combinatorial analysis is extensively 
cultivated, and that, consequently, those works in which it is 
adopted are excluded from general perusal, in consequence 
of their not being written in that peculiar form of symbolical 
language with which our mathematical associations are indis- 
solubly connected. 

Trigonometry.—The term Trigonometry sufficiently indicates 
the primitive object of this science, which was the determina- 
tion, from the requisite data, of the sides and angles of trian- 
gles: it was in fact considered in a great degree as an inde- 


* See Eytelwein’s Grundlehre der hohern Analysis, a very voluminous work, 
which contains the principal results of modern analysis and of the theory of 
series exhibited in the language and notation of this analysis. 

+ Professor Jarrett, of Catherine Hall, Cambridge, in some papers in the 
Transactions of the Philosophical Society of Cambridge, and in a Treatise on 
Algebraical Developement, has attempted to introduce the use of the lan- 
guage of the combinatorial analysis. The great neglect, however, which has 
attended those speculations, which are very general and in some respects 
extremely ingenious, is a sufficient proof of the difficulty of overcoming those 
mathematical habits which a long practice has generated and confirmed. 


; 
. 
q 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 289 


pendent science, and not as auxiliary to the application of al- 
gebra to geometry. It is to Euler* that we are indebted for 
the emancipation of this most important branch of analytical 
science from this very limited application, who first introduced 
the functional designations sin x, cos z, tan z, &c., to denote the 
sine, cosine, tangent, &c., of an are x, whose radius is 1, which 
had previously been designated by words at length, or by simple 
and independent symbols, such as a, b, s, c, t, &c. The intro- 
duction of this new algorithm speedily changed the whole form 
and character of symbolical language, and greatly extended 
and simplified its applications to analysis, and to every branch 
of natural philosophy. 

The angles which enter into consideration in trigonometry 
are generally assumed to be measured by the arcs of a circle 
of a given radius, and their sines and cosines are commonly de- 
fined with reference to the determination of these arcs, and not 
with reference to the determination of the angles which they 
measure. It is in consequence of this defined connexion of 
sines and cosines with the arcs, and not immediately with the 
angles which they measure, that the radius of the circle upon 
which those ares are taken must necessarily enter as an element 
in the comparison of the sines and cosines of the same angle 
determined by different measures: and though they were ge- 
nerally, at least in later writers, reduced to a common standard, 
by assuming the radius of this circle to be 1, yet formule were 
considered as not perfectly general unless they were expressed 
with reference to any radius whatsoever}. In the application, 
likewise, of such formule to the business of calculation, the 
consideration of the radius was generally introduced, producing 
no small degree of confusion and embarrassment; and even in 
the construction of logarithmic tables of sines and cosines the 


* Introductio in Analysim Infinitorum, vol. i. cap. viii. ‘Quemadmodum 
logarithmi peculiarem algorithmum requirunt, cujus in universa analysi summus 
extat usus, ita quantitates circulares ad certam quoque algorithmi normam 
perduxi: ut in caleulo eque commode ac logarithmi et ipsze quantitates alge- 
braicee tractari possent.’’—Extract from Preface. 

+ We may refer to Vince’s Trigonometry, a work in general use in this 
country less than a quarter of a century ago, and to other earlier as well as 
contemporary writers on this subject, for examples of formule, which are uni- 
formly embarrassed by the introduction of this extraneous element. Later writers 
have assumed the radius of the circle to be 1, and have contented themselves 
with giving rules for the conversion of the resulting formulz to those which 
would arise from the use of any other radius. It is somewhat remarkable that 
the elementary writers on this subject should have continued to encumber their 
formule with this element Jong after its use had been abandoned by Euler, 


Lagrange, Laplace, and all the other great and classical mathematical writers 
on the Continent. 


1833. U 


290 THIRD REPORT—1833. 


occurrence of negative logarithms was avoided by a fiction, 
which supposed them to be the sines and cosines of arcs of a 
circle whose radius was 10”, 

A very slight modification of the definition of the sine and 
cosine would enable us to get rid of this element altogether. 
In a right-angled triangle, the ratio of any two of its sides will 
determine its species, and conse- © 
quently the magnitude of its angles. P 
If we suppose, therefore, a point P 
to be taken in one (A C) of the two 
lines A C and A B containing the 
angle B A C (6), and P M to be 
drawn perpendicular to the other 
line (A B), then we may define the A M 


sine of 6 to be the ratio aos and the cosine of 6 to be the 
By such definitions we shall make the sine and 


ti AM 
ratio |p: 
cosine of an angle depend upon the angle itself, and not upon 
its measure, or upon the radius of the circle in which it is taken: 
and upon this foundation all the formule of trigonometry may 
be established, and their applications made, without the neces- 
sity of mentioning the word radius*. 

If we likewise assume the ratio of the are which subtends an 
angle to the radius of the circle in which it is taken, and not 
the arc itself, for the measure of an angle, we shall obtain a 
quantity which is independent of this radius. In assuming, 
therefore, the angle @ to be not only measured, but also repre- 
sented by this ratio, we shall be enabled to compare sin # and 
cos 6 directly with 6, and thus to express one of them in terms 
of the other. It is this hypothesis which is made in deducing 
the exponential expressions for the sine and cosine, and the 
series which result immediately from them f. 

* See A Syllabus of a Course of Lectures upon Trigonometry, and the Appli- 
cation of Algebra to Geometry, published at Cambridge in 1833, in which all 
the formulz of trigonometry are deduced in conformity with these definitions. 

+ If we should attempt to deduce the exponential expressions for sin @ and 
cos 6 from the system of fundamental equations, Re 


cos? § + sin? @= 1 (1.) 
cos § = cos (— @) (2.) 
sin §@ = — sin (— @) (3.) 
we should find, 
pAdV—1 4 .—Abv—1 hal AOSD abl 
cos § = + ah eracear WARP aaa and sin §@= Mets ee 


in which the quantity A, in the absence of any deferminate measure of the 


aie 


— se 


————- =! 


’ of the equation 


Jaen See ae eee ee eee 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 291 


The sines and cosines and the measures of angles defined 
and determined as above, are the only essential elements in a 
system of trigonometry, and are sufficient for the deduction of 
all the important formule which are required either in algebra 


angle 6, would be perfectly indeterminate. It is the assumption of the measure 
of an angle which is mentioned in the text which makes it necessary to re- 
place A by 1. 

The knowledge of the exponential expressions for the sine and cosine would 
furnish us immediately with all the other properties of these transcendents. 
Thus, if the sines and cosines of two angles be given, we can find the sines 
and cosines of their sum and difference; and from hence, also, we can find 
the sine and cosine of any multiple of an angle from the values of the sine 
and cosine of the simple angle; and also through the medium of the solution 
of equations the sine and cosine of its submultiples. In fact, as far as the 
symbolical properties of those transcendents are concerned, it is altogether 
indifferent whether we consider them to be deduced primarily from the 
assumed functional equations (1.), (2.), (3.), or from the primitive geome- 
trical definitions of which those equations are the immediate symbolical con- 
sequences. 


¢ x dex y dy 
If we should denote the integrals and >=, (com- 
0 0 vl—y 


a/1 — 22 


mencing from 0 respectively) by @ and 6’ respectively, then the integral of 
the equation 


dx Oya 
/1 — 2 Ji —y2 (a.) 
would furnish us with the fundamental equation 
sin (9 + 6’) = sin 4 cos 6’ + cos 6sin 6’, (B.) 


if we should replace x by sin 6, /1 — x by cos 6, y by sin 6’, and 1 — y 
by cos @’. If the formulz of trigonometry were founded upon such a basis, 
they would require no previous knowledge either of circular arcs considered 
as the measures of angles, or of the geometrical definitions of the sines and 
cosines, except so far as they may be ascertained from the examination of the 
values and properties of the transcendents which enter into the equation (a.). 
In a similar manner, if we should suppose 4 and 6! to represent the integrals 
d 
1 


ie 
of the transcendents , ot and if, Wie ey then the integral 


y 
Tt 


dx dy 


: ——*__ =9 : 
vate) * Ja+ey ce 

would be expressed by the equation 
hsin (6 + 6’) =hsin 6 X hcos 6’ + hcos 6 X Asin #, (8.) 


if we should make « = h sin 6 (the hyperbolic sine of 4), and 4/(1 + 2?) 

=h cos 4 (the hyperbolic cosine of 6), y= h sin 6’, and V1 + y? = hos 6’, 

adopting the terms which Lambert introduced, and which have been noticed 

in the note in p. 231; and it is evident that it would be possible from equa- 

tion (0.), combined with the assumptions made in deducing it, to frame a 

system of hyperbolic trigonometry (having reference to the sectors, and not 
u2 


292 THIRD REPORT—1833. 


or in its applications to geometry. The terms tangent, co- 
tangent, secant and cosecant, and versed sine, which denote 
very simple functions of the sine and cosine, may be defined by 
those functions and will be merely used when they enable us to 
exhibit formule involving sines and cosines, in a more simple 
form. By adopting such a view of the meaning and origin of 
the transcendental functions, the relations and properties of 
which constitute the science of trigonometry, we are at once 
freed from the necessity of considering those functions as lines 
described in and about a circle, and as jointly dependent upon 
the magnitude of the angles to which they correspond and of the 
radius of the circle itself. It is this last element, which is thus 
introduced, which is not merely superfluous, but calculated to 
give erroneous views of the origin and constitution of trigono- 
metrical formule and greatly to embarrass all their applications. 


to the arcs of the equilateral hyperbola), whose formule would bear a very 
striking analogy to the formule of trigonometry, properly so called. 

Abel, in the second volume of Crelle’s Journal, has laid the foundation, of 
a species of elliptic trigonometry, (if such a term may be used,) in connexion 
with a remarkable extension of the theory of elliptic integrals. If we denote 
the elliptic integral of the first species 


oldie 
Be / (1 — & sin? ) 
by 4, and replace sin W by «, we shall get 
dx 


0 if { a _ a) — a a) } 


= 


or more generally 


dx 
= STATE GPa GaRTTa eT ata 
« v {0 +ea%) (1— 2 a} 
If we now suppose «= 96, ./(1 — c? a?) > f band / (1+ 22?) = F 8, it 
may be demonstrated that 
n — 9Osf 0. Fe +08. fH). FA 
COT 14 ee ee.ge’ 
rn _f4-f4'—20b.00.F).F# 
fO+ “= Ite ee. es , 
FO.FV+29060.900.fO.f 0 
1+ec¢?s.g f 
or if, for the sake of more distinct and immediate reference to these peculiar 
transcendents, we denote 
Q 4 by sin 4 (elliptic sine of 4), 


F6@+@%)= 


f 4 by cos 6 (elliptic cosine of 4), and 
F é by (sur 4 (elliptic sursine of @), 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 293 


The primitive signs + and —, when applied to symbols de- 
noting lines, are only competent to express the relation of lines 
which are parallel to each other when drawn or estimated in dif- 
ferent directions; but the more general sign cos + “—1 sin 4, 
which has been noticed in the former part of this Report, when 
applied to such symbols, is competent to express all the rela- 
tions of position of lines in the same plane with respect to each 
other. It is the use of this sign which enables us to subject 
the properties of rectilinear figures to the dominion of algebra: 
thus, a series of lines represented in magnitude and position by 


Mp, (cos 6, + VW —1 sin 6,) ay, {cos (6, + 6) + VW —1 sin (6, + 45) } ata, 
w+ {cos (6, + 65+... On1) + YW = sin (0,4 05+... Oni) } Qos 


will be competent to form a closed figure, if the following equa- 
tions be satisfied : 


then these fundamental equations will become 
sin 6 cos 4' surs 6! + sin 6’ cos 6 surs 6 


. e e e e e e€ 
sin (6+ 6')= : - 
2 SAPP fl 1+ e ¢? sin? 6 sin? 6’ ’ 
e e 


cos 6 cos 6’ — c2 sin @ sin 6’ surs 4 surs 6' 


Aappetany e ile aa Re na Nat oat 
cos (6+64= 1 + & c2 sin? 6 sin? 6’ 
e e 


surs @ surs 6! + e2 sin 6 sin 6’ cos 6 cos 6! 


8 nyt e € e é 4 e 
Piaaly ld dic 1 + e2 c2 sin? @ sin? 6! 
e € 


If we add, subtract and multiply, the elliptic sines, cosines and sursines of 
the sum and difference of 4 and 4! respectively, reducing them, when necessary, 
by the aid of the fundamental relations which exist amongst these three tran- 
scendents, we shall obtain a series of formulz, some of which are very remark- 
able, and which degenerate into the ordinary formule of trigonometry, when 
e= 0 and c= 1: we shall thus likewise be enabled to express sin » 6, cos n 6, 


é 
surs n6, in terms of sin 6, cos 6, sursé. The inverse problem, however, to express 
e e é e 
sin 8, cos 6, surs 6, in terms of sin n 6, cos n 4, surs n 6, is one of much greater 


e é e 4 e e 

difficulty, requiring the consideration of equations of high orders, but whose 
ultimate solution can be made to depend upon that of an equation of (m + 1) 
dimensions only. It is in the discussion of these equations that Abel has dis- 
played all the resources of his extraordinary genius. 

It would be altogether out of place to enter upon a lengthened statement 
of the various properties of these elliptic sines, cosines, and sursines ; their 
periodicity, their limits, their roots, and their extraordinary use in the trans- 
formation of elliptic functions. My object has been merely to notice the ru- 
diments of a species of elliptic trigonometry, the cultivation of which, even 
without the aid of a distinct algorithm, has already contributed so greatly to 
the enlargement of the domains of analysis. 


294 THIRD REPORT—1833. 


dy +a, cos 6, + aq cos (6; +03) +. - An, COS (6, + 4+.» On—1)= 0 (1.) 
a, sin 6; +a sin (6, +45) +... @p_; sin (6,4 6)+.. 6,1) =0 (2.) 
6 t+b+..-41.=(n—Q2r)e (3.) 

The first two of these equations may be called equations of 
figure, and the last the equation of angles, and all of them must 
be satisfied in order that the lines in question may be capable 
of being formed into a figure, along the sides of which if a point 
be moved it will circulate continually. If the values of 4,, 
6, — 9), 6; — 95. » bn-1 — 4n—2 be all positive, and if r = 1, then 
the equation of angles will correspond to those rectilineal figures 
to which the corollaries to the thirty-second proposition of the 
first book of Euclid are applicable, and which are contemplated 
by the ordinary definitions of rectilineal figures in geometry. 
If we should suppose r = 2 or 3 or any other whole number 
different from 1, the equation would correspond to stellated 
figures, where the sum of the exterior angles shall be 8, 12, or 
4 r right angles. The properties of such stellated figures were 
first noticed by Poinsot in the fourth volume of the Journal 
de 1 Ecole Polytechnique, in a very interesting memoir on the 
Geometry of Situation*. 

All equal and parallel lines drawn or estimated in the same 
direction are expressed by the same symbol affected by the 
same sign, whatever it may be: and it is this infinity of lines, 
geometrically different from each other, which have the same 
algebraical representation, which renders it necessary to con- 
sider the position of lines, not merely with respect to each other, 
but also with respect to fived lines or axes, through the medium 
of the equations of their generating points. In other words, it 
is not possible to supersede even rectilineal geometry by means 
of affected symbols only. We are thus led to the consideration 
of a new branch of analytical science, which is specifically de- 
nominated the Application of Algebra to Geometry, and which 
enables us to consider every relation of points in space and the 
laws of their connexion with each other, whatever those laws 
may be. It is not our intention, however, to enter upon the 
discussion of the general principles of this science, or to notice 
its present state or recent progress. 

A great number of elementary works on trigonometry have 
been published of late years in this country, many of which are 
remarkable for the great simplicity of form to which they have 
reduced the investigation of the fundamental formule. Such 
works are admirably calculated to promote the extension of 


® See also Peacock’s A/gebra, p. 448. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 295 


mathematical education, by placing this most important branch 
of analytical science, the very key-stone of all the applications 
of mathematics to natural philosophy, within the reach of every 
student who has mastered the elements of geometry and the 
first principles of algebra. 

We have before had occasion to notice the work of the late 
Professor Vince upon this subject, which was generally used 
in the Universities of England for some years after the com- 
mencement of the present century. Its author was a mathema- 
tician of no inconsiderable powers, and of very extensive know- 
ledge, but who was totally destitute of all feeling for elegance 

‘in the selection and construction of his formule, and who had 
no acquaintance with, or rather no proper power of appreciating, 
those beautiful models of symmetry and of correct taste which 
were presented by the works of Euler and Lagrange. But 
though this treatise was singularly rude and barbarous in its 
form, and altogether inadequate to introduce the student to a 
proper knowledge either of the objects or of the powers of this 
science, yet it was greatly in advance of other treatises which 
were used and studied in this country at the period of its pub- 
lication. Amongst these may be mentioned the treatise on Tri- 
gonometry which is appended to Simson’s Euclid, which was 
more adapted to the state of the science in the age of Ptolemy 
than at the close of the eighteenth century *. 

The Plane and Spherical Trigonometry of the late Professor 
Woodhouse appeared in 1810, and more than any other work 
contributed to revolutionize the mathematical studies of this 
country. It was a work, independently of its singularly oppor- 
tune appearance, of great merit, and such as is not likely, not- 
withstanding the crowd of similar publications in the present 
day, to be speedily superseded in the business of education. 
The fundamental formule are demonstrated with considerable 
elegance and simplicity ; the examples of their application, both 
in plane and spherical trigonometry, are well selected and very 
carefully worked out; the uses of trigonometrical formule, in 
some of their highest applications, are exhibited and pointed 


* Similar remarks might be applied to treatises upon trigonometry which 
were published both before and after the appearance of Professor Wood- 
house’s Trigonometry. The author of this Report well recollects a treatise of 
this kind which was extensively used when he was a student at the Univer- 
sity, in which the proposition for expressing the sine of an angle in terms of 
the sides of a triangle, was familiarly denominated the black triangle, in con- 
sequence of the use of thick and dark lines to distinguish the primitive tri- 
angle amidst the confused mass of other lines in which it was enveloped, for 
the purpose of obtaining the required result by means of an incongruous 
combination of geometry and algebra. 


296 THIRD REPORT—1833. 


out in a very clear and striking form; and, like all other 
works of this author, it is written in a manner well calculated 
to fix strongly the attention of the student, and to make him 
reflect attentively upon the particular processes which are fol- 
lowed, and upon the reasons which lead to their adoption. 
The circumstances attending the publication and reception of 
this work in the University of Cambridge were sufficiently re- 
markable. It was opposed and stigmatized by many of the older 
members, as tending to produce a dangerous innovation in the 
existing course of academical studies, and to subvert the pre- 
valent taste for the geometrical form of conducting investiga- 


tions and of exhibiting results which had been adopted by - 


Newton in the greatest of his works, and which it became us, 
therefore, from a regard to the national honour and our awn, 
to maintain unaltered. It was contended, also, that the primary 
object of academical education, namely, the severe cultivation 
and discipline of the mind, was more effectually attained by 
geometrical than by analytical studies, in which the objects of 
our reasoning are less definite and tangible, and where the 
processes of demonstration are much less logical and complete. 
The opposition, however, to this change, though urged with 
considerable violence, experienced the ordinary fate of attempts 
made to resist the inevitable progress of knowledge and the 
increased wants and improving spirit of the age. In the course 
of a few years the work in question was universally adopted. 
The antiquated fluxional notation which interfered so greatly 
with the familiar study of the works of Euler, Lagrange, La- 
place, and the other great records of analytical and philoso- 
phical knowledge, was abandoned*; the works of the best 
mathematical writers on the continent of Europe were rapidly 
introduced into the course of the studies of the University; and 
the secure foundations were laid of a system of mathematical 
and philosophical education at once severe and comprehensive, 
which is now producing, and is likely to continue to produce, 
the most important effects upon the scientific character of the 
nation. 

Theory of Equations. 1. Composition of Equations.—The 
first and one of the most difficult propositions which presents 
itself in the theory of equations is to prove “that all equations 
under a rational form, and arranged according to the method 


* The continental notation of the differential calculus was first publicly 
introduced into the Senate House examinations in 1817. Though the change 
was strongly deprecated at the time, it was very speedily adopted, and in 
less than two years from that time the fluxional notation had altogether dis- 
appeared. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 297 


of Harriott, the significant terms forming one member, and 
zero the other, are said to be resolvible into simple or quadratic 
factors.” It is only another form of the same proposition to 
say, ‘that every equation has as many roots as it has dimen- 
sions, and no more; those roots being either real* or ima- 
ginary ;” that is, being quantities which are expressible by 
symbols denoting real magnitudes affected by such signs as 
are recognised in algebra. 

We have before said that it is impossible to assign before- 
hand an absolute limit to the possible existence of signs of 
affection different from those which are involved in the sym- 
bolical values of (1)” and (—1)"; and when it is said that every 
equation is resolvable into factors of the form x — a, we presume 
that a is either a real magnitude, or of the form « + 6 Woah) 
where « and # are real magnitudes. If we should fail in esta- 
blishing this proposition, it would by no means necessarily fol- 
low that there might not exist other forms of factors like x — a, 
where a denoted a real magnitude affected by some wnknown 
sign different from +, —, or cos 6+ “—1 sin 8, which might 
satisfy the required conditions: at the same time its demonstra- 
tion will show that our recognised signs are competent to de- 
note all the affections of magnitude which are subject to any 
conditions which are reducible to the form of an equation. 

If we assume in the first instance the composition of equa- 
tions to be such as we have stated in the enunciation of the 
fundamental proposition, we can at once ascertain the composi- 
tion of the several coefficients of the powers of « in the equa- 
tion 

x — pw 4+ poem -3— 2... + py = 0; 


and we can complete the investigation of all those general pro- 
perties of equations which such an hypothesis would lead to. 
All such conclusions, when established upon such a foundation, 
are conditional only. It is not expedient, however, to make 
the fate of any number of propositions, however consistent with 
each other, and however unquestionable their truth may appear 
to be from indirect or from a posteriori considerations, depend- 
ent upon an hypothesis, when it is possible to convert this hypo- 
thesis into a necessary symbolical truth. Using such an hypo- 
thesis, therefore, as a suggestion merely, let us propose the 


* It is convenient in the theory of equations, for the purpose of avoiding 
repetition, to consider symbols denoting arithmetical magnitudes and affected 
with the signs + or —, as real; and quantities denoted by symbols affected 
with the sign cos 6 + “—1 sin 6, as imaginary. 


298 THIRD REPORT—1833. 


following problem, and examine all the consequences to which 
its solution will lead. 

“To find n quantities x, 21, 2, ... %,_,, such that their sum 
shall be equal to p,, the sum of all their products two and two 
shall be equal to p,, the sum of all their products three and 
three shall be equal to p,, and so on, until we arrive at their 
continued product, which shall be equal to py.” 

The quantities x, x,,... a,-1, are supposed to be any quan- 
tities whatever, whether real or affected by any signs of affec- 
tion whether known or unknown. It is our object to show that 
the only sign of affection required is cos @ + “—1 sin 4, taken 
in its most general sense. 

It is very easy to show that the solution of this problem will 
lead to a general equation, whose coefficients are p, Po, +++ Pn: 
for if we suppose the first of these quantities a to be omitted, 
and P,, P.,... P,_, to be the quantities corresponding to p,, 
Po, » ++ fn when there are (n — 1) quantities instead of m, then 
we shall get 


z+P,=p,, 
xP, + Po = pa, 
xP, + P3= ps, 


x Py_5 ar PR) = Pn-1s 
ax ier = Pn 


If we multiply these equations from the first downwards by the 
terms of the series #"~!, 2"-?,... x?, x, 1, and add the first, 
third, fifth, &c., of the results together, and subtract the second, 
fourth, sixth, &c., we shall get the general equation 


a — p, a" 4+ pa? —... + (— 1) pp = 9. (1.) 


In as much as p,, po, --- Pn May represent any real magni- 
tudes whatever, zero included, it is obvious that we may consi- 
der this equation as the result of the solution of the problem in 
its most general form. And in as much as # may represent any 
one of the » quantities involved in the problem, we must equally 
obtain the same equation for all those m quantities: it also fol- 
lows that every general solution of this equation must compre- 
hend the expression of all the roots. 

By this mode of presenting the question we are authorized in 
considering the symbolical composition of the coefficients of 
every equation as known, though the ultimate symbolical form 
of the roots is not known ; and our inquiry will now be properly 
limited to the question of ascertaining whether symbols repre- 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 299 


senting real magnitudes affected by the recognised and known 
signs of affection only, are competent, under all circumstances, 
to answer the required conditions of the problem. 

If the value of one root can be ascertained, and that root be 
real, the problem can be simplified, and the dimensions of the 
equation depressed by unity; for the coefficients of the reduced 
equation P,. P,. Pa-1, which are also real, can be successively 
determined. If more real roots than one can be found, the 
dimensions of the equation can be depressed by as many wnities 
as there are real roots. If the root determined be not real, and if 
a similar process for depressing the dimensions of the equation 
be adopted, the coefficients of the new equation would not be 
real, and the conditions of the problem with respect to the re- 
maining roots would be changed. But if we could ascertain a 
pair of such roots, such that their sum = x + x, and their pro- 
duct = 2 x, should be real, then the dimensions of the equation 
might be depressed by two unities, without changing the con- 
ditions of the problem with respect to the remaining roots; for 
if we supposed Q,, Q,, Q;, &c., to represent the coefficients of 
the reduced equation, we should find, 


e+a,+Q,= p 
XX, + (x + x,)Q, + Q2 = pa 
x t,Q) + (@ + a) Q, + Qs = ps; 


xX, Q,-4 => (x + 2) Qn-3 + Q,-2 = Pn-1s 
x Ly Qn-2 = Pn 
from which equations we can determine successively rational 
values of Q,, Q,, . - - Qr-s It remains to show, therefore, 
that in all cases we can find pairs of roots which will answer 
these conditions. 

If the number of quantities x, 2,,. . . ,, be odd, it is very 
easy to prove that there is always a real value of one of them, «, 
which will satisfy the conditions of the general equation (1.) *, 
and that consequently the dimensions of the equation may be 
depressed by unity, and our attention confined therefore to 
the case where the dimensions of the equation are even. If m, 
therefore, be any odd number, the form of x may be either 2 m, 
2? m, 22 m, 2" m, and so on. Let us consider, in the first place, 
the first of these cases. 
~ The number of combinations of 2 m, things taken two and two 
together, is m (2 m — 1,) and therefore an odd number: these 


* This may be easily proved without the necessity of making any hypothesis 
respecting the composition of the equation. See the Article ‘ Equarions’ in 
the Supplement to the Encyclopedia Britannica, written by Mr. Ivory. 


300 THIRD REPORT—1833. 


combinations may be either the swms of every two of the quanti- 
ties, @, @,...X,_1, Such as x + 2, x + X29, &c., or their products, 
such as w 2, or other rational linear functions of those quanti- 
ties, involving two of them only, such as « + a, + ra,,2 + 2, 
+2xx,,0rx+ 2, +x x, where k may be any given num- 
ber whatsoever. If we take any one of these sets of combina- 
tions, we can form rational expressions for their sum, for the 
sum of their products, two and two, three and three, and so on, 
in terms of the coefficients p,, 2, ..- Pn, of the original equa- 
tion (1.), by means of the common theory of symmetrical func- 
tions *, and consequently, we can form the corresponding equa- 
tions of m (2m— 1) dimensions which will have rational and 
known coefficients. - Such equations being of odd dimensions 
must have at least one real roct ; or, in other words, there must 
exist at least one real value of one of the sums of two roots, 
such as # + x,, of one of the products, such as x 2, of one 
of the functions, x + 2, + «a,orx +a,+hkeau,. If the 
symbols which form the real sum 2 + 2, are the same with those 
which form the real value of the product x x,, then, under such 
circumstances, x and x, are expressible by real magnitudes af- 
fected with the ordinary signs of algebra. We shall now pro- 
ceed to show that this must be the case. 

If we form the equations successively whose roots are x + 2, 
+ kx x,, corresponding to different values of 4, we shall have 
one real root at least in each of them. If we form more than 
m (2m — 1), such equations for different values of 4, we must 
at least have amongst them the same combination of x and a, 
forming the real root, in as much as there are only m (2 m — 1) 
such combinations which are different from each other. Let & 
and #, be the values of * which give such combinations, and 
let a! and 6! be the values of the real roots corresponding ; then 
we must have 

r+a,+khea =e’ 
eee a a, ay Se 

* The formation of symmetrical combinations of any number of symbolical 
quantities 2, 2,,...#n—1, and the determination of their symbolical values 
in terms of their sums (p,), their products two and two (pg), three and three 
(ps), and so on, involves no principle which is not contained in the direct 
processes of algebra, and is altogether independent of the theory of equations. 
The theorems for this purpose may be found in the first chapter of Waring’s 
Meditationes Algebraice, in Lagrange’s Traité sur la Résolution des Equations 
Numériques, chap. i. and notes 3 and 10, and with more or less detail in 


nearly all treatises on Algebra. 
+ Ife+a,=aandxex,=8., where wand Aare real magnitudes, then 


= 4/ { os — af the values of which are either real or of the form 


(cos 6+ VST sin 6) /B, where the modulus /f is real. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 301 


and therefore 


haba 
ices 
he —kB 


diddul ia dane a 


There are therefore necessarily two roots of the equation or two 
values of the symbols x, 2}, %o,-.- 1, Such that x + x, and 
x x, are real; and therefore it is always possible, in an equation 
whose dimensions are impariter par, to depress them by two 
unities, so that the reduced equation may still possess rational 
coefficients. 

If the number of symbols involved in the original problem be 
22m, then the number of their binary combinations must be 
2 m (2? m — 1) or impariter par. It will immediately follow, from 
what we have already proved, that there are two values of the 
sum and product of the same symbols, which are either real or 
of the forma + 6 “—1; and consequently the symbols them- 
selves will admit of expression under a similar form *. 

If the dimensions of the original equation be 2? m or 24 m, or 
any one in an ascending series of orders of parity, it may be re- 
duced down to the next order of parity ina similar manner: and 
under all circumstances it may be shown that there must be two 
roots which are reducible to the forma + 6 / —1, where «and 6 
real or zero; and also in any equation of even dimensions, we 
can reduce its dimensions successively by two unities, thus pro- 
ducing a series of equations of successive or decreasing orders of 
parity, in which we can demonstrate the existence of successive 
pairs of roots of the required form until they are all exhausted. 

This mode of proving the composition of equations differs 
chiefly from that which was noticed by Laplace, in his lectures 
to the Ecole Normale in 1795}, in the form in which the ques- 
tion is proposed. A certain number of symbols, representing 
magnitudes with unknown affections, are required to satisfy 


~ * Let vt a2! =r (cosé+ /—1 sin 6) 
x x' = e(cos@+ V.— 1sinQ) 
x + al? —4e2' = R?(cos2y + V/—T sin 2) 
orz— x'=R (cos + V—1 sin) 
oa neon tS Cad aye 
= r' (cos % + V1 sin x) 


x' =r! (cosy — VW —I1sinx). 
+ Legons de l’Ecole Normale, tom. ii. 


302 THIRD REPORT—1833. 


certain real conditions: those conditions are found to be iden- 
tical with those which the unknown quantity, or, in other words, 
the root in an equation of m dimensions, is required to satisfy. 
The object of the proof above given is to show that it is always 
possible to find ” real magnitudes with known affections which 
are competent to satisfy these conditions ; and those quantities, 
therefore, are of such a kind that the equation, whose roots 
they are, is always resolvible into real quadratic factors ; a most 
important conclusion, which the greatest analysts have laboured 
to deduce by methods which have not been, in most cases at 
least, free from very serious objections. 

There are two classes of demonstrations which have been 
given of this fundamental proposition in the theory of equations. 
The first class comprehends those in which the form of the 
roots is determined from the conditions which they are required 
to satisfy; the second class, those in which the form of the 
roots is assumed to be comprehended under different values of 
p and 4 in the expression p (cos § + “—1 sin 6), and it is shown 
that they are competent to satisfy the conditions of the equa- 
tion. To the first class belongs the demonstration given above; 
those given by Lagrange in notes ix. and x. to his Résolution des 
Equations Numériques; the first of those given by Gauss in the 
Gottingen Transactions for 1816*; and by Mr. Ivory in his 
article on Equations in the Supplement to the Encyclopedia 
Britannica. 'To the second class belongs the second demon- 
stration given by Gauss in the same volume of the Gottingen 
Transactions; by Legendre in the 14th section of the first 
Part of his Théorie des Nombres; by Cauchy in the 18th 
cahier of the Journal de Ecole Polytechnique; and subse- 
quently under a slightly different form in his Cours d’ Analyse 
Algébrique. 

The first of the demonstrations given by Gauss, like many 
other writings of that great analyst, is extremely difficult to 
follow, in consequence of the want of distinct enunciations of 
the propositions to be proved, and still more from their not 
always succeeding each other in the natural order of investi- 
gation. It requires the aid likewise of principles, or rather of 
processes, which are too far advanced in the order of the re- 
sults of algebra to be properly employed in the establishment 
of a proposition which is elementary in the order of truths, 
though it may not be so in the order of difficulty. If we may 


* There is another demonstration by Gauss, published in 1799, which I 
have never seen. In his Preface to his Demonstratio Nova Altera he speaks 
of its being founded partly on geometrical considerations, and in other re- 
spects as involving very different principles from the second. 


y 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 303 


be allowed, however, to consider it apart from such considera- 
tions, it would appear to be complete and satisfactory, and 
very carefully guarded against any approach to an assumption 
of the proposition to be proved, a defect to which most of the 
demonstrations of this class are more or less liable*. It extends 
to equations whose dimensions involve different or successive 
orders of parities, nearly in the same manner as in the demon- 
stration which we have given above. 

The demonstration given by Mr. Ivory is different from any 
other, and the principles involved in it are such as naturally 
present themselves in such an investigation; and it will be re- 
commended to many persons by its not involving directly the 
use, or supposing the necessary existence of, imaginary quan- 
tities. It is not, however, altogether free from some very serious 
defects in the form under which it at present appears, though 
most of them admit of being remedied without any injury to 
the general scheme of the demonstration, which is framed with 
great skill, and which exhibits throughout a perfect command 
over the most refined and difficult artifices of analysis. 

Lagrange has devoted two notes to his great work on the 
Resolution of Numerical Equations to the discussion of the 
forms of the roots of equations. In the first of these notes, 
after examining the very remarkable observations of D’Alem- 
bert on the forms of imaginary quantities, he proceeds to con- 
sider the case of an equation such as f (x) + V = 0, where 
J (x) is a rational function of x; if for different values a and } 
of the last term of this equation, where a 2 6, we may suppose 
a root which is not real for values of V between those limits, to 
become real at those limits, he then shows that for values 
of V between those limits, and indefinitely near to them, the 
corresponding root of the equation must involve “—1, or 

=1, or Y—1, and so on; or, in other words, that the roots 
of the equation in the transition of their values from. real 
to imaginary (whatever may be the affection of magnitude 
which renders them imaginary), will change in form from « to 
m +n —1. He subsequently shows that the same result will 
follow for any values of V between a and 6, and consequently, 


* I do not yenture to speak more decidedly; for though I have read it en- 
tirely through several times with great care, I do not retain that distinct and 
clear conviction of the essential connexion of all its parts which is necessary 
to compel assent to the truth of a demonstration. It is unfortunately fre- 
quently the character of many of the higher and more difficult investigations 
connected with the general theory of the composition and solution of equa- 
tions to leave a vague and imperfect impression of their truth and correctness 
even upon the minds of the most laborious and best instructed readers. 


304. THIRD REPORT—1833. 


that in every instance, when roots of equations cease to be real, | 
they will assume the form m + n VW —1. 

This demonstration is not merely indirect, but it does not _ 
arise naturally from the question to be investigated. It seems 
likewise to assume the existence of some algebraical form which 
expresses the value of the root in terms of the coefficients of 
the equation, an assumption which, as will afterwards be seen, 
it would be difficult to justify by any & priori considerations. 
The illustrious author himself seems to have felt the full force 
of these objections, and he proceeds therefore in the following 
Note to prove that every polynomial of a rational form will ad- 
mit of rational divisors of the first or second degree. The de- 
monstration which he has given is founded upon the theory of 
symmetrical functions, and shows that the coefficients of such 
a divisor may be made to depend severally upon equations all 
whose coefficients are rational functions of the coefficients of 
the polynomial dividend. Whatever be the degree of parity of 
the number which expresses the dimensions of this polynome, 
he shows the possibility of the coefficients of this quadratic di- 
visor, which is the capital conclusion in the theory. It ought 
to be observed, however, that the whole theory of the compo- 
sition of equations is so much involved in the different steps of 
this investigation, or, at all events, that so little provision is 
made in conducting it to guard against the assumption of 
this truth, that we should not be justified in considering this 
demonstration as perfectly independent or as furnishing an 
adequate foundation for so important a conclusion. If we view 
it, however, simply with reference to the problem for exhibiting 
the nature of the law of dependence which connects the coefli- 
cients of the polynomial factor with those of the original poly- 
nomial dividend, it must still be considered as an investigation 
of no inconsiderable importance, as bearing upon the general 
theory of the solution and depression of equations. 

The second of the proofs given by Gauss, the proof of Le- 
gendre, and both of those which have been given by Cauchy, 
belong to the second class of demonstrations to which we 
have referred above. Assuming the root to be represented 
by p (cos 6 + “—1 sin), the equation is reduced to the form 
P + QV-=1, or /(P? + Q?). (cos¢ + “—1 sin 9); and the 
object of the demonstration is to show that there exist neces- 
sarily real values of p and 6, which make P? + Q* =0. This 
is effected by Gauss by processes which are somewhat syn- 
thetical in their form, and such as do not arise very natu- 
rally or directly from the problem to be investigated ; and the 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 305 


essential part of the demonstration requires a double integra- 
tion between assigned limits, a process against which serious 
objections may in this instance be raised, independently of its 
involving analytical truths and principles of too advanced an 
order. 

_ The demonstration of Legendre depends upon the possible 
discovery, by tentative or other means, of values of @ and 4, 
which render P and Q very small; and subsequently requires 
us, by the application of the ordinary processes of approxima- 
tion, to find other values of e and 4, subject to repeated correc- 
tion, which may render P and Q smaller and smaller, and ulti- 
mately equal to zero. The objection to this demonstration, if 
so it may be called, is the absence of any proof of the necessary 
existence of values of e and 4; and if they should be shown to 
exist, it seems to fail in showing that the subsequent correc- 
tions of their values which this process would assign would really 
and necessarily increase thé required approximations. 

The demonstrations of Cauchy are formed upon the general 
scheme of that which is given by Legendre, at the same time 
that they seem to avoid the very serious defects under which 
that demonstration labours: he shows that (P* + Q*) must ad- 
mit of a minimum, and that this minimum value must be zero. 


_ The second of the demonstrations differs from the first merely 


in the manner of establishing the existence and value of this 
minimum: they both of them appear to me to be quite com- 
plete and satisfactory. 

It is not very difficult to establish this fundamental propo- 
sition by reasonings derived from the geometrical representa- 
tion of impossible quantities. This was done, though imper- 
fectly, by M. Argand, in the fifth volume of Gergonne’s An- 
nales des Mathématiques*, and has been since reconsidered by 
M. Murey, in a very fanciful work upon the geometrical in- 
terpretation of imaginary quantities, which was published in 
1827. It seems to me, however, to be a violation of propriety 
to make such interpretations which are conventional merely, 
and not necessary, the foundation of a most important symbo- 
lical truth, which should be considered as a necessary result of 
the first principles of algebra, and which ought to admit of de- 
monstration by the aid of those principles alone. 

General Solution of Equations.—The solution of equations 
in its most general sense would require the expression of its 
roots by such functions of their coefficients as were competent 


i In the fourth volume of the same collection there are demonstrations of 
this fundamental proposition, given by M. Dubourguet and M. Encontre, 
which do not appear, however, to merit a more particular notice. 

. x 


306 THIRD REPORT—1833. 


to express them, when those coefficients were general symbols, 
though representing rational numbers. Such functions also 
must equally express all the roots, in as much as they are all of 
them equally dependent upon the coefficients for their value ; 
and they must express likewise the values of no quantities which 
are not roots of the equation. 

The problem, in fact, is the inverse of that for the formation 
of the equation which is required to satisfy assigned condi- 
tions. And as we have shown that there always exist quanti- 
ties expressible by the ordinary signs of algebra which will fulfil 
the conditions of any equation with rational coefficients, so like- 
wise we might appear to be justified in concluding that there 
must exist explicable functions of those coefficients which in all 
cases would be competent to represent those roots. 

A very little consideration, however, would show that sucha 
conclusion was premature. In the first place, such a function 
must be irrational, in as much as all rational functions of the 
coefficients admit but of one value; and they must be such ir- 
rational functions of the coefficients as will successively insulate 
the several roots of the equation,—for they must be equally ca- 
pable of expressing all the roots,—and they must be capable 
likewise of effecting this insulation without any reference to the 
specific values of the symbols involved, or to the relation of the 
values of the roots themselves; for otherwise they could not be 
said to represent the general solution of any equation whatever 
of a given degree. ‘The question which naturally presents it- 
self, after the enumeration of such conditions, is, whether we 
could conclude that any succession of operations which are, pro- 
perly speaking, algebraical, would be competent to fulfil them. 

If it be further considered that those successive operations 
must be assigned beforehand for every general equation of an 
assigned degree; that every one of these operations can give 
one real value only, or at the most two; and that the result of 
these operations, which must embrace all the coefficients, must 
express the » roots of the equation and those roots only; it 
will readily be conceded that the solution of this great pro- 
blem is probably one which will be found to transcend the 
powers of analysis. 

The solutions of cubic and biquadratic equations have been 
known for nearly three centuries; and all the attempts which 
have hitherto been made to proceed beyond them, at least in 
equations in which there exists no relation of dependence 
amongst the several coefficients, and no presumed or presuma- 
ble relation amongst the roots, have altogether failed of success: 
and if we consider that this great problem has been subjected to 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 307 


the most scrutinizing and laborious examination by nearly all 
the greatest analysts who have lived in that period, we may be 
justified in concluding that this failure is rather to be attributed 
to the essential impossibility of the problem itself than to the 
want of skill or perseverance on the part of those ho have 
made the attempt. But in the absence of any compete and 
uncontrovertible proof of this impossibility, the question cannot 
be considered as concluded, and will still remain open to spe- 
culations upon the part of those with whom extensive and well- 
matured knowledge, and a deep conviction founded upon it, have 
not altogether extinguished hope. 

The different methods which have been proposed for the 
resolution of cubic and biquadratic equations, and the conse- 
quences of the extension of their principles to the solution of 
equations of higher orders, have been subjected to a very de- 
tailed analysis by Lagrange, in the Berlin Memoirs for 1770 
and 1771, and in the Notes xiii. and xiv. of his Traité sur la 
Résolution des Equations Numériques ; and it would be diffi- 
cult to refer to any investigations of this great analyst which are 
better calculated to show the extraordinary power which he 
possessed of referring methods apparently the most distinct to. 
a common principle of a much higher and more comprehensive 
generality. In the subsequent remarks which we shall make, 
we shall rarely have occasion to proceed beyond a notice of the 
general conclusions to which he has arrived, and to show their 
bearing upon some later speculations upon the same subject. 

A very slight examination of the principles involved in the 
solution of the equations of the third and fourth degrees will 
show them to be inapplicable to those of higher orders.. A no- 
tice of a very few of such methods will be quite sufficient for 
our purpose. 

Thus, the ordinary solution of the cubic equation 

e—3qae+2r=0* | 
is made to depend upon that of the following problem: __. 
_ “To find two numbers or quantities such that the sum of 
their cubes shall be equal to 2 r and their product equal to g.” 

If we represent the required numbers by wu and », we readily 
obtain the equation of reduction 


w&—2ruv4+q=0, 


* This equation may be considered as equally general’ with 
e&—Asxv®+Ba—C=0, . 
in as much as we can pass from one to the other by a very easy transforma- 
tion; and the same remark may be extended to equations, of higher orders. 
Such a change of form, however, will determine the applicability or inappli- 
cability of many of the methods which are proposed for their solution. 
x2 


308 THIRD REPORT—1833. 


which gives, when solved as a quadratic equation, 
War + 7 (r= 9g), 
and consequently, 
wa {rt VP @¥8, 
and therefore 
ih Wifi ee Sates 
ua {r+ v—¢ys 
If we call 1, a, a, the three cube roots of 1, or the roots of 
the equation =? — 1 = 0, and if we assume a to represent the 


arithmetical value of uw, we shall obtain the following three 
values of w + v, which are 


v= 


at Lae La bit pe 
a a aa 
These values, though derived from the solution of an equation 
of six dimensions *, are only three in number, and form, there- 
fore, the roots of a cubic equation. A little further inquiry will 


show that they are the roots of the cubic equation 
w—S8qx+2r=0: 

for it may readily be shown, in the first place, that their sum 

= 0; that the sum of their products two and two = — 3 q; and 

that their continued product = 27; or in other words, that 

they are the roots of an equation which is in every respect iden- 

tical with the equation in question +. 

* There are siz values of vu, in as much as the values of w and v are inter- 
changeable, from the form in which the problem was proposed ; but there are 
only three values of u + v. 

+ Since qd 

{r 2b A (9? — q3) }3 
it is usual to express the roots of the equation 28 —~3qz+2r=0, by the 
formula . : 

:= {r+ A (r? — 3) }3 + {r— V (1? — 8) }3, (1.) 
which is in a certain sense incorrect, in as much as it admits of nine values 
instead of three. The six additional values are the roots of the two equations 

w—3agx+2r=—0, 
we—3e2¢x+2r=0, 
and the formula (1.) expresses the complete solution of the equation 
(a8 — 27r)? — 27 a? = 0, 
which is of 9 dimensions. It is the formula 


+ ~~, where u= {r+ V(r q) }3, 


and has the same value in both terms of the expression, which corresponds to 
the equation a—3qx+2r=0. 


= {r— V(?— 9}, 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 309 


This mode of effecting the solution of a cubic equation would 
altogether fail if the original equation possessed all its terms: 
and though the absence of the second term of a cubic equation 
cannot be said, in a certain sense at least, to affect the gene- 
rality of its character, yet it would lead us to expect that the 
method which we had followed was of so limited a nature as not 
to be applicable to general equations of a higher order. ‘Thus, 
if it was proposed to find two quantities, w and v, the sum of 
whose x‘ powers was equal to 2 r, and whose product was equal 
to g, we should find 


ua {r+ V2 —Q}r; 

. le Se ee 
Prev e——}* 
where uw + v is the root of the equation 


n(n—3 if n(n — 3) (n—4) 4 
“Sera e: amas girs, 


utov={rt flee Higa ob 


a —ng ar? + 


Hh yeecs S08 HF. 

The form of this equation is of such a kind as to prevent its 
being identified with any general equation whatever, beyond a 
cubic equation wanting the second term; a circumstance which 
precludes all further attempts, therefore, to exhibit the roots of 
higher equations by radicals + of this very simple order: but 
it is possible that there may exist determinate functions of the 
roots of higher equations (not symmetrical functions of all of 
them, which are invariable as far as the permutations of the roots 
amongst each other are concerned,) which may admit of triple 
values only, and which will be expressible, therefore, by means 
of a cubic equation, and consequently by the general formula 
for its solution. — 

Thus, if 2, 2, 3, %4, were assumed to represent the roots of 
a biquadratic equation 


* This equation was first solved by Demoivre in the Philosophical Trans- 
actions for 1737, and it was readily derived from the theorem which goes by 
his name. It was afterwards shown to betrue, by a process, however, not al- 
together general, by Euler, in the sixth volume of the Comment. Acad. Petrop., 

. 226. See also Abel’s ‘‘ Mémoire sur une Classe particuliére d’Equations 
résolubles algébriquement,”’ in Crelle’s Journal, vol. iv. 

+ Abel has used the term radicality to designate such expressions. To 
say, therefore, that the root of an equation is expressible by radicalities, is 
the same thing as to say that the equation is solvable algebraically. It is 
used in contradistinction to such transcendental functions, whether of a known 
or unknown nature, as may, possibly, be competent to express those roots, 
when all general algebraical methods fail to determine them. 


310 THIRD REPORT—1833. 


v—prPigx—rx«+s=0, (1.) 

such functions would be 2, x, + 23x, and (a, + xv, — x3 — 24), 
which admit but of three different values, and which may seve- 
rally form, therefore, the roots of cubic equations, whose coeffi- 
cients are expressible in terms of the coefficients of the original 
equation. Such a function also would be (a, + «,)°, if we should _ 
suppose p or the coefficient of the second term of equation (i.) 
to be zero*. The function (x, + x9) (x3 + 24) would give 
three values only under all circumstances. The functions x, 
+ # + #3, and 2, xx; are capable of four different values, 
and therefore do not admit of being expressed by a determina- 
ble equation of lower dimensions than the primitive equation. 
Functions of the form 2, x, admit of six values, and require for 
their expression equations of six dimensions, which are reduci- 
ble to three, in consequence of being quasi recurring equations +. 
Innumerable functions may be formed which admit of 12 and of 
24 values, and one alternate function which admits of two values 
only f. 

The success of such transformations m reducing the dimen- 
sions of the equation to be solved, would naturally direct us to 
the research of similar functions of the roots of higher equa- 
tions than the fourth, which admit of values whose number is 
inferior to the dimensions of the equation. We may presume 
that, if such functions exist, they are rational functions, for 
if not, their érrationality would increase the dimensions of 
the reducing equation, and would tend to distribute its roots 
into cyclical periods; and what is more, it has been very 
clearly proved that if equations admit of algebraical solution, 
all the algebraical functions which are jointly or separately in- 
volved in the expression of their roots, will be equal to rational 


* The first of these transformations involves the principles of Ferrari’s, some- 
times called Waring’s, solution of biquadratic equations ; the second that of 
Euler; and the third that of Des Cartes. See the third chapter of Meyer 
Hirsch’s Sammlung von Aufgaben aus der Theorie der algebraischen Gleichungen, 
which contains the most complete collection of formule and of propositions 
relating to symmetrical and other functions of the roots of equations with 
which I am acquainted. The combinatory analysis receives its most advan- 
tageous and immediate applications in investigations connected with the 
theory of such functions. See also Peacock’s Algebra, note, p. 619. 


+ The form of its roots being « and =, they are reducible by the same me- 


thods as are applied to recurring equations. 

t See Cauchy, Cours d’ Analyse, chap. iii. and noteiv. The use of such al- 
ternate functions in the elimination of the several unknown quantities from 
simultaneous equations of the first order, involving » unknown quantities, 
will be noticed hereafter. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. $11 


functions of these roots; and consequently, if irrational func- 
tions of those roots are employed in the formation of the re- 
ducing equation, the roots of the equation must enter into the 
final expression of the required roots, in a form where that ir- 
rationality has altogether disappeared *. If we assume, there- 
fore, that such functions are in all cases rational, the next ques- 
tion will be, whether they are discoverable in higher equations 
than the fourth. 

This inquiry was undertaken by Paolo Ruffini, of Modena, 
in his Teoria delle Equazione Algebraiche, published at Bo- 
logna in 1799, and subsequently in the tenth volume of the 
Memorie della Societa Italiana, in a memoir on the impossibi- 
lity of solving equations of higher degrees than the fourth. 
He has demonstrated that the number of values of such func- 
tions of the roots of an equation of” dimensions must be either 
equal to1.2.3...m, or to some submultiple of it; and that 
when z = 5, there is no such function, the alternate function 
being excluded, which possesses less than 5 values. The pro- 
cess of reasoning which is employed by the author for this pur- 


* This proposition has been proved by Abel, in his Beweis der Unméglich- 
keit algebraische Gleichungen von hoheren Graden als dem vierten allgemein 
Aufzulésen, in the first volume of Crelle’s Journal: the same demonstration 
was printed at Paris, in a less perfectly developed form, during his residence 
in that capital. This proof applies to algebraical solutions only, excluding 
the consideration of the possibility of expressing such roots by the aid of un- 
known transcendents. After defining the most general form of algebraical 
functions of any assigned degree and order ; and after demonstrating the pro- 
position referred to in the text, and analysing the demonstrations of Ruffini and 
Cauchy, and showing their precise bearing upon the theory of the solution 
of equations, he proceeds to show that the hypothesis of the existence of 
such a solution in an equation of five dimensions will necessarily lead to an 
equation, one member of which has 120 values and the other only 10; an ab- 
surd conclusion. It is quite impossible to exhibit this demonstration in 
a very abridged form so as to make it intelligible ; and though some parts of 
it are obscure and not perfectly conclusive, yet it is, perhaps, as satisfactory, 
upon the whole, as the nature of the subject will allow us to expect. 

It is impossible to mention the name of M. Abel in connexion with this 
subject, without expressing our sense of the great loss which the mathematical 
sciences have sustained by his death. Like other ardent young men, he com- 
menced his career in analysis by attempting the general solution of an equa- 
tion of five dimensions, and was for some time seduced by glimpses of an 
imagined success ; but he nobly compensated for his error by furnishing the 
most able sketch of a demonstration of its impossibility which has hitherto 
appeared. His subsequent discoveries in the theory of elliptic functions, 
which were almost simultaneous with those of Jacobi, have contributed most 
materially to change the whole aspect of one of the most difficult branches of 
analytical science, and furnish everywhere proofs of a most vigorous and in- 
ventive genius. He died of consumption, at Christiania in Norway, in 1827, 
in the 27th year of his age. 


312 THIRD REPORT—1833. 


pose is exceedingly difficult to follow, being unnecessarily en- 
cumbered with vast multitudes of forms of combination, and 
requiring a very tedious and minute examination of different 
classes of cases; and it was, perhaps, as much owing to the 
necessary obscurity of this very difficult inquiry as to any im- 
perfection in the demonstration itself, that doubts were ex- 
pressed of its correctness by Malfatti * and other contemporary 
writers. The subject, however, has been resumed by Cauchy in 
the tenth volume of the Journal de l’ Ecole Polytechnique, who 
has fully and clearly demonstrated the following proposition, 
which is somewhat more general than that of Ruffini: “ That 
the number of different values of any rational function of x 
quantities, is a submultiple of 1 . 2.3... ,and cannot be re- 
duced below the greatest prime number contained in n, without 
becoming equal to 2 or to 1.” Ifwe grant, therefore, the truth 
of this proposition, it will be in vain to seek for the reduction 
of equations of higher dimensions than the fourth, by transfor- 
mations dependent upon rational functions of the roots. 

The establishment of this proposition forms an epoch in the 
history of the progress of our knowledge of the theory of equa-~ 
tions, in as much as it so greatly limits the objects of research 
in attempts to discover the general methods for their solution. 
And if the demonstration of Abel should be likewise admitted, 
there would be an end of any further prosecution of such in- 
quiries, at least with the views with which they are commonly 
undertaken. 

Lagrange, in his incomparable analysis of the different me- 
thods which have been proposed for the solution of biquadratic 
and higher equations, has shown their common relation to each 
other, and that they all of them equally tend to the formation o 
a reducing equation, whose root is 


ty +4%+ex,+ ex,+ &e. 


where 2, %, %3, &c., are the roots of the primitive equation, 
and where «¢ is a root of the equation 
Eien Gai 78 tpt tae si Oy 

where z expresses the dimensions of the equation to be solved. 

He then reverses the inquiry, and assuming this form as 
correctly representing the root of the reducing equation, he 
seeks to determine its dimensions. The beautiful process which 
he has employed for this purpose is so well known { that it is 
quite unnecessary to describe it in this place; and the result, 


* Memorie della Soc. Ital., tom. xi. 
t+ Resolution des Equations Numériques, Note xiii. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 3138 


as might be expected, perfectly agrees with the conclusions 
which are derived from more direct, and, perhaps, more ge- 
neral considerations. If nm, or the number of roots x,, Xo, gy 
&c., be a prime number, then the dimensions of the final re- 
ducing equation will be 1.2... (m — 2); and ifn be a compo- 
site number = mp, then the dimensions of the final reducing 
equation will be : 
ee ria 12 fim 
(m— 1)m.(1 -2...pr (p —1)p.(1.2...m)?’ 
according as we arrive at it, by grouping the terms of the ex- 
pression 
H, +a%,+ oe 4,+ &e. 
into m periods of p terms, or into p periods of m terms. It thus 
appears, that for an equation of 5 dimensions, the final reducing 
equation is of 6 dimensions; for an equation of 6 dimensions, 
the final reducing equation is of 10 dimensions in one mode of 
derivation and 15 in the other ; and the higher the dimensions 
of the equation are, the greater will be the excess of the dimen- 
sions of the final reducing equation. And in as much as there 
exist no periodical or other relations amongst the roots of these 
reducing equations, it is obvious that the application of this 
process, and therefore also of any of those primary methods 
which lead to the assumption of the form of the roots of the 
reducing equation, must increase instead of diminishing the 
difficulties of the solution which was required to be found. 

It was the imagined discovery of a cyclical period amongst 
the roots of this reducing equation which induced Meyer Hirsch, 
a mathematician of very considerable attainments, to believe 
that he had discovered methods for the general solution of equa- 
tions of the fifth and higher degrees. Amongst the different 
methods which Lagrange has analysed in the Berlin Memoirs 
is that which Tschirnhausen proposed in the Acta Eruditorum 
for 1683. It proposed to exterminate, by means of an auxiliary 
equation, all the terms of the original equation except the first 
and the last, and thus to reduce it to a binomial equation. 
Thus, in order to exterminate the second term of x? + ax 
+ 6=0, we must employ the auxiliary equation y + A +a 
= 0, and then eliminate x. To exterminate simultaneously 
the second and third terms of the cubic equation 2° + aa? 
+ bx + ¢c=0, we must employ the auxiliary equation y +A 
+ Bez + «? = 0, and then eliminate x; and more generally, to 
destroy all the intermediate terms of an equation of » dimen- 
sions, 

Bt Ay 0) 4 dg, BOA bo on An = 0, 


314 ; _ THIRD REPORT—1833. 


we must employ the auxiliary equation 
y+tA+A 27+ Aga?+...a*'=0, 


whose dimensions are less by | than those of the given equation. 

Such a process is apparently very simple and uniform and 
equally applicable to all equations; and so it appeared to its 
author. But it wili be found that the equations upon which 
the determination of A, A,, A,, depend, in an equation of the 
fourth degree, will rise to the sixth degree, which are subse- 
quently reducible to others of the third degree; and that for 
an equation of the fifth degree, it will be impossible to reduce 
them below the sixth degree. Such was the decision of La- 
grange, who has subjected this process to a most laborious 
analysis, and who has actually calculated one of the coefficients 
of the final reducing equation, and shown the mode in which 
the others may be determined *. 

Meyer Hirsch, however, though fully adopting the conclu- 
sions of Lagrange to this extent, attempted to proceed further; 
and, deceived by the form which he gave to his types of combina- 
tion, imagined that he had discovered cyclical periods amongst 
the roots of this final equation, by which it might be resolved 
into two equations of the third degree. If such a distribution 
of the roots was practicable in the case of the final equation cor- 
responding to equations of the fifth degree, it would be practi- 
cable in that corresponding to equations of higher degrees. 
But some consequences of this discovery, and particularly the 
multiplicity of solutions which it gave, would have startled an 
analyst whose prudence was not laid asleep by the excitement 
consequent upon the expected attainment of a memorable ad- 
vancement in analysis, which had eluded the grasp even of 
Lagrange. Its author, however, was too profound an analyst 
to continue long ignorant at once of the consequences of his 
error and of the source from which it sprung. In the Preface 
to his Integraltafeln, an excellent work, which was published 
in 1810, within two years of the announcement of his discovery, 
he acknowledges with great modesty and propriety, that he 
had not succeeded in effecting general solutions of equations 
in the sense in which the problem was understood by Euler, 
Lagrange, and the greatest analysts. 

The well known Hoéne de Wronski, in a short pamphlet pub- 
lished in 1811, announced a method for the general resolution of 
equations. He assumes hypothetical expressions for the roots of 
the given equation in terms of the z roots of 1, and of the (m—1) 


* Inthe Berlin Memoirs for 1771, p. 170: it forms a work of prodigious 
labour, such as few persons would venture to undertake or to repeat. 


—_— 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 315 


roots of a reduced equation of (x — 1) dimensions, and employs 
in the determination of the coefficients of this reduced equation 
n"-! fundamental equations, designated by the Hebrew letter x, 
and n"-? others designated by the Greek letter 2. It is un- 
necessary, however, to enter upon an examination of the truth 
of processes which the author who proposes them has left un- 
demonstrated ; and in as much as the application of his method 
to an equation of 5 dimensions would require the formation of 
625 fundamental equations of the class Aleph and 125 of the 
class Omega, and the determination of the greatest common 
measure of 2 polynomials of 24 and 30 dimensions respectively, 
it was quite clear that M. Wronski might in perfect safety retire 
behind an intrenchment of equations and operations of this 
formidable nature. And this was the position which he took 
in answer to M. Gergonne, who, in the third volume of the An- 
nales de Mathématiques, in the modest form of doubts, showed 
that the form of the roots which he had assumed was not essen- 
tially different from those which Waring, Bezout, and Euler, 
had assumed, and which Lagrange had shown to be incompa- 
tible with the existence of a final reducing equation of the di- 
mensions assigned to it*. 

The process given by Lagrange for determining the dimen- 
sions and nature of the final reducing equation has been the 
touchstone by which all the methods which have been hitherto 
proposed for the solution of equations have been tried, and will 
probably continue to serve the same purpose for all similar at- 
tempts which may be hereafter made. Its illustrious author, 
however, hesitated to pronounce a decisive opinion respecting 
the possibility of the problem, contenting himself with demon- 
strating it to be so, with reference to every method which had 
been suggested, or which could be shown to arise naturally out 


* The works of Hoéne de Wronski were received with extraordinary favour 
in Portugal, where the Baron Stockler, a mathematician of considerable at- 
tainments, and other members of the Academy of Sciences became converts 
to his opinions. There is, in fact, a bold and imposing generality, and appa~ 
rent comprehensiveness of views in his speculations, which are well calculated 
to deceive a reader whose mind is not fortified by the possession of an extensive 

. and well digested knowledge of analysis. In the year 1817, the Academy of 
Sciences at Lisbon proposed as a prize, ‘‘ The demonstration of Wronski’s 
formule for the general resolution of equations,’’ which was adjudged in the 
following year to an excellent refutation of their truth by the academician 
Evangelista Torriani: it chiefly consists-in showing, and that very clearly, 
that the coefficients of the reducing equation of (n — 1) dimensions, assuming 
the form of the roots of the equation which Wronski assigned to them, can- 
not be symmetrical functions of those roots, and therefore cannot be expressed 
by the coefficients of the primitive equation, whatever be the number, nature 
and derivation of the fundamental equations & and © which are interposed. 


316 THIRD REPORT—1833. 


of the conditions of the problem itself. But even if we should 
assume the impossibility of the problem, to the full extent of 
Abel’s demonstration, it is still possible that there may exist 
solutions by means of undiscovered transcendents. It is, in fact, 
quite impossible to attempt to limit the resources of analysis, or 
to demonstrate the nonexistence of symbolical forms which may 
be competent to fulfil every condition which the solution of this 
problem may require. In conformity with such views, we may 
consider the numerical roots of equations as the only discover- 
able values of such transcendental functions; but it is quite 
obvious that such values will in no respect assist us in deter- 
mining their nature or symbolical form, in the absence of any 
knowledge of the course of successive operations upon all the 
coefficients of the equation which were required for their de- 
termination. 

Though we may venture to despair, at least in the present 
limited state of our knowledge of transcendental functions, of 
ever effecting the general resolution of equations, in the large 
sense in which that problem is commonly proposed and under- 
stood, yet there are large classes of equations of all orders 
which admit of perfect algebraical solution. ‘The principal pro- 
perties of the roots of the binomial equation #* — 1 = 0, had 
long been ascertained by the researches of Waring and La- 
grange, and its general transcendental solution had been com- 
pletely effected. Its algebraical solution, however, had been 
limited to values of » not exceeding 10; and though Vander- 
monde in some very remarkable researches *, which were con- 
temporary with those of Lagrange, had given the solution of 
the equation «” — 1 = 0, as a consequence of his general me- 
thod for the solution of equations, and had asserted that it 
could be extended to those of higher dimensions, yet his solu- 
tion contained no developement of the peculiar theory of such 
binomial equations, and was so little understood, that his dis- 
covery, if such it may be termed, remained a barren fact, which 
in no way contributed to the advancement of our analytical 
knowledge. 

The appearance of the Disquisitiones Arithmetice of the 


* Mémoires de l Académie de Paris for 1771. The result only of this solu- 
tion was given, the steps of the process by which it was obtained being omitted. 
This result has been verified by Lagrange in Note xiv. to his Traité sur la 
Résolution des Equations Numériques. Poinsot, in a memoir on the solution 
of the congruence a” — 1 = M (p), which will be noticed in the text, has at- 
tempted to set up a prior claim in favour of Vandermonde for Gauss’s memo- 
rable discovery ; in doing so, however, he appears to have been more influ- 
enced by his national predilections in favour of his countrymen, than by a 
strict regard to historical truth and justice. 


a“ 


REPORT ON CERTAIN BRANCHES OF ANALYSIS, 317. 


celebrated Gauss, in 1801, gave an immense extension to our 
knowledge of the theory and solution of such binomial equa- 
: . ae—1 
tions. It was well known that the roots of the equation chy =0, 
where » is a prime number, could be expressed by the terms of 
the series ~ 


Pp? ap Po rt}; 


where r represented any root whatever of the equation, and 
where, consequently, the first term 7 might be replaced by any 
term of the series. But in this form of the roots there is pre- 
sented no means of distributing them into cyclical periods, nor 
even of ascertaining the existence of such periods or of determin- 
ing their laws. It was the happy substitution of a geometrical 
series formed by the successive powers of a primitive root * of x, 
in place of the arithmetical series of natural numbers, as the in- 
dices of r, which enabled him to exhibit not merely all the dif- 
Le 
ferent roots of the equation - - = 0, but which also made 
manifest the cyclical periods which existed amongst them. 
Thus, if a was a primitive root of n, and nm — 1 = mf, then in 


the series ‘ 
2 3 k—-1 mk—1 
Ma aN am CaS RE EENY Mo, cule oe 


é] 


the m successive series which are formed by the selection of 
every i term, beginning with the first, the second, the third, 
and so on successively, or the % successive series which are 
formed in a similar manner by the selection of every m™ term, 
are periodical; and if the number m or £ of terms in one of 
those periods be a composite number, they will further admit of 
resolutions into periods in the same manner with the complete 
series of roots of the equation. The terms of such periods will 
be reproduced in the same order, if they are produced to any 
extent according to the same law, it being understood that the 
multiples of » which are included in the indices which succes- 
sively arise, are rejected, for the purpose of exhibiting their 
values and their laws of formation in the most simple and ob- 
vious form. If two or more periods also are multiplied together, 
the product will be composed of complete periods or of 1, or of 
multiples of them, the rules for whose determination are easily 


* There are as many primitive roots of as there are numbers less than 
m—1 which are prime to it. Euler, who first noticed such primitive roots 
as determined by Fermat’s theorem, determined them by an empirical pro- 
cess. Mr. Ivory, in his admirable article on Equations, in the Supplement 
to the Encyclopedia Britannica, has given a rule for finding them directly. 


318 . THIRD REPORT—1833. 


framed * ; and it arises from the application of such rules that 
we are enabled to determine the coefficients of an equation of 
which those periods are the roots, and thus to depress the 
original binomial equation to one whose dimensions are the 
greatest prime number, which is a divisor of m — 1. 

It follows, therefore, that if the highest prime factor of m — 1 
be 2, the resolution of the binomial equation 2 — 1 = 0 will 
be made to depend upon the solution of quadratic equations 
only, and consequently to depend upon constructions which 
can be effected by combinations of straight lines and circles, 
and therefore within the strict province of plane geometry : 
this will take place whenever m is equal to 2* + 1 and is also 
a prime number. Thus, if k = 4 we have nm = 17, a prime 
number, and therefore the solution of the equation «” —1=0 
will be reducible to that of four quadratic equations. Similar 
observations apply to the equations 


Px | gern ene and iA pe 


The same principles which enable us to solve algebraically 
binomial equations, under the circumstances above noticed, will 
admit of extension to other classes of equations, whose roots 
admit of analogous relations amongst each other. Gauss} has 
stated that the principles of his theory were applicable to func- 


tions dependent upon the transcendent f°", which de- 
fines the arcs of the lemniscata, as well as to various species of 
congruencies ; and he has also partially applied them to certain 
classes of equations dependent upon angular sections, though 
in a form which is very imperfectly and very obscurely deve- 
loped. Abel, however, in a memoir} which is remarkable for 
the generality of its views and for its minute and careful ana- 
lysis, has not merely completed Gauss’s theory, but made most 
important additions to it, particularly in the solution of exten- 
sive classes of equations which present themselves in the theory 
of elliptic transcendents §. Thus he has given the complete 


* Symmetrical functions of these periods will be multiples of the sum (—1) 
of these periods and of 1. This conclusion follows immediately from the re- 
placement of the arithmetical by the geometrical series of indices, according 
to the general process of Lagrange, without any antecedent distribution of 
the roots into periods. See Note xiv. to the Résolution des Equations Numé- 
riques. It follows from thence that the coefficients of the reducing equations 
will be whole numbers. 

+ Disquisitiones Arithmetice, pp. 595, 645. 

{ “Sur une Classe particuliére d’Equations résolubles algébriquement,”— 
Crelle’s Journal, vol. iv. p. 131. 

§ Crelle’s Journal, vol. iv. p. 314, and elsewhere. 


Sous 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 319 


algebraical resolution of an equation whose roots can be repre- 
sented by 


we, bay Peen eer me, 


where 6” x = x, and where @ is a rational function of “x and of 
known quantities; and also of an equation where all the roots 
can be expressed rationally in terms of one of them, and where, 
if x and 6, x express any other two of the roots, we have like- 
wise 

Go,7 = 6,0 2, 


It is impossible, however, within a space much less than that 
of the memoir itself, to give any intelligible account of the pro- 
cess followed in the demonstration of these propositions, and 
of many others which are connected with them. We shall con- 
tent ourselves, therefore, with a slight notice of their applica- 
tion to circular functions. 


2 ; 
If we suppose a = a the equation whose roots are cos a, 


cos 2a, cos3a,... coswais 


ee a ne ee 
J. rae 2 eats CN 4 —— 

x qe + 76: ito @ isha 0 (1.) 
which may be easily shown to possess the required form and 
properties ;—for, in the first place, cos m a = 4 (cos a), where @ 
is, as is well known, a rational function of cos a or x; and, 
in the second place, if 9 = cos ma and 4, x = cos m, a, then 
likewise 66, 2 = cosmm,a = cosm,ma = 6,6 x, which is the 
second condition which was required to. be fulfilled. 

Let us suppose » = 2” + 1, when the roots of the equation 
(1.) will be 

Qa 7 4na 9 
Int cos On+ 1 « « « COS Qn+l’ COS a7, 
of which the last is 1, and the x first of the remainder equal to 
the v last. The equation (1.) may be depressed, therefore, to 
one of z dimensions, which is 


cos 


n 1 n—1 1 n—2 1 n—3 


1 @—2)(»—3) 94, 1 @—-3)(@—4) sg 
er 2 ee ie, eee 
whose roots are 

Qa 4 x za Qnn 


3820 THIRD REPORT—18353. 


Qu 2m 
= x = cos a, and Cos = = 6x =cosma, 


2Qn+1 2Q2n+1 
then these roots are reducible to the form 


If cos 


5.0 yO &3. 0). OF aes 
or, 
COs @, COS ma, cos m2 a,... cos ma: 


and if we suppose m to be a primitive root to the modulus 
2n + 1, then all the roots 


COs a, COS Ma, COS mM? a, ... Cosma 


will be different from each other, and cos m" a = cos a; con- 
sequently it will follow, since the roots of the equation (2.) are 


of the form 
Co, ean a «On 


where 6" x = x, they will admit, in conformity with the preceding 
theorems, of algebraical expression. 

Abel has given the general form of the expression for these 
roots, which in this case are all real; and their determination 
will involve the division of a circle into 2” equal parts, the 
division of an assigned or assignable arc into 2 equal parts, 
and the extraction of the square root of 2m + 1; a conclusion 
to which Gauss had also arrived, though he has not given the 
steps of the process which he followed for obtaining it*. If we 
suppose 2 = 2%, we shall get the case of regular polygons of 
2+! + 1 sides, which admit of indefinite inscription in circles 
by purely geometrical means. It will follow from the same re- 
sult that the inscription of a heptagon will depend upon that 
of a hexagon, the trisection of a given angle, and the extraction 
of the square root of 7. 

Poinsot + has given a very remarkable extension to the theory 
of the solution of the binomial equation 2” — 1 = 0, by showing 
that its imaginary roots may be considered in a certain sense 
as the analytical representation of the whole numbers which 
satisfy the congruence or equation 


a” —1=M(p), 
whose modulus (a prime number) is p: thus, the imaginary 
cube roots of 1, or the imaginary roots of «7 —1=0, are 
ae Wf 8: — 1 = OS 
2 ; 2 


‘ 


, and the whole numbers 4 and 2, 


* Disquisitiones Arithmetice, p. 651. , 
+ Journal de l’ Ecole Polytechnique, cahier 18. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 321 


which satisfy the congruence 
2—l1=M x 7, 


whose modulus is 7; are expressed by patel Ms Sukd 


and pedis -innerah <2: which arise from adding 7 to the 
parts without and beneath the radical sign. 

The principle of this transition from the root of the equation 
to that of the congruence is sufficiently simple. We consider 
the roots of 2” — 1 = 0 as resulting from the expression for 
those of the congruence 2” — 1 = M (p), when M = 0; and 
we thus are enabled to infer, in as muchas M (p), its multiples 
‘and powers, are involved in those formule, whether without 
‘or beneath the radicals, and disappear, therefore, when M = 0, 
‘that some such multiples, to be determined by trial, or other- 
wise, are to be added when M (p) is restored, or when 1 is 
replaced by 1 + M(p). When the congruence admits of in- 
‘tegral values of x, which are less than p, then they can be found 
by trial: when no such integral values exist, then, amongst the 
irrational values which thus arise, those values will present them- 
selves which will satisfy the congruence algebraically, though 
they can only be ascertained by a tentative process. 

The equation of Fermat, 


2? —]|=M (p)s 
where p is a prime number, will be satisfied by all the natural 


numbers 1, 2, 3,.. as far as (pv — 1): and it follows, therefore, 
that all the rational roots of the equation 


a” — 1 = M (p) 
will be common to the equation 
1 =M(p), 
the number of them being equal to (d), the greatest common 


divisor of m and of p—1. If d be 1, then all the roots except 
1 are irrational. If we suppose the equation to be 


a —1 = M(p), 


then all the roots will be equal to each other and to 1. It is 
unnecessary, however, to enter upon the further examination 
of such cases, which are developed with great care and sin- 
gular ingenuity in the memoir referred to. 
These views of Poinsot are chiefly interesting and valuable as 
ily ai the theory of indeterminate with that of ordmary 
3. Y 


322 THIRD REPORT—1833. 


equations. It has, in fact, been too much the custom of analysts 
to consider the theory of numbers as altogether separated from 
that of ordinary algebra. The methods employed have generally 
been confined to the specific problem under consideration, and 
have been altogether incapable of application when the known 
quantities employed were expressed by general symbols and not 
by specific numbers. It is to this cause that we may chiefly attri- 
bute the want of continuity in the methods of investigation 
which have been pursued, and the great confusion which has 
been occasioned by the multiplication of insulated facts and 
propositions which were not referable to, nor deducible from, 
any general and comprehensive theory. 

Libri, in his Teoria det Numeri, and in his Mémoires de 
Mathématique et de Physique, has not merely extended the 
views of Poinsot, but has endeavoured to comprehend all those 
conditions in the theory of numbers, by means of algebraical or 
transcendental equations, which were previously understood 
merely, and not symbolically expressed. He has shown that 
problems which have been usually considered as éndeter- 
minate are really more than determinate, and he has thus been 
enabled to explain many anomalies which had formerly embar- 
rassed analysts, by showing the necessary existence of an equa- 
tion of condition, which must be satisfied, in order that the 
problem required to be solved may be possible. By the aid of 
such principles the solutions of indeterminate equations, at 
least within finite limits, may be found directly, and without 
the necessity of resorting to merely tentative processes. 

A great multitude of new and interesting conclusions result 
from such views of the theory of numbers; but the limits and 
object of this Report will not allow me to discuss them in de- 
tail, or to point out their connexion with the general theory of 
equations, and with the properties of circular and other func- 
tions. The reader, however, will find, in the second of the 
memoirs of Libri above referred to, a general sketch of the 
nature and consequences of these researches, which is unfor- 
tunately, however, too rapid and too imperfectly developed to 
put him in full and satisfactory possession of all the bases of 
this most important theory. 

On the Solution of Numerical Equations.—The resolution 
of numerical equations formed the subject of a truly classical 
work by Lagrange, in which this problem, one of the most im- 
portant in algebra, is not only completely solved, but the imper- 
fections of all the methods which had been proposed for this 
purpose by other authors are pointed out with that singular 
distinctness and elegance which always distinguish his reviews 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 823 


of the progress and existing state of the different branches of 


the mathematical sciences. In the following report we shall 
commence by a general account of the state in which the pro- 
blem was left by him, and of the practical difficulties which 
attend the use of his methods, and we shall then proceed to 
notice the important labours of Fourier and other authors, with 
a view to bring its solution within the reach of arithmetical 
processes which are at once general and easy of application. 
The resolution of numerical equations involves two principal 
objects of research: the first of them concerns the separation 
of the roots into real and imaginary, positive and negative, and 
the determination of the limits between which the real roots 
are severally placed; the second regards the actual numerical 
approximation to their values, when their limits and nature have 
been previously ascertained. Many different methods have been 
proposed for both these objects, which differ greatly from each 
other, both in their theoretical perfection and in their practical 
applicability. We shall begin with a notice of the first class of me- 
thods, which have been proposed for the separation of the roots. 
If the coefficients of an equation be whole numbers or rational 
fractions, their real roots will be either whole numbers or ra- 
tional fractions, or otherwise irrational quantities, which will be 
generally conjugate* to each other and which will generally pre- 
sent themselves, therefore, in pairs. The method of divisors 
which Newton proposed, and which Maclaurin perfected, will 
enable us to determine roots of the first class, and they are also 
determined immediately and completely by nearly all methods 
of approximation. It will be to roots of the second class, there- 
fore, that our methods of approximation will require to be ap- 
plied, though such methods will never enable us to assign them 
under their finite irrational form, nor would our knowledge of 
their existence under such a form in any way aid us, unless in a 
very small number of cases, in the determination of their ap- 
proximate numerical values. ' 
The equal roots of equations, if any exist, may be detected 
by general methods; and the factors corresponding to them 
may be completely determined, and the dimensions of the equa- 


* An irrational real root may be conjugate to the modulus of a pair of im- 
possible roots; and there-may exist, therefore, as many irrational real roots 
which have no corresponding conjugate real roots as there are pairs of im- 
possible roots in the equation. It is not true, therefore, generally, as is some- 
times asserted, that such irrational roots enter equations by pairs. It would 
not be very difficult to investigate the different circumstances under which 
roots present themselves, and the different conditions under which they can 
be conjugate to each other; but the inquiry is not very important, in as much 
as the knowledge of their form would not materially influence the application 
of methods for approximating to their values. 

Y¥2 


824. THIRD REPORT—1833. 


tion depressed by a number of units equal to the number of 
such factors. We might suppose, therefore, in all cases, that 
the roots of the equation to be solved were unequal to each 
other; but if it should not be considered necessary to perform 
the previous operations which are required for the detection 
and separation of the equal roots, the failure of the methods of 
approximation or other peculiar circumstances connected with 
the determination of the limits of the roots, would indicate their 
existence, and at once direct us to the specific operations upon 
which their determination depends. 

If we suppose, therefore, the equal roots to be thus separated 
from the equation to be solved, and if we assume a quantity 
A which is less than the least difference of the unequal roots, 
then the substitution of the terms of the series 

Ay (k — TAs. 2, a, 0, — A, 2 oy ee ee me a, 
where k 4is greater than the greatest root, and — /, 4 less than 
the least root*, will give a series of results, amongst which the 
number of changes of sign from + to — and from — to + will 
be as many as the number of real roots, and no more ; and where 
the pairs of consecutive terms of the series of multiples of 4 
which correspond to each change of sign are limits to the seve- 
ral real roots of the equation. ‘This is the principle of the me- 
thod of determining the limits of the real roots which was first 
proposed by Waring, and which has been brought into practical 
operation by Lagrange and Cauchy. It remains to explain the 
different methods which have been proposed for the purpose of 
determining the value of 4. 

Waring first, and subsequently Lagrange, proposed for this 
purpose the formation of the equation whose roots are the 
squares of the differences of the roots of the given equation. If 
we subsequently transform this equation into one whose roots 
are the reciprocals of its roots, and determine a limit / greater 


than the greatest root of this transformed equation}, then —, 


Vo 


* A negative root is always considered as less than a positive root, unless 
the consideration of the signs of affection is expressly excluded. 

+ Newton proposed for this purpose the formation of the equation whose 
roots are x — e, and where e is determined by trial of such a magnitude that 
all the coefficients of the equation may become positive. In sucha case e is the 
limit required. Maclaurin proved that the same property would belong to the 
greatest negative coefficient of the equation increased by 1. Cauchy, in his 
Cours d’ Analyse, Note iii., and in his Exercices des Mathématiques, has shown 
that if the coefficients of the equation, without reference to their sign, be 
A, As, .. Am, and if x be the number of such coefficients which are different 
from zero, then that the greatest of the quantities 


1 
n Aj, (n A,)?, (n As)", .. (a Am)” 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 325 


will be less than the least difference of any two of the real roots 
of the primitive equation, and will consequently furnish us with 
such a value of 4 as will enable us to assign their limits. The 
extreme difficulty, however, of forming the equation of dif- 
ferences, which becomes nearly impracticable in the case of 
equations beyond the fourth degree*, renders it nearly, if not 
altogether, useless for the purposes for which this transforma- 
tion was intended by the illustrious analysts who first proposed 
it; in other words, it is only ina theoretical sense that it can be 
said to furnish the solution of the problem of determining the 
limits of the real roots of an equation. 

Cauchy has succeeded in avoiding the necessity of forming 
the equation of the squares of the differences of the roots, by 
showing that a value of 4 may be determined from the last term 
of this transformed equation, combined with a value of a limit 
greater than the greatest root of the primitive equation. If we 
suppose H to represent this term, & to be the superior limit 
required, and a and 6 to represent any two roots of the equa- 
tion, whether real or imaginary, then he has shown that their 
difference a — 6, or the modulus of their difference, will be 


will be a superior limit to the roots. An inferior limit (without reference to 
algebraical sign) may be readily found by the same process by the formation 
of the equation whose roots are the reciprocals of the former. 

M. Bret, in the sixth volume of Gergonne’s Annales des Mathématiques, has 
investigated other superior limits of the roots of equations, which admit of 
very easy application, and which likewise give results which are generally not 
very remote from the truth. One of these limits is furnished by the following 
theorem : ‘‘If we add to wnity a series of fractions whose numerators are the 
‘successive negative coefficients, taken positively, and whose denominators 
are the sums of the positive coefficients, including that of the first term, the 
greatest of the resulting values will be a superior limit of the roots of the 
equation.” Thus, in the equation 


2 a7 + 11 «7° — 10 a — 26 at + 31 a3 + 72 a — 2304 — 348 = O, 
the number 4, which is equal to the at sa of the Dian: 


10 
Le 
iF iz} += hia ~~ Bisposs — 


is a superior limit required ; and re we change the signs of the alternate terms, 
we shall have 1 + cat or 7, a superior limit of the roots of the resulting 


oer : it will follow, therefore, that all the real roots of the first equation 

will be included between 4 and — 7.. Other methods are proposed in the 
same memoir which are not equally new or equally simple with the one just 
given, and which I do not think it necessary to notice. 

* Waring, as is well known, gave the transformed equation of the 10th de- 
gree, whose roots were the squares of the differences of the roots of a general 
equation of the fifth degree, wanting its second term: it involves 94 different 
combinations of the coefficients of the original equation, many of them ‘eee 
large numerical coefficients. 


326 THIRD REPORT—1833. 


Hi 
greater than ap? if nm denote the dimensions of the 
2 


equation; and in as much as H is necessarily, when the coeffi- 
cients are whole numbers, either equal to or greater than 1, it 


will follow that ipightala will furnish a proper value of 4, 
2 


where / has been determined by the methods described above, 
or in any other manner. The chief objection to the use of a 
value of 4 thus determined arises from its being generally much 
too small, and from the consequent necessity of making a much 
greater number of trials for the discovery of the limits of the 
roots than would otherwise be necessary. 

Lagrange has proposed different methods of determining the 
value of 4, which, though much less laborious, at least for 
equations of high orders, than the equation of the squares of 
the differences, are still liable to great objections, in conse- 
quence of their being indirect, difficult of application, and likely 
to give values of 4 so small and so uncertain as greatly to mul- 
tiply the number of trials which are necessary to be made*. It 
is for this and other reasons that such methods have never been 
reduced to such a form as to be competent to furnish the re- 
quired limits by means of processes which are expressible in 
the form of arithmetical rules, like those which are given for the 
extraction of the square and cube root in numbers. In this re- 
spect, therefore, they have failed altogether in satisfying the 
great object proposed to be attained by their author, who con- 
sidered the resolution of numerical equations as properly consti- 
tuting a department of common arithmetic, the demonstration 
of whose rules of operation must be subsequently sought for in 
the general theory of algebraical equations +. 

The basis of all methods of solution of numerical equations 
must be found in the previous separation of the roots; and the 
efforts of algebraists for the last two centuries and a half have 
been directed to the discovery of rules for this purpose. The 
methods, however, which have been proposed have been chiefly 
directed to the separation of the roots into classes, as positive 
and negative, real and imaginary, and not to the determination 
of the successive limits between which they are severally placed. 
The celebrated theorem of Des Cartest gave a limit to the 
number of positive and negative roots, but failed in deter- 

* Résolution des Equations Numériques, Note iv. 

+ Ibid., Introduction. 

+ The proper enunciation of this theorem is the following: “ Every equa- 


tion has at /east as many changes of sign from + to — and from — to + 
as it has real and positive roots, and at Jeas¢t as many continuations of sign 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 327 


mining the absolute number either of one class or of the other, 
in the absence of any means of ascertaining the number of ima- 
ginary roots. If the roots of the equation were all of them real, 
and could be shown to be so by any independent test, it would 
be easy to determine the limits between which the roots were 
severally placed; for the number of changes of sign which are 
lost upon the substitution of 2 + ¢ for 2 would show the number 
of roots which are included between 0 and e; and if, therefore, 
we should assume a succession of values of e, whether positive 
or negative, such as to destroy one change of signs and no more, 
upon the substitution of any two of these successive values, we 
should be enabled to obtain the limits of every root of the 
equation. 

It was chiefly with a view to this consequence of Des Cartes’s 
theorem that De Gua investigated and assigned the conditions 
of the reality of all the roots of an equation. If we suppose 
X = 0 to be the equation, and Xi, X"/, Xili, Xiv, Xv, &c., to 
denote the successive differential coefficients of X, then, if all 
the roots of X = 0 be real, the roots of the several derivative 
equations Xi= 0, Xii'=0, X#i# = 0, &c., must be real like- 
wise; and if the roots of any one of these equations X = 0 
be substituted in X°-” and X°*”, it will give results affected 
with different signs. If we form, therefore, a succession of 
equations in y by eliminating successively x from the equations 


y= X™ . XX) and X—) = 0, 
y= Xe | Xe ) and X*-) = 0,0... .°. 
y = Xi Xiliand Xi = 0, y = X Xl and X'=0, 
the coefficients of all these equations must be positive, forming 


from + to + and from — to — as it has real and negative roots.’”’ It is very 
doubtful, notwithstanding the assertions of some authors, whether Des Cartes 
himself was aware of the necessary limitation of the application of this theorem, 
which is required by the possible or ascertained existence of imaginary roots. 
The demonstration which was given by De Gua of this theorem in the Mé- 
moires de l’ Académie des Sciences for 1741, founded upon the properties of the 
limiting equation or equations, has been completed by Lagrange with his 
usual fullness and elegance, in Note viii. to his Résolution des Equations Nu- 
mérigues. 'The most simple and elementary, however, of all the demonstra- 
tions which have been given of it, and the one, likewise, which arises most 
naturally and immediately from the theory of the composition of equations, is 
that which was given by Segner in the Berlin Memoirs for 1756, The few im- 
perfections which attach to this demonstration, as far as the classification of 
the forms which algebraical products may assume is concerned, have been 
completely removed in a demonstration which Gauss has published in the 
third volume of Credle’s Journal. 
_ This theorem is included as a corollary to Fourier’s more general theorem 
for the separation of the roots, as we shall have occasion to notice hereafter. 


328 THIRD REPORT—1883. 


a collection of conditions of the reality of the roots of an equa- 


i i : ‘ n(n—1). 
tion of x dimensions which are segs in number~*. 


These speculations of De Gua were well calculated to show 
the importance of examining the succession of signs of these 
derivative equations, with a view to the discovery of their con- 
nexion with the nature of the roots of the primitive equation. 
The changes in the succession of signs of the coefficients of the 
equations which resulted from the substitution of 2 + a and 
x + b, gave no certain indications of the nature and number of 
the roots included between a and 8, unless it could be shown 
that all the roots of the primitive equation were real, a case of 
comparatively rare occurrence, and which left the general pro- 
blem of the separation of the roots, as preparatory to their 
actual calculation, nearly untouched. It was the conviction that 
all attempts to effect the solution of this problem by the aid of 
Des Cartes’s theorem would necessarily fail, which led Fourier, 
one of the most profound and philosophical writers on analysis 
and physical science in modern times, to the examination of the 


* Résolution des Equations Numériques, Note viii. The equation of the 
squares of the differences of the roots of an equation will indicate the reality 


n(n — 1) 
2 


nately positive and negative. The succession of signs of the coefficients very 
readily furnishes the indications of the number of impossible roots in all equa-. 
tions as far as five dimensions, as has been shown by Waring and Lagrange. 

The number of conditions of the reality of the roots of an equation of five 
dimensions would appear from the formula in the text to be 10; but some of 
these conditions, as Lagrange has intimated, may, and indeed are, included 
in the system of the others, so as to reduce them to a smaller number. La- 
grange has assigned two conditions (not three) of the reality of the roots of 
a cubic equation ; but the first of these is necessarily included in that of the 
second, so as to reduce the essential conditions to one. Similar consequences 
are found to present themselves in the examination of these conditions for 
an equation of the fourth degree, which are three in number, and not six, as 
the formula would appear to indicate. 

Cauchy, in the 17th cahier of the Journal de l’ Ecole Polytechnique, has suc- 
ceeded, by a combined examination of the geometrical properties of the curve 
whose equation is y = X (where X is a rational function of x of the form 
am + p,am-1 + ... . pn), and of their corresponding analytical charac- 
ters, in the discovery of general methods, not merely for the determination of 
the number of real roots, but likewise of the number of positive and negative 
roots, as distinguished from each other. These methods are equally appli- 
cable to literal and numerical equations. He has applied his method to ge- 
neral equations of the first five degrees, and the results are in every respect, 
as far at least as they have been examined in common, equivalent to those 
which are derived from the equation of the squares of the differences. It is 
impossible, however, in the space which is allowed to me in this Report, to 
give any intelligible account of this most elaborate and able memoir, and I 
must content myself, therefore, with this general reference to it. 


of all the roots, if its coefficients have changes of sign, or be alter- 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 329. 


succession of signs of the function X and its derivatives, upon 
the substitution of different values of 2. The conclusions 
which have resulted from this examination, which we shall now 
proceed to state, have completely succeeded in effecting the 
practical solution of this most difficult and important problem, 
as far, at least, as real roots are concerned. 

If we suppose 


X = 2" 4+ 4,27 + a,x"? +...a,=9, 
and if we write X and its derivatives in the following order, 
ee a te Oe a. ie 


alts L, © sseagl gers ‘ 
then the substitution of he and — —,, will give two series of re- 


0 0 
sults, the terms of the first series being all of them positive, 
and those of the second being alternately positive and negative. 
The same will be the case if, in the place of 9: We put any 
limit («) greater than the greatest root of the equation X = 0,and 
if in the place of —>o we substitute any negative value of 
x (— P) (to be determined by trial or otherwise) which will 
make the first terms of X, X', X¥, &c., considered with regard 
to numerical value only, severally greater than the sum of all 
those which follow them. In the course of the substitution of 
values of x intermediate to those extreme values — 6 and a, all 
the m changes of sign of X and its derivatives, from + to — 
and from — to +, will disappear, in conformity with the fol- 
lowing theorems, which are capable of strict demonstration. 
Ist. If, upon the substitution of any value of 2, one or more 
changes of signs disappear, those changes are not recoverable 
by the substitution of any greater value of x. é 
2nd. If upon the substitution of two values a and 6 of x, 
_ one change of signs disappears, there is one real root and no 
more included between a and 6. If under the same circum- 
stances an odd number 2 p + 1 of changes of sign have disap- 
eared, there must be at least one, and there may be 2p! + 1 
where p’ is not greater than p) real roots between @ and 6; 
but if an even number 2 p of signs have disappeared in the in- 
terval, there may be 2p — 2p! real roots, and p! pairs of ima- 
ginary roots corresponding to it, where p’ is not greater than p. 
If no change of sign disappears, upon the successive substi- 
tution of a and 6, then no root whatever of the equation X = 0 
can be found between the limits @ and 0b. 
drd. If the substitution of a value a of a makes X = 0, then 
a isa root of the equation. If the substitution of the same 
value of « makes at the same time X = 0 and X' = 0, then 


330 THIRD REPORT—1833. 


there are two real roots equal to a; and generally, as many of 
the final functions X, X', X", &c., will disappear, under the same 
circumstances, as there are roots equal to a. 

4th. If the substitution of a value of a makes one intermediate 
function X equal to 0, and one only, and if the result 0 be placed 
between two signs of the same kind, whether + and + or — and 
—, then there will be one pair of imaginary roots corresponding to 
this occurrence ; but if 0 be placed between two unlike signs, + 
and — or — and +, then there will be no root corresponding to it, 
unless at the same time X = 0. Ifthe substitution of a makes any 
number of consecutive derivative functions equal to 0, then, if 
there be an even number 2 p of consecutive zeros, there will be p 
or (p — 1) pairs of imaginary roots corresponding, according as 
they are placed between the same or different signs; and if there 
be an odd number 2p + | of consecutive zeros, then there will 
be p + 1 or p pairs of imaginary roots corresponding, according 
as they are placed between the same or different signs *. 

The preceding propositions may be easily shown to include 
the theorem of Des Cartes ; for it is obvious that the substitution 
of 0 for x in X and its derivatives will give a succession of signs 
identical with those of the successive coefficients of X, deficient 
terms being replaced by 0. If the extreme values « and — 6 
be substituted, there will be m permanences in one case and m 
changes in the second ; it will follow therefore that the number 
of real and therefore positive roots between « and 0 cannot ex- 
ceed the number of changes of sign corresponding to « = 0, or 
amongst the successive coefficients of the equation ; and that the 
number of real and therefore negative roots between — 6 and 0 
cannot exceed the number of permanences corresponding to 
x = 0, or of changes between 0 and — 8, which is also identical 
with the number of successive permanences of sign amongst the 
coefficients of the equation. 


* | have stated this rule differently from Fourier, whose rule of the double 
sign appears to me to be superfluous. If we consider the zeros as possessing 
arbitrary signs, the nature and extent of the ambiguity which they produce 
will always be determined by the circumstances of their position with respect 
to the preceding and succeeding sign. 

The rule of the double sign, when one of the derivative functions Xi, Xi, Xi, 
&c., becomes equal to zero, is made use of in a memoir by Mr. W. G. Horner, 
in the Philosophical Transactions for 1819, upon a new method of solving nu- 
merical equations. This memoir, though very imperfectly developed, and in 
many parts of it very awkwardly and obscurely expressed, contains many 
original views, and also a very valuable arithmetical method of extracting the 
roots of affected equations. It makes also a very near approach to Fourier’s 
method of separating the roots of equations. It is proper to state that 
Fourier’s proposition was known to him as early as 1796 or 1797, as very 
clearly appears from M. Navier’s Preface to his Analyse des Equations Déter- 
minées, a posthumous work, which appeared in 1831. 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 331 


In order to render the preceding propositions more easily in- 
telligible, we will apply them to two examples. 

Let X = 24 — 423 —32 + 23=0, and underneath X”, 
Xiii, Xi, Xi, X, let us write down the signs of the results of the 
substitution of 0, 1, 2, 3, 10, in the place of x, in conformity 
with the following scheme: 


be ee b, tah X', XxX, 
(0) radon pont): Dx sosmie snk 
(1) epsom wh caghee g 
(2) ae 0 — + 
(3) Fabs cries lt wngic ict 
(10) eel Ws bae te 


For x = 0, there is a result 0 placed between two similar 
signs; there is therefore a pair of imaginary roots correspond- 
ing to it. Every value of z less than 0 will give results alter- 
nately + and —, and there is therefore no real negative root. 

For x = 1, there is a result 0 placed between two dissimilar 
signs: there is therefore no pair of imaginary roots corre- 
sponding ; and since there is no loss of changes of sign in pass- 
ing from 0 to 1, there is no real root between those values. 

For x = 2, there is a result 0 placed between two dissimilar 
signs ; there is therefore no pair of imaginary roots correspond- 
ing, and there is no root between | and 2. 

For z = 8, there is a loss of one change of sign, and there is 
therefore one real root between 2 and 3. 

For x = 10, there is a loss of one change of signs and all the 
resulting signs are positive; there is therefore one real root 
between 3 and 10. 

The limits of the real roots are thus completely determined, 
and the substitution of the successive whole numbers, from 3 
upwards, will show the nearest whole numbers 3 and 4, between 
which the greatest root is situated. 


Let X = «®— 122° + 6024 + 123.2% + 4567 x — 89012 = 0 
xX", » oa , GP Bi bd Xi, ».¢ 


(—10) apy be Mored baerechp he asia 
(—1) 2 Spihiite o2 Dye we einbniceat Se = 
(0) Ep <ARGieaL, Di, if ee 
(1) Sa Bi: REI Se aa Ms 
(10) 2 aren i ae A iat i 


All the real roots of the equation are included between the 
extreme values — 10 and 10. 


332 THIRD REPORT—1833. - 


One change of sign is lost in the transition from — 10 to —1, 
and there is therefore one real root between them; the sign of 
the last term is therefore necessarily changed from + to —. 

For x = 0, there is a result 0 between two similar signs ; 
there is therefore a pair of imaginary roots corresponding, and. 
consequently a loss of two changes of sign. ) 

There is no root of the equation between 0 and 1. 

There is a loss of three changes of sign in the transition from 
1 to 10, and therefore there are three roots corresponding, one 
or all of which may be real: the application of a subsequent 
rule will show that two of them are imaginary. 

It is obvious, in a series of derivatives, X™, X-,... X@ 
... X, that X™, X‘-) may be considered as the derivatives 
of the (m — r —1)™ and (m — r — 2)™ order from X, as well 
as the m and (m — 1) derivatives from X, and that the same 
rules may be applied to the separation of the roots of these de- 
rivatives when they become equations, whether they-be consi- 
dered as belonging to the inferior or to the superior order. The 
substitution, therefore, of a and 6 successively for x, will show 
the number of roots of the successive derivative equations which 
are found in this interval, which will be equal successively to 
the number of changes of sign which have disappeared in the 
transition from one value of x to the other. If we now place 
under the several results of the substitution of a and 8, a series 
of zeros or numbers as indices to signify that no change, or an 
indicated number of changes of signs, have disappeared, then in 
passing from the left to the right, we shall find first zero, and sub- 
sequently, whether immediately or not, the numbers, |, 2, &c., 
which will indicate the number of roots which must be sought 
for, in that interval, in the derivative or other functions, consider- 
ed as equations, which are severally placed above them. ‘Thus, 
f xX = at — a2 +42? + « —4=0, then from the scheme 


ota i, Sa BS xi, X, 
fee, an eccleee der lad oF Situ hall te 
fre a Np erenigcanenny? FES 
(— 1) Hepat nn ort thd corte ae 
Titel pea ree 
(0) pk Seance: EE 7S ApS ee 
0 1 2 aa 
(1) aS». PDair Slope i, Pigs ae 


we infer that there is one root of X = 0, and no root of any of 
the several derivative equations situated between — 10 and — 1; 


ee 


Pa. 
s 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 333 


that there is one root of X' = 0, and no root of X = 0, between 
— 1 and 0; that there is one root of X= 0, two roots of 
Xii = 0, two roots of Xi = 0, and'three roots of X = 0, situated 
between 0 and 1. It remains to determine whether these three 
roots are all of them real, or two of them imaginary, and also 
to assign the limits, in the first case*, between which they are 
placed. 

In the first place, if imaginary roots exist in the derived, 
they will exist also in the primitive equation: The converse of 
this proposition is not necessarily true. 

If the succession of indices be 0, 1, 2, then the succession of 
signs corresponding to 


X#, X!, X, or Mitek 5 X®; X¢-), 


will be 
(a) + — +or- + - 
Oot Liw 2 Os [doivi 
(d) - ies We 


There will be one real root between a and 3b in the equation 
Xi = 0 or X = 0, and two roots, whether real or imaginary, 
corresponding to this interval, in X = 0 or X°-Y = 0. 

In the first case, if there be two real roots between a and 6, 
then the curve whose equation 


isy = X =f (x), where 0 a=a, \ “ 

0b6=b,an=f (a), bm = 

F (6), will cut the axis at the 

points «and 6 between aand b. to = atl Lp 
(i 


The curve will have no point Aiton tex 
of inflection between a and 6, ce 

since X# preserves the same sign, whether + or — ; and there 
will be a point ¢, where the tangent is parallel to the axis, 
since Xi‘, in the same interval, changes from + to —, or con- 
versely, and therefore becomes equal to zero between those 
limits. In this case, the sum of the subtangents (considered 
without regard to algebraical signs) will be necessarily less than 
ab; and if the interval a b be subdivided sufficiently, so as to 
furnish new limits a’ and 0/, then one or both of these points 
will sooner or later be found between the points of intersection 
a and #, and therefore f(a’) and f(6’) will one or both of them 
change their signs. The analytical expression of those geo- 
metrical conditions, and therefore of the existence of two real 


f @) 
(a) 


* * We seek for the limits of the real roots only ; we have no concern with 
those of the imaginary roots or of their moduli. 


roots, will be, that the sum of the subtangents or quotients 


334 THIRD REPORT—1833. 


+ far Z b —a, when no regard is paid to the sign of f’ (a). 
and f' (6). In this case new limits must be taken successively, 
intermediate to a and 8, until f (a’) and f (6') one or both of 
them change their sign. 

In the second case, if there be two imaginary roots cor- 
responding to the interval Fie. 2 
between a and 3B, then the an 
curve whose equation is y= X 
though similar in its other ge- 
ometrical properties to fig. 1, 
will not eut the axis between” “ os eae 6 
aand b. In this case the sum of the subtangents a n’/ and 
b m! will either exceed the interval a 6, or will ultimately ex- 
ceed it, when the interval a 6 is sufficiently diminished. The 


corresponding analytical character will be that a + ae 
is either greater than 6 — a, or that it may ultimately be made 
to exceed it *. 

Thus, in the example referred to above, p. 332, write down 


the following scheme : 


Des, ae, yal ae X, 


6 8 
(0) Sc aes Peppa in 
0 1 BY SBuCKE 
(1) 7 Aide Fa Recttvay ain 
18 14 


and place above and below the indices 1 and 2, in the succes- 
sion of indices 0, 1, 2, the values of X™ and X® respectively, 
without regard to sign, corresponding to x = 0 and x = 1; then 


we shall find & 7 1 and, @ fortiori, therefore 2 + a also 


greater than 1, which is the interval between which the roots 

required are to be sought for: it consequently follows that two 

of the roots corresponding to this interval are imaginary, and 

there remains, therefore, only one real root between 0 and 1. 
If we suppose 


K=2e 4+ att 2 — 227° +27-1=0, 
* The new values a’ and b' of aandb may be made a +50 and d— FQ, 


which are 0 n! and 0m! respectively : a second trial will generally succeed. 


_ oe ae ee ee a. = 
RE OAD 


Ose 


™ 


ot: 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 335 


the corresponding scheme will be as follows : 


sh a", pga 2 ae bes X, 


96 _ 42 
Pe Bs ehinliyneats ui te pbociyee 
369 
fog) sat ro be nt 
at i: ny al ip tae Og tt 
EA) camisoes bdr sats culate eee” siledtee’ xt BG eC Ase 
Bd 9c) Bien! oho yi 
Oo ft Oita ghee! “Ora ap 
(3) Hee - 
36°) 10 


(Det ey eethe octitee thos Mchia F 

If we take the interval from (— 1) to 0, we find two roots in- 
cluded within it; but since hs + a is less than the interval, 
no certain conclusion can be drawn with respect to the nature 
of the corresponding roots. If we now consider the interval 
from — 3 to 0, which includes the same roots, we shall find 
RS 
36 1 24 ~ 
are consequently authorized in concluding that the correspond- 
ing roots are imaginary. In a similar manner, we find the in- 


2 , a quantity equal to the whole interval, and we 


ee ; é 1 
.dication of the existence of two roots between 0 and 33 and 


: 2 1 ‘ : 
in as much as mara, ess the whole interval, we at once con- 


clude that the two roots in question are imaginary*. 
It thus appears that we are enabled, by the processes just 


_ described, to separate all the real roots of an equation and to 


* When we speak of the existence of imaginary roots between two limits, 
we do not mean that such limits comprehend the moduli of these roots, but 
merely that the real roots which would be found between those limits, if cer- 
tain conditions were satisfied, are wanting, and that there are as many ima- 
ginary roots of the equation which may be said to correspond to them which 
are sufficient to complete the required number of changes of sign which are 
lost. The theory of Fourier as given in his work, determines nothing con- 
cerning the values or limits of the moduli, or of the peculiar nature of the 
signs of affection, of such imaginary roots. 


336 THIRD REPORT—1833. 


assign their limits, and thus to prepare them for the certain ap- 
plication of methods of approximation. They constitute a most 
important element in the theory of numerical equations; and 
though they do not enable us to assign the limits of the moduli 
of the pairs of impossible roots nor to determine their signs of 
affection, yet they at once indicate both their existence and their 
number, and thus form the proper preparation, at least for the 
application of methods, whether tentative or not, for the deter- 
mination of their values. 

Lagrange, in the fifth chapter of his Résolution des Equa- 
tions Numériques, has shown in what manner the equation of 
the squares of the differences may be applied to the deter- 
mination of these imaginary roots; and the methods which 
thence arise are equally complete, in a theoretical sense, with 
those which are made use of, by the aid of the same equation, 
for the determination of the limits of the real roots; and Le- 
gendre, also, has furnished tentative methods of approximating 
to their values. But all such methods are more or less nearly 
impracticable for equations of high orders; and the invention 
of a ready and certain method of separating the imaginary 
roots of equations, as the basis of processes for approximating 
to their values, must still be considered as a great desideratum 
in algebra. 

The method of approximating to the roots of numerical equa- 
tions, when their limits are assigned, which Lagrange has given, 
by means of continued fractions, is so well known that it is quite 
unnecessary to enter upon a detailed examination of its princi- 
ples. If there is only one real root, included between two con- 
secutive whole numbers, there will be only one positive root in 
the several transformed equations, which is greater than 1, and 
methods which are certain and sufficiently rapid may be applied 
to the determination of the several quotients which form the 
converging fractions. If, however, there are two or more roots 
included between two consecutive whole numbers, there will be 
two or more roots of the first transformed equation, and possi- 
bly, likewise, of the transformed equations which follow which 
are greater than 1, and which may be placed between two con- 
secutive whole numbers. ‘The separation of such roots may be 
effected by the methods of Fourier, which have been explained 
above; but when we have once arrived at a transformed equa- 
tion which has two or more roots greater than 1, no two of which 
are included between two consecutive whole numbers, then we 
‘shall find the same number of sets of successive transformed 
equations, which will furnish the several sets of quotients to the 
continued fractions, which represent the roots of the primitive 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 337 


equation, which are included between two limits which are con- 
secutive whole numbers. The formation, however, of these 
transformed equations, and the determination of the next infe- 
rior integral limit of their roots, even when no further separation 
of the roots is required, is excessively laborious, and Lagrange 
has pointed out methods by which the operations required for 
both these objects may be greatly simplified. Legendre also, 
in the 14th section of the first part of his Theory of Numbers, 
has given a considerable practical extension to these methods 
of Lagrange. If we combine their processes for finding the 
nearest inferior limit of the root with the theorems of Budan * 
for the formation of the transformed equations, we shall proba- 
bly have arrived at the greatest simplification which the practi- 
cal solution of numerical equations, by means of continued frac- 
tions, is capable of receiving. 

Lagrange has pointed out the principal defects of the me- 
thod of approximation to the roots of numerical equations which 
was given by Newton}. It is only under particular conditions 
that it is competent to attain the object proposed, and in no 
case does it immediately furnish a measure of the accuracy of 
the approximation. But notwithstanding these objections to 
this method, in the form under which it has been commonly 
applied, it is unquestionably that which most naturally arises 
out of the analytical conditions of the problem, and which is also 
capable of the most immediate and most simple application in 
almost every department of analysis. Lagrange had demon- 
strated that this method could only be applied with safety to 
find the greatest and least roots of an equation, and in those 
cases only in which the moduli of the imaginary roots, if any ex- 
isted, were included in value between such roots. But Fourier 
has shown, by considering the superior and inferior limits of 
every real root, and by a proper examination of certain condi- 
tions which those limits may be made to satisfy, and by insti- 
tuting the approximation simultaneously with respect to both 
those limits, that all sources of ambiguity may be removed and 
the accuracy of the approximation determined t. We shall now 
proceed to give a short notice of these researches. 


~* Nouvelle Méthode pour la Résolution des Equations Numériques.. It con- 
tains the exposition of exceedingly simple and rapid rules for the formation 
of the transformed equation whose unknown quantity is «—e, where e is 
any integral or decimal number. In other respects, however, this publication, 
though announced with great pomp and circumstance, is a very superficial 
production, and is only remarkable for having received the charitable notice 
and approbation of Lagrange. 
+ Reésolution des Equations Numériques, Note v. 
eae des Equations déterminées, livr. ii., Calcul des Racines. 
. Z 


338 THIRD REPORT—1833. 


1. If f (x) =0, or X = 0 be the equation, f’ (x), f” (x), or 
X’, X” its first and second derivatives, then the limits a and } 
of one of the roots will be sufficiently near for the application of 
this method of approximation, if the three last indices (p. 332) 
be 0, 0,1. If this be not the case, the interval between a and 6 
must be further subdivided until this last condition is satisfied. 

Under such circumstances there will be no root of the equa- 
tions f' («) = 0 and f” (a) = 0, included between a and 6; and 
if we suppose y = f (x) to be the equation of a parabolic curve 
CAB,whereOa=a,O0b=b,am=f (a), bn =f (6), then 


there will be no point of inflection between a and 4, and no tan- 
gent parallel to the axis. The analytical conditions above men- 
tioned would show that f(a) and f(6) must necessarily have 
different signs. 

2. If we suppose 6 to represent the superior limit of the root 
(«), then the Newtonian approximation gives us the new su- 


perior limit 6’ = 6 — £3. a new inferior limit will be found 
tobe a =a vate en these limits are still superior and inferior 


limits of the root a, and are both of them nearer to it than the 
primitive limits 6 and a. 

If the same operation be repeated by replacing 6 and a in 
F (6) and f (a) by 8’ and a’, nearer limits will be obtained, and 
it is obvious that the same process may be repeated as often as 
may be thought necessary. And in as much as we obtain both 
the inferior and superior limits corresponding to each operation, 
the difference between them will always be greater than the 
error of each approximation. If we refer to the above figure, and 
suppose x 0’ to be a tangent to the curve at 2, and a’ m to be 


f (4) J (@) 


drawn parallel to n 6’, then 5b’ = f(b)’ and aad’ = 7(by’ 


since f’ (b) = tan n b’b= tanma‘a. It follows, therefore, that 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 339 


O b’ and O a’ are the new limits 5’ and a’: and if ordinates b’ n! 
and a’ m’ be drawn to the curve, and n’ 4’ be drawn a tangent, 
and m’ a” parallel to n’ b’, then O b" and O a” will be the new 
values 6” and a” of b’ and a’. The progress of the approxima- 
tion, upon the continued repetition of this process, will now be 
sufficiently manifest. 

3. If we consider the different arrangements of the signs of 
St” (x), f' (x), f (x), in the transition from the inferior limit a to 
the superior limit 4, they will be found to be the following, it 
being kept in mind that the sign of f(x) alone changes from 


+ to —, or conversely. 
J" (x) fl (@) f() 


se amenet ace 
Bi gonie' soma. aon 
crip she 
Berane Mactan iiy 


In the first two cases, the formule of approximation are 


b= 4 o and a — aon and commence therefore with the su- 
perior limit. In the last two cases, the formule of approxima- 


tion are a — ae and 6 — A sand commence therefore with 
the inferior limit. In other words, that limit must in all cases 
be selected which gives the same sign to f" («) and f (x), whe- 
ther + or —. The construction of the portions of the corre- 
sponding parabolic curves included between a and 6 in these 
several cases, will at once make manifest the reason of the selec- 
tion of the superior or inferior limit and likewise the progress 
of the approximation itself*. © " 


'* Tf, inthe figure p. 338, we join the extremities m and of the ordinates am 
and 6 n by the chord m N , which cuts the axis of ‘i in the point N, we shall 
Mow = a 702 ©), 2 LO). O SD), sain igavadian ae 
fat (a) F@ atlaer Wwe ner 
proximate inferior limit in the first two cases considered in the text, and anew 
superior limit in the last two. Other constructions are noticed by Fourier, 
which give similar results. 

‘In the Mémoires de V Académie Royale de Bruxelles for 1826, there is a 
memoir on the resolution of numerical equations by Dandelin, in which the 
analytical conditions which must be satisfied by the limit, towards which the 

: Z2 


340 THIRD REPORT—1833. 


4. In the application of these rules some precautions are oc- 
casionally necessary. Thus, if f" (a) and f (a) have a common 
measure ¢ (x), and if a root («) of ¢ (a) = 0 be included between 
a and 3, then there is a point of inflection of the paraboli¢ arc 
between a and 6 at the point of its intersection with the axis. 
Under such circumstances, the method of approximation must 
be applied to the equation ¢ («) = 0, and not to the primitive 
equation, f (x) = 0, for the purpose of determining the value of «. 
Again, if there exists a common measure of f! (x) and f (x), which 
becomes equal to zero, for a value of x between a and 6, then 
there are two or more equal roots of f(x) = 0 in that interval, 
and the final succession of indices is no longer 0, 0, 1. Other 
precautions connected with the subdivision of the interval b — a 
are sometimes required, which the limits of this Report will not 
allow me to notice in detail. 

It remains to add a few remarks upon the rapidity of the ap- 
proximation, and upon the means by which it may be ascer- 
tained. If we express the primary and secondary intervals 
b — aand D! — a’ by i and 7, it may be very easily proved that 


j— 7 fen ® 

gps” gl) iat 
where f" (a...) denotes some value which f" (a) assumes when 
we substitute for x a quantity between a and 6: and if we form 
the quotient (C) which arises from dividing the greatest value 
of f" (a) and f" (b)* by the least value of 2 f/ (a) and 2f’ (4), 
and suppose & the order of the greatest articulate or subarticu- 


approximation in Newton’s method must be made, are established by a com- 
bination of analytical and geometrical considerations, and in which also the 
new limits 5! and a! are respectively found by what he terms the rule of tan- 
gents in one case, and by the rule of chords in the other. The first is the 


subtraction of the subtangent b 0! or 5 @ from 4, as involved in the or- 


dinary Newtonian approximation when the proper limit is selected. The 
second is the determination of the value of ON, ora oP 


or 
Ps f (6) —f (a 
oe by the method taught at the beginning of this Note. It is 
evident that these conclusions involve all that is important in Fourier’s re- 
searches upon this part of the subject. 
is memoir of M. Dandelin, which contains a very full and a very clear. 


exposition of the whole theory of the Newtonian method of approximation, 
preceded by five years the publication of M. Fourier’s work. 

* Since no root of f'” (x) = 0 is included between a and 4, it follows that 
either f” (a) or f" (6) will be the greatest value of f" (a...): the same. 
remark applies likewise to f’ (a) and f! (2). 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 341 


late number * immediately greater than this quotient, and 7 the 
order of the articulate or subarticulate number which is not less 
than the difference of the limits 6 — a, then if we divide f(b) 
by f' (), and continue the operation as far asthe (2 + h)™ 
decimal, and increase the last digit by 1, the quotient which 
arises being subtracted from or added to, b, according as f (0) 
and f! (6) have the same or different signs, will give a result 
which will differ from the true value of the root by a quantity 


an+kh 
less than Ga . And if the same operations be repeated, 


forming successively new limits by means of the results thus 
obtained, we shall obtain a series of limits which are correct as 
far as the (4n+3 h)", the (8n+7 kh), &c., decimal place f. 

The processes of approximation which have been described 
above, as well as those which belong to all other methods, re- 
quire divisions and other operations with numbers which are 
sometimes beyond the reach of logarithmic tables, and which it 
is extremely important to abbreviate as much as possible, con- 
sistently with the determination of the accurate digits of the 
results which are required to be found. Such processes were 
taught by Oughtred and other algebraists of the seventeenth 
century, but both their theory and applications have been 
greatly and, perhaps, undeservedly, neglected in later times. 
The consideration, however, of such methods has been partially 
revived by Fourier and some other writers, the first of whom 
has given examples of what he terms ordinate division (division 
ordonnée,) the principle of which is to conduct the division by the 
employment of a small number of the first digits of the divisor 
only, and to correct the successive remainders, augmented by 
the successive digits of the original dividend, in such a manner 
as to bring into operation the successive digits of the divisor 
when they are required for the determination of the correct 
digit of the quotient, and not before. Such processes, however, 
are incapable of being briefly described, and we can only refer to’ 
the original work t for the developement of the rule and for ex- 
amples of its application. 

_* An articulate number is one of the series 1, 10, 200, 7000, &c., where 
the first digit is followed by zeros only. A subarticulate number is one of 
the series “1, ‘02, 003, &c., and the number which designates the place of 
the first significant digit is supposed to be negative. 

_ + The course of the approximation, in order to be perfectly regular and rapid, 
would require that 2 x + k should be greater than n, or that n should be 


greater than — k, a circumstance which might occur if k or n was negative. 
In such a case it will be necessary, or rather expedient, to subdivide the in- 


. ‘ - 3 
terval 6 — a, until the difference of the two limits does not exceed (+) F 


where 7 is equal to, or greater than, 1 — k. 
t Analyse des Equations determinées, livr. ii. p. 188. 


342 THIRD REPORT—1833. 


Similar processes, also, have been investigated and applied 
with remarkable ingenuity and success by Mr. Holdred*, Mr. 
Horner}, and Mr. Nicholson{. The first of these writers, a 
mathematician in humble life, who had formed his taste upon 
the study of the older algebraical writers of this country, gave 
very ingenious rules for finding the roots of numerical equa- 
tions. The method proposed by Mr. Horner was founded upon 
much more profound views of analysis and of the relation which 
exists between the processes of algebra and arithmetic, and he 
has not only succeeded in making a very near approximation to 
the true principles upon which the limits of the roots of numerical 
equations are assigned, but by considering the rules for extract- 
ing the roots of numbers and of affected numerical equations 
as founded upon common principles, he has reduced the rules 
for these purposes to a form which admits of very rapid and ef- 
fective, though not perhaps of very easy, application. Mr. Ni- 
cholson, by a combination of the methods of Mr. Holdred and 
Mr. Horner, has greatly simplified them both, and reduced them 
to the form of practical rules, which are not much more compli- 
cated than those which are commonly given for the extraction 
of the cube and higher roots of numbers. 

The Newtonian method of approximation, which we have 
hitherto considered, may be termed Jinear, in as much as the 
equations of a straight line combined with the general equation 
of the parabolic curve are competent to express all the circum- 
stances which characterize it. But methods of approximation 
of higher orders than the first, involving the second or higher 
powers of the unknown quantity to be determined, have likewise 
been considered by Fourier and other writers. That of the se- 
cond order, viewed with reference to the properties of curve 
lines, may be said to result from the contact of arcs of a conical 
parabola. The superior and inferior limits, thus determined, 
converge with great rapidity, the error corresponding to each 
operation being the product of a constant factor with the cube 
of the preceding error. Such methods, however, if viewed with 
reference to the facility of their practical applications, are incom- 
parably less useful than those which are founded upon linear 
approximations ; but there is much which is instructive in their 
theory, and particularly as furnishing the means of determining 
immediately the nature of two roots of an equation included in 
a given interval, which the application of the methods for the 


* This method is particularly noticed in Mr. Nicholson’s Essay on Involu- 
tion and Evolution. I have never seen the original tract published by Mr. 
Holdred. 

+ Philosophical Transactions for 1819. 

t Essay on Involution and Evolution. 1820. 


rh CET 


~ REPORT ON CERTAIN BRANCHES OF ANALYSIS. 343 


separation of the roots which we have previously described 
may have left in the first instance uncertain. We refer to the 
end of the second book of Fourier’s Analyse des Equations 
determinées, for a very complete examination of the theory of 
such approximations *. 

It has been a question agitated on more than one occasion, 
whether the tests of the reality of the roots of equations of finite 
dimensions which De Gua established, or rather the principles 
of the much more general theorem of Fourier, were applicable 
likewise to transcendental equations. In a discussion of the 
transcendental equation 

92 93 4 
yal—O+ a ae te grre — Bes 


which presents itself in the expression of the law of propagation 
of heat in a solid cylinder + of infinite length, Fourier ventured 
to apply the principles in question to show that all its roots were 
real; but M. Poisson} has disputed the propriety of such an 
application, both in this case and in others: thus, if we suppose 


XxX = e — be*’, 


‘we shall find 


* The rule for the determination of the nature of two roots included in a 
given interval, which is given in page 333, is merely the expression of a con- 
sequence of the application of the method of linear approximation to the di- 
stinction of those roots; and whatever difficulties in certain extreme cases 
may attend the successful application of that rule, will necessarily present 
themselves likewise in the application of the linear approximation under the 
same circumstances. This character, however, is not confined to the Newto- 
nian or linear method of approximation. If the interval of the roots be deter- 
mined, by the application of Fourier’s theorem of the succession of signs of 
the original function X and its derivatives, so that no more than two roots 
may be said to exist in that interval, whose nature is unknown, whether real 
or imaginary, then the application of the method of continued fractions, as 
well as of other equivalent modes of approximation, will be competent to de- 
termine the values of those roots when real, and their nature, when imaginary. 
Such, at least, is the assertion of Fourier, who refers to the third book of his 
work on equations for its demonstration. It is unfortunate, however, that 
only two books of this work, which is full of such remarkable researches upon 
the theory of equations, were fully prepared for publication at the time of his 
death. Our knowledge of the contents of the other five books, which were 
left unfinished, is derived from an Exposé Synoptique prefixed to those which 
are published, and which contains a general review and analysis of their prin- 
cipal contents. It is to be hoped, however, that the materials which he has 
left behind him will be found to be sufficient at least for their partial, if not 
for their complete restoration. 

+ Théorie de la Chaleur, p. 372. 

{ Journal de V Ecole Polytechnique, cahier xix. p. 381; Mémoires de I’ In- 
stitut, tom. ix. p. 92. 


344. THIRD REPORT—1833. 


ma e* — ba" e** 
Barer’ : 
n+l 
a = e? — ba’ *, 
where z is any whole number, or zero. If we now suppose 
d*1X 
dani = 9 
and eliminate, by means of this equation, e”, we shall get 
OX __ ya aarer, 
d™*2 X 
dant = b(1 — a) a" e*, 


and therefore 
aX ax 
d a . d ant2 


a quantity which is negative for every real value of x. The 
conclusion which should be drawn, in conformity with Fourier’s 
principles, is, that all the roots of the equation e” — be** = 0 
are real, as well as those of its successive derivatives; whilst 
the fact is, that each of those equations has one real root, and 
an infinite number of imaginary roots, which are included un- 
der the formula 


log ba® + 2ixnVv—1 
2= 
l—a 


= — 8 (1 — aa ae, 


In reply to this objection, it has been urged by Fourier that 
Poisson has not very accurately stated the terms of the propo- 
sition in question as applicable to such a case*, and also that 
he has neglected to take into consideration all the roots of the 
equation. For if we suppose that the substitution of two limits 
a and 6, in a function f (x) and its derivatives, gives results 
which present the same succession of signs between f+” (a) 
and f (x), then those extreme derivative functions, and those 


* This inaccuracy of statement is rather chargeable upon Fourier himself 
than upon Poisson, who has certainly failed to notice the necessary limitation 
of this proposition upon the occasion which gave rise to its application in 
page 373 of the Théorie de la Chaleur. 


REPORT ON. CERTAIN BRANCHES OF ANALYSIS. ‘845 


also which are included between them, when considered as 
equations, will contain the same number of roots, or none, be- 
tween those limits. This proposition is true, whether the num- 
ber of derivative functions be finite, as in the case of algebraical 
equations, or infinite, as in the case of transcendental equations. 
In the first case, however, it admits of absolute application, in 
consequence of our arriving at a final derivative, from which 
the comparison of the signs of the two series of results com- 
mences. In the second case we can draw no conclusion, in the 
absence of any difference in the signs of the series of results, 
in the transition from one derivative function to another, with 
respect to the number of roots of any of those functions which 
are included between the given limits: thus, if f(x) = sin x, 
we shall have the same series of signs of sin x and of its deri- 
vatives, however far continued, upon the substitution of the 
limits a and a + 27, although it is manifest that there are two 
real roots of sn « = 0 between those limits. The general pro- 
position, therefore, will, in such a case, authorize us in con- 
cluding merely that whatever number of roots the equation 
sin x = 0 includes between the limits a and a + 2 =, will be 
possessed likewise by all its derivative equations between the 


same limits *. 


There is another point of view, likewise, in which the objec- 
tion advanced. by Poisson may be considered as not altogether 
applicable to the example which he puts forward. In considering 

d™X dtiX qre2X 
da” dart? Gari he 
has not included those of the factor e*, which those functions 


e. \ 00 
severally involve. Since e” = (1 oh =) = 0, it follows that 


there are an infinite number of equal roots (where « = — «) 
of e” = 0, which equally reduce three or any number of conse- 
cutive derivative functions to zero, and to which, therefore, 
the test of De Gua is no longer applicable. It would follow, 
therefore, that the existence of imaginary roots in the equation 
X = 0 is no longer contradictory to Fourier’s proposition, even 


the roots of the derivative functions 


* If the transcendental function denoted by f (x) be a determinate function, 
it will always be possible to assign an interval 3, such that the derivative 
function f” (x) = 0 contains no root, or a determinate number of roots, be- 
tween a anda + 9. If such an interval or succession of intervals can be de- 
termined for any one derivative function, such as f ) (x), it will become a 
point of departure for the determination of the number and nature of the roots 
corresponding to the same interval or intervals for all the other derivative 
functions which form the superior or inferior terms of the series. In the case 
ofalgebraical functions, the point of departure is that derivative function which 
is a constant quantity. 


346 THIRD KHErURT—10u0vu. 


admitting the correctness of that form of it which Poisson ha. 
assigned *. 


5 1 ‘ 

ia we transform e” by replacing « by — zie we shall get the expression 
e *””, which may be easily shown as above, and also by other means, to be 
equal to zero when 2’ is equal to zero, and equal to 1 when x' is equal to in- 
finity. 

Professor Hamilton of Dublin, in a paper in the Irish Transactions for 1830, 

1 ; 

has quoted the expression e #" as possessing some very peculiar properties, 
which are inconsistent with the universality of a very commonly received 
principle of analysis. It is commonly assumed that ‘if a real function of a 
positive variable x approaches to zero with the variable, and vanishes along 
with it, then that function can be developed in a real series of the form 


Aa®+Ba° + C27 + &. (1.) 


where «, 8, y, &c., are constant and positive, A, B, C, &c., constant, and all 
those coefficients different from zero: but if we put the equation under the 
form 
alt 
ae @ = A+ BaP—-* 4 CaxY-@4 &., 


supposing « the least of the several indices &, 6, y, &c., then if a = 0, we 
1 


= ! 1 
shall find e—“e * =0 or A equal to zero; for if we replace aa by y, we 
shall get 


all whose terms are positive, and which, when z=0ory= , will necessarily 
2: 


become equal to infinity : it follows, therefore, that the function e 2” is not ca- 
pable of developement in a series of the assumed form (1.). The same ex- 
pression, as has been remarked by Professor Hamilton, has been noticed by 
Cauchy as an example of the vanishing of a function and of all its differential 
coefficients, for a particular value of the variable, without the function va- 
nishing for other values of the variable, thus forming an exception to another 
principle generally received in analysis. In his Legons sur le Calcul Infinite- 
simal, Cauchy has produced this last anomaly as a sufficient reason for not 
founding the principles of the differential calculus upon the developement of 
functions, as effected by or exhibited in, the series of Taylor. 

It is possible that more enlarged views of the analytical relations of zero 
and infinity, and of the interpretation of the circumstances of their occurrence, 
as well as of the principles and applications of Taylor’s series, may enable 
us to explain these and other anomalies, and to show that they arise natu- 
rally and necessarily out of the very framework of analysis; but it must be 
confessed that there are many other difficulties, which are yet unexplained, 
which are connected with the developement of e* when « is negative or ima- 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 347 


Another method of approximation to the roots of equations 
by means of recurring series was proposed by Daniel Ber- 
noulli*, and very extensively illustrated and applied by Euler +. 
If we write down m arbitrary numbers to form the first m terms 
of the series, and if we assume, for the scale of relation, the 
coefficients of an equation of m dimensions, and form by means 
of it and the assumed terms the other terms of the series which 
may be indefinitely continued, and if we also form a series of 
quotients by dividing each succeeding term (after the arbitrary 
terms) by that which precedes it, then the terms of the se- 
ries of quotients which thence arise, will converge continually 
towards the value of the greatest root of the equation; and if 
we form the equation whose roots are the reciprocals of those 
of the original equation, and proceed in a similar manner, we 
shall obtain a series of quotients which will converge to the 
greatest root of this equation, whose reciprocal will be the least 
root of the original equation, considered without reference to 
its algebraical sign. 

Lagrange, in the 6th Note to his Résolution des Equations 
Numériques, has analysed the principles of this method, and 
has shown that its success will depend upon the greatest real 
root, without reference to algebraical sign, being greater than 
the modulus of any of the imaginary roots. If this condition be 
not satisfied, the quotients will not approximate to the value of 
any root of the equation, a consequence which Euler had also 
pointed out. 

The recurring series which is formed by dividing the first 
derivative function f’ (x) by f (x), which is equal to 


ginary. Some of these have been noticed in the note to p. 267, in connexion 
with our observations upon Mr. Graves’s researches upon the theory of loga- 
rithms ; another is noticed by M. Clausen of Altona, in the second volume of 
Crelle’s Journal, p. 287; it is stated as follows :—Since ene V1 1, 
when 7 is a whole number, we get el t+2n7V—1 — e¢ and therefore 


fl +2neV—1)? _ 1+ 2nxV—1 L4naVv—1— 41? 2? 


2 . * 
consequently e 4 “tie 1, whenever 7 is a whole number,—a conclusion 
which M. Clausen characterizes as absurd. Its explanation involves no other 


difficulty than that which is included in the equation Be ntora = I, and 
must be sought for in the circumstances which accompany the transition from 
a function to its equivalent series, when a strict arithmetical equality does not 
exist between them. It must be confessed, however, that these difficulties 
are of a very serious nature, and are in every way deserving of a more care- 
ful examination and analysis than they have hitherto received. 
~* Comment. Acad. Petrop., vol. iii, 4 
+ Introductio in Analysim Infinitorum, vol. i, cap. xvii. 


and 


=—e—e 


348 THIRD REPORT—1835. 


1 1 1 
Aya mgr haps pints 


when «, 6, y, &c., are the roots of the equation, whether real or 
imaginary, has been shown by Lagrange to be the series fur- 
nished by this method which is most easily formed, and to be 
likewise that which converges most rapidly and certainly to a 
geometrical series in the case of equal roots. In every case the 
terms of the series of quotients are alternately greater and less 
than the root to be determined, and consequently furnish a mea- 
sure of the accuracy of the approximation. 

This method of approximation is generally less rapid and 
certain than those which have already been considered, and, 
as commonly stated, is extremely limited in its application. It 
is true, as has been shown by Lagrange, that a knowledge of 
the limits of the roots would enable us to apply it to the deter- 
mination of all the real roots by means of a series of transformed 
equations equal to their number, such as is required in the New- 
tonian method of approximation, and also in that of Lagrange; 
but under such circumstances, and with such data, it is more 
convenient and more expeditious to employ those methods in 
preference to the one which we are now considering. 

Fourier has shown in what manner this method may be ap- 
plied to determine all the roots of an equation, whether ima- 
ginary or real. Let us suppose a, ), c, d, e, &c., to represent 
the roots of the equation arranged in the order of magnitude, 
the magnitudes of imaginary roots being estimated by the mag- 
nitudes of their moduli; and let A, B, C, D, E, &c., be the 
terms of the recurring series, whose quotients furnish the value 
of the greatest root,when that root is real. Form, in the se- 
cond place, a series whose terms are A D — BC, BE — CD, 
CF — DHE, &c., which is also a recurring series, whose quo- 
tients may be easily shown to approximate to the sum of the 
two first roots a + 5. Again, form a series whose terms are 
AC — B?, BD — C’, CE — D*, &c., which is also a recurring 
series, whose successive quotients will approximate to the value 
of the greatest product a 6. Ina similar manner, we may de- 
duce from the primitive recurring series three other recurring 
series, the terms of the convergent series formed by whose quo- 
tients will form, in the first series, the sum a + 6 + ¢ of the 
three first roots; in the second, the sum of their products two 
or two, or ab + ac + bc; and in the third their continued 
product a 6c: and similarly for four or a greater number of 
roots. If, therefore, we suppose the first root a to be imaginary, 
the first series will give no result ; but the values of a + 6 and 


a Pe 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 349 


of a b, which are given by the two first recurring series de- 
rived from the primitive recurring series, will enable us to de- 
termine their separate values: in both cases the series of quo- 
tients is convergent. 

If the third root be real, the third series of derived quotients 
is convergent; if not, the fourth series will be so, and so on as 
far as we wish to proceed. 

These propositions have been merely announced by Fourier 
in his Introduction. _ The chapter of his work, which contains 
the demonstrations, has not yet been published. 

If the root of an equation be determined approximately, the 
equation may be depressed, and the general processes of solu- 
tion or of approximation may be applied to find the roots of 
the quotient of the division. Thus, in the equation 


x? — 3x + 20000001 = 0, 


one of the roots is very nearly equal to 1, if we divide the 
equation by x — 1, and neglect the small remainder. which re- 
sults from the division, we shall get the quotient 

xv —x—2=(«—-—1)(«#+2)=0, 
whose roots are 1 and — 2; or we may suppose one of the 
roots to be 1:0001, the second °9999, and the third — 2; or 
we may suppose two of the roots to be imaginary, namely, 
1 + -0001 “—1. All these roots are approximate values of 
the roots of the equation, which different processes, whether 
tentative or direct, may determine: and it is obvious that when 
two roots are equal, or nearly so, an inaccuracy of the approxi- 
mation to those roots which are employed in the depression of 
the primitive equation may convert real roots into imaginary, 
or conversely. Such consequences will never follow when the 
limits and nature of the roots are previously ascertained, and 
every root is determined independently of the rest; but it is 
not very easy to prevent their occurrence when methods of ap- 
proximation are applied without any previous inquiries into the 
nature and limits of the roots, though the resulting conversion 
of imaginary roots into real, and of real roots into imaginary, 
may not deprive them of the character of true approximations 
to the values of the roots which are required to be determined. 
. If the limits of the roots of an equation F x = 0 be assigned, 
and if the Newtonian method of approximation be applied con- 
tinually to one of these limits a, we should obtain, for the value 
of the root, the series * 


q'' gill 
a= a’ Fa + T.9 Fo - Tra 7g Fm + &e., 


* Lagrange, Résolution des Equations Numériques, Note xi. 


350 THIRD REPORT—1833. 


where 
Pap 
Fa 
iy rsee tie aM ini tes 
CEB) Sirarecin brs 
ql! mach as a! ud a 3 a! (F" a) 


~ aye Pare 
RE SY oy 
~~ Fay * Wap” 


This series was first assigned by Euler, and the observations 
which we have had occasion to make in the preceding pages 
upon linear approximations will at once explain the circum- 
stances under which it may be safely applied: it cannot be 
viewed, however, in any other light than as the analytical ex- 
pression for the result of the application of such linear approxi- 
mations, repeated as many times as there are terms of the series 
succeeding the first. 

The celebrated theorem of Lagrange, which is so extensively 
used in the solution of the transcendental equations which pre- 
sent themselves in physical and plane astronomy, will enable 
us to assign, likewise, a series for the Jeast root, or for any 
function of the least root of an equation in terms of its coeffi- 
cients. Mr. Murphy, in a very able memoir in the Transac- 
tions of the Philosophical Society of Cambridge for 1831, has 
shown the mode in which such series may be determined, by 
means of a very simple rule, which admits of very rapid and 
very extensive application. The rule is as follows: 

** To find the series for the least root of the equation ¢ (x) = 0, 
divide the equation by 2, and take the Napierian logarithm of 


. ‘ , : i evichinell 
the quotient which arises; then the coefficient of rg with its 


sign changed is the series which expresses the least root re- 
quired.” 
Thus, to find the series for the least root of the quadratic 
equation 
v’+tax+b=0, 
(a?+ax+b) 


find the coefficient, withits sign changed, of = in log = 


b 
or log (: eager =) which is 
a 
2 b3 
@ 


b b 4, 6. She SB. 1.10 
\et+ete +o-3-a tecera wt Sh, 


on 


=) 


REPORT ON CERTAIN BRANCHES OF ANALYSIS. 351 


and therefore identical with that which arises from the deve- 
2 
lopement of — 3 + (+ - b). If 6 be greater than me the 


roots of the equation are impossible, and the series becomes 
divergent, and gives no result. 

Any function / (x) of the least root of an equation ¢ (x) = 0 
may be found “ by subtracting from f (0) the coefficient of 


- in f’ (x) log = This more general theorem evidently 


includes the former. 
_ © The sum of any assigned number (m) of roots of the equa- 
tion ¢ (x) = 0 is equal to the coefficient, with its sign changed, 


yal is $ (2) ,, 
of 7 in log rae 

The expression for the sum of m roots of an equation which 
is thus obtained gives the arithmetical value of the sum of the 
m least roots. In estimating the order of magnitude of such 
roots no regard is paid to their signs of affection. 

Mr. Murphy has shown in what manner the same general 
proposition which is employed in the deduction of the results 
just given may be applied to the investigation of some of the 
most general theorems which have been employed in analysis 
for the developement and transformation of functions. Amongst 
many others the following very remarkable theorem seems to 
merit particular notice. 

If x,, 2%, Lg,» » Xm be the m least roots of the equation 


(a — a)” —hF (x) = 0, 


FS (41). + f (%2) + f (#3) + ++» fm) 


q@-) | FE (a 
SF ieheh barman eae 


Pree Ve. ea — Didar 


If in this very general theorem we make m = 1, it becomes 
the theorem of Lagrange; and if we make m equal to the di- 
mensions of the equation, or greater than any power of 2 in- 
volved in F (x), then it becomes the theorem which Cauchy 
has given, without demonstration, in the ninth volume of the 

new series of the Memoirs of the Institute, for the expression 
of the sum of the different values of f(«), when x is succes- 
sively replaced by every root of the equation. 

The preceding conclusions, so very remarkable for their 

great generality, and for the very simple means employed in 


then, 


+ &e. 


352 ‘| THIRD REPORT—1833. 


their derivation, will be sufficient to direct the attention of the 
reader to the other contents of this very original and valuable 
memoir. 

There are some other most important departments of the 
general theory of equations which it was my intention to have 
noticed, and without which no report upon the present state 
and recent progress of algebra can be said to be complete. 
Amongst these may be particularly mentioned the theory of 
elimination and the solution of simultaneous equations, and also 
the theory of the solution of literal and of implicit equations. 
The very undue length, however, to which this Report has 
already extended, and the arrangements which have been made 
connected with the publication of this volume, compel me, 
though most reluctantly, to omit them. I venture to indulge 
a hope, however, that I may be allowed upon some future oc- 
casion to add a short supplemental Report upon this extensive 
department of analysis, in which I may be enabled to supply 
some of the numerous deficiencies of the preceding sketch. 


— 
~ 


ERRATA IN THE FOREGOING REPORT. 
Page 197, line 21, dele not 


— 215, — 3, for T(r) Pr) read T(r) T (1 — 1) 
— 215, — 11 from the bottomj:gor a* 4+ 2+ #1 reada?+22+1 
— 221, — 15, forcos } 4 read cos—} 7 
226 a gatadie Tat sueadtdaage OF Pre pen ae 
= _— rom the bottom, for Jat Veto 
— 234, — 7, for (0) read (0)" 
— 240, — 18, for In this last case (a — 5)" read If we suppose to be 


an even whole number or a fraction in its lowest terms 
with an even numerator, then (a — b)” 

— 240, — 10 from the bottom, for fraction read function 

— 248, — 6, for diventical read diacritical 


— 251, — 1, for cos {2— 2+ +) read cos {2— Gt}, 
— 259, — 23, dele or the greatest of the two quantities # and 6 
th 261, - 7 14, for 4 read 4 


— 263, — 15, for Xi read X 
— 316, — 29, for «” —1=0 read v1 —1=0. 


en Lakad 
¥ 


~ Report of the Brit. Assoc, Val.2. p.3583. 


i 


WBrown, Jeulp ; 


nfo r gagea] 


TRANSACTIONS OF THE SECTIONS. 


$$. 


“1. MATHEMATICS AND PHYSICS. 


On the Compressibility of Water. By Professor Qirstep. 
(From a Letter to the Rev. WitL1AMWHEWELL, dated Copen- 
4 - hagen, June 18, 1833.) 


[With an Engraving. | 


“Were I not withheld by official duties, I should certainly not 
omit so excellent an opportunity of renewing the very interest- 
i ing and useful acquaintance 1 made during my last visit to 
_ England and Scotland, and of forming new ones with those 
_ distinguished scientific characters that I was not fortunate 
enough to meet with at that time, or such as have risen to emi- 
nence of late years. But though I must now forgo this ad- 
_ vantage, I will not let this opportunity pass without giving the 
illustrious assembly some mark of my high esteem, and of my 
_ desire to keep up the friendly intercourse which I have main- 
__ tained with the British philosophers since my acquaintance with 
your happy country. 
~“ You are, perhaps, aware that I have published several no- 
_ tices upon the compressibility of water, the first as early as 
1818, and the first description of the improved method in 1822. 
Since that time I have still gone on improving my methods, and 
am now preparing a paper on the subject for the Transactions 
of the Royal Society of Sciences at Copenhagen. I will endea- 
_ vour to give you a succinct account of my method and its re- 
_ sults. It has been found that the apparatus for compressing 
; _ Water, a description of which I published in 1822, can give very 
br _ aécurate results; so that the results it has given in the hands of 
_ philosophers in different countries, have agreed more than 
_ might have been expected. Next to the accuracy of the mea- 


x, 


= han ap however, one of the most important requisites ot 


__ such an apparatus is, that the experiments be performed with 
the greatest celerity possible. When the experiment is pro- 
a ted, the change of temperature produces great variations in 


the volume of the water, +h of the thermical measure. (1° cenr 
1833. 2a 


354 THIRD REPORT—1833. 


tigrade *) causing at high temperatures the volume of the water 
to vary more than the pressure of 3, 4, or even 5 atmospheres. 
“The improved apparatus is represented in the diagram fig. 1. 
Its principal parts are the same as in the earlier; in each of them, 
however, some change is introduced. A BC D isa strong 
glass cylinder, having at the top a cylinder E F G H, contain- 
ing a piston J mno, moved by a screw K K, as in the first 
apparatus; but the handle II is now arranged in such a man- 
ner that the screw can be turned without interruption; by this 
means the effect is accelerated, and subitaneous strokes avoided. 
The bottle ¢cc, with its capillary tube a a, is different from 
the earlier only so far, that the tube is not soldered to the bot- 
tle, but merely adjusted by grinding. This alteration is not 
necessary except when solid bodies are to be compressed. The 
scale ef gh is divided into parts of 3, inch. In order to ex- 


clude the water with which the large cylinder is filled, from com- 


munication with that of the bottle, the top of the tube aais 
covered with a small diving-bell, or rather diving-cap, pp p; 
whose conical shape has the advantage of preventing the water 
from reaching the top of the tube aa, even when the air is com- 
pressed to a tenth or twelfth part of its first volume. Its margin 
is loaded with a ring of lead or brass. cd is a glass tube with 
proper divisions, containing air, whose compression measures 
the pressure; its inferior part is loaded with some lead or a 
ring of brass. ¢wuz is a siphon; P, a vessel containing water ; 
zz are two buoys of cork for lifting up the bottle and the glass 
‘tube cd; sr is a tube of brass, which can be stopped by a 
screw. In the beginning and at the conclusion of the experi- 
ment it serves to introduce water into the space E F G H, or to 
get it out again. Before the experiments the calibre of the 
two tubes must be exactly ascertained, and the relative capaci- 
ties of the bottle and its capillary tube determined by the quan- 
tities of mercury they can admit. I have had some tubes.in 
which 35 of an inch (making one division) held only 2 mil- 
lionths of the capacity of the bottle, in others they have held 
more, in some even as much as 7 millionths. The capacity of 
the bottle was not less than 14 pound, often 2 pounds of mer- 
cury. It is next filled with water, which must be boiled in the 
bottle in order to expel the air, which might be suspected of 
having a great influence in these experiments, though Canton 


* The unit of thermical measurement is the distance between the freezing 
and the boiling point. I think that the most natural expression for the tem- 
peratures would be this unit and its fractions. Thus, the temperature 0°50 
would be the same as 50° centigrade, 19°30 the same as 1930° centigrade. 
I will mark this metrical measure by Th. If this innovation should not please, 
I wish that it might be suppressed, and centigrade degrees put in the place, 
which is an easy change. 


TRANSACTIONS OF THE SECTIONS. 855° 


has already observed that this is not the case. When the large 
cylinder is filled with water, the bottle is to be immersed in it. 
If the tube aa is full of water, a little of it must be expelled, 
which can be done by heating it gently with the hand, or better, 
by introducing a wire into it. As the bottle may be considered 
as a water thermometer, it is easy to ascertain whether it is in 
thermical equilibrium with the water in the cylinder. The air 
in the tube ¢ d must likewise be brought to the same tempera- 
ture with the water, before it is ultimately immersed. When 
the pumping cylinder shall be placed in its box, the piston 
must be at GH. If the large cylinder is full of water, part of 
it will be expelled through the siphon ¢uz. Now the piston is 
to he lifted up by means of the screw, whereby the pumping 
cylinder is filled with water. When this is done, the siphon is 
taken away, and the tube rs is stopped by a screw appertaining 
to it. The experiment is most conveniently performed by three 
persons ; one turning the screw, the second observing the height 
of the water in the tube a a, and the third observing the volume 
of the air in the tube cd: the last writes down the numbers ob- 
served. Now, the point where the water stands in the tube of 
the bottle is to be noted. The descending piston having re~ 
duced the volume of the air in the tube ¢ d to the point desired, 
the observer of it takes hold of the handle of the screw, and 
keeps the volume unchanged until the other observer has settled 
the point to which the water is brought down, and writes down 
the observation. When the piston is lifted up to its first place, 
the screw at r is to be opened, and the state of the water in the 
capillary tube again noted down. I commonly make ten or more 
such observations, one after another, which is performed in less 
than ten minutes when the operators are accustomed to work to- 
gether. An example will illustrate the use of the observations. 


Height of the Water in the Capillary Tube. 


Length of descent |Point to which the 
inthe capillary |pressure has driven 
tube. the water down. 


248-9 248-95 249-0 =| 50°35 1986 

_ 249-0 249-2 (249-4 50:20 1990 
249-4 249°7 250-0 49-90 199°8 
250°0 250°5 251-0 50°10 200°4 
251-0 Q51°5 252-0 49-90 201-6 
252-0 25265 2533 49°85 202°8 
253°3 2541 254°9 49-90 204-2 
254-9 2555 2561 | 49°70 205:8 
2561 25695 | 2578 49:95 207-0 
257-8 2584 259-0 49°80 208°6 
259.0 259°5 2600 | 49°70 209°8 


Mean of descents = 49:96. 


Before the pres- 
sure, 


‘When the pressure 


Mean. has ceased. 


356: ' THIRD REPORT—1833. 


The height of the mercury in the barometer, reduced to the 
freezing point, was at the same time 332°36 French lines. The 
volume of the air was in each experiment reduced to 5-264, or 
the pressure added to that of the atmosphere was 4°264 atmo- 
spheres. The pressure reduced to lines of mercury is thus, 
33236 x 4°264 = 1417°18; yet this reduction was not produced 
by the united pressure of the atmosphere and the piston alone, 
but was aided by a pressure of 40 lines of water, whose effect 
is equal to that of 2°94 lines of mercury, which is to be de- 
ducted, leaving then a pressure of 1414°24. Now, when a 
pressure of 1414-24 produces a descent of 49°96 parts, a pres- 
sure of 336 must produce a descent of nearly 11°87 parts. 
Each part makes in the instrument here employed 3-497 mil- 
lionths of the whole capacity. 11°87 x 3°497 gives ultimately 
41°51. The temperature of the water was at the beginning 
0:20 Th., at the end 0°2025 Th., by the thermometer. The 
water stood 10:1 parts, about 55 millionths, higher at the 
end of the experiments than at the beginning. This gives 
0°202 Th., which is as perfect an agreement as could be de- 
sired, the difference being only 00005 of the thermical mea- 
sure, or 0:09 degree of the scale of Fahrenheit. During the 
last three months I have not made use of the tube e d for mea- 
suring the compression of air, but I have employed a glass tube 
LMN (fig. 2.), whose shape is better seen in the diagram than 
it can be described. The capacity of the part above the line y, 
and that of the whole, are measured by weights of mercury. When 
the instrument is sunk in the water, the liquid mounts in the tube 
which has the scale O, whose parts are likewise measured by 
mercury: This has the double advantage of giving a more ac- 
curate measure, and of showing whether or not the volume of 
air has changed. In the series of experiments above mentioned 
this measure has been employed. By a considerable number - 
of experiments, I have found that the compressibility of water 
is not so great in high temperatures as in lower. Canton had 
already obtained this result, but some doubts might remain, be- 
cause his experiments were made by means much more trouble- 
some to make use of, and at a time when all instruments were 
less perfect. Here, as well as in the whole research into the 
compressibility of water, the new experiments prove the great 
skill and acute judgement of this distinguished philosopher. 
My experiments are much more numerous than his, and have 
been extended to a greater range of temperatures. Their re- 
sults may be expressed by supposing that the pressure of one 
atmosphere equivalent to 336 French lines’ height of meréury 
develops a heat 0:00025 Th. = 0:045° Fahr. In calculating 


TRANSACTIONS OF THE SECTIONS. 357 


this I have made use of the tables of Professor Stampfer at 
Vienna, who finds the highest contraction of water at 0°0375 
Th:, or 38°75° Fahr.* At this point the recession of water by 
the pressure of one atmosphere is 46°77 millionths. At 0:09125 
Th. the volume of the water augments 71°75 millionths, having 
its temperature augmented 0:01 Th. The heat developed by the 
1°75 
pressure thus augments its volume 0-00025 . st = 1:79 mil- 
lionth, or the recession is 46°77 — 1°79 = 44°98 millionths. 
Actual experiments have given it 44°89, or 0°09 millionth 


* PART OF STAMPFER’S TABLE, 


Temperatures ac- 
cording to the Volumes of the 
centigrade ther- water. 
mometer. 


an 1:000373 
1-000269 
1-000182 
1-000113 
1-000061 
1:000025 
1-000005 
1-000000 
1-000001 
1-000012 
1:000038 
1-000079 
1-000135 
1-000205 
1-000289 
sped 1-000387 
: 1:000497 
6. 1-000620 

1:000757 
1-000906 
1-001066 
1-001239 
1-001422 
1:001617 
1-001822 
1-002039 
1-002265 
i 1-002502 
os: 1:002749 
ee 1-003005 
Ht 1-003271 
1-003545 
1-003828 
1-004119 
1:004418° 


Differences. 


WOWIA CBO WOH SM r*~ 


358 THIRD REPORT—1833. 


greater. The coincidence is often less perfect. At0°1775 Th. 
the quantity calculated is 42°65, the quantity given by experi- 
ment 43°03, a difference of 0°38 millionth. The experiment 
mentioned above gave a recession = 41°51 at 0°20125 Th. 
(mean of the temperatures of the beginning and end of the se- 
ries), The calculation gives 41°63, or a difference of 0°12 mil- 
lionth. At 0°005 Th. the change of volume produced by one 
0-01 is 60°5 millionths, but inversely, as the water at low tem- 
perature loses in volume by augmented heat; thus an addition 
is to be made equal to + .0°00025 = 1:5. Now 46°77 + 1:5 
gives 48°27, experiment 48°02. At 0-019 the quantity calcu- 
lated is 47°72, that given by experiment 47°97. I have not yet 
finished the tedious discussion of all the experiments, but as 
far as I have proceeded the agreement of the hypothesis with 
facts is satisfactory. Messrs. Colladon and Sturm have in the 
calculation of their experiments introduced a correction founded 
upon the supposition that the glass of the bottle in which the 
water is compressed should suffer a compression so great as to 
have an influence upon the results. Their supposition is, that 
the diminution of volume produced by a pressure on all sides 
can be calculated by the change of length which takes place in 
a rod during longitudinal traction or pression. Thus, a rod of 
glass, lengthened by a traction equal to the weight of the atmo- 
sphere as much as 1°1 millionth, should by an equal pression on 
all sides lose 3°3 millionths, or, according to a calculation by the 
illustrious Poisson, 1°65 millionth. As the mathematical cal- 
culation here is founded upon physical ‘suppositions, it is not 
only allowable, but necessary, to try its results by experiment. 
Were the hypothesis of this calculation just, the result would 
be, that most of the solids were more compressible than mer- 
cury. For this purpose I have procured cylinders of glass, of 
lead, and of tin, which filled the greater part of a cylinder, to 
which a stopple of glass, perforated by a capillary tube, was 
adjusted by grinding. I have not yet exactly discussed all the 
experiments on this subject, but the numbers obtained are such 
as to show that the results are widely different from those cal- 
culated after the supposition above mentioned. The quantity 
assigned by this calculation to the glass is very small indeed, 
yet the experiment gives it much less. Lead, which extends, 
according to Tredgold, 20°45millionths by a weight equal to that 
of the atmosphere, and thus much more by the pressure on all 
sides, does not change one millionth. Tin is not more com- 
pressible. The inverse experiment is, perhaps, still more 
striking. I published it some years ago: however, as I have now 


TRANSACTIONS OF THE SECTIONS. 359 


repeated these experiments, and as they appear hitherto not to 
have satisfied philosophers, I shall here mention, that in all my 
experiments upon the subject, I have invariably found that the 
recession of the water in the capillary tube is about 1-5 millionth 
greater in bottles of lead or tin than in those of glass. Sup- 
posing the compressibility of the solid bodies to be so small 
that it cannot be observed in those experiments, yet the heat 
developed by the compression, feeble as it is, produces a small 
augmentation of the recession of the water in the capillary tube. 
If the dilatation of a rod of glass by 1 Th. is 00009, its cubical 
dilatation is 0°0027, and the dilatation by an increase of 0:00025 
is 0000000675, or nearly 7 ten millionths. The dilatation of lead 
is about 3 times greater, and the bottle containing it must get an 
increase of 0:00000225, which exceeds the former by more than 
1:5 millionth. The dilatation of tin should give only one mil- 
lionth more than glass, but it seems to give alittle more, yet 
the quantity is not great. After all this, I think that the true 
compressibility of water is about 46°1 millionths, and that the 
apparent compressibility depends upon the effect of the heat 
developed by the compression, by which the liquid and the bot~ 
tle are dilated. 

“« My continued experiments have confirmed my earlier re- 
sult, that the differences of volume in the compressed water are 
proportionate to the compressing power. Ido not know if the 
method I have made use of to try the effects of high compression 
has been published in England. These experiments cannot be 
made in a cylinder of glass; one of metal is required. As, in 
this case, the opacity prevents direct observations being made, 
an index, nearly like that in Six’s register-thermometer, is 
placed in the capillary tube of the bottle. This tube is dilated 
a little at the top, so as to form a minute funnel. Some drops 
of mercury are poured into it, which being pressed, pushes the 
index forward ; thus the recession may be seen when the bottle 
is taken out of the large cylinder. The compression of the air 
is measured in another way: a bent tube, of the form shown in 
fig. 3, is fixed in a glass vessel FG HI containing mercury, and 
exposed to the pressure together with the bottle. ‘The pressure 
of the piston upon the water in the cylinder is communicated to 
the mercury, and pushes it into the wide part of the tube, as far 
as the resistance of the air will permit. ‘The weight of the mer- 
cury driven into the wide part ABCD, together with that 
which has filled D E, and which may be computed, compared 
with the weight of mercury which the whole tube can admit, 
gives the volume of the air compressed. By this kind of ex- 
periment I have found that the decrease of volume produced by 


360 THIRD REPORT—1833. 


pressure preserves the same proportion to the pressing power 
as far as the pressure of 65 atmospheres, and probably much 
further ; but how far, I have not hitherto been able to try, my 
apparatus not having resisted a greater pressure. 

**T have thus given you a short abstract of my researches into 
the compressibility of water. They may be considered as a 
continuation of those of Canton. I should feel much flattered 
if they should obtain the approbation of the philosophers of the 
country where the first good experiments upon the subject have 
been made.” 


On some Results of the View of a Characteristic Function in 
Optics. By Witu1aM R. Hamitton, M.R.I.A., Royal As- 
tronomer of Ireland. 


The author gave a statement of some optical results, deduced 
from the view which he had explained in the preceding year at 
Oxford. 

His general method, for the study of optical systems, consists 
in expressing the properties of any optical combination by the 
form of ONE CHARACTERISTIC FUNCTION, one central or radical 
relation. In order to investigate the properties of the systems 
of rays, produced by any object-glass, or atmosphere, or other 
optical instrument, or combination of surfaces and media, ordi- 
nary or extraordinary, he has proposed, as a fundamental pro- 
blem, to express for any such combination, the laws of depend- 
ence of the final and initial directions of a linear path of light 
on the final and initial positions or points, and on the colour. 
And the solution which he has offered for this fundamental pro- 
blem consists, Ist, in reducing by uniform methods (analogous 
to the methods of discussing the equation of a curve or surface,) 
these several laws of dependence (of the four extreme angles of 
direction of a curved or polygon ray on the six extreme coordi- 
nates and on the colour,) to that one law, different for different 
combinations, according to which his one characteristic function 
depends on the same seven variables. And 2ndly, in establish- 
ing uniform processes for the research of the form of this func- 
tion, namely, the action or time of propagation of the light, for 
any proposed combination. 

For example, in the case ofa single plane mirror, supposed to 
coincide with the plane of x y, we may propose to determine the 
laws of the two extreme directions of the linear path by which 
light goes to an eye (« yz) from an object (a! y’ 2'), or (ex- 
pressing the same thing more fully,) to determine the final co- 
sines «#8 y, and the initial cosines «’ 6! y', of the inclinations of 


TRANSACTIONS OF THE SECTIONS. 361 


this bent path to the positive semiaxes of coordinates, as func- 
tions of x y 2, 2! y'2', that is, of the six extreme coordinates 
themselves, the colour being here indifferent. And Mr. Ha- 
milton’s general solution, for this and for all other questions 
respecting combinations of ordinary reflectors,—-a solution 
which is itself a particular case of a more general result, extend- 
ing to all optical combinations,—is expressed by the following 


equations ; 
DSi WAN 4 Mahe hid 
Ses ied 8 nde SM uci Be? 
jes Ni tdsnichts risa a) 
8a” Sy” Y he] 


the characteristic function V representing, in all questions re- 
specting combinations of reflectors, the length of the bent path 
of the light, and being for the present mirror of the form 


VaV@-ty-oFtetee 


but being different in other cases. Thus, for a reflecting sphere, 
or for a Newtonian telescope, the length of a bent path of light 
would depend differently on the extreme points of that path, 
and we should have a different form for the characteristic func- 
tion V ; but by substituting this new form in the equations (1.), 
we should still deduce the connected forms of the six dérection- 
functions or direction-cosines, «8 y, «’ B! y', and so might deduce 
all the other properties of the telescope; at least, all the pro- 
perties connected with its effects upon systems of rays. 
_ It may be perceived from what has been said, that Mr. Hamil- 
ton divides mathematical optics into two principal parts: one 
part proposing to find in every particular case the form of the 
characteristic function V, and the other part proposing to use 
it: as in algebraical geometry, it is one class of problems to 
determine the equations of curves or surfaces which satisfy as- 
signed conditions ; and it is another class of problems to discuss 
these equations when determined.. The investigations which 
, the author has printed in the fifteenth, sixteenth, and seven- 
teenth volumes of the Transactions of the Royal Irish Academy, 
contain examples of both these inquiries, although they relate 
chiefly to the second part, or second class of problems, namely, to 
the using of his function, supposed found. He has endeavoured 
to establish, for such using,a system of general formule, and has 
leduced many general consequences and properties of optical 
systems, independent of the particular shapes and positions 
and other peculiarities of the surfaces and media of any optical 


$62 THIRD REPORT—1833. 


combination. A few results less general than these, and yet 
themselves extensive, may not improperly, perhaps, be men- 
tioned here. 
When we wish to study the properties of any object-glass, 
or eye-glass, or other instrument én vacuo, symmetric in all re- 
spects, about one axis of revolution, we may take this for the 
axis of x, and we shall still have the equations (1.), the charac- 
teristic function V being now a function of the five quantities, 
e+y, cr + yy, 2? + y%, z,3', involving also, in general, 
the colour, and having its form determined by the properties of 
the instrument of revolution. Reciprocally, these properties 
of the instrument are included in the form of the characteristic 
function V, or in the form of this other connected function, 


Ta=art+Ppytyzez—aa’— ply —y2—V, (3) 


which may be considered as depending on only three inde- 
pendent variables besides the colour; namely, on the inclina- 
tions of the final-and initial portions of a luminous-path to each 
other and to the axis of the instrument. Algebraically, T is in 
general a function of the colour and of the three quantities, 
a? + 6, ac! + BP’, a? + 6%; and it may usually (though not 
in every case) be developed according to ascending powers, po- 
sitive and integer, of these three latter quantities, which in 
most applications are small, of the order of the squares of the 
inclinations. We may therefore in most cases confine our- 
selves to an approximate expression of the form 


T=T + T° + To, (4.) 


in which T is independent of the inclinations : T® is small of 
the second order, if those inclinations be small, and is of the 
form 


T® =P (a? h 6?) ae P, C a! + B p') 4+ Pp! (al? a 6!) ; (5.) 
and T“ is small of the fourth order, and is of the form 


TO=Q (2+ °° +Q, (a? +6) (wa! +B) + QUa?-+ 6%) (@P+ a 6.) 
ie Q, (a a! +6 py? + Q! (a a! + BB’) (ce? + 6'?) +. Q' (c!? 48 [ota \e 


the nine coefficients, P P, P’Q Q,Q'Q, Q’, Q", being either 
constant, or at least only functions of the colour. The optical 
properties of the instrument, to a great degree of approxima- 
tion, depend usually on these nine coefficients and on their 
chromatic variations, because the function 'T may in most cases 
be very approximately expressed by them, and because the 
fundamental equations (1.) may rigorously be thus transformed ; 


TRANSACTIONS OF THE SECTIONS. 363 


‘ ¢T B / en 
soa = Foy Se = ap! 
Y (i (7.) 
x! nay ATi ay Blige AE 
sing ts alias Bede irda 3B" 


The first three coefficients, P P, P', which enter by (5.) into 
the expression of the term T®, are those on which the focal 
lengths, the magnifying powers, and the chromatic aberrations 
depend: the spherical aberrations, whether for direct or in- 
clined rays, from a near or distant object, at either side of the 
instrument (but not too far from the axis), depend on the six 
other coefficients, Q Q, Q!’ Q, Q', Q", in the expression of 
the term TIT. Here, then, we have already a new and remark- 
able property of object-glasses, and eye-glasses, and other 
optical instruments of revolution; namely, that all the circum- 
stances of their spherical aberrations, however varied by di- 
stance or inclination, depend (usually) on the values of six 
RADICAL CONSTANTS OF ABERRATION, and may be deduced from 
these six numbers by uniform and general processes. And as, 
by employing general symbols to denote the constant co- 
efficients or elements of an elliptic orbit, it is possible to deduce 
results extending to all such orbits, which can afterwards be 
particularised for each; so, by employing general symbols for 
the six constants of spherical aberration, suggested by the 
foregoing theory, it is possible to deduce general results re- 
specting the aberrational properties of optical instruments of 
revolution, and to combine these afterwards with the pecu- 
liarities of each particular instrument by substituting the nu- 
merical values of its own particular constants. ‘The author 
proceeds to mention some of the general consequences to which 
this view has conducted him, respecting the aberrational pro- 
perties of optical instruments of this kind. 

When a luminous point is placed on the axis of an object- 
glass, or eye-glass, or other instrument of revolution, and when 
its rays are not refracted or reflected so as to converge exactly 
to, or diverge exactly from, one common focus, they become, 
as it is well known, all tangents to one caustic surface of revo- 
lution, and they all intersect the axis, at least when they are 
prolonged, if necessary, behind the instrument. But if the 
uminous point be anywhere out of the axis, the arrange- 
ment of the final rays becomes less simple than before. They 
are not now all tangents to the meridian of a surface of revolu- 
tion, nor do they all intersect the axis of the instrument ; they 
become, by another known theorem, the tangents to two caustie 


364 THIRD REPORT—1833. 


surfaces, and to two sets of caustic curves, and compose two 
series of developable pencils, or ray surfaces ; so that each ray 
of the final system may be considered as having, in general, two 
foci, or points of intersection with other rays, indefinitely near. 
The theorem here alluded to, namely, that of the general exist- 
ence of two foci for each ray of a system proceeding from any 
surface according to any law, was first discovered by Malus. 
Mr. Hamilton also obtained it independently, but later, in 1823. 
It appears to be, as yet, but little known; but it is, he thinks, 
essential to a correct view of the arrangement of rays in space, 
for which the analogy of rays in a plane seems quite inade- 
quate. Combining this theorem of the two foci with his view of 
the characteristic function, and of the six constants of spherical 
aberration, for the final system produced by oblique inci- 
dence on an instrument of revolution, the author has found 
that the two foci of a ray of this final system do not in 
general close up into one, except for TWO PRINCIPAL RAYS, 
having each its own PRINCIPAL Focus. The interval between 
the two foci of any other ray is proportional, very nearly, to the 
product of the sines of its inclinations to the two principal rays; 
and the tangent planes of the two developable pencils, passing 
through any variable ray, bisect (very nearly) the two pairs of 
supplemental dihedrate angles formed by the two planes which 
contain this variable ray and are parallel to the two principal 
rays; in such a manner that all the rays of any developable 
pencil of one set have (very nearly) one common swm, and all the 
rays of any developable pencil of the other set have (very 
nearly) one common difference, of inclinations to the same two 
principal rays, or axes of the final system. These latter axes 
always intersect each other, and their plane is either the 
diametral plane of the instrument (containing the luminous 
point or focus of incident rays), or a plane perpendicular 
to that diametral plane, according to the sign of a cer- 
tain quantity, which vanishes when the two axes happen to 
coincide in one principal ray, round which the whole final 
system has then a very perfect symmetry ; and, in general, the 
angle of the two principal rays, whether in or out of the dia- 
metral plane of the instrument, is bisected (very nearly) by a 
certain intermediate ray in that plane, which may be called the 
CENTRAL RAY of the system, because the other final rays are 
disposed about it with a certain symmetry of arrangement, 
less perfect than the symmetry about an axis of revolution, but 
resembling that of the normals to an ellipsoid about one of its 
three axes, when unequal; and accordingly the author finds 
that the final rays from an instrument of revolution (when the 


- 


TRANSACTIONS OF THE SECTIONS. 865- 


incident rays are oblique) are very nearly normals to a portion of 
such an ellipsoid, having the central ray for one of its three un- 
equal axes, and having the two principal rays for its two umbili- 
cal normals, at two out of the four points where the ellipsoid has 
complete contact of the second order with an osculating sphere. 
The centres of the two osculating spheres at these two points 
are the two principal foci of the system; and the centres of the 
two extreme osculating spheres at any other point of the ellipsoid 
are the two foci of the corresponding ray, or the points at which | 
that ray touches the two caustic surfaces. These latter sur- 
faces are, in the present approximation, the surfaces of centres 
of curvature of the ellipsoid: they have a curve of intersection 
with each other, which contains the two principal foci; every 
point upon the curve, except these two, being the first focus of 
one ray and the second focus of another. A plane may be 
drawn perpendicular to the central ray, and passing through 
the two principal foci; and this plane will cut the two caustic 
surfaces in sections which compose a kind of little lozenge, con- 
sisting (very nearly) of two curvilinear equilateral triangles, 
having the principal foci for two common corners: the quadra- 
ture of these curvilinear triangles, and of the other sections 
of the caustic surfaces, depending on elliptic integrals. In all 
the foregoing remarks, it is supposed, for greater generality, 
that the aberrations do not vanish with the obliquity of the 
incident rays ; but when the instrument is aplanatic for direct 
incident rays, it is easy to apply the same theory of the charac- 
teristic function and the six radical constants of aberration, 
and to determine, for this particular case, the components of 


spherical aberration which arise from obliquity only. 


' This theory of the aberrations of oblique rays, for an optical 
instrument of revolution, may admit of practical applications. 
For the mathematical symmetry of arrangement of the final 
rays about the central ray of their system, and the intensity of 
the two principal foci, may perhaps affect our sight, and have 
some appreciable influence on the practical performance of an 
instrument ; but of this Mr. Hamilton speaks with diffidence, 
because experiments directed expressly to the question appear 


to be required for its decision. If the mathematical properties 


which he has determined by theory in the, arrangement and 


aberrations of a system, shall be found in practice to have any 
sensible influence on the phenomena of oblique vision, it will 


become necessary to alter some of the received rules for the 


-tonstruction of telescopes and microscopes; or, at least, it will 
be possible to improve those rules by following the indications 


366. THIRD REPORT—1833. 


of this theory. A new track seems to be opened thus to ma- 
thematical and practical opticians. 

The principle of the characteristic function, from which have 
been deduced the foregoing results, among others not yet pub- 
lished, respecting optical instruments of revolution, may be 
applied to every part of mathematical and perhaps of physical 
optics ; and an analogous function and method may be intro- 
duced in other sciences, especially in dynamical astronomy*. 
But the author confines himself to mentioning the application 
which he has made of the principle to the study of the laws of 
extraordinary refraction in the crystals called biaxal. The 
general laws of reflection and refraction, ordinary and extraor- 
dinary, at any point of any surface, are expressed by his 
function as follows, when the normal to the reflecting or 
refracting surface at the point of incidence is taken for the 


axis of 2: a 

8 tv 

Fp Age AO: (8.) 
and in the language of the undulatory theory they may be 
enunciated by saying, that if the normal slowness of propa- 
gation of a luminous wave, at any point of incidence on any 
reflecting or refracting surface, be decomposed in any direction 
parallel to this surface at this point, the component of normal 
slowness is not altered by reflection or refraction. Tn the case 
of ordinary refraction, this comes to saying, that if on the in- 
cident ray prolonged, and on the refracted ray, we measure 
from the point of incidence lengths represented by the indices 
of the first and second media, those lengths will have one 
common projection on the refracting surface or on its tangent 
plane ; which is a form for the law of Snellius. For extra- 
ordinary refraction, we must in general construct the normal 
slowness of a wave by a variable length not always coinciding 
with the ray; but the two lengths thus substituted for the two 
successive indices will still have one common projection on the 
refracting face of the crystal, if plane, or on its tangent plane, if 
it be curved. If now we seek the locus of the end of the line, 
which represents in length and direction the normal slowness 
of a wave, for all possible directions of this slowness, we get for 
ordinary media a sphere, but for extraordinary media (on 
Fresnel’s principles) a certain double surface, which is not the 


* See the Dublin University Review for October 1833... Mr. Hamilton has 
since developed the dynamical application of his principle, in an essay On a 
General Method in Dynamics, which has been presented to the Royal Society, 
and ordered to appear in the Philosophical Transactions for 1834. 


4 


TRANSACTIONS OF THE SECTIONS. 367, 


same as Fresnel’s curved wave-surface, propagated in all di- 
rections from a point, but is connected therewith by several 
remarkable relations of reciprocity, and may be called the 
surface of components, since its coordinates are themselves the 
components of normal slowness of propagation. They are 
equal to the partial differential coefficients of the first order 
of the author’s characteristic function V, and are connected 
by a partial differential equation of the form 


fV 8V BV 
0=a(5, ee bar ge (9.). 


which may be regarded as the equation of the surface. And 
the general equations of reflection or refraction (8.), when put 
under the form 
Pe Patan yee OMe We gly ' 
Ha ee Gels ta Sun te ee 


express that the corresponding points on the two surfaces of 
components, before and after any reflection or refraction, or- 
dinary or extraordinary, are on one common ordinate to the 
reflecting or refracting surface, or to its tangent plane ; which 
gives a new and general construction for the direction of a- 
reflected or refracted wave, and therefore for that of a re- 
flected or refracted ray, simpler in many cases than the con- 
struction proposed by Huygens. Thus, if it were required to 
determine by this new construction the direction and the un- 
dulatory velocity of an extraordinary ray, refracted in Iceland 
spar, being given the direction of the incident ray in air, we 
should have to construct first the two successive surfaces.of 
components, which would be here a sphere for the air, and a 
spheroid (not the Huygenian) for the crystal, the common 
centre of both being at the point of incidence; and then, after 
determining the point of the hemispheroid within the crystal, 
which is on the same ordinate to the refracting face as the 
point where the incident ray prolonged meets its own interior 
hemisphere, we should only have to draw a tangent plane to 
the spheroid at the point thus determined, and to let fall a per- 
pendicular on this plane from the point of incidence ; for this 
perpendicular is, in length and direction, the radius vector of 
the Huygenian spheroid, and therefore represents the undu- 
latory velocity and the direction of the extraordinary ray. And 
other more complicated cases may be treated in a similar 
manner, either by using a construction of this kind, or by the 
Man formule derived from the characteristic function. 
hen the author proceeded to apply this general method to 


368 _ THIRD REPORT—1833. 


Fresnel’s principles respecting biaxal crystals, he arrived at the 
curious result that the surface of components, in such a crystal, 
has not at every point a determined tangent plane, but that at 
each of four cusps, opposite, two by two, it is touched by an 
infinite number of such planes, or by a tangent cone; and 
hence he immediately concluded, by his general method, that 
if a ray in air fall so upon a biaxal crystal as to make the point 
upon the air-sphere correspond (by the rule already explained) 
to one of those cusps on the surface of components of normal 
slowness in the crystal, his construction would give no unique 
refracted ray, nor even a pair or other finite number of such 
rays. within the crystal, but an infinite number of refracted 
rays, namely, all the perpendiculars which can be let fall from 
the point of incidence on the tangent cone at the cusp. The 
author saw also that these rays must terminate in some curve 
of plane contact on Fresnel’s double wave, in the whole extent 
of which curve the wave must be touched by one plane, and 
that there must be four such curves, which he afterwards found 
to be circles; a curious property of this wave, which Fresnel 
himself had not noticed. But the most remarkable part of this 
result was, the new and delicate experimental test which it 
offered for Fresnel’s principles, since the INTERNAL CONICAL 
REFRACTION which it indicated, for certain cases of incidence 
on a biaxal crystal, had not only not been hitherto observed, 
but seemed contrary to all former analogies of observation; so 
that if this theoretical consequence of Fresnel’s principles, 
which he had not himself perceived, should be verified by 
subsequent experiment, the principles would receive a new 
and striking confirmation; and if, on the contrary, after all 
due care employed in experiments directed expressly to the 
question, the small but finite conical dispersion in biaxal 
crystals, which the author had thus theoretically concluded, 
should not be found in fact to take place, the principles them- 
selves would require to be abandoned or modified. Professor 
Lloyd was applied to by the author to undertake this expe- 
rimental inquiry. After some unsuccessful trials with crystals 
of insufficient size and purity, he obtained a fine piece of arra- 
gonite from Mr. Dollond, and at length completely succeeded in 
exhibiting the phenomenon which Mr. Hamilton had expected. 
The rays of the internal cone emerged, as they ought, in a cy- 
linder from the second face of the crystal; and the size of this 
nearly circular cylinder, though small, was decidedly perceptible, 
so that with solar light it threw on silver paper a little luminous 
ring, which seemed to remain the same at different distances 
of the paper from the arragonite. Professor Lloyd describes 


TRANSACTIONS OF THE SECTIONS. 369 


‘the appearance as very beautiful when he employed a lamp, 
and received the emergent rays on a lens: he seemed to see 
the two points of light, which the double refraction usually 
produced, spread out on a sudden, when the critical incidence 
was obtained, into a ring of gold viewed on a dark ground. 
His account is contained (with Professor Hamilton’s theoretical 
investigation) in the First Part of the seventeenth volume of 
the Transactions of the Royal Irish Academy; a shorter state- 
ment was also published in the numbers of the London and 
Edinburgh Philosophical Magazine for the months of February 
and March 1833. : 
_» From the connexion of the surface of components with the 
wave-surface propagated from a point, the author saw that the 
existence of four conoidal cusps on the one surface in Fresnel’s 
theory involved the existence of four such cusps upon the 
other, namely, at the points of intersection of Fresnel’s circle 
and ellipse in the plane of the extreme axes of elasticity: and 
thus he was led to expect an EXTERNAL CONICAL REFRACTION, 
corresponding to the internal incidence of a cusp-ray when 
emerging into air from a crystal with two axes. On this point 
also he requested Professor Lloyd to undertake a series of ex- 
periments; and on this point also (indeed, somewhat sooner 
than on the other,) he obtained a complete verification. His 
experimental determinations of the size and position of this 
emergent cone, as of the former emergent cylinder, and of the 
laws of polarization in each, for the same large piece of arra- 
gonite, agreed with the theoretical results deduced from the 
rinciples of Fresnel by the method of the Characteristic 
Function. 
Although this method appears likely to be adopted by 
analysts at some future time in the researches of theoretical 
optics, the author does not pretend that its results cannot be 
obtained in other ways; and with respect to the two kinds of 
conical refraction, in particular, Mr. MacCullagh (F.T.C.D.) 
has published, in the London and Edinburgh Philosophical 
Magazine for the months of August and September in the 
‘present year, an elegant geometrical investigation, together 
‘with some account of the progress of his thoughts upon the 
‘subject. The surface which Professor Hamilton has called the 
surface of components, (of normal slowness of propagation,) and 
to which he was conducted some years ago, as constructing a 
fundamental equation between .the partial differential coefficients 
of his Characteristic Function, occurred to Mr. MacCullagh 
-also, as he has informed the author, independently from consi- 
. siete of a geometrical kind. The same important surface 
. 2B . 


370 THIRD REPORT—1833. 


presented itself to M. Cauchy, likewise, in his dynamical re- 
searches respecting a system of attracting or repelling points. 


On Conical Refraction. By the Rev. H. Luoyn, Professor 
of Natural and Experimental Philosophy in the University 
of Dublin. 


Professor Lloyd gave a brief account of the experiments by 
which he established the existence of conical refraction in 
biaxal crystals, in conformity with the theoretical anticipations 
of Professor Hamilton. 

The substance employed in these experiments was arrago- 
nite, which was selected chiefly on account of the magnitude 
of its biaxal energy. The specimen was one of remarkable 
purity, procured by Mr. Dollond. Its thickness was °49 of an 
inch, and its parallel faces were perpendicular to the line 
bisecting the optic axes, being cleavage planes of the crystal. 

The first case of conical refraction examined by the author 
was that called by Professsor Hamilton external conical re- 
Jraction. It was expected to take place when a single ray 
passes within the crystal in the direction of the line connecting 
two opposite cusps on the wave-surface. When this is the 
case, Professor Hamilton has shown that there should be a 
cone of rays without, the magnitude of which will depend on 
the biaxal energy of the crystal. In the case of arragonite, the 
angle of the cone, calculated from the elements of the crystal 
as determined by M. Rudberg, amounts to 3° very nearly. 

A thin metallic plate, perforated with a minute aperture, 
was placed on each face of the crystal, and these were adjusted 
in such a manner that the line connecting the apertures should 
coincide nearly with one of the optic axes. The flame of a 
lamp was then brought near the first surface of the crystal, 
and in such a position that the central part of the beam con- 
verging to the aperture should have an incidence of between 
15° and 16°. When the adjustment was completed, there 
appeared, on looking through the aperture on the second sur- 
face, a brilliant luminous circle with a small dark space around 
its centre; and in this central dark space were two bright 
points, separated by a well defined dark line. When the plate 
on the second surface was slightly shifted, so that the line 
connecting the two apertures no longer coincided accurately 
with the line joining the cusps on the wave-surface, the phzeno- 
mena rapidly changed, and ultimately resolved themselves into 
the two pencils into which a single ray is divided under ordi- 
nary circumstances. : 


TRANSACTIONS OF THE SECTIONS. 371 


The incident converging cone was sometimes formed by a 
lens of short focus, placed at the distance of its focal length 
from the surface. In this case the lamp was removed to a con- 
siderable distance, and the plate on the first surface dispensed 
with. The same experiment was repeated with the sun’s light, 
instead of that of a lamp, and the emergent cone of rays thus 
formed was of sufficient intensity to be reflected from a screen 
at some distance. In this manner the section of the cone was 
observed at various distances from its summit. 

When these phenomena were examined in detail, and com- 
pared with the results of theory, they appeared to differ in two 
important particulars. In the first place, the observed cone 
was very nearly a solid cone of rays, while that of theory was 
- but a conical surface. Secondly, the two cones differed widely 
in magnitude, the angle of the experimental cone being nearly, 
double of that of the theoretical one. This discordance between 
the results of experiment and those of theory, the author con- 
ceived to arise from the rays which were inclined to the optic 
axis at small angles, and which were transmitted through the 
aperture on the second surface in consequence of its sensible 
magnitude.. To examine this point he proceeded in the next 
place to try the effects of apertures of various dimensions. 
The effects of these variations in the resulting phenomena cor- 
responded exactly with his preconceived views. ‘The rays: 
which in the first experiments filled the whole of the conical 
space, parted in the centre, when the aperture was much di- 
minished ; and the section of the cone, instead of a complete 
luminous circle, was reduced to a luminous annulus, whose 
breadth diminished with the aperture. Simple theoretical con- 
siderations showed that the angle of the true cone in this case 
must be, very nearly, half the sum of the angles of the observed 
interior and exterior cones; and when this correction was ap- 
plied to the measurements, the resulting angle agreed, as nearly 
as could be expected, with that deduced from theory. 

A remarkable variation of the phenomenon was obtained by 
substituting a narrow linear aperture for the small circular one, 
in the plate next the lamp in the first-mentioned mode of per- 
forming the experiment; and by adjusting it so that the plane 
passing through it and the aperture on the second surface 
should coincide with the plane of the optic axes. In this case, 
according to the received theory, all the rays transmitted 
through the two apertures should be refracted doubly in the 
plane of the optic axes, so that no part of the line ‘should ap- 
pear enlarged in breadth on looking through the aperturé on 
the second surface. But if Professor Hamilton’s conclusion 

2B2 


872 THIRD REPORT—1833. 


be physically exact, the ray which proceeds in the direction of 
the line joining two opposite cusps on the wave-surface should 
be refracted in every plane. This was accordingly found to be 
the case. In the neighbourhood of each of the optic axes the 
luminous line swelled out on either side of the plane of the 
axes in an oval curve. This curve is the conchoid of Nico- 
medes, whose asymptot is the line on the first surface; and 
its variations of form, as the plane passing through the two 
apertures deviated from the plane of the optic axes, were highly 
curious and remarkable. 

_ Examining the state of polarization of the rays composing 
the emergent cone, Mr. Lloyd discovered that they observed 
the following law, namely, that “the angle between the planes 
of polarization of any two rays of the cone is half the angle 
contained by the planes passing through the rays themselves 
and its axis.” This law, it is easy to show, is in perfect ac- 
cordance with theory. 

_ Internal conical refraction should take place, according to 
Professor Hamilton, when a single ray has been incident ex- 
ternally upon a biaxal crystal in such a manner that one of the 
refracted rays may coincide with an optic axis. The ray, in 
this case, ought to be divided into a cone of rays within the 
crystal, the angle of which in the case of arragonite is 1° 55’. 
The rays forming this cone will be refracted at the second sur- 
face of the crystal in directions parallel to the ray incident on 
the first; so that they will form a small cylinder of rays in air, 
whose base is the section of the cone formed by the surface of 
emergence. . 

. The minuteness of this phenomenon, and the perfect accu- 
racy required in the incidence, render it much more difficult of 
detection than the former. A very fine ray of light proceeding 
from a distant lamp was suffered to fall upon the crystal, and 
the position of the latter altered with extreme slowness, so as 
to change the incidence very gradually. When the required 
position was attained, the two rays suddenly spread into a con- 
tinuous circle, whose diameter was apparently equal to their 
former distance. The same experiment was repeated with the 
sun’s light, and the emergent cylinder received on a small 
screen of paper at various distances from the crystal. No sen- 
sible enlargement of the section was visible on increasing the 
distance. 

The magnitude of the angle of the cone of rays within the 
crystal was ascertained experimentally, and agreed within very 
narrow limits with that deduced from theory. The rays com- 
posing the cone were all polarized in different planes, and the 


TRANSACTIONS OF THE SECTIONS. 373 


law connecting the planes of polarization was the same as that 
already found in the case of external conical refraction*. 


‘On the Absorption of Light by coloured Media, viewed in con- 
‘nexion with the undulatory Theory. By Sir Joun F. W. 
Herscuet, F.R.S. 


The absorption of light by coloured media having been of 
late regarded as offering peculiar difficulties when attempted 
to be reconciled with the undulatory theory, the object of this 
paper is to point out considerations which appear capable of 
reconciling most of the phenomena with that theory, at least 
in so far as to show that they involve nothing contrary to sound 
dynamical principles and obvious analogies when regarded as 
particular cases of the general doctrine of undulatory motion. 

_ The extinction of light in general is first considered. In the 
corpuscular hypothesis, light,to be extinguished, must be anni- 
hilated, or transformed either into caloric or some other form 
of matter. On the undulatory hypothesis, the extinction of 
light resolves itself into the more general and highly interest- 
ing dynamical question, ‘‘ What becomes of Motion?” 'This 
question is therefore considered, and by tracing the progress 
of an undulation through a body imperfectly elastic or a system 
of bodies placed in communication, it is shown that a continual 
subdivision of the original undulation, and a perpetual internal 
reflection of the subdivided portions, will speedily have the 
effect of agitating every molecule of the system at one and the 
same instant, with vibratory motions, in every possible phase 
and direction. The dynamical state of a molecule so agitated 
is identical with a state of perfect rest, so that the general fact 
of the extinction or absorption of light within a medium imper- 
fectly transparent is so far from being repugnant to dynamical 
principles, that it is one of their immediate and most natural 
consequences ; imperfect transparency, in this view of the sub- 
ject, consisting in the juxtaposition of parts unequally elastic, 
or of portions of the etherial medium wnequally loaded, or re- 
' strained from their natural aptitude to motion by the gross par- 
ticles of material bodies with which they are connected. 
- This connexion of the etherial particles with the grosser 
molecules which constitute our solids, liquids, and gases, may 
be considered as giving rise to compound vibrating systems, 
each of which, it is easy to conceive, may have a greater apti- 


“A full account of these experiments is given in a memoir read before the 
Royal Arish Academy, on the 28th January 1833, and ordered to be pub- - 
lished in the 17th volume of its Transactions. 


374 THIRD REPORT—1833. 


tude to propagate vibrations of some determinate degree of 
frequency, than of a greater or less degree, as being in unison, 
or an approach to unison, with that in which they themselves 
would vibrate if existing alone, and agitated by an external im- 
pulse. But this greater disposition to propagate some vibra- 
tions does not render them ¢ncapable of transmitting others. 
In illustration of this position Sir John Herschel exhibited an 
experiment where the column of air in a closed pipe, being main- 
tained in a state of forced vibration by two tuning-forks held 
over it, differing materially in pitch, yielded at the same instant 
both their sounds, being actually out of unison with itself, and 
uttering distinct beats*. It is on this principle of forced vibra- 
tions that, according to the view here taken of the subject, the 
phznomena of absorption, or, as it should rather be called, of 
obstructed transmission, depend. ‘The degree of obstruction 
which an undulation experiences from a medium will be in pro- 
portion to the number and extent of its subdivisions, im travers- 
ing the molecular systems of which that medium consists, and 
the latter will be greater the more completely out of unison 
those systems happen to be with the vibrations thus forced 
through them. To assign the law of degradation may be dif- 
ficult, but it is easy to satisfy ourselves that it may depend on 
relations quite sufficiently varied and different in differently 
constituted molecules, to give rise to all the apparently capri- 
cious phenomena of absorption. 


On the Dispersive Powers of the Media of the Eye, in con- 
nexion with its Achromatism. By the Rev. Bapen Powe tt, 
M.A., F.R.S., Sav. Prof. of Geometry, Oxford. 


The principal experimental results bearing on the question 
of the achromatism of the eye are as follows: 

At distances Jess than the limit of distinct vision the image 
of a luminous point is coloured; or if a small opake object be 
interposed, its shadow is coloured. 

To near-sighted eyes, a small bright object on a dark ground 
appears edged with colour, and the effect is rendered more con- 
spicuous by using a blue glass, which allows the extreme rays 
of the spectrum to pass, and stops or weakens the middle rays. 


* This experiment, which was published by Sir John Herschel in his Essay on 
Sound, was then supposed by him to be new. It appears, however, to have been 
previously made by Mr. Wheatstone. See a very ingenious paper by that 
gentleman, “ On the Resonance of Columns of Air,” Quarterly Journal of 
Science, N.S. No. 5. : 

+ The paper, of which the above is an abstract, has since been communicated to 
the London and Edinburgh Philosophical Magazine and Journal, vol. iii. p. 401. 


TRANSACTIONS OF THE SECTIONS. 375 


Thus, when from any cause the rays do not converge accu- 
rately upon the retina, the dispersion is sensible. 

To some eyes, however, it is scarcely appreciable, and in all 
cases is but small. 

So long as direct centrical pencils have their mean rays 
ald accurately «pon the retina, no colours are percepti- 

es 

When. the incident rays are partially intercepted by an 
opake body, the image is formed by eccentrical pencils, and 
the colours become very conspicuous. 

Fraunhofer admitted the different prismatic rays successively 
into a telescope, and found it necessary, in passing from the 
red to the violet ray, to adjust both the eye-glass to the object- 
glass, and the eye-glass to the micrometer-wire, in order to see 
the wire distinctly in the different sorts of light. 

The whole displacement must be the sum of the chromatic 
aberrations of the object-glass, the eye-glass, and the eye. 

Fraunhofer does not notice the first; says that the second 
nts allowed for; and takes the residuum as the dispersion of 
the eye. 

He elsewhere states that the telescope was not perfectly 
achromatic; and as the data are not stated, we cannot regard 
the inference as conclusive, and Fraunhofer admits that it is 
not precise. 

The author of this paper has tried similar experiments, but 
found the displacement so small that he is quite in doubt whe- 
ther any was requisite: and considering that the aberrations 
of the lenses may be uncertain to a larger amount than the 
quantity sought, it cannot be satisfactorily deduced by this 
method. 

The theories which have been proposed are by no means 
satisfactory. 

D’Alembert conceives that the agitation (ébranlement) occa- 
sioned on any one point of the retina, extends itself to the 
neighbouring points, and thus each point is influenced by the 
sum of the effects due to all the coloured rays at once, and 
simple vision results, 

Euler’s explanation is grounded on hypothetical assumptions 

as to the dispersive powers. 
__ This is remarked by Dr. Maskelyne, who by calculation 
finds the aberration so small as to be insensible. He, however, 
assumes the dispersions only by analogy from the proportions 
assigned by Newton. 

Dr. Wells leaves it as a point yet to be explained. 

Mr. Coddington, admitting a considerable difficulty in the 


376 THIRD REPORT—1833. 


‘subject, suggests that a compensation may take place between 
the refractions at the cornea and the crystalline. 

Several writers observe, that the sensible achromatism is 
difficult to understand, because the refractions are all performed 
one way. 

With a view to the theory, it is to be observed that although 
the refractive powers of the several media of the eye have 
been accurately determined, and the dispersive indices of the 
aqueous and vitreous humours, yet no observations of the se- 
paration of the different prismatic rays in these media have 
been given, and none for the dispersion of the crystalline lens. 

The author has accordingly tried such experiments by form- 
‘ing portions of these media (from the eye of an ox) into prisms 
‘between glass plates, and measuring the proportions, both of 
the whole extent, and of the several parts, of the spectra which 
they produced. 

In the vitreous humour this was easily done, as the whole 
spectrum was very distinct, and the principal lines well defined. 
With the crystalline it was extremely difficult to make the ob- 
servations satisfactorily, and the results are less certain. 

The measurement was made micrometrically by the rotation 
in its own plane of a plate-glass prism of very small angle, so 
as to give an image of the spectrum sensibly unaltered, but - 
deviated in position, and any definite part of this image could 
be brought by rotation into coincidence with any part of the 
fixed image: the angular separation of the different parts or 
lines of the spectrum could be thus readily deduced from the 
arc of rotation observed. 

For comparison of the prismatic dispersion of the vitreous 
and crystalline, the mean results of several sets of observations 
were as follows: 

Angle of the prism, 75°. 


Angular separation of the extreme red form. 


The Boundary Extreme 
mean of green violet. 

Crystalline lens : yellow. and blue. 
Central portion... .0° O1'), .. 0° 407". Cie aa 
Outer portion 3" OB. 0 a5, eee 
Vitreous humour. .... Tan. O. al ot. ke 


As far as such determinations can be relied on, it hence ap- 
pears: 

Ist, That the dispersion of the several portions of the cry- 
stalline are nearly equal. 

2ndly, That the separation of the different rays in the ery- 
stalline is greater than in the vitreous humour ; but 


TRANSACTIONS OF THE SECTIONS. 877 


»8rdly, That they are as nearly as possible in the same ratio. 
In'theory, the principles on which achromatism might be 
produced in a combination similar to the eye are easily under- 
stood by reference to the common formula for the dispersion 
of a single lens. Let m, m,,m,, &c., be the absolute indices, 
relative to vacuum, of the medium of the lens, for the mean, 
red, violet, &c., rays. 
The common formula applies to the focus for parallel rays 
formed in vacuo. Let us now suppose it formed in some other 
medium, whose absolute indices (as before) are m,, mM, mj, &C., 
and f the principal focal length of the lens for mean rays in this 
medium. Then, instead of the common formula for the chro- 
matic aberration (a), we shall have the analogous one 


[ita 
ita yn menus 
Lomo) 


Here the numerator may be = 0, or the aberration be de- 
stroyed, whatever be the absolute values of the indices, pro- 
vided only the indices of corresponding rays are in the same 
ratio in the two media. 


- By a construction of the course of the rays it will be easily 


seen, that with certain dispersive powers and curvature the 
rays will reunite at given distances. 
This appears to be the same principle as that on which Sir 


‘David Brewster proposed an achromatic microscope. (Phil. 


Instruments, p.408.) 


On the Power of Glass of Antimony to reficct Light. By 


R. Porter, Jun. 
In an abstract of a paper which is published in the preceding 


‘volume of these Reports, Mr. Potter gave the results he had 


obtained of the reflective power of diamond for light, by the 


‘method of photometry by comparison. Since then he had 


experimented with a large larke diamond belonging to Sir 
David Brewster, and obtained results nearly identical. He 
found this latter diamond to reflect at 2° to 3° incidence 
equally with crown-glass at 64° incidence, from which the va- 
rious trials differed very little. 

' The ordinary surfaces on diamond being too small and the 


planes not sufficiently true to determine the reflective power at 


high incidences, he had recourse to glass of antimony, a sub- 
stance of very high refractive power, to which he succeeded 


378 THIRD REPORT—1833. 


in giving a plane surface and high polish. Comparing such 
a surface with crown-glass, he obtained measures which 
enabled him to calculate the light reflected for all requisite 
incidences, from the known reflective power of crown-glass. 
These show that glass.of antimony follows the same law in the 
reflection of light as low refracting substances, and that the 
number of rays reflected of every 100 incident may be ealcu- 
lated from the same formula, namely, y = a x ae where 
y is the number of rays reflected of every 100, x the sine of 
incidence to radius (r), as 100; a, 6, and e¢ being constants, 
and having the following values for this substance, a= 7*4, 
6= 1:25, and e = 9. The maximum polarizing angle was 
found to be about 65°, so that the refractive index of the 
specimen would be about 2°1 to 2°2.. The undulatory formula 


Y — : . . . 
(yay by using this value of », shows that according to 


the received principles of that theory, the reflection of glass of 
antimony at a perpendicular incidence should have been about 
13°33 rays of every 100 incident, whilst experiment shows that 
only 8-20 such rays are really reflected. 


On a Phenomenon in the Interference of Light hitherto un- 
described. By R. Porter, Jun. 


When the image of a luminous point is viewed in a Newtonian 
microscope, of which the mirror is of a spherical figure, and 
the aperture large in proportion to the radius of the sphere, 
there is a large circle of aberration visible. This appearance 
is best studied when the point, where the rays reflected near 
the edge of the mirror cross the axis, is in the focus of the eye- 
glass. The circle of aberration then appears to be composed 
of dark and bright rings, which are broad and distinct near 
the edge, but from the mixture of colours become soon lost 
in white light. 

To produce the luminous point, Mr. Potter used a small 
mercurial globule, attached by some glutinous matter on a slender 
wire. When the image of the sun given by this globule is 
viewed in the microscope, the appearance of the rings formed 
by interference is one of the most beautiful of this class of 
phenomena. It may be seen by candlelight, but to much less 
advantage. Mr. Potter states, that on studying the theory of 
this case of interference he found that, according to the undu- 
latory theory, the outer ring should be brightest on its outside 
edge, whilst, according to the theory of interference which he 


TRANSACTIONS OF THE SECTIONS. 379 


has proposed, it should shade away gradually into darkness on 
this outside edge; and he adds that the phenomenon corresponds 
exactly to this latter deduction. 


Explanation of the Principle and Construction of the Actino- 
meter. By Sir Joun F. W. Herscuet, F.R.S. 


This is an instrument, invented by Sir John F. W. Herschel, 
for measuring at any instant the direct heating power of the 
solar rays. 

In the ordinary methods of estimating the sun’s calorific 
effect, the heating power of the sun is put in equilibrimn with 
the cooling influence of external causes, and the elevation of 
temperature in the body (usually a thermometer,) thus heated, 
so as to maintain the equilibrium, is noted and set down as a 

‘measure of the sun’s calorific effect. But the principle of such 
a proceeding is obviously faulty: the temperature maintained 
is just as properly a measure of the cooling as of the heating 
influences, and, in fact, is a measure neither of one nor of the 
other. This objection applies to all statical measurements of 
dynamical effects, taking the word generally to mean cases 
where power is transformed into its immediate effect, and ex- 
hausted by such transformation. The actinometer is intended 
‘to afford a dynamical measure of the solar radiation, @. e. one 
in which the actual quantity of heat received per second or in 
a given infinitely small time is the object of measurement, upon 
a given surface directly exposed to the sun. ‘The instrument 
itself is nothing more than a very large cylindrical thermometer- 
bulb with a scale greatly enlarged, so as to render the smallest 
possible increase of temperature distinctly measurable. In 
consequence, the scale is not divided into degrees, but into ar- 
bitrary equal parts; and as avery slight elevation of tempe- 
rature suffices to carry the liquid up the whole column, the 
bulb itself is furnished with a screw, by withdrawing which its 
capacity may be enlarged, and the power of reading off restored. 
The bulb is of colourless glass, and is filled with an intensely 
blue liquid. Into the interior of this the calorific rays pene- 
trate, and are absorbed at some sensible depth within the bulb, 
so that the liquid is heated from within, and the whole of the 
calorific rays (not stopped by the glass) go to dilate the con- 
tents, and are extinguished in producing that effect. To make 
an observation with the actinometer, the instrument must be 
freely exposed in the shade for one minute, and the variation 
of the reading noted; then freely exposed to sunshine for the 
same time, and the variation again noted; and lastly, once more 


380 THIRD REPORT—1833. 


in the shade. The mean of the two variations in the shade 
being subducted from the variation in the sun, the excess gives 
the dilatation per minute due to the sun’s rays, the quantity sub- 
ducted being the effect of the other causes in action at the time. 

Many years’ experience and repeated trial of the instrument 
under a great variety of circumstances have. demonstrated the 
complete dependence which may be placed on its indications, as 
well as its extreme sensibility. It has been used by its inventor, 
Sir John Herschel, on the Puy de Dome; in an imperfect state 
also on Etna by him, and since by Mr. Lunn, in its improved 
form, on the same volcano; and by M. Nicollet and Professor 
Forbes on the Alps in Switzerland. In a series of experiments 
on the absorptive powers of glasses on heat, and on the reflec- 
tive powers of metallic and other polished surfaces, as well as 
for the purpose of ascertaining the rate of increase and dimi- 
nution of solar radiation throughout the day, from sunrise to 
sunset, it has been found of great utility. In the latter inquiry, 
the increase or diminution of altitude of the sun near the ho- 
rizon is marked by the actinometer nearly as well as by actual 
measurement with a sextant*. . 


Sir John F. W. Herschel also communicated some results 
obtained by Professor Forbes on the diminution of the intensity 
of the solar rays in traversing the atmosphere by means of 
the actinometer. He pointed out by direct comparison with 
two instruments the degree of dependence to be placed on these 
results, which, from experiments made by Professor Forbes at 
the observatory of Paris, seem to indicate not more than —+.. of 
the whole effect of radiation as the probable error under fa- 
vourable circumstances. But the point particularly deserving 
of attention, and upon which observations strictly comparative 
have now been made for the first time, is the numerical estimate 
of the loss of intensity suffered by the radiant heat of the sun 
in passing through a definite thickness of the atmosphere. Out 
of twenty series, each consisting of several (often many) indi- 
vidual comparisons made by Professor Forbes, with the assist- 
ance of M. Kamtz, of Halle, at the top and bottom of the 
Faulhorn, an elevated mountain of the Bernese Oberland, and 
under favourable circumstances, not one failed to indicate the 
diminution of heat alluded to. The difference of height at the 
different stations at which simultaneous observations were made, 
varied from 5000 to near 7000 English feet, and these obser- 


* The actinometer, as used by Sir John Herschel, is made and sold by 
Mr. Robinson, optician, Devonshire-street, Portland-place, London. i: 


TRANSACTIONS OF THE SECTIONS. 38P 


vations were made on different days and at different hours of 
the same day. Professor Forbes has not yet finally reduced 
his observations; but from the approximate numbers communi- 
cated by Sir John Herschel to the Meeting, it appears that even 
in the clearest weather (in which alone these experiments were 
made,) the force of solar radiation is diminished by about one 
Jifth in passing through a column of 6000 feet of the purest 
air. : 


ccount of some recent Experiments on Radiant Heat. By 
~~ M. Mettont. Communicated by Professor Forses. 


_ The following truths at which M. Melloni has arrived are of 
capital importance. The quantity of calorific rays which tra- 
verse a screen, derived from sources of high or low tempera- 
ture, is proportional to that temperature ; but the difference 
constantly diminishes as the thickness of the screen becomes 
less, till at last, with excessively thin amine of mica, it is insen- 
sible.. This remarkable result proves that the resistance to 
passage which affects the transmission of heat depending on 
the temperature of its source, is not exerted at the surface, but 
in the interior of the mass.’ 

_ The extension of this observation to the solar rays gives the 
following result. M. Melloni operated with various thicknesses 
of sulphate of lime, water, and acids. He found that the quan- 
tity of heat intercepted by increasing the thickness of the trans- 
parent medium is greater for the less refrangible than for the 
more refrangible rays; that is, that while for example an 
evanescent pellicle of mica would permit as much per cent. of 
the heat accompanying the red ray to pass through it as of the 
violet ray, if the thickness be increased, a much larger per- 
centage of the former will be stopped than of the latter; whence 
the author concluded that the refrangibility of a heating ray is 
coordinate with, and a measure of, its intensity. Thus, a plate 
of glass 2™ in thickness stops, out of 100 incident rays of heat, 
45 when they proceed from a lighted candle, 70 from a mass of 
copper at 950° Cent., 92 from a vessel of boiling mercury, and 
100 from a vessel of boiling water. 

Hence the impossibility, as he observes, by the use of lenses 
of glass and suchlike materials of concentrating heat of low 
temperature. This admits, however, of one most important 
exception, the discovery of which forms an epoch in the history 
of the science of heat. In a previous memoir, M. Melloni had 
compared the transmissive powers of a great number of sub- 
stances. For uncrystallized compounds and fluids, he found 


382 THIRD REPORT—1833. 


the ratio of transmission for the heat of a lamp to be propor- 
tional to their refractive powers; but in crystallized solids the 
strangest anomalies appeared. The most remarkable of these 
was, that the transmissive power of rock-salt for heat was six or 
eight times greater than that of an equal thickness of alum, 
which had almost the same transparence and refractive powers. 
But the extraordinary property discovered by M. Melloni in 
rock-salt is this, that it is equally transparent for heat of all 
temperatures or of all degrees of refrangibility ; that the heat 
of the hand loses as little in passing through a given thickness 
of it, as the same quantity of solar or any other kind of heat. 
This singular fact places rock-salt in a still more important posi- 
tion in relation to heat than the discovery of Sir David Brewster 
has already placed it in regard to light. 

A plate of rock-salt 7™™ (-28 in.) in thickness transmits equal 
proportions of heat, radiating from red-hot iron, from the flame 
of oil or alcohol, boiling water, and water even below 120° Fahr. ; 
and in all these cases the transmission was 92 per 100 of the in- 
cident rays. A piece about an inch in thickness was equally 
constant in the ratio of transmission. 

To all who have been in the slightest degree occupied with 
the study of radiant heat, the value of this property will at once 
be apparent. It gives the means of making experiments 
hitherto physically impossible. The more delicate of M. Mel- 
loni’s experiments are made with the thermo-multiplier; but 
many are capable of repetition by common instruments. ‘The 
different transmissibility of alum and salt, for example, may be 
shown with the rudest apparatus. By means of a common three- 
sided prism, formed of rock-salt, M. Melloni can refract a pen- 
cil of heat derived from boiling water. 


The following is an abstract of the later discoveries of 
M. Melloni, communicated by him to Professor Forbes, in 
order to be laid before the Association. 

The colouring matter of glass has been proved to diminish its 
power of transmitting heat. The question is, Does this ab- 
sorptive force stop all the radiant heat, except that having a 
definite degree of refrangibility, in the same way as it acts on 
the rays of light ? 

The character of colour, which always accompanies and 
indicates the refrangibility of light, is wanting in regard to 
heat; and the operation of measuring the intensity of an 
emergent pencil of the latter would be, it is easy to see, 
extremely difficult. It has formerly been shown that the power 
of heat to traverse diaphanous bodies diminishes with the 


TRANSACTIONS OF THE SECTIONS. 383 


refrangibility of the heat. From this we derive almost as easy 
a mean for estimating the refrangibility of a degree of heat as 
in the corresponding case of light. M. Melloni took a clear 
glass, and coloured glasses of various tints of the spectrum, and 
adapted each glass to the opening in a screen, through which 
rays fell from a lamp upon the sensible part of the thermo- 
multiplier. The distance of the lamp was varied till, with each 
glass successively, a deviation of 40° was produced in the index- 
needle of the galvanometer. In each case the rays transmitted 
by the glass were then caused to traverse a plate of sulphate of — 
lime. The needle always indicated a smaller thermal effect 
than before, and its new position was the same for the white, 
violet, indigo, blue, yellow, orange, and red glasses; only in 
the green, instead of the angular deviation being reduced 
from 40° to 18°, as in the former cases, the needle stood at from 
7° to 10°. When alum was substituted for sulphate of lime; 
the deviation for green glass was from 1° to 1°6, whilst for all 
the others it was 8°. Hence we conclude that red, orange, 
yellow, blue, indigo, and violet, as well as clear glasses, pro- 
duced no elective action on heat, since the rays issuing from all 
these traverse an interposed plate with equal facility, which 
would not be the case were they differently refrangible. Green 
glass, on the other hand, transmits rays easier stopped than 
any of the others; and therefore, from what has been already 
said, admits the passage of the least refrangible rays, whilst it 
stops the more refrangible, which we know to traverse the sul- 
phate of lime or alum most easily. To confirm this deduction 
was the object of a second series of experiments, since the 
transmission of the first and the stoppage of the second ought 
to be made manifest. 

The author took an Argand lamp, and a spiral of platinum 
wire placed over a lamp of alcohol, which, it is easy to show, 
had a much lower temperature, and therefore its heat had a 
lower degree of refrangibility than that of the Argand burner. 
The differences of the quantity of heat from these two sources, 
transmitted by all the kinds of glass except the green, showed 
that the transmitted rays, in the case of incandescent platinum, 
were almost exactly half those transmitted when the source was 
an Argand burner. In the case of green glasses, the difference 
was little or nothing. Hence the conclusion that, as was antici- 
pated, the green glass was more transparent for the rays of 
small refrangibility than for those of greater refrangibility (or 
temperature), 

Another experiment was performed with the same object, 
and with similar results. 


884 THIRD REPORT—1833. 


Since the rays which have been transmitted by citric acid and 
some other substances are those only of greatest refrangibility, 
these have, relatively to most substances, a great penetrating 
power. In the remarkable case of green glass, where rays of 
a low degree of refrangibility are most easily transmitted, the 
reverse should be the case. Accordingly, out of 100 parts of 
heat transmitted by citric acid, white-, violet-, red-, orange-, yel- 
low-, blue-, and indigo-coloured glass transmitted various quan- 
tities of heat, varying from 89 to 28 parts ; whilst two specimens 
of green glass only transmitted 6 and 2 parts. What is most re- 
markable is, that while for all the glasses but green the trans- 
missive power is three times greater than when the heat of in- 
candescent platinum was directly incident upon them, the effect 
with that particular glass descended from 23 or 24 per cent. 
nearly to 0. The reason is obvious; for the whole of the rays 
of low refrangibility emitted by the platinum, and for which 
alone the green glass was transparent, had been stopped by the 
interposition of the plate of citric acid, which had, as it were, 
sifted it free from these rays. 

Hence M. Melloni concludes that green glass is the only kind 
which has coloration for heat, if we may use the expression ; 
the others acting upon it only as a more or less transparent 
glass of uniform tint does upon light. 


On Thermo-Electricity. By Joun PripEavux. 


In examining the theory of Thermo-electricity, Mr. Prideaux 
satisfied himself— 

1. That thermo-electricity is conveyed through long wires 
with the same promptitude as common or voltaic electricity ; 
and that it does not contain caloric as an element, differently 
from those principles :— 

2. That bringing two heated wires into contact in the bulb 
of a large air thermometer, so as to generate a thermo-electric 
current there, does not accelerate the cooling of the wires, and 
thus that caloric is not consumed in its production; nor does a 
thermo-electric current become developed when caloric is ren- 
dered latent, as in the fusion of a metal :— 

3. That making the surfaces of a thermo-electric pair iden- 
tical, by tinning them all over before contact, does not impair 
their powers (but rather the reverse); and that large plates of 
copper and tin, or tin plate, heated together, separated from 
metallic contact only by fine gauze, or a few threads spread over 
the surface, will not produce the slightest deviation of the 
needle, although metallic communication being made between 


TRANSACTIONS OF THE SECTIONS. 385 


them by means only of a slip of tinfoil, immediately throws off 
the needle 25°; which experiments, he conceives, can hardly 
be reconciled with M. Becquerel’s views respecting radiation :— 
(Ann. de Phys. et. de Chim., Aout 1829.) 

4, That copper is, under ad/ circumstances, a better con- 
ductor both of heat and electricity than iron; wherefore the 
electro-positive quality of iron to copper cannot be due to supe- 
riority of conduction; and, by analogy, conduction should not 
generally be the positive principle in this particular develop- 
ment of electricity :— 

5. That iron, antimony, and zinc become positive by heating; 

that is, a bar of either of these metals hot is positive to a bar of 
the same metal cold; a quality which he denominates thermo- 
positive. 
_ That platinum, silver, copper, lead, tin, bismuth, become 
negative by heating; that is, a hot bar is negative to a bar of the 
same metal cold, which he calls thermo-negative ; and thus the 
inetals are resolved into two thermo-electric classes. 

That the thermo-positive quality is possessed by the above- 
named. three metals in the following order : 

: Antimony, 

Tron, 

Zinc. . 
That the thermo-negative order is as follows : 

Bismuth, 

Silver, 

Platinum, 

Copper, 

Tin, 

Lead. 

And that the order of thermo-electric reaction seems due to 
the idio-positive or negative quality, compounded with that of 
conduction, taking conduction directly in the thermo-negative 
class, inversely in the thermo-positive. That is to say, anti- 
mony and iron are not very different in thermo-positive power ; 
but antimony is the worse conductor, and takes the lead. 

Iron and zine differ little in conducting power; but iron is 
the stronger thermo-electric, and takes the lead. 

--In the thermo-negative class bismuth is the worst conductor, 

oo thermo-negative, and therefore takes the lowest 
ce. ' 

Silver is a better conductor, but stronger thermo-negative, 

than copper. ‘The one quality should raise it, the other lower it, 

in the table, and accordingly the difference between those two 

metals is small; and Mr. Prideaux frequently finds copper 

1833. 2c 


nearly equal, 


386 THIRD REPORT—1833. 


wires + to silver, though the latter generally stands highest in 
the tables. Platinum is a worse conductor, and a stronger 
thermo-negative, than copper: both properties cooperate to 
lower it in the table, and it falls considerably. 

Platinum conducts a little better than tin, but is much more 
thermo-negative. The latter quality being predominant, tin 
stands above platinum in the thermo-electric table. 

Lead conducts worse than either of those metals, but has 
much less thermo-negative quality; hence they both range 
below it in thermo-electric reaction. 

Zinc has a peculiarity deserving the notice of experimenters 
on this subject: 

Above 250° it is thermo-positive. 
Below 200° it is thermo-negative. 


And two bars—one heated to about 230°, the other-at the 
atmospheric temperature (say 60°)—have no reaction on the 
needle ; that is, there is between 200° and 250° a neutral point. 
The investigation of this thermo-electric change, at such ma- 
nageable temperatures, seems to offer an opening to new and 
advanced ground. Zinc becomes ductile at about 250°. 

6. That the currents in masses of bismuth cannot be easily 
drawn out by better conductors, and do not seem analogous to 
currents in the same or other metals acting in pairs :— 

7. That foreign metals brought into contact with a homoge- 
neous circuit, near a heated point, enter into the thermo-electric 
communication, and determine a current conformably to their 
position in the table. 


On some new Phenomena of Electrical Attraction. By 
W. Snow Harris, F.R.S. gc. 


*. The author exhibited and explained to the Section the con- 
struction and use of some new electrical instruments, by which 
he has been enabled to investigate more rigorously than has 
been hitherto done the phenomena of electrical attraction. 
These instruments consist of an electroscope, an electrometer, 
a unit of measure, and an adaptation of the common balance to 
the measurement of electrical force. : 

The electroscope acts on the principle of divergence, and con- 
sists of a long index of reed, easily moveable on a delicate axis 
set on points in an elliptical ring of brass. This index diverges 
over a graduated circle from two fixed arms of brass when a 
slight degree of electricity is communicated to the ring. . 

The electrometer measures the force of electrical attraction, 
under given conditions, by means of. two opposed conductors, 


—_—_——- 


TRANSACTIONS OF THE SECTIONS. 387 


one of which is fixed, the other moveable, the latter being sus- 
pended from the periphery of a wheel. The distance between 
the conductors is either constant or variable, according to the 
conditions of the experiment, and the force is estimated on a 
graduated arc in terms of a known standard of weight or other- 
wise in degrees. 

The unit of measure consists of a small electrical jar, having 
a discharging electrometer, and inverted on the prime conductor 
of the machine. The quantity accumulated is measured by its 
repeated explosions, in consequence of electricity passing from 
the outer coating. 

When it is required to communicate comparative quantities 
of electricity to simple insulated conductors, sparks are drawn 

by means of an insulated transfer plate from an insulated jar 
' charged by the preceding method with a known quantity. 

By the aid of these instruments, the author has arrived at 
some new results in electricity, which appear to be not un- 
worthy of notice. 

In disposing different quantities of electricity on insulated 
conductors of similar form and dimensions, he finds that the at- 
tractive force is directly as the square of the quantity, the sur- 
face being constant; and inversely as the square of the surface, 
the quantity being constant; being similar to the results which 
he had already arrived at in examining the disposition of elec- 
tricity on coated jars*. 

On comparing those results with the striking distance of a 
given accumulation, he finds that the quantity of electricity re- 
quisite to produce a discharge is invariably as the square of the 
distance directly; that is to say, when the distance is twice as 
great, there is four times the quantity of electricity accumulated 
at the instant of the discharge, all other things remaining the 
same; and this is true, whether the accumulation take place on 
simple insulated conductors or on coated jars. 

The above laws are general when the conductors or jars are 
in every respect similar, but they do not obtain under every 
possible kind of extension of the same surface. The author 
finds, with Professor Volta, that extension in length increases 
the capacity of a givenconductor. In examining the capacities 
of plane surfaces for electricity, he finds the capacity to be as 
the linear dimensions of the plate inversely, the area being con- 
stant; and if the linear dimensions be the same, it varies in an 
inverse ratio of the area. The extent of edge or boundary of 
the plate, however, has not in itself any influence on the capa~ 


* Memoirs of the Plymouth Institution. 
2c2 


388 THIRD REPORT—IS833. 


city of a conductor: the extent of edge is merely a function of 
the peculiar kind of extension to which the given area has been 
subjected, and by which the electrical particles become placed 
in respect of each other, in such way as to diminish their ope- 
ration on external bodies, that is to say, their intensity. 

This view seems confirmed by numerous experiments made 
by the author. The same plates turned into cylinders, both in 
the direction of their lengths and breadths, were found to have 
still the same capacities, that is to say, whether the conductor 
was simply a plate or an opencylinder. Neither did any differ- 
ences arise in turning a given plate into other figures approach- 
ing cylinders, such as triangular and hexagonal prisms. The 
capacity of a sphere, also, is the same as a plane circular area of 
equal superficies. Hence the capacity of a cylinder or sphere 
for free electricity, may be conceived to be the same as that of 
the plane area into which it may be supposed to be rectified. 

These results seem of some consequence to the theory of 
electrical action. It has been found, for example, that the in- 
terior surface of a sphere charged with free electricity does not 
electrify a conducting substance when placed completely within 
it; it has been hence inferred, that the electrical accumulation is 
disposed altogether upon its exterior surface. Now the inten- 
sity of a sphere being the same as that of a plane area of equal 
superficial extent, it should follow that the electrical distribution 
is In each case the same, since it is difficult from any known fact 
to suppose a given quantity of electricity expanded over twice 
the surface, as in the plane area, and yet maintain the same 
intensity. ‘The redundant electricity, therefore, if the above 
deduction be true, should only be disposed on one side of the 
plate. 

Since the capacity of plane conductors varies in an inverse 
ratio of their area and linear dimensions, it is requisite, if we de- 
sire to obtain plane surfaces whose capacities for electricity shall 
be double, treble, &c., of each other, to construct the plates so 
that their areas and linear dimensions shall be likewise double, 
treble, &c., of each other respectively. 

The law according to which the force of electrical attraction 
varies at different distances is an important object of physical 
research. It may be easily and satisfactorily arrived at by the 
methods of experiment above mentioned. The author exhibited. 
to the Section, by the means of his balance, the resulting law 
of electrical force when exerted between two circular plane 
areas directly opposed to each other, at different distances, and 
which was found to be in an inverse ratio of the squares of the 
respective distances exactly. 


TRANSACTIONS OF THE SECTIONS. 389 


» This law is immediately apparent when the opposed surfaces ° 
are parallel planes or rings; but in the case of spheres, or 
bodies of other forms, the experiment is of a somewhat more 
complicated character. 

With a view of reducing the experiments to a more simple 
form, the author has been led to some further inquiries into the 
peculiar mode of action of the force under examination, which 
merit an attentive consideration. He finds, 

1. The force exerted between an electrified and insulated 
neutral conductor is not at all influenced by the form and dis- 
position of the unopposed portions. Thus the force is precisely 
the same, whether the opposed bodies are merely circular plane 
areas, or are otherwise backed by hemispheres or cones. 

2. The force is as the attracting surface directly, and as the 
squares of the distances inversely. Hence the attractive force 
between parallel plane circles being found, the force between 
any other two similar planes will be given. 

_ 3. The attractive force between two unequal circular areas is 
no greater than that exerted between two similar areas each 
equal to the lesser. ror 

4. The attractive force between a mere ring on a circular 
area is no greater than that between two similar rings. 

5. The force between a sphere and an opposed spherical seg- 
ment of the same curvature is no greater than that of two similar 
segments each equal to the given segment. 

These results have been arrived at by the instruments already 
mentioned, the electrical intensity in each series of experiments 
being supposed the same. 

A careful induction from the above facts has led the author 
to consider the attractive force between two equal spheres as 
made up of a system of parallel forces operating between the 
homologous points of the opposed hemispheres, the total force - 
being as the number of attracting points directly, and as the 
squares of the respective distances inversely. 

These simple elements enable us to determine a point q 7 
within each hemisphere, in which the whole force may be sup- 
posed to be concentrated, and to be the same as if emanating 
from every point of the hemisphere. The locus of this point 
within the surface will depend on the distance between the near- 
est points of the spheres, and may be readily found by the ex- 
(a? + 2ar)—a 


pression s = 3 


, a being the distance between the 


nearest points of attraction, and r the radius of the sphere, 


and 2 the distance of the point within the hemisphere. 
The points q q' being thus found, the whole force is observed 


390 THIRD REPORT—1833. 


to vary between them according to the general law already 


arrived at. It varies also as , that is, inversely as the 


$ 
a.(a+ 2r) 
distance between the nearest points multiplied into the distance 
between the centres. 

The author found these deductions accord very completely 
with subsequent experiments, so that the weight requisite to 
counterpoise the force between two equal spheres at given di- 
stances may be invariably predicted with great precision. Some 
of these results were exhibited to the Section by means of the 
balance. 

When two plane circular areas, each equal to the areas of the 
hemispheres rectified into plane surfaces, are opposed to each 
other, and placed so as to pass through the points q g' above 
mentioned, the attractive force is extremely near that obtained 
by means of the opposed spheres, the distances being estimated 
from the same points. 

The author believes that many of the results thus arrived at 
have not been hitherto included in the received theories of elec- 
tricity, and is disposed to think that we may yet arrive at easier 
views of electrical action, and hence bring the phenomena more 
completely under the dominion of analysis. ° 


On Electricity. By the Rev. Joun G. MacVicar, of St. 
Andrew's. (Extract of a Letter from the Author to the Rev. 
W. Wuewe 1, F.R.S., &c.) 


As a basis for his theory of electrical phenomena, Mr. Mac- 
Vicar assumes that those parts of bodies which are capable of 
displaying electric phenomena possess a filamentary or acicu- 
lar structure, an hypothesis which he is not disposed to rest 
upon such a consideration as M. Ampére makes it rest, when 
advancing his hypothesis of magnetic action, (namely, that by 
making such a supposition one may explain many phznomena,) 
but upon these two principles : First, The action of the first law 
of motion, which must ever tend to distribute the matter in 
which impulses are embodied in a filamentary or acicular ar- 
rangement, or at all events in an arrangement in whicha linear 
continuity of matter is maintained, which (since every material 
medium is known to be very porous,) can only give rise to such 
a structure. Secondly, Upon the known intimate structure of 
bodies, so far as that can be observed by the microscope, du- 
ring solidification, by cleavage or otherwise. 

This granted, he inquires what are the phenomena of trans- 
verse vibration that must be displayed by such filamentary 


ee 


TRANSACTIONS OF THE SECTIONS, 891 


tissue, the length of each filament being regarded as indefinite 
in comparison with the smallness of the elementary impulses to 
which it is supposed to be subjected during the excitement of 
electric phenomena. And from the well known phznomena 
_of extended chords, he infers that each must undulate in a ser- 
pentine manner, the point where the exciting impulse is ap- 
plied being the centre of a vibrating portion, which may be 
called the primary; and this accompanied on both hands by 
ulterior vibrating portions, which may be called reciprocal or 
secondary, and whose phases of motion must ever tend to be 
exactly opposite to that of the primary or central part, and 
would of course succeed:in being so could the filament. consist- 
ing of the primary and two secondary vibrating portions be 
regarded as an elastic chord stretched between two fixed nuts 
‘at its extremities. But since the filament, as has been stated, 
is in the present case to be viewed as of indefinite length, and 
instead of being stopped by two nuts at two definite points, is 
to be viewed as equally prepared for being stopped at every 
point, and so is analogous to an extended chord possessing 
many moveable nuts which spontaneously shift their position so 
as to be always adapted to the impulse, he concludes that the 
central vibratory portion, and the two reciprocal or secondary 
vibratory portions, will not be in unison, but the latter shorter 
and consequently more rapid in their movements, and taken 
together equivalent to the central or primary one in the quan- 
tity of motion they embody. Such is the theory from which 
he thinks all the phznomena of electric action may be deduced, 
without calling in the aid of electric fluids. 

Mr. MacVicar then proceeds to apply his theory to the ex- 
planation of electric polarity and induction, and points out the 
analogy between the production of light by undulations in the 
universal «ther, and the excitement of electrical phenomena 
by undulations in denser media. 


Inquiry into the Cause of Endosmose and Exosmose. By the 
. Rev. J. PowEr. 


_ The author investigates these phenomena upon La Place’s 
principles of capillary attraction. 

~ The two fluids are supposed to communicate by means of a 
capillary tube, which has a stronger attraction for one than the 
other. The attraction between the particles of the opposite 
fluids is supposed to be greater than the attraction of the par- 
ticles of either fluid for particles of their own kind. Also the 
attraction between the particles of the tube and of the more 


392 THIRD REPORT—1839. 


strongly attracted fluid in contact with it, is supposed to be 
greater than the attraction of the latter particles for themselves. 
By reason of this last supposition the fluid will, according to La 
Place, form an internal coating of the capillary tube, which in- 
ternal coating will itself act as a capillary tube upon the fluid 
next in contact. This new capillary tube, attracting the par- 
ticles of the opposite fluid more strongly than its own, will in 
turn be coated by a thin stratum of that fluid, which may again 
be regarded as a fresh capillary tube; and this alternation may 
extend itself to the very axis of the original tube, the opposite 
fluids thus forming a series of interlacing cylinders one within 
another. 

That cylinder of fluid which lies in immediate contact with 
the tube, must be supposed to become mixed with the opposite 
fluid, either before it issues from the tube, or at a distance be- 
yond it less than the sphere of sensible attraction. Were this 
not the case, the tube acting equally at its two extremities 
would produce no effect; and the action of endosmose (so far 
at least as it depends on the tube,) would cease, allowing the or- 
dinary mixing process time to extend itself within the tube; 
after which the motion would recommence. 

The maximum force is obtained by supposing this mixture to 
be completed within the tube, at a distance from the issuing ex- 
tremity not less than the sphere of sensible attraction ; and this 
is the force which it is our object to compute, for it is plain that 
at the moment when the endosmose ceases from the increase of 
pressure within the vessel where the fluid accumulates, the or- 
dinary mixing process will be at liberty to extend itself within 
the tube; and, in experiment, it is by means of these ultimate 
pressures that the forces are compared. , 

The principle of the computation is this :—the process being 
supposed to have gone on for some time, the effective attractions 
are found by taking the sums of the attractions due to the dif- 
ferent substances regarded as coexisting (each with a dimi- 
nished density,) within the same identical space. Expressions 
for the forces are thus obtained in terms of these partial densi- 
ties. The fluids being supposed to suffer no penetration of di- 
mensions in consequence of mixture, the partial densities, which 
are four in number, are readily expressed in terms of the two 
actual or total densities, and the constants depending on the 
initial state of the fluids ; and thus we are able to express the 
forces in terms of the actual densities and the constants above 
mentioned. 

The expression for the maximum force of endosmose is thus 
found to be composed of two terms; the first being propor- 


TRANSACTIONS OF THE SECTIONS. 393 


tional to the difference of densities, and the latter to the square 
of this difference. 

The last term is the force of exosmose. It is proportional . 
to the effort with which the fluid cylinders tend to penetrate 
each other, independently of the action of the tube, and seems to 


_. bea proper measure for the force with which the opposite fluids 


tend to unite in the ordinary case of mixture. It is positive 
when the attraction between the particles of the opposite fluids 
exceeds the arithmetic mean between the attractions of the par- 
ticles of the two fluids for particles of their own kind; is negative 
when less than this mean, and of course vanishes in the case of 
equality. 

When the force expressed by this last term is negative, the 
fluid cylinders, instead of penetrating further into the opposite 
fluid, will retreat into their respective fluids, and no mixture 
will take place. This is probably the reason why some fluids, 
such as oil and water, refuse to mix. 

The substances with which Dutrochet made his interesting 
experiments were saccharine or gummy solutions on the one 
hand, and water on the other. These fluids have a sufficient 
tendency to mix for the purpose of carrying on the process ; 
but that tendency béing a very slow one (as any one may con- 
vince himself by dropping a little treacle or a solution of gum 
arabic into a glass of water), the last term is probably very 
small; in which case the force ought to be very nearly propor- 
tional to the difference of densities, which is the law observed 
by Dutrochet. 

In support of that part of this theory which supposes the in- 
terlacing of the internal cylinders, a passage is quoted from the 
Mécanique Céleste, in which La Place contemplates such an 
arrangement*. But in whatever way the fluids be supposed to 
mix in the interior of the tube, the same force of endosmose will 
result, provided that the extremities of the tube are totally im- 
mersed, one in one fluid and the other in the other, to a distance 
within the tube not less than the sphere of sensible attraction. 


On Electro-chemical Decomposition. By Micuart Farapay, 
_D.C.L. F.R.S. Professor of Chemistry in the Royal In- 


~ stitution. 


In explanation of the present report of the subject brought 
forward by Mr. Faraday at this time, it may be needful to say 
that it forms part of his fifth series of Experimental Researches 


* Supplem. au X¢ Livre, pp. 28, 29, 


394 _ THIRD REPORT—1835. 


in Electricity. In the first and second were developed the 
theory and facts of magneto-electricity; in the third, the 
identity of voltaic, common, and other electricities, was proved, 
so as to allow of the use of one for the other; in the fourth, 
a new law connected with the conduction and decomposition of 
bodies was developed; and in the fifth, by the application of 
the third and fourth, an attempt is made to ascertain the true 
nature of electro-chemical decomposition. 

This important part of chemical or electrical science has been 
expressly treated of by different philosophers, and viewed in 
various lights. The opinions formed of it differ very much 
from each other, but agree generally in referring it to the at- 
traction of the poles or metallic terminations of the pole, for 
the principles either proximate or ultimate present in the de- 
composing body. 

Having proved to his own satisfaction that common elec- 
tricity had the same power of effecting chemical decomposition 
as that of the voltaic battery, and even shown that when mea- 
sured out, a quantity of it does the same chemical work as an 
equal quantity of voltaic electricity, Mr. Faraday resorted to 
it for the purpose of taking advantage of its high intensity in 
effecting decomposition under circumstances in which the 
voltaic battery would be quite inefficient, and so to accumu- 
late facts and gain knowledge which might bear upon and 
explain the usual mode of action of the latter. He found that 
decomposition could be effected without the use of any poles, 
or at least without the use of those metallic terminations usually 
called poles; for when salts and iodides were submitted to the 
action of electricity, passing to and from them only through 
the air, still decomposition took place. For instance, a piece 
of paper being cut into a lozenge-shape, moistened with solution 
of sulphate of soda, and placed between two points, one con- 
nected with the electrical machine, and the other with a dis- 
charging train, formed of a very extensive metallic system, 
communicating with the ground, soon showed evidence of the 
evolution of free acid and alkali, as the electricity of the ma- 
chine passed through it, though it was two inches distant from 
either of the points. The acid was evolved at that point where 
the positive electricity entered the moistened paper, and the 
alkali at that point where the electricity left the paper; i.e. 
considering the paper as a little conductor surrounded by air, 
the acid was evolved at the negative extremity, and the alkalt 
at the positive extremity. Many other facts were referred to, 
in which, as in this, air poles were successfully used; but the 
word pole must in these researches be understood as having 


TRANSACTIONS OF THE SECTIONS. 395 


no other meaning than the substance or surface which limits 
the decomposing body in the direction of the electric current. 

It then became an object to produce a water pole, and by 
much care this was effected. A strong solution of sulphate of 
magnesia was so adjusted as to have pure water floating upon 
it over one half of its surface; a platina plate was dipped. hori- 
zontally into this water, and another placed vertically in the 
half of the solution of magnesia uncovered by water; so that 
the order of things were, platina, sulphate of magnesia solution, 
pure water, platina: the first platina was rendered positive, 
and the latter negative, by contact with a voltaic battery. Im- 
mediately there was decomposition both of the water and of 
the salt present; the water yielded its elements against the 
platina pole, but the salt could not do so. It is true the acid 
appeared against the positive pole, but the earth magnesia 
appeared in the plane where the solution and water were in 
contact; in fact, upon the surface of water next the decom- 
posing solution, and not upon the negative platina pole. Here, 
therefore, a true water pole was obtained, 7. e. a surface of 
water upon or against which the earth magnesia evolved in a 
case of electro-chemical decomposition was deposited. 

These were considered as very important facts in determining 
the true character of the action taking place in cases of electro- 
chemical decomposition generally. 

Another correction of commonly received opinions was con- 
sidered as essential in assisting to determine the nature of 
electro-chemical action. It has been generally stated by Sir 
Humphry Davy and others, that water is essential to decom- 
position by the voltaic battery; but Mr. Faraday has shown that 
such is not the case, and that an immense number of other bodies 
have, not merely equal powers, but powers far surpassing those 
of water in that respect. The cause of this mistake has been 
our ignorance of a law developed in the fourth series of the 
Experimental Researches, under which an extensive class of 
compound bodies, such as oxides, chlorides, iodides, salts, &c., 
which are either very bad or nearly non-conductors when solid, 
become most excellent conductors when melted, and then are 
decomposed with extreme facility. Water happens to be the 
only one of this class of bodies existing naturally in the fluid 
and therefore decomposable state: it loses this power when it 
becomes a solid, and other bodies gain it when they become 
fluid. Electro-chemical decomposition does not therefore, of 
necessity, involve the presence of water. 

For the purpose of making his own view of matters clear 
and distinct, Mr. Faraday gave a brief view of the theories of 


596 THIRD REPORT—1833. 


electro-chemical decomposition advanced by others, as far as 
they could be gathered from their published papers. 

Grotthuss published on the subject in the year 1805. He 
considers the pole as an attractive agent; the poles have at- 
tractive and repelling powers: the negative pole attracts 
hydrogen and repels oxygen, whilst the positive pole attracts 
oxygen and repels hydrogen. The powers of each pole upon 
any one particle are inversely as the squares of the distances; 
hence giving, as he says, (though erroneously,) for any one 
particle, a constant force. He first put forth the view of suc- 
cessive decompositions and recompositions, and so explains 
the appearance of the elements at a distance from each other. 
He thinks the elements about to separate at the poles, unite 
to the two electricities there, and so become gases*. 

Sir Humphry Davy wrote on the subject in 1806. His ex- 
pressions are very general, so as to accord with many specific 
views of decomposition; but when more particular, he refers 
the decomposing effects to the attraction of the poles}. He 
agrees with Grotthuss in supposing successive compositions and 
decompositions throughout the fluid; supposes that the attrac- 
tive and repellent agencies may be communicated from the 
metallic surtaces throughout the whole of the menstruum, being 
communicated from one particle to another particle of the same 
kind, and diminishing in strength from the place of the poles 
to the middle part, which is necessarily neutral. In 1826, Sir 
Humphry Davy said he had nothing to alter in the fundamental 
theory just referred to, and uses the terms attraction and re- 
pulsion in the same sense as before. 

MM. Riffault and Chompré in 1807 came to the conclusion 
that not merely decompositions, but actual separations of the 
elements took place throughout the whole of the humid con- 
ductors. They consider the negative current as collectitig and 
carrying the acids, &c., to the positive pole, and the positive 
current as doing the same thing with bases, leaving them at 
the negative pole. . 

M. Biot speaks very philosophically and cautiously, but refers 
the effects to the attraction and repulsion exerted by the poles 
and the parts in their vicinity. He does not appear to admit 
the successive recompositions and decompositions of Grotthuss 
and Davy, but seems to consider the particles under transfer 
as attached for the time to the electricity, and carried on by it, 


* Annales de Chimie, 1806, tom. lviii. p. 64. 
+ Philosophical Transactions 1807, pp. 28, 29, 30; Elements of Chemical 
Philosophy, p. 160. 


Ca a le aS aS a nae 
: 


‘ TRANSACTIONS OF THE SECTIONS. 397 


Decomposition is supposed to take place at both poles, and not 
at all in the intermediate parts, which merely act as imperfect 
conductors *. 

M. A. De la Rive experimented and wrote on this subject 
in 1825. He says, those who refer the effect to the attraction 
of the poles only express the fact, and do not explain it. He 
thinks the results are due to a combination of the elements with 
the electricities at the poles by virtue of a species of play of 
affinities between the substances and the electricities. ‘The 
current at the positive pole unites with the hydrogen, combus- 
tibles, metals and bases it may find there, leaving the oxygen 
and the acids at liberty to escape: it passes across to the ne- 
gative pole, on the surface of which it leaves the ponderable 
bodies, entering itself into the metal, and passing away}. The 
electricity at the negative pole does the same thing with oxygen, 
chlorine, acids, &c., and carrying them across to the positive 
pole, there leaves them. Thus half the elements on each side 
are ejected by a species of superior affinity, and the other half 
are brought from the opposite pole in combination with the 
electricity of that pole. Successive decompositions and recom- 
positions are not admitted by M. De la Rive. Decomposition 
takes place at both poles, and only at the poles, and the inter- 
vening parts of the humid conductor are themselves unaltered. 

Mr. Faraday contended that the experiments he had de- 
scribed opposed generally all those views which referred electro- 
chemical decomposition to the attractions and repulsions of the 
poles. Thus, when the elements were evolved against air, the 


’ poles as attractive agents were absent; or if it be theoretically 


said that the air then acted as poles, and attracted the element, 
which, however, could not pass across it to the superior metallic 
poles on the outside, then he quoted the water pole, where the 
element might have gone to the metal, according to the theory, 
but was found not to do so; and as the evolved elements can 
be made to appear against air, which is not a conductor, nor is 
decomposed; or against water, which is a conductor, and can 
be decomposed ; as well as against metals, which are excellent 
conductors, but undecomposable, there appears but little reason 
to consider the phznomena generally as due to the attraction 
or attractive powers of the latter, when used in the ordinary 
way, since similar attractions can hardly be imagined in the 
former instances. 

_ Mr. Faraday’s view is, that the effect is produced by an zz- 
ternal corpuscular action, exerted according to the direction of 


-* Précis Elémentaire de Physique, 1824, tom. i. pp. 637, 641, 642, &c. 
} Annales de Chimie, tom. xxviii. pp. 190, 200, 202. 


598 THIRD REPORT—1833. 


the electrical current, and that it is due to a force either super- 
added to, or giving direction to, the ordinary chemical affinity 
of the bodies present. ‘The decomposing body may be consi- 
dered as a mass of acting particles, all those which are included 
in the course of the electrical current contributing to the final 
effect: and it is because the ordinary chemical affinity is re- 
lieved, weakened, or partly neutralized by the influence of the 
electric current in one direction parallel to the course of the 
latter, and strengthened or added to in the opposite direction, 
that the combining particles have a tendency to pass in oppo- 
site courses. 

In this view the effect is considered as essentially dependent 
upon the mutual chemical affinity of the particles of opposite 
kinds. A particle could not travel from the negative towards 
the positive pole unless it found other particles in its course 
inclined to travel from the positive to the negative pole. As far 
as regards two compound particles, the case may be considered 


as analogous to one of ordinary chemical decomposition; but . 


as all the compound particles in the course of the electric cur- 
rent, except those actually in contact with the poles, act con- 
jointly, the case becomes more complicated, though not more 
difficult of comprehension. 

. Many facts were then quoted in illustration of the manner in 
which particles acting under the force of their ordinary che- 
mical affinities, governed in a greater or smaller degree for the 
time by the passing current, according to its force or quantity, 
would render themselves, or be excluded at the opposite extre- 
mities of the decomposable mass in the direction of the currents. 
Dilute sulphuric acid of a certain strength was found, when sub- 
mitted to the action of the pile, to yield after a time a stronger 
acid at the positive than at the negative side; @. e. a certain 
quantity of acid had been transferred from the negative towards 
the positive pole: but when the acid was previously neutralized 
by an alkali, then a far greater quantity of acid was transferred 
by the same current of electricity. These and many other ex- 
perimental results will appear in the fifth series of Experimental 
Researches, to be published in the Philosophical Transactions. 

All the phenomena of transfer of acids through alkalies, &c. 
&c., which appeared so extraordinary to Davy and others when 
they were discovered, are the necessary consequence of the 
theory of Mr. Faraday ; and, in fact, such transferences are by 
him considered as essential to electro-chemical decomposition. 

The manner in which the evolved elements are set free at the 
poles, inexplicable upon the old theories, are necessary conse- 
quences of his theory; or if they combine, still the effects are 


pe Ae > eer ae 


Cee 


TRANSACTIONS OF THE SECTIONS. 399. 


natural results due to the influence of the chemical affinity, 
modified by the current. The refusal of elements or substances 
present to collect at the poles unless they are in relation by 
solution or chemical affinity with the substances present, also 
finds its natural reason in the theory. 

The fact that an element will sometimes go to one pole and 
sometimes to the other, according to the substances with which 
it is in association at the time, is an immediate result-of the 
theory. Nitrogen is said to do this: it will go freely to the po- 
sitive pole, and doubtfully to the negative pole. Water will go 
either to the positive or negative pole, and sulphur will also do 
so: from oxygen it will go to the negative pole, from silver to 
the positive pole. 

Want of time and apparatus prevented a further develop- 
ment of this view and its consequences, but it will appear in de- 
tail in the next Part of the Philosophical Transactions. 


Experiments on Atomic Weights. By Dr. Turner. 


Dr. Turner reported to the Meeting that he had continued 
his researches into atomic weights, and had to his own convic- 
tion determined the points which had induced him to undertake 
the inquiry. ‘These were, first, to form an opinion of the re- 
lative accuracy of the tables of equivalents employed in this 
country and on the continent; and, secondly, to ascertain whe- 
ther there existed any trustworthy evidence in proof of the 
hypothesis that the equivalents of bodies are multiples by whole 
numbers of the equivalent of hydrogen. To examine these 
questions he endeavoured to ascertain by careful and often- 
repeated experiments the equivalents of silver, chlorine, lead, 
barium, thercury, and nitrogen, in relation to oxygen. These 
were selected in consequence of their frequent use in analysis. 
An error in these could not exist without affecting the equiva- 
lents of nearly all the other elementary substances. The re- 
searches on this subject had been lately read before the Royal 
Society, and would probably, ere long, be published in some 
form or other. The general result is, that the atomic weights 
current in this country are much less exact than those given 
by Berzelius; that though they had been recommended to 
British chemists as rigidly correct, they were often very inexact, 
and had been determined by methods which in some important 
cases were defective. Further, he finds that as far as experi- 
mental evidence at present goes, the hypothesis above alluded 
to is unsupported. In some instances the equivalents are so 
nearly simple multiples of that of hydrogen that they may be 
taken as such without appreciable error; but in many other 


400 THIRD REPORT—1833. 


cases the numbers given by experiment cannot. be reconciled 
with the hypothesis. The following are the numbers which he 
is disposed to believe very nearly correct :—lead, 103-6; silver, 
108 ; chlorine, 35°42; barium, 68-7; mercury, 202, perhaps 
slightly higher, but not higher than 202°3; nitrogen, 14-2. Dr. 
Turner states that his methods for ascertaining nitrogen were 
not so advisable as that in which Dr. Prout is occupied by 
weighing the gases. This weight should be kept in abeyance 
for the present. He conceives that it does not fall below 14, 
nor exceed 14°2. During these researches he incidentally ob- 
tained some facts for inferring the equivalent of silver; and. 
from these it appears that the equivalent of sulphur is nearer 
16:1 than 16. He would not venture, however, to make a 
positive statement without further inquiry. 

He then mentioned that Dr. Prout had kindly informed him 
of a fact which he conceived analytical chemists in general to be 
ignorant of, and which he thought might have had an influence 
on these researches. The fact is, that chloride of silver, however 
white and well washed, gives out a little muriatic acid at the 
moment of fusing: This fact Dr. Turner has examined, and can 
confirm. It especially ensues when fusion takes place before 
the chloride has been well dried ; but in the event of the chloride 
of silver being first well dried at 300° (when no acid is given 
out), and then, without exposure to the atmosphere while cold, 
fused, the loss of acid is not appreciable in weight, though it is 
still sufficient to redden delicate litmus paper. In two experi- 
ments about fifty grains of chloride of silver were fused, (pre- 
viously dried at 300°, introduced while hot into a dry bottle 
furnished with a tight cork, and weighed in that state,) and the 
loss was inappreciable. From this circumstance, taken in con- 
junction with the mode in which he habitually weighs the 
chloride of silver, he is satisfied that the fact observed by Dr. 
Prout does not necessarily produce any error in the determi- 
nation of chlorine by means of silver. 


Notice of a Method of analysing Carbonaceous Iron. By 
Professor JOHNSTON. 


Professor Johnston gave an account of a new mode of deter- 
mining the amount of charcoal in the carbonaceous irons, by » 
which he hopes to obtain results more precise and trustworthy 
than those arrived at by any former mode. This method con- 
sists essentially in reducing the iron to fine powder in a steel 
mortar, and burning it with oxide of copper. Mr. Johnston 
expects to be able to lay a series of results before the next 
Meeting of the Association. 


‘ews eS re 


TRANSACTIONS OF THE SECTIONS. 401 


‘Communication respecting an Arch of the Aurora Borealis. . 
By R. Porter, Jun.* 

A very luminous arch of an aurora borealis was observed at 
Edinburgh by Professor Forbes on the evening of the 21st of 
March 1833. It was observed at Athboy in Ireland by the 
Earl of Darnley, by Dr. Robinson at Armagh, and also by a 
correspondent of one of the Carlisle newspapers. 

‘The observations demonstrate the view of the symmetrical 
arches being similar to parallels of latitude round the magnetic 
axis, the arch being seen in those positions at Edinburgh, 
Armagh, and Athboy which such a direction requires. 


Report of Experiments on the Quantities of Rain falling at dif- 
ferent Elevations above the Surface of the Ground at York, 
undertaken at the request of the Association, by Wiua1aAM 
Gray, Jun., and Joun Puttuirs, F.R.S. G.S., Secretaries of 
the Yorkshire Philosophical Society ; with Remarks on the 
Results of these Experiments, by Joun Putuutrs, F.R.S. G.S. 


I. Report of the Experiments.—York, the site of these 
experiments, stands in the centre of perhaps the most uniform 
and extensive vale in England, reaching from the mouth of the 
Tees to the estuary of the Humber, a length of 70 miles, with 
a breadth of from 15 to 25 miles. In this vast space no ground 
rises more than 100 or 150 feet above the level of York ; and 
the Minster, elevated 200 feet from the ground, looks down upon 
an area of above 1000 square miles, in which hardly any object, 
whether of nature or art, rises to within 100 feet of its summit. 

On the east the vale is bordered by the range of the Wolds, 
whose extreme height is 805 feet, and the escarpments of the 
oolitic system, which swell to 1485 feet. On the west, the 
distant hilly regions of the coal and limestone tract appear 
above the low plateau of magnesian limestone. 

- These circumstances of situation give an importance to the 
moderate height of York Minster which is denied to many 
loftier buildings in England. From its summit the course of a 
aed storm may be well traced from even the distant hills of 
ichmond ; and the deflections occasioned by the attraction of 
the sides of the vale, the rushing of the air, the sudden fall 
of temperature, and many other curious phznomena accom- 
panying the precipitation of rain, may be well observed. 
_ It is, probably, to the peculiarity of its geographical situation 
that we are to attribute the remarkable general regularity of the 
curves of mean temperature at York; for the deviation of the 


* See Lond. and Edinb. Philosophical Magazine, Third Series, vol. iii. p. 422. 
1833. 2D 


402 THIRD REPORT—1833. 


daily mean temperature at this place from the annual mean is 
pretty exactly proportional to the sines of the sun’s declination 
25 days before the day of observation. The mean temperature 
of the year is 48°-2; of July 62°, of January 34°°5. Average 
quantity of rain 24 inches. Prevalent winds W. and S.W.: 
about the vernal equinox N.E. winds are frequent. 

The Yorkshire Museum lies nearly west of the Minster, en- 
tirely beyond the city, which here encircles the Minster by a 
narrow belt of houses. Its roof is the highest in the immediate 
vicinity: it stands free from other buildings, and is surrounded 
on every side by the grounds of the institution. 

In these grounds, south-west of the Museum, the third sta- 
tion is taken, in the midst of a large grass-plat. ‘The second 
and third stations are nearly equidistant from the Minster: 
the intervening distances are, Pace 

Between the Minster and the Museum gauges . 1100 © 
the Museum and garden gauges ... 186 


Elevation of the gauges above the river, which is nearly 


level with high water in the Humber: PE a. 
Minster top gauge, raised on 9 ft. pole . . . 241 104 
Museum top Bade Vy.) ceil ee oY 72 8 
Gauge in grounds ...... Ve? Ae ee 29° 0 


From these data it will appear that it would be difficult to 
select three points more remarkably embracing the desired 
conditions of gradation of altitude, openness of sky, and con- 
tiguity of position. : 

The gauges employed are of the simplest construction. A 
cubical box of strong tin, exactly 10 inches by the side, open 
above, receives, at an inch below its edge, a funnel sloping to 
a small hole in the centre. On one of the lateral edges of the 
box, close to the top of the cavity, is soldered a short pipe, 
in which a cork is fitted. The whole is well painted. This is 
the gauge. The water which enters it is poured through the 
short tube into a cylindrical glass vessel graduated to cubic 
inches and fifths of cubic inches. Hence one inch depth of 
rain in the gauge will be measured by 100 inches of the gra- 
duated vessel, and +,!5;th of an inch of rain may be very easily 
read off. All the gauges were made on the same mould, and 
the same glass jar has been used in every observation. 

The gauge in the ground has its edge nearly level with the 
grass ; that on the Museum projects 11 inches above the stone- 
work; and the Minster gauge is supported on a pole nine feet 
above the level of the battlements of the great tower, whose 
top is 70 feet square. ; 


TRANSACTIONS OF THE SECTIONS. 


Table of Results for Twelve Months. 


1832, 1833. 


Feb. 4 to Feb. 13. 


September 


October . 
November 


December . 


January . 
February . 


Total of 12 months} 15-910 
Ditto, exclusive of 
snow in Feb. 1833 


Minster. 
Inches, 

‘060 

20. -010 
27. 017 
5. 174 
12. *198 
19. 052 
041 

2. “005 
9. ‘701 
16. 013 
23. “249 
30. 1-113 
ve 375 
14. 133 
21. 088 
002 

5. 557 
12. "958 
2. “908 
9. +351 
16. “999 
113 

6. ‘711 
13. 033 
27. 1-639 
3. 1:388 
17. *376 
8. 439 
12. 1°459 
30. 1:019 
17. “708 
14. “836 
1, Snow *195 


} 15715 


Museum. 


Inches. 


7119 
010 


7020 


“251 
273 
062 
116 
-006 


756 


“015 
"353 
1:574 
442 
203 
141 
010 
“719 
1-138 
1-166 
397 
1-115 


133 | 


*785 


*050 


1911 
1:747 

500 

*605 
2-080 
1308 
1:012 
1165 


Snow'279 


20°461 
20°182 


Ground. 


Inches. 
"147 
-008 


018 


+366 
*386 
093 
*238 
004 


Snow ‘616 drift- { 
ed. 


24:401 
23°785 


Q2p2 


Abundance of Aphodii 
in the Minster gauge. 


Remarks. 


Violent gales. 


{ Perpendicular 


without a trace of 
wind, in large drops. 


Thunder storm. 


Abundance of small 
Hymenoptera in the 
Minster gauge, but 
not in the others. 

ditto. 

(This occurrence has 
been frequently noticed 
in the summer and au- 


Ditto 


tumn.) 


Drifted snow on the top 
of the ground gauge. 


403 


rain, 


404 THIRD REPORT—1833. 


II. Remarks on the Results of the Experiments.—The pre- 
ceding table of results appears sufficient, when combined with 
some other data which I have obtained, to authorize some in- 
teresting inductions concerning the law and the cause of the 
remarkable inequality of the quantities of rain at different ele- 
vations above the ground. 

1. The notion which is most generally entertained of the 
cause of this inequality is, that wind, blowing horizontally, 
causes fewer drops of rain to fall upon the more elevated 
gauges. That this notion is a mere fallacy, the least acquaint- 
ance with mechanics is sufficient to prove; for certainly the 
number of drops of rain which fall, under the joint influence 
of gravitation and ordinary wind, upon horizontal surfaces, 
will be, ceteris paribus, exactly the same at all elevations be- 
low the point from which the rain descends. 

2. It is supposed by some that eddy winds, produced by 
the sides of buildings and rising upwards, may deflect the rain 
so as to prevent much of it from falling on those buildings. It 
is certainly conceivable that this irregular action against gra- 
vity may, when very violent, under particular circumstances, 
produce a sensible effect, and such appears to be recognised 
by our experiments, in one instance, during the equinoctial 
period of March 1832. 

But it is evident that in the majority of cases the effect of 
the eddying wind is quite unimportant. I have noticed in se- 
veral instances the fact, that the wind which accompanies the 
fall of rain takes the line of the rain-drops themselves ; and 
on the Minster, in particular, this was very strikingly illus- 
trated, when, with my friends Mr. Jonathan Gray and Mr. 
William Gray, junior, I watched the progress of a storm for 30 
miles down the vale of York. The wind was insensible except 
during the fall of rain, and then it came downward with the 
drops. There is no need of further remarks on this subject, 
because the results recorded are too regular, considerable, and 
consistent with known properties of the atmosphere, to be 
explained by such fluctuating and inadequate agency. 

3. With respect to the observations on the ground, I have 
procured several registers of the rain which fell in and about 
York, for comparison with the observations in the Museum 
garden. By this comparison it is abundantly evident that the 
situation of the gauge, its exposure to eddy winds, and other 
irregularities, have very little influence upon the mean results. 
While the gauge in the Museum garden is remarkably open on 
all sides, and set devel with the ground, Mr. J. Gray’s is raised 
three feet above, and placed ina small garden, surrounded by 


TRANSACTIONS OF THE SECTIONS. 405 


high walls and buildings. Yet in the Museum garden we have 
for 12 months 23°785 inches of rain, and Mr. J. Gray’s results 
for the same period are 23:020. (The snow which fell in this 
period is excluded from both these numerical statements.) ‘The 
depth of rain appears pretty nearly uniform over the broad 
vale of York, and even beyond it. Thus at Ackworth, 25 
miles S.W., the quantity collected in 1832 .... . =24°94 
At Brandsby, 12-miles N. (station nearly level with the 

top of York Minster) .. 1... + +++ ee ee + 25°69 
a ere CE CO ow ne ai a lan i se LOT 


4. I shall now proceed to arrange the numerical results of 
the experiments, in relation to mean temperature and the 
season of the year, and thence to énfer the ratios of quantity 
at the several stations. The quantity of snow which fell is 
always deducted, because it was found to drift into the lower 

auge. This quantity was, however, very small and only sen- 
sible in February 1833. = 


On On On 
Periods. Mean Minster. Museum. Ground. Ratios, 
Temp. | In. of Rain.| In. of Rain.| In. of Rain. 


Whole year... . . | 48:20| 15°715 | 20-182 | 23-785 

7 coldest months, 
Oct., Nov., Dec., j : y : 
Jan., Feb., Mar., 40°8 7:089 9-725 | 12:079 
April 

7 warmest months, 
April, May, June, 
July, Aug., Sept., 
October .... 

5 coldest months, 
Nov., Dec., Jan.,, 
Feb., March 

5 warmest months, 
May, June, July, 
Aug., September 

Winter quarter, 
Dec., Jan., Feb. 

Spring quarter, 
Mar., April, May 

Summer quarter, 
June, July, Aug. 

Autumn quarter, 


Sept., Oct., Nov. 
* In 1833, The Museum gauge gave. . . - + - 22959 inches. 
Mr. J. Gray’s 23-060 


Dr. Wasse, at Moat Hall, (10 miles from 


York)... 23-895 


406 THIRD REPORT—1833. 


5. The first remark which I shall make on these results is, 
that the diminution of the quantity of rain received at different 
heights above the ground, as compared with that received on 
the ground, is very accurately represented by a simple formula 
involving one constant, viz. the square root of the height of 
the station above the ground, and one variable coefficient. 

Thus, m / h = the diminution of rain at the given height. 
In these experiments 

WV h for the Minster gauge =° / 212°833 = 14°5885 
for the Museum gauge = / 43°666 = 6°6080 


Taking m = 2°29, we have for the whole year, 
: By calculation . . .. 66°5 84°97) 
Ratios : 
By observation .. 66:1 we 


For the 7 coldest months (m = 2°88,) 
By calculation ... 58 81 
By observation .. 58°6 80°5 
For the 7 warmest months (m = 1:97,) 
By calculation ... 71°3 87-0 . +, 100 
By observation .. 71:2 SS ORE Aer 
For the 5 coldest months (m = 3:06,) 
By calculation... 55°4 — 79°8 
By observation .. 562 79 
For the 5 warmest months (m = 1°75,) 
By calculation... 745 88:4 
By observation .. 73°7 89°2 J 


In these, which are the longest averages attainable from the 
experiments, there is an almost exact agreement between 
the calculated and the observed results, the greatest error 
being +3,th. 

‘In shorter averages of three months, and, indeed, though 
less exactly, in every single month when much rain fell, we 
may recognise the same constant relation. Thus we have 

For the summer quarter (m = 1°43,) 7 
By calculation . . . 79-0 90°5 
By observation .. 771 92°5 
For the winter quarter (m = 3°79,) tel 
By calculation... 446 747 
By observation .. 49°3 70°5 ‘ to 100 
For the spring quarter (m = 2°84,) ; 
By calculation... 58°6 811 
By observation .. 59°8 80:0 
For the autumn quarter (m = 2°19,) 
By calculation ... 68:1 85°4: 
By observation .. 65°8 87°7 J 


TRANSACTIONS OF THE SECTIONS. 407 


The three most rainy months of the year 1832 were June, 
August, and November. 


For June we have, 7 
By calculation... 77°6 90:1 
By observation .. 74° 5 93°2 

For August, 
By calculation ... 77°6 90:1 
By observation .. 77:9 89:8 | 

For November, 

By calculation .. . 70°2 86°8 

By observation .. 663 90°7 


6. From these comparisons it appears to follow, that though 
the exact relation between the diminution of rain and the 
height of the station can hardly be considered as satisfactorily 
determined by the experiments of twelve months, the nature 
of this relation is so far ascertained that we may conclude it 
to be constant for all periods of the year, and that the form 
WV h is a good first approximation. 

7. The account of Dr. Heberden’s experiments on West- 
minster Abbey does not state the elevation of the stations ; 
yet if we take the height of the square part of the roof at 
about 120 feet, and from this infer, according to the formula, 
the height of the house-top which was the middle station above 
the point below the house-top which was the lowest station, 
we shall still be able to use those experiments as a check upon 
the law of the ratio now given. In this case, fA = 11:0 
and 4°6, and we have : 


For the whole year, qoin 
By calculation (m = 4°23)... 53°5 80°5 
By observation... ..... 53°5 80:5 


For the 7 coldest months, 
By calculation (m = 4°26) . 53:1 80:4 
By observation ....... 53 80°5 
For the 7 warmest months, 
By calculation (m = 3:90) . 57:1 is D 
By observation ....... 57 
For the 5 coldest months, 
By calculation (m= 4°70) . 48:3 78°53 
By observation ....... 48°6 78°0 
For the 5 warmest months, 
By calculation (m = 4:19) 53°9 ms °| 
By observation ...... 54°5 


to 100. 


CH 


408 THIRD REPORT—1833. 


For the winter quarter, 7 
By calculation (m = 4°42) . 51°4 79°6 
By observation ....... 51°5 79°5 
For the summer quarter, 
By calculation (m = 4°14) . 55°5 808 
By observation ¢ it! sa. Wee 52°7 82°6 
For the spring quarter, Mee 
By calculation (m = 4°92) . 45:9 17:2 | 
*By observation. ;... 00°20. 48 75°1 


For the autumn quarter, 
By calculation (m = 3°71) . 59:2 82:9 
By observation ....... 57°8 84°3 J 
It is therefore probable that the Westminster results obey the 
-same constant relation to height as those of York. 

8. But it is evident that the values of the variable coefficient 
are very different ; that its maxima and minima are perhaps 
not quite in the same periods of the year as at York; and 
that ‘the range of variation in its value is very much less. 
From M. Arago’s determination of the relative quantities of 
rain falling on the Observatory at Paris and in the court 28 
metres below, as given by Professor Forbes, in his Report on 
Meteorology, 50:47 : 56°37, the relative mean value of m at 
Paris = 1°24, while at Westminster it was 4°23, at York 2°29. 
It must be owned that these discrepancies with other obser- 
vations as respects the quantity of the diminution of rain up- 
wards are somewhat discouraging, and probably will, for a 
considerable time, deprive the most exact local determinations 
of this quantity of a general application. This, indeed, could 
hardly be expected, since the whole quantity of rain is so va- 
riously dependent on circumstances of physical geography, 
that centuries have been found insufficient to determine the 
general law and ascertain the numerical constants of local 
climate. Yet, on account of the remarkable regularity of the 
progress of monthly temperature at York, and some very ob- 
vious relations between the quantities of rain collected and 
the mean temperature of the period, I will venture to state 
what seem unavoidably to suggest themselves as probable in- 
ferences. 

9. First, it is obvious that the diminution at the upper sta- 
tions is greatest in the cold season, and least in the warm 
season, and therefore the coefficient is in some way znversely 
dependent on the temperature, or on some effect of this tem- 


* March very anomalous. 


ee ee ee ee eee 


TRANSACTIONS OF THE SECTIONS. 409 


perature. If we consider it in relation to the mean tempera- 
ture, we shall find a near coincidence between the following 
t i 
; —~ta-, 
formula, and observation: m =* 7 ik ¢*, where a = the 


ascertained value of m, for the whole year, ¢ the mean tempe- 
rature of the year, and ?¢' that of the particular period under 
consideration. 


Value of m| Value of m Difference: 


For the 7 colder months .. 40'S |= 2°98 


7 warmer months..| 55°5 1°87 
5 colder months ..| 39:3 3°16 
5 warmer months..| 58°5 1:74 
Winter quarter....| 363 3°57 
Spring quarter....| 47°6 2°35 
Summer quarter ..| 60°8 1-64 


Autumn quarter ..| 48:3 2°30 


10. Secondly, it is obvious that the relation between the 
values of m and the dryness of the air is inverse. This dry- 
ness is usually expressed by the difference between the mean 
temperature and the mean dew point, and where the latter is 

erfectly determined, no better plan perhaps can be suggested. 
But this is the case for very few places in Great Britain. There 
is, however, another mode of expressing the dryness of the air, 
which is fortunately applicable to the present purpose; the 
mean range of daily temperature, or mean difference of maxima 
and minima, is a good approximation to an accurate expression 
of the relative dryness of the air. The following table of the 
mean ranges of temperature near York has been determined 
with the greatest nicety, by long averages, from the careful 
and continued observations of Francis Cholmeley, Esq., of 
Brandsby. 


January mean range.... 8°0 July: mean range ....:. 19°6 

BEDURAIRe js \<:- 30s apnteyl ae 10-1 AME USbic wiped «AF di 218 17-7 

ES an re es 13°1 Senter Ber oy. a uicisadetieds 16:0 

To nahin sean 16:2 9 Fee) 17 a ee nT 118 

Ea 19°7 INOVEMIDER ws. nicsnteg ass, « 9:0 

Nr ct cin nin 20°1 DE CARA DEE pies non crac ion 
General mean Tage... ws os 14:08 


On comparing these numbers with Mr. Daniell’s estimates of 
the dryness of the air in London, they will be found analogous 


410 THIRD REPORT—1833. 


in general proportions. ‘They may also be compared with an 
excellent series of dew point observations in the Manchester 
Memoirs by my triend Mr. John Blackwall, whose results in 
other respects | have found remarkably in accordance with my 
own inferences concerning the climate of York. 

Now, let m be taken inversely as the mean range of tempe- 


= we shall have the following 


rature, (r) orm=a 


comparison between the calculated and observed values. 
(a = 2°29.) 


Value of m | Value of m 


Pet es ae | Difference. 
For the 7 coldest months...... = 2°98 = 2°88 0:10 
7 warmest months .... 1°86 1:97 O11 
5 coldest months...... 3°36 3°06 0°30 
5 warmest months .... 1°73 L- 73 0-00 
Winter quarter ...... 3°74 3°79 0:05 
Spring quarter........ 2°48 2°84 0°36 
Summer quarter ...... 1°68 1°43 0°25 
Autumn quarter ...... 2°63 2°19 0:44 


Na a a 


So remarkable and continued an accordance between the co- 
efficients fixed by observation and those derived by two methods 
from a very simple view of the condition of the air as to heat 
and moisture, appears to me decisive of the question as to 
the general cause of the variation of the quantity of diminution 
of rain at any one height above the ground. It has already 
been shown how strictly the observations warrant the con- 
clusion that the ratio of diminution at different heights is con- 
stant through the whole year. It is therefore rather as a 
matter of very probable inference than a plausible speculation 
that I offer the hypothesis, that the whole difference in the 
quantity of rain, at different heights above the surface of the 
neighbouring ground, is caused by the continual augmentation 
of each drop of rain from the commencement to the end of its 
descent, as it traverses successively the humid strata of air at 
a temperature so much lower than that of the surrounding me- 
dium as to cause the deposition of moisture upon its surface. 
This hypothesis takes account of the length of descent, because 
in passing through more air more moisture would be gathered ; 
it agrees with the fact that the augmentation for given lengths 
of descent is greatest in the most humid seasons of the year ; 
it accounts to us for the greater absolute size of rain-drops in 


ee) 


TRANSACTIONS OF THE SECTIONS. 411 


the hottest months and near the ground, as compared with 
those in the winter and on mountains; finally, it is almost an 
inevitable consequence from what is known of the gradation of 
temperature in the atmosphere, that some effect of this kind 
must necessarily take place. The very common observation 
of the cooling of the air at the instant of the fall of rain, the 
fact of small hail or snow whitening the mountains, while the 
very same precipitations fall as cold rain in the valleys where 
the dew point may be many degrees above freezing, is enough 
to prove this. A converse proof of the dependence of the 
quantity of rain at different heights on the state of the air at 
those heights, is found in the rarer occurrence of a shower 
falling from a cloud, but dissolving into the air without reaching 
the ground. Lastly, I cannot forbear remarking, that this hy- 
pothesis of augmentation of size of the elementary drops 
agrees with the result that the increase of quantity of rain for 
equal lengths of descent is greatest near the ground; for 
whether the augmentation of each drop be in proportion to its 
surface or its bulk, the consequence must be an increasing 
rate of augmentation of its quantity as it approaches the 
ground. 

The direct mathematical solution of this problem, now that 
the laws of cooling and of the distribution of temperature have 
undergone such repeated scrutiny, may perhaps be attempted 
with success; but for the purpose of eliminating the effects of 
periodical or local modifying causes, it is desirable that obser- 
vations on the same plan should be instituted at many and di- 
stant places,—both along the coasts and in the interior,—in the 
humid atmosphere of Cornwall and in the drier air of the mid- 
land counties. Always, at least three stations should be chosen, 
as open as possible, one of them very near to the ground: 
their relative heights, the mean temperatures, the mean 
ranges of temperature, and the mean dew point for each 
month should be ascertained. It would be useful to measure 
‘the size of the rain-drops, and, if possible, their own tempe- 
rature. The height of clouds, according to the plan of Mr. 
Dalton, in his Meteorological Essays, and the direction and 
force of wind should be noted, and distinctions made between 
snow; hail, and rain. Some of these data I have not yet found 
the means of procuring, partly in consequence of the great 
labour and time required, and partly from the difficulty of well 
arranging the experiments themselves. But since it is now 
ascertained that the general results follow some settled laws, 
and that the effects may be very well appreciated at moderate 
_ heights, I hope not only to procure these, but also several 


412 THIRD REPORT—1833. 


other data towards the completion of the theory of this curious 
subject, the patient investigation of which cannot fail to give 
us new and penetrating views into the constitution of the at- 
mosphere. 


II. PHILOSOPHICAL INSTRUMENTS AND 
MECHANICAL ARTS. 


On a peculiar Source of Error in Experiments with the Dip- 
ping Needle, By the Rev. Wituiam Scoressy, F.R.S. 


Certain discrepancies, at the time apparently unaccountable, 
in observations made with a beautiful dipping-needle, by Dol- 
lond, entrusted to Mr. Scoresby’s care by the Board of Lon- 
gitude, in an arctic voyage, led him, after a considerable interval 
of time, to reflect upon the cause. Whatever might be the 
apparent consistency of any particular series of observations 
in the ordinary use of the instrument, the differences perceived 
when the poles of the needle were changed indicated that 
the preceding results were not accurate. But as the different 
results thus obtained were capable of being combined for ob- 
taining the true position of the needle, the formula of Professor 
J. Tobias Meyer, given in his treatise De Usu accuratiori 
Acts Inclinatorie Magnetice, was adopted for this purpose. 
To verify the position thus obtained, another series of obser- 
vations on the same spot was then made, with one of the arms 
of the needle weighted, so as to render its position more de- 
cisive from being the resultant of two forces, gravity and 
magnetic attraction. The results, however, of the different 
series were again anomalous ; but the cause of the discrepancies 
thus observed not being fully apprehended at the time, the 
consequence was that the observations were set aside for want 
of consistency. 

Subsequently, however, it occurred to the author that the 
cause of the discrepancies was to be found in the alteration of 
the magnetic intensity of the needle when the polarity was 
changed; a circumstance furnishing a new element in these 
calculations, not hitherto, he believes, taken into account. For 
when the poles of the dipping needle are changed, (unless 
magnetized with extraordinary care and some perseverance in 
repeating the process,) the magnetic intensity of both positions 
will not be the same. It is indeed a fact which the author fre- 
quently observed, that the capacity of a properly tempered 
steel bar for the magnetic power (provided it have been kept a 
considerable time in the same magnetic condition,) is the great- 


TRANSACTIONS OF THE SECTIONS. 413 


est when the polarity is preserved in the usual way, and of 
consequence loses in intensity of directive force whenever the 
polarity is changed. This effect, then, if the needle be not 
perfectly balanced and suspended,—which, strictly speaking, it 
scarcely can be expected to be,—must necessarily follow, that 
the calculated position, by the usual formule, derived from 
observations made with the needle in opposite conditions of 
polarity, will not give the true direction of the magnetic force. 
For wherever there is any defect in accuracy of suspension, 
balancing, or adjustment, then the needle being acted upon 
by two forces,—that of magnetic direction, depending on its 
magnetic intensity, and gravitation, depending on the horizontal 
leverage of the centre of gravity of the needle in respect to the 
axis,—its position will, of course, be the resultant of the two 
operating influences. But if with one direction of the poles 
the magnetic intensity be less than in the other, then the re- 
sultant of the combined forces must change its direction from 
the preponderating of gravitation there; and, consequently, 
the effect of gravitation will not be neutralized by the ordinary 
mode of mutual correction, because of the relations of that 
force to the directive being changed. Of this, an example 
taken from arithmetical means will be sufficient for illustration. 
Suppose the dip, as determined by the mean of various ob- 
servations in the usual position of the needle, to be, say 70°, 
and the result on inversion of poles be 72°, then the actual 
dip will evidently not be the arithmetical mean of 71°, neither 
the mean of the tangents of 70° and 72°, thatis, 71° 3!, unless 
the magnetic intensity be in both cases precisely the same. But 
as the intensity in the original direction of the polarity would, 
probably, be very considerably greater than the other*, the 
real dip might be 70° 50’, or even 70° 40’, rather than 71°! 
Hence, for accuracy of result in such cases, a new element 
seems to be requisite—that of the relative magnetic intensities 
or powers of the poles of the needle under each condition of 
polarity—and by observing the number of vibrations of the 
needle in a given time, in each state of polarity, these reduced 
to actual intensities would afford an element as a corrective for 
the source of error herein under consideration. 


* The difference of intensity on changing the poles will be the most consider- 
able in the hardest-tempered needles, or in cases where the fixedness of the axis 
renders the best modes of magnetizing impracticable. In soft bars, or where 
very powerful magnets are used, the differences from this cause become com- 
paratively trifling, and sometimes altogether disappear; but still there is no 


security without verification, that in any case the intensities of the changed 
poles will be the same. ' 


414 THIRD REPORT—1833. 


On the Construction of a New Barometer. By the Rev. W.H. 
Mutter, F.G.S., Professor of Mineralogy, Cambridge. 


This barometer consists of two tubes, of equal diameter, a 
little longer than the greatest height and greatest range of the 
barometric column respectively, terminating in a small cistern, 
the bottom of which can be elevated or depressed by a screw. 
The long tube is bent so that the upper part of it, which is 
closed at the end and has a fine point of glass or steel fixed 
in its axis, may coincide with the prolongation of the short 
tube, which is open at the end. A graduated scale slides along 
a vernier attached to the frame of the instrument, in such a 
manner that a steel point fixed to the lower end of the scale 
may move in the axis of the short tube. 

In making an observation with this instrument, the bottom 
of the cistern must be elevated or depressed till the surface of 
the mercury in the long tube touches the fixed point therein : 
the moveable point on the scale being then brought down till 
it touches the surface of the mercury in the short tube, the 
height of the barometric column is indicated by the division of 
the scale opposite to zero on the vernier. The barometer may 
be rendered portable by placing a stopcock between the short 
tube and the cistern. 


On a Barometer with an enlarged Scale. By Wiuu1am L. 
WuartTon. 


In this barometer a light fluid is introduced upon the top of 
the mercurial column of the common barometer, the tube of 
the instrument being enlarged at the point of junction of the 
two fluids, by which device an instrument of equal or superior 
extent of scale may be obtained, without the expense and dif- 
ficulty attendant upon the construction and erection of a baro- 
meter of which the whole tube is occupied by the light fluid. 
Mr. Wharton has employed an instrument of this construction 
for twelve years without perceiving that its sensibility is at all 
impaired. , 


On the Construction of a new Wheel Barometer. By 
Wii1aMm Snow Harris, F.R.S., §e. 


The tube of this instrument is 0°5 of an inch in diameter 
within, bent in a siphon form at one end, and at the other ex- 
panded into a flattened spheroidal bulb, whose diameter is 
four inches, and axis, in the direction of the tube, two inches. 


eo 


a 


TRANSACTIONS OF THE SECTIONS. 415 


The straight part of the tube, exclusive of the bulb, is 32 
inches ; inclusive of the bulb, 34; the recurved end is bent 
twice at right angles, so as to project from the tube 3 
inches, and rise parallel thereto 7°5 inches. The tube is at- 
tached to a mahogany support, the spheroidal bulb being up- 
wards; and the quantity of mercury is so adjusted in the tube 
that at mean pressures the upper level is nearly coincident 
with the greatest diameter of the spheroid, and the lower is 
near the middle of the shorter leg. 

There is a circle of brass, divided into 1000 parts, fixed to 
the front of a light copper drum or case, having a glass front 
and back, the centre of which circle is placed just over the 
orifice of the glass tube: a small frame of brass is fixed to the 
circle behind, so as to carry a light horizontal axis bearing two 
small pulleys. The extremities of this axis are turned to ex- 
tremely fine pivots, and are set in small jewels: the front one 
projects forward so as to carry a light index of straw, which is 
sustained on a small brass ring, placed by means of a socket on 
the extremity of the axis, in the manner of the hand of a watch. 
The two small pulleys above mentioned carry, by means. of 
fine untwisted silk threads, two small cones of glass or wood, 
one of which rests on the surface of the mercury in the recurved 
tube, the other hangs freely on the outside of the tube. These 
cones are nearly equal in weight, that resting on the mercury 
being rather the heavier of the two. ‘This slight difference of 
weight, setting aside the inertia and friction of the axis, is the 
amount of the resistance which the rising or falling of the mer- 
cury in the tube has to contend with; and this is all extremely 
little, so little that the index moves by the unequal action of 
the wind during a light gale, and is put into a state of oscil- 
‘lation of some considerable duration by the mere opening of 
the door of a room. These pulleys measure very nearly one 
inch in circumference, so that if the mercury moves an inch 
the index is carried once round the brass circle, and hence one 


division thereon corresponds to ath of an inch, a correction 
being made on the pulley according to the relative capacity of 
the tube and the bulb. 

The index is made in three parts, of light straws, a centre 
piece and two extreme pieces inserted into it: one extremity 
is cut after the manner of a pen to a somewhat short and very 
fine point, which is turned edgeways. The whole is carefully 
equipoised by a short piece of straw sliding on one of the ex- 
treme pieces, so that when attached to the axis it takes indif- 
ferently any position in the circle, and, consequently, follows 
exactly the movement of the mercury. 


416 THIRD REPORT—~1833. 


A varnished paper is pasted on the front of the tube, marked 
27, 28, 29, 30, &c., to denote the height of the mercurial co- 
lumn in inches ; these measures being taken with care from the 
surface of mercury in the bulb to that in the tube. The index 
is set as nearly as possible when the altitude corresponds to 
fixed divisions of the scale or measure. 

There is a thermometer close to the mercurial column, the 
bulb of which is placed in a small cistern of mercury, to indi- 
cate the temperature, and a hygrometer to measure the change 
which may be supposed to happen in the silk line, to which 
the cone resting on the mercury is attached; but the author 
has found that by employing fine unspun silk the changes are 
quite unimportant. 

The quantity of mercury in the instrument is about 15 lbs. 
In order to fill the tube clear of air, the following process was 
adopted as a substitute for boiling. A small iron cap, polished, 
was first cemented air-tight upon the end of the tube, and into 
this was screwed an iron stopcock: a long glass tube was then 
cemented to the stopcock, furnished with iron caps, &c., so 
that by reverting the instrument and steadying the tubes by 
cross-bars of wood, tied with silk-ribbon band, the whole may 
be screwed into the plate of a good air-pump. 

The air being withdrawn as completely as a good air-pump 
will effect, the cock is closed, the whole is detached from the 
pump, the long tube removed, and the barometer tube trans- 
ferred to a cistern of mercury, under the surface of which the 
stopcock is fairly immersed, whilst the tube is inclined as much 
as possible. The operator, being placed in a convenient position, 
supports the ball of the tube in his hand, and turns the cock 
gradually, so as to allow the mercury to be pressed up in an 
extremely small stream into the tube, and to flow down without 
violence into the ball. During this process the ball is gently 
moved about with an easy circular motion, which allows of the 
more speedy union of the mercury and displacement of the air. 
An assistant should be ready to close the cock occasionally, for 
the purpose of examining the state of the mercurial mass 
within the tube. 

In this way the barometer tube may be filled with great 
nicety, so as to show a most resplendent surface, equal in ap- 
pearance to that produced by boiling even under a powerful 
magnifying glass. 

When the tube is complete to the point required, the stop- 
cock is again closed, the whole is reverted, and the tube is 
placed in its intended place; the cock is then opened by de- 
grees, and the mercury will gradually descend to the level of 


pe 


TRANSACTIONS OF THE SECTIONS. 417 


the atmospheric pressure. The iron stopcock and cap may 
now be removed, by cautious application of first a warm and 
then a hot iron rod to the cement. 

The mercury intended for the purpose of the barometer 
should be first distilled, and then well agitated, about an ounce 
or less at a time, in phials capable of holding one ounce and a 
half. Previously to introducing the mercury into the tube it 
should be well boiled in a crucible, of porcelain or Wedgwood 
ware, and should be used just before getting cold, at a tempe- 
rature of 90° or 100°, the tube of the barometer being also 
made a little warm by careful exposure to a charcoal fire. 

The process now described is believed by the author to be, 
when carefully performed, in every respect equal to that of 
boiling. The wheel-barometer made in this way has been com- 
pared with other instruments with boiled tubes and of un- 
doubted excellence, amongst others with a fine mountain baro- 
meter on Gay-Lussac and Renard’s principle, which had been 
compared with the standard of the Royal Society in Somerset 
House, and with that in the observatory at Paris. The dif- 
ferences from this instrument, when both were placed in the 
same room, were very minute. It is found to be more sensible 
than a very finely boiled tube, carefully prepared by that emi- 
nent maker Mr. Cox, of Plymouth, with a scale and vernier 


divided to awth of an inch, set up in an adjoining room. 


On a new Method of Constructing a Portable Barometer. 
By Joun Newman, Mathematical Instrument Maker. 


The object of this construction is to make barometers port- 
able without the use of a leather bag, which has always been 
a defective part of the instrument. 

The method adopted is to have a cylindrical cistern of iron 
in two parts, rather longer than usual, the upper part, or 
chamber, or that to which the cap is fastened, which connects 
to the tube, being about three times the length of the lower 
part, of the same diameter, moving round upon a pin, and 
secured by a screw and collar. The two chambers thus formed 
communicate internally in one situation by means of holes in 
the divisions, through which the mercury flows upon invert- 
ing the instrument. The vacant space, or that intended to 
receive the mercury from the tube when the barometer falls, is; 
when the instrument is in use, in the upper part of the upper 
cistern, the lower one being full. Upon inverting the instru- 
ante mercury flows from the latter into the former, which 


418 THIRD REPORT—1833. 


becoming filled, is by a quarter turn of the one now upper- 
most cut off from communication with it, and the instrument 
is rendered portable with the end of the tube dipping into a 
cistern of mercury, which is perfectly secure. 

By this method Mr. Newman is enabled to make portable 
mountain barometers with very large tubes, for sufficient room 
can be left in the cistern to receive the mercury which flows 
from the tube into the cistern in high situations, notwithstand- 
ing the increased diameter of the tube. Barometers, therefore, 
can be made and transported, which when put up may be de- 
pended upon as standard instruments with perfect security. 


On an Instrument for measuring the total heating Effect of 
the Sun’s Rays for a given time. By the Rev. James 
Cummine, V.P.R.S., F.G.S., Professor of Chemistry, 


Cambridge. 


It has appeared to Professor Cumming that the information 
conveyed to us by the ordinary instruments for measuring the 
heating power of the sun’s rays is, in one respect, imperfect, 
in as much as these instruments indicate only the momentary 
energy of the rays: he was therefore led to devise a process 
which should measure the total result of their action in a given 
time. The process employed is to expose to the sun a retort 
with a blackened bulb containing ether, and to note the quan- 
tities of this liquid distilled over in different days. In some 
cases, a second bulb of plain glass has been used to increase 
the condensing surface, and the apparatus has been otherwise 
modified. With instruments on this plan Professor Cumming 
has registered the daily effects of the sun’s radiation for more 
than two years, and he hopes soon to publish his results in a 
connected form. 


On some Electro-magnetic Instruments. By the Rev. James 
Cummine, V.P.R.S., Professor of Chemistry, Cambridge. 


The instruments exhibited and explained by Professor Cum- 
ming consisted of: ; 

1. A galvanometer of four spirals, similar to that described 
in his translation of Demonferrand, (pl. v. fig. 86,) but formed 
of flattened copper wire with silk ribbon interposed, each spiral 
being fixed upon a graduated slide. 

2. A Breguet’s thermometer, with a conducting wire passed 
through its axis, for the purpose of measuring either the heat 
evolved by different galvanic arrangements in passing through 
a given wire, or that evolved in different wires by the same 
battery. 


TRANSACTIONS OF THE SECTIONS, 419 


On the Thermostat, or Heat-governor, a self-acting physical 
Apparatus for regulating Temperature. By ANDREW 
Ure, M.D., F.R.S., &c. 


This instrument acts by the unequal expansion of different 
metals in combination: it admits of many modifications of ex- 
ternal form, but, im all, the metallic bars must possess such force 
of flexure in heating or cooling as to enable their working rods 
or levers to open or shut valves, stopcocks, and ventilating 
orifices. 

»- Steel and zinc are the two metals employed: they possess a 
great difference of expansiveness, nearly as two to five, and 
are sufficiently cheap to enter into the composition of 'ther- 
mostatic apparatus; but zinc has in reference to the present 
object one property which should be corrected. After being 
many times heated and cooled, a rod of that metal remains 
permanently elongated. This property may, however, be in 
a great measure destroyed, and considerable rigidity acquired 
by alloying it with four or five per cent. of copper and one of 
tin. Such an alloy is hard, close-grained, elastic, and very 
expansible, and therefore suits pretty well for making the more 
expansible bar of a thermostat. 
-» Let a bar of zine or of this alloy be cast, about an inch in 
breadth, one quarter of an inch thick, and two feet long, and 
let it be firmly and closely riveted along its face to the face of 
a similar bar of steel of about one third the thickness. The 
product of the rigidity and strength of each bar should be 
nearly the same, so that the texture of each may pretty equally 
resist the strains of flexure. Having provided a dozen such 
compound bars, let them be united in pairs by a hinge-joint at 
each of their ends, having the steel bars inwards. At ordinary 
temperatures the steel plates of such a pair of compound bars 
will be parallel and nearly in contact, but when heated they 
will bend outwards, receding from each other at their middle 
parts, like two bows tied together at their ends. Supposing 
this recession to be one inch for 180° Fahrenheit, then six such 
pairs of bows, connected together in an open frame with 
rabbeted end plates, and with a guide rod playing through a 
hole in'the centre of each, will produce an effective aggregate 
motion of six inches, being half an inch for every 15° Fahr., 
or 843° C.\ ‘Instead of limiting ourselves to half a dozen such 
pairs of compound bars, we may readily lodge in a slender iron 
frame a score or two of them, so as to furnish as great a range 


of motion as can be desired for most purposes of heat regu- 
2E2 


420 THIRD REPORT—1833. 


lation; and the power of pressure or impulsion may be in- 
creased, if necessary, by increasing somewhat the thickness of 
each component lamina. One extremity of the series must 
obviously be firmly abutted against a solid fulerum or bearing, 
while the opposite extremity gives motion to a working-rod of 
a suitable kind. 

The author of the communication then describes in detail 
the various mechanical adjustments by which the apparatus 
may be applied to maintain any determined rate of ventilation 
through the casements of church windows; to give alarm and 
open yalves in water-cisterns in case of fire; to preserve a cer- 
tain rate of combustion in furnaces, a uniform temperature in 
baths and stills, and to act as a safety-valve for a steam- 
engine. 


Ona Reflecting Telescope. By Tuomas Davison, of Low 
Brunton, near Alnwick. 


The author of the invention described in this communication 
is a weaver of linen, who has devoted himself with great per- 
severance and ingenuity to the construction of telescopes and 
other instruments. ‘The modification of the ordinary con- 
struction of a reflecting telescope which Thomas Davison has 
executed is intended to improve the performance of the in- 
strument by diminishing the false or aberrant light which in- 
terferes with the distinctness of the image. 

From the nature of its construction, the Gregorian telescope 
is most exposed to this defect, and it is, therefore, to that form 
of the instrument that the invention more particularly applies. 
To each reflector tubes are adapted, having their axes coin- 
cident with that of the mirrors, and their free ends directed 
towards each other. The tube connected to the great speculum 
enters the hole of that speculum, and is of a slightly conical 
form, diminishing cuban; and prolonged to such a degree as, 
without stopping many of the rays which should meet in the 
image, to prevent nearly all the false light from entering the 
eye-tube. The tube connected with the small mirror is pro- 
longed so as to meet the extreme rays which converge from 
the great speculum towards the ibaa Constructed in this 
manner, Thomas Davison’s telescope was found more effective 
than one upon the ordinary plan. By simple contrivances the 
instrument can be converted to the Cassegrainian or Newtonian 
form. 


TRANSACTIONS OF THE SECTIONS. 421 


On a Steam-engine for pumping Water. ByW.L.Wuarton. 


In this engine the steam is admitted from the boiler upon a 
deep float, occupying the top of a column of water contained in 
a metallic cylinder, placed in the flue of the boiler fire. The 
lower part of the column of water is connected by pipes to the 
under side of a piston, moving water-tight in a much smaller 
cylinder, fixed immediately above the pumps of any mine, to 
the rods of which is affixed the piston rod. By this arrange- 
ment the steam always acts upon a heated surface, and its 
power is applied to the pump rods without the intervention of a 
main beam, parallel motion, &c., and, consequently, without any 
expense for frame-work and buildings requisite for their support 
in other engines. The friction of this engine, moreover, is very 
trifling, a stratum of oil being introduced both above and be- 
low the piston. A rod or wire is attached to the float, and, 
passing through a stuffing box in the top of the large cylinder, 
works the hand gear at the proper periods after the admission 
and escape of the steam, and consequent depression and ele- 
vation of the water and float, within that cylinder. A condensing 
apparatus may be added, by which the atmosphere may be 
rendered available, in addition to the weight of the pump rods, 
to force down the piston in the small cylinder, and, conse- 
quently, the water and float to the top of the large cylinder, 
after each stroke of the engine. 


On the Application of a glass Balance-spring to Chrono- 
meters. By Enwarp Joun Dent. 


Mr. Dent described the various difficulties in the con- 
struction of chronometers dependent on the imperfection of 
metallic balance-springs, whether made of gold, or of soft or 
hardened steel; and explained the advantages which may be 
expected to arise from the substitution of some substance pos- 
sessed of greater and more regular elasticity. Glass appeared 
a substance likely to answer this condition, and when formed 
into a cylindrical spring, it promised, from the trials that had 
been made, to be both accurate and durable. An instrument 
was exhibited with the glass spring in movement. 


ost 
On the Effect of Impact on Beams. By Karon Hopexinson. 


The author gave the results of some inquiries into the power 
of beams to resist impulsive forces. The experiments were 


422 THIRD REPORT—1833. 


made by means of a cast-iron ball, 44 Ibs. weight, suspended 
by a cord from the top of a room with a radius of 16 feet. The 
ball, when hanging freely, just touched laterally an uniform 
bar of cast-iron, sustained at its ends in a horizontal position 
by supports under it and behind it, four feet asunder. The 
intention was to strike the bar, sometimes in the middle and 
sometimes half-way between the middle and one end, with im- 
pacts obtained by drawing the ball and letting it fall through 
given arcs, shifting the bar when the place of impact was to 
be changed, and obtaining the deflections of the bar at that 

lace by measuring the depth which a long peg, touching the 
back of the bar, had been driven by the blow into a mass of 
clay placed there. The results were: 

1. The deflections were nearly as the chords of the arcs 
through which the weight was drawn, that is, as the velocities 
of impact. 

2. ‘The same impact was required to break the beam, whe- 
ther it was struck in the middle, or half-way between the middle 
and one end. 

3. When the impacts in the middle and half-way between 
that and the end were the same, the deflection at the latter 
place was to that at the former nearly as three to four, which 
would be the case if the locus of ultimate curvature, from suc- 
cessive impacts in every part, was a parabola. 

The preceding deductions the author had found to agree 
with theoretical conclusions, depending on the suppositions, 
(1.) that the form of a beam bent by small impacts was the 
same as if it had been bent by pressure through equal spaces ; 
and (2.) that the ball and beam, where struck, proceeded to- 
gether after impact as one mass. ‘These suppositions likewise 
gave as below: 

4, The power of a heavy beam to resist impact is to the 
power of a very light one, as the sum of the inertias of the 
striking body and of the beam is to the inertia of the striking 
body. 

rag The time required to produce a deflection, and conse- 
quently the time of an impact, between the same bodies, is 
always the same, whether the impact be great or small. The 
time, moreover, is inversely as the square root of the stiffness 
of the beam. 

6. The results of calculations, comparing pressure with im- 
pact, gave deflections agreeing with the observed ones, within 
an error of about one eighth or one ninth of the results. 


TRANSACTIONS OF THE SECTIONS. 428. 


On the direct tensile Strength of Cast Iron. By 
E. Hopexinson. 


The absolute strength of this metal, notwithstanding the 
extensive use made of it in the arts, is still a matter of doubt. 
If we turn for information to authors, we find Mr. Tredgold 
and Dr. Robison making it nearly three times as great as Mr. 
Rennie or Captain Brown, and the advocate of the greater 
strength (Tredgold) attributing the less strength, as found by. 
the others, to the straining force not having been kept in the 
centre of the prism. For supposing the extensions and com- 
pressions to continue always equal from equal forces (which they 
are under slight strains), a small deviation from a central strain 
would make a great reduction in the strength; and if the force 
were applied along the side of a square piece, the strength 
would be reduced to one fourth. (Tredgold, Art. 61, 62. 234.) 

The above contrariety of opinion was the cause of the fol- 
lowing experiments, in which the utmost care was taken to _ 
keep the straining force along the centre of the castings, which 
had their transverse sections of the form +, except in the 
last two experiments, in which the section of the castings was 
rectangular, and the force applied exactly along the side. The 
iron was of a strong kind, the same as in the author’s expe- 
riments on beams(Manchester Memoirs, vol. v.), and was broken 
— machine on Captain Brown’s principle for testing iron 
cables. 


Force up the middle. 


Area of section 


— in parts of an i 7 int Strength per square inch in tons. © 
inch. 
1 3:012 22°5 747 Tons. 
2 2°97 21:0 7:07 -mean 7°65. 
3 3031 25°5 8°41 
4 2:95 19°5 6:59 Different quality of iron. 


Force along the side. 


». Experi- 


ents. Area of section. | Breaking weight. 


Strength per inch. 


| 
4°83 11°5 2°38 
2°855 


Tons. 
4-815 13°75 } mean 2°62. 


424 THIRD REPORT—1833. 


Whence the strength of a rectangular piece of cast iron 
drawn along the side is rather more than one third of 73 
tons, its strength, as above found, to bear a central strain 


2°62 
( for G5 s) ; but from the preceding. remarks it ought 


only to be one fourth; and, therefore, it would appear that a 
shifting of the neutral line had made the pieces capable of 
bearing a greater force along the side than in their natural 
state. 


An Investigation of the Principle of Mr. Saxton’s locomotive 
differential Pulley, and a Description of a Mode of pro- 
ducing rapid and uninterrupted Travelling, by means of a 
Succession of such Pulleys, set in Motion by Horses or by 
stationary Steam-engines. By Joun Isaac Hawkins. 


In order to a clear understanding of the operation of this 
differential pulley, in the propelling of carriages or vessels, it 
will be convenient to view the principle under three cases. 

Case Ist. Let the bottom spoke or radius of a wheel, rolling 
on a horizontal plane, be considered as a lever. 

Let the point of contact of the wheel with the plane be the 
fulcrum of the lever. 

If a cord be fastened at one fourth of the length of the 
lever above the fulcrum, and it be pulled a given distance, (say 
one inch,) then the top of the lever or axis of the wheel will 
be moved in the same direction four times the distance, or four 
inches, agreeably with the common doctrine of the lever. 

If now a pulley be concentrically affixed to the wheel, and 
the circumference made to meet the point in the lever where 
the cord is fastened ; in other words, if the pulley be three 
fourths of the diameter of the wheel, and the cord be wound 
around the pulley, and be drawn horizontally in the vertical 
plane of the pulley, then the wheel will run along the hori- 
zontal plane in the direction of the pull with a velocity equal 
to four times the speed of the cord, because every point of 
the circumference of the wheel as it comes in centact with the 
plane becomes a new fulcrum, and the perpendicular line from 
that point to the axis becomes a new lever, upon which the 
cord acts at one fourth of the length of the lever above the 
fulcrum, and thus a repetition of such leverage is continually 
brought into action as the cord is drawn along. 

Case 2nd. Let the point where the periphery of the pulley 
meets the spoke or lever, and where the cord of Case 1. was 
attached, be considered the fulcrum; and let another cord be 


TRANSACTIONS OF THE SECTIONS. 425 


applied to the bottom end of the lever. If this lower cord be 
drawn horizontally in the same vertical plane, but in the op- 
posite direction to that in which the former one was pulled, 
then the top of the lever or axis of the wheel will be moved 
in the same direction as before, three times the distance that 
the cord passes through: thus, if the cord be pulled one inch, 
the axis will be moved three inches, because the leverage is 
in this case as three to one. Let the pulley be made to roll 
along a horizontal plane, and the cord be passed around a 
wheel concentrically attached to the pulley by the side of the 
plane, the radius of which wheel is equal to the whole lever, 
as was the wheel of Case 1.; then the cord being passed around 
that wheel, and pulled, the pulley will run along the plane with 
three times the velocity of the cord that draws the wheel, but 
the motion will be in a direction opposite to the pull. 

Case 3rd. Let both the cords of Case 1. and Case 2. be 
pulled at the same time, (say each one inch,) then the fulcrum 
will necessarily be removed to a point exactly half-way between 
the two cords, which fulcrum will be at one eighth of the length 
of the lever from the bottom end; and the top of the lever or 
common axis of the wheel and pulley will, in this case, be 
moved seven inches, being seven times the distance through 
which the cords pass. The ratio of the velocity of the axis to 
the cord is as the sum of the two radii of the pulley and wheel 
divided by their difference. 

Now fix a spindle in the axis, and support it on a four- 
wheeled travelling carriage, or on a vessel afloat upon water, 
and make a groove in the wheel to constitute it a pulley, and 
pass a cord around each pulley in opposite directions, and 
pull both cords with equal speed, then the carriage or floating 
vessel will be propelled with seven times the velocity of the 
cords, in the direction in which the cord of the smaller pulley 
is drawn, because the axis of the pulleys or top of the lever 
is seven eighths of the whole length of the lever above the 
fulcrum, and the two cords act at one eighth of the length of the 
lever above and below the fulcrum, which, in every part of the 
revolution of the pulleys, remains perpendicularly under the axis; 
at a height half-way between the bottom ends of the radii of 
the two pulleys. But if instead of the two cords being at- 
tached to the pulleys, an endless cord be stretched around two 
riggers, placed at some considerable distance from each other, 
and one side of the cord be made to take one turn around one 
pulley, and the other side of the cord one turn also around the 
other pulley, then the cord being drawn at either side or either 
end will cause the pulleys to revolve, and the carriage or vessel 


426 . THIRD REPORT—1833. 


in which they are hung to be propelled the whole length of the: 
space between the riggers, with a speed seven times greater 
than the motion of the cords. 

In applying this admirable invention of Mr. Saxton to the 
propelling of carriages to great distances, Mr. Hawkins pro- 
poses to place a number of endless ropes in a line, each rope 
stretched between two riggers, from a quarter of a mile to four 
miles apart, the rope lying upon several rollers to keep it off 
the ground, and passing around a pair of differential pulleys, 
supported on a light four-wheeled truck, running upon a pair of 
slight rails about 30 inches apart; the diameter of one pulley 
to be about 22 inches, and of the other about 26 inches, giving 
a velocity of 12 to 1: the diameter of the wheels on which 
the truck runs to be about 30 inches. Each rope to be set in 
motion from one of the two riggers being placed on a shaft 
passing under the rails and extending a few feet outside the 
railway, where the shaft may be turned either by a horse or 
horses, by an ox or oxen, or by a stationary steam-engine, ac- 
cording to the quantity of travelling or traffic on the road, or 
to other circumstances.. The coach for passengers, or wagon 
for goods, to be placed upon four wheels, of about four feet 
diameter, running upon a pair of rails, placed five feet apart, 
parallel with and lying on each side the pair of truck rails, and 
also a little above their level; so that the axletrees of the coach 
and wagon wheels shall pass over the rims of the truck 
wheels; or the same effect may be produced by placing the 
four rails on a level, and cranking the axletrees to raise them 
over the truck wheels. A pawl in the frame of the carriage 
or wagon being let fall upon a post arising from the frame of 
the truck, will enable the truck to draw or drive the carriage 
the length of its rope; but on the truck being stopped near 
the end of its rope, the momentum of the carriage will continue 
its motion until it pass over and beyond the truck of the next 
rope, which truck being set in motion, its post catches against 
the pawl of the carriage, and drives or draws it on until 
it reach the third truck, which again operates in the same 
manner. 

In this way 388 horses, each acting, at their most effective 
or walking pace of two miles and a half per hour, on a mile of 
rope, might easily drive a coach containing eight persons from 
London to Edinburgh in 13 hours, at the rate of 30 miles an 
hour, the coach passing from truck to truck without stopping, 
and the truck returning to take another coach every five 
minutes: 500 passengers a day for the whole distance would 
be very moderate labour for that number of horses. 


TRANSACTIONS OF THE SECTIONS. ; 427 


Account of the Depths of Mines. By Joun Taytor, F.R.S., 


Se. 


Mr. Taylor exhibited a section, showing the depths of shafts 
of the deepest mines in the world, and their position in relation 
to the level of the sea. 


The absolute depths of the principal ones were : 
eet. 


1. The shaft called Roehrobichel, at the Kitspuhl mine 
inithé: Byvolctoara goin in oes hs bas «e1 RIG 
. At the Sampson mine, at Andreasberg in the Harz 2230 
. At the Valenciana mine, at Guanaxuato, Mexico 1770 
. Pearce’s shaft, at the Consolidated mines, Cornwall 1464 
. At Wheal Abraham mine, Cornwall... . . . .. 1452 
. At Dolcoath mine, Cornwall . . . . . . . 1410 
. At Ecton mine, Staffordshire . . . . . . ~ 1880 
. Woolf’s shaft, at the Consolidated mines .. . 1350 

These mines are, however, very differently situated with re- 
gard to their distance from the centre of the earth, as the last 
on the list, Woolf’s shaft, at the Consolidated mines, has 1230 
feet of its depth below the surface of the sea, while the bottom 
of the shaft of Valenciana in Mexico is near 6000 feet in abso- 
lute height above the tops of the shafts in Cornwall. The bot- 
tom of the shaft at the Sampson mine in the Harz is but a few 
fathoms under the level of the ocean; and this and the deep 
mine of Kitspthl form, therefore, intermediate links between 
those of Mexico and Cornwall. 

Mr. Taylor stated, that taking the diameter of the earth at 

8000 miles, and the greatest depth under the surface of the sea 
being 1230 feet, or about 3th of a mile, it follows that we 
have only penetrated to the extent of 35355 part of the earth’s 
diameter. 
_ Some:account was then given of the mines to which the shafts 
referred to belong. 2 
~ Of the deepest, at Kitspiihl, as it has long ceased to work, 
we donot know much.  Villefosse, in his great work on the 
Richesse Minérale de l’ Europe, states that this was a copper 
mine, which passed for being the deepest in Europe ; and that 
in 1759, it was reported on, amongst other mines, by MM. Jars 
and Duhamel, and it was then proposed to abandon the work- 
ing, the water having been already suffered to rise near 200 
fathoms. 

The Sampson mine in the Harz is one of the most celebrated 
in that district: it has been working since the middle of the 


00 ID Orb 5 2 


428 THIRD REPORT—1833. 


sixteenth century, and produces silver ores of superior quality. 
The principal shaft is sunk about 6 feet deeper every year, by 
which ground enough is drained for a regular extraction of the 
ores. The mine is one of the oldest in Germany, and has al- 
ways been profitable : it employs from 400 to 500 men. It is 
the property of shareholders, who are very numerous, the in- 
terest having been much subdivided in the course of time. 

The mine of Valenciana at Guanaxuato was one of the most 
renowned in Mexico. It produced annually, about the end of 
the last century, 360,000 ounces of silver, worth about £600,000 
sterling, and then employed 3100 persons. The shaft referred 
to in the section was commenced in 1791, the mine having been 
long previously worked by other shafts: it had attained its pre- 
sent depth in 1809, when the mine was stopped by the Revo- 
lution. It is octagonal, and more than 30 feet in diameter, a 
great part of its depth being walled with beautiful masonry, and 
is probably the most magnificent work of the kind. The ex- 
pense of forming this shaft is estimated by Humboldt at the 
enormous sum of £220,000. The mine was so little troubled 
with water that it was considered almost a dry one: during the 
suspension of the works it, however, gradually filled. In 1825 
one of the English companies undertook to drain it, which 
was, after great labour and expense, accomplished; but the 
mine has not been sufficiently productive since to make it worth 
while to continue the working. 

The Consolidated mines form the most extensive concern in 
Cornwall, embracing what were formerly several distinct mines, 
which, as the name indicates, were connected in one under- 
taking. , 

This was arranged in 1818, and the mines which had remained 
unwrought for many years were drained by very powerful steam- 
engines, and were put into a state of active working. The ma- 
nagement was confided to Mr. Taylor and the late Captain 
William Davey : an outlay of £73,000 was incurred, which has 
since been repaid with ample profit. The present produce is 
20,000 tons of ore a-year, yielding about 1920 tons of fine cop- 
per, being one seventh of the whole quantity raised in Great 
Britain. The mines employ about 2400 persons, of whom about 
1400 are miners working underground. The water raised to 
the adit level is about 2000 gallons per minute: the height to 
which this is lifted is more than 220 fathoms, or 1320 feet; the 
aggregate weight of the columns of water in the pumps being 
512,000 pounds, or about 230 tons, and the whole is put in 
motion by eight immense steam-engines, four of which are the 
largest ever made. 


TRANSACTIONS OF THE SECTIONS. 429 


The depth of the mines has been increased 100 fathoms 
since the period of the drainage being completed, being at the 
rate of about 8 fathoms a year. 

There are, in the whole concern, 95 shafts, besides other 
perpendicular communications’ from level to level underground 
called winzes. The depths of the whole added together make up 
‘about 22,000 fathoms, or 25 miles ; and the levels, or galleries, 
will make up, in horizontal distance, a length of 38,000 fathoms, 
or about 45 miles. 

Wheal Abraham is an old copper mine, the working of which 
was abandoned a few years since, the vein having ceased to be 
productive in depth. It was, until very lately, the deepest 
mine from the surface in Cornwall, but is now surpassed by 
the Consolidated mines. 

Dolcoath mine was formerly called Bullen Garden, and a 
section of it as it was at that time will be found in Dr. Pryce’s 
work, Mineralogia Cornubiensis, published in 1778. It wasthen 
rather more than 90 fathoms deep, and probably one of the 
deepest mines at that time. It has, therefore, been sunk 140 
fathoms since; but, like all the great mines, it has not been in 
constant work. It has now been actively prosecuted for many 
years, and at present stands third in the list of copper mines in 
Cornwall, arranging them according to the value of their pro- 
duce. That of Dolcoath, however, does not amount to one half 
of that of the Consolidated mines. 

Ecton mine is celebrated in most books on mineralogy as one 
of the principal copper mines in England ; and it was so at one 
period, though the produce is now inconsiderable. It is situ- 
ated in Staffordshire, on the borders of Derbyshire, and is 
very curious, from being in limestone and having no regular 
vein. The ore has been found in large masses, irregularly de- 
posited, and is generally taken to be an example of contempo- 
raneous formation. ‘The mine has been regularly worked for a 
long series of years, and is now nearly exhausted. It is the 
property of the Duke of Devonshire, and very large profits 
were given by it in the latter part of the last century, some of 
which, it is said, were applied by the late Duke to the erection 
of the beautiful Crescent at Buxton. The mine is not far di-— 
stant from this place, and is in a very picturesque situation on 
the banks ‘of the river Manifold. 

_ Mr. Taylor gave some account of the extent to which steam 
power is at present employed in Cornwall in draining the mines 
which penetrate so far beneath the level of the sea, showing 
the influence that the great improvements, which have from time 


430 THIRD REPORT—1833, 


to time been made, and many of them even recently, must have 
upon the production of some of the most useful metals. 

The number of steam-engines used in pumping water from 
the mines in Cornwall in December 1882 was altogether 64. 

Some of these are of immense size and power: there are five 
in the county, of which the diameter of the cylinder is 90 inches, 
the pistons making a stroke of 10 feet. Four of these are at 
the Consolidated mines, and the first constructed of this size 
was planned and erected there by Mr. Woolf. The beam of 
such an engine weighs 27 tons; the pump rods are of mast 
timber, 16 inches square, connected by iron strapping plates of 
enormous weight. ‘The column of water lifted, the rods and 
beam, make up a weight of more than 100 tons, and this is kept 
in motion at the rate of from 5 to 10 strokes per minute. 

The quantity of coal consumed in drawing water in the same 
month in all the mines of Cornwall was 84,034 bushels, and the 
quantity of water delivered, about 19,279 gallons per minute. 
‘The weight of water actually poised by all these engines to 
produce this effect amounts to about 1137 tons. 

From calculations carefully made in Mexico as to horse power 
employed in draining mines, and deduced from a large scale of 
operations, it is found that the performance is equal to 19,000 
Ibs. raised one foot high per minute for each horse. 

According to this rate, the coal consumed in Cornwall in a 
month being 84,000 bushels, or 2800 per day, and taking the 
duty of the engines at 55,000,000 pounds lifted 1 foot by each 
bushel, which is very nearly the fact, it will be found that the 
sixteenth part of a bushel does as much in raising water in 
Cornwall as a horse does in Mexico, (working 3 hours out of 
24,) and that thus the number of horses required to drain the 
mines of Cornwall would be 44,800. 


On Naval Architecture. By JEREMIAH OWEN, 


A great deal has been done by mathematicians towards at- 
tempting to establish a general theory of resistances, and con- 
siderable expense has been incurred in conducting experiments, 
some of which have been made on the Continent under the su- 
perintendence of eminently scientific men. D’Alembert, Bos- 
sut, Romme, and several others were employed at different 
times in experiments of this nature. Don Juan in Spain, and 
Chapman, the great Swedish naval architect, also made several 
experiments on the same subject; as did also the Society for 
the improvement of Naval Architecture, which was established 
in England some years ago, but which has now ceased to exist, 

These experiments have always been made upon models, the 


SR a are ee 


TRANSACTIONS OF THE SECTIONS. 431 


largest of which, it is believed, have never exceeded 14 feet in 
length. They were generally much smaller. The results which 
have been thus obtained on small bodies have not been found 
to agree with results similarly obtained on larger bodies; and 
not only has this been the case, but experiments conducted ap- 
parently with equal care by different individuals have even led 
to different results. 

Naval architecture has, consequently, gained but little from 
the labour that has been bestowed upon these experiments, 
and the forms which have been given by different individuals 
to ships have depended rather upon the fancy and general ex- 
perience of those individuals than upon any facts which this 
branch of experimental science has furnished. 

In order to discover that form of the body of a ship which 
shall oppose the least resistance to its passage through the 
water, the author recommends that experiments be made on 
ships themselves, under all the ordinary circumstances of 
sailing. 

These experiments must be conducted, not in the mode in 
which experimental squadrons have hitherto been, viz. by 
comparing together the sailing qualities of ships that have va- 
ried in every particular. We are no more justified in saying 
that results obtained in this way have proved which form of 
body is best calculated for velocity, than we are in saying which 
ship has been the best managed. 
~ If it be desirable to discover by experiment which of two or 
more forms is best adapted for velocity, it is, of course, neces- 
sary that the form shall be the only variable element ; the ships 
in every other respect must be exactly similar. 

By the aid which the mathematics afford, we shall be able 


_ most completely to accomplish this. Let a ship be given which 


sails well, and which is in all other respects an efficient man-of- 
war as regards capacity, stability, &c. &c. Let this ship be 
docked, and let the most complete drawings of her form be 
made, from which we shall be able to calculate exactly every 
necessary element, such as displacement, area of midship sec- 
tion, of load water section, stability, &c.; and let the surface 
of sail, the position and rake of masts be also acertained. 

‘Then let one or more ships be constructed, having exactly 
the same principal elements as the given one, with whatever 
difference of form it may be thought proper to select, and let 
the same surface of sail be given to them all. We shall thus 
have the same weights to be moved, and the same propelling 
force to move them; the result will, of course, show which form 
is best calculated for velocity. 


432 THIRD REPORT—1833. 


These ships may be made to sail against each other under 
ever possible circumstance of sailing, taking care always to ap- 
ply the propelling force in the same way, that is, by bracing the © 
corresponding yards of all the ships to the same angle with a fore 
and aft line, during every comparison, and by raking the masts 
to the same degree. 3 

It is of importance that during the experiments the surface 
of sail in each ship should be presented as nearly as possible 
at the same angle to the action of the wind ; and this is perfectly 
practicable, for it is easy to measure the angles of the yards by 
an instrument for that purpose; and the officer who commands 
the squadron can take care, by means of frequent signals, to 
have the yards of all the ships braced to the same angle at the 
same time. 

These experiments would not be: limited in their result to the 
discovery merely of that form of a ship which is best calculated 
for velocity, although this of itself is so important as to justify 
almost any expense, but we might also be able to discover how 
far the angle of leeway is affected by the form, -which is also a 
very important question connected with the sailing of ships ; 
and after having, by repeated and careful trials, discovered the 
order of superiority of ships in respect of velocity, we might 
then, by varying the angle of bracing the yards, discover also 
the trim of the sails and the course of the ship by which to gain 
most on a wind, a question which is not by any means satisfac- 
torily settled, in as much as it involves all the uncertainty of our 
present knowledge of the resistance of fluids. 

Experiments to determine this latter question might, however, 
be made immediately on sister ships, of which there are several 
at present in His Majesty’s service. 

Let two ships be selected of the same form, and let the ut- 
most pains be taken to make the position and rake of the 
masts, the seat in the water, the stowage of the ballast, and of 
all the heavy weights, exactly the same in both ships, and let 
them be compared together in sailing both on the wind and at 
various points off the wind; the angles of bracing the yards 
being constantly varied, we should doubtless, from a series of 
experiments of this nature, succeed in discovering the best trim 
of the sails for every direction of the wind on every course. 

It may, perhaps, be urged against a series of experiments like 
those which have been here recommended, that the expense of 
building a ship is so great as to render it advisable not to run 
the risk of building a bad one for the sake of experiment merely. 
But the author suggests that our knowledge of naval architec- 
ture is such as to enable us to construct ships which we can 


ae 


TRANSACTIONS OF THE SECTIONS. 433 


certainly predict will be good, notwithstanding they may not 
be the best that might be produced ; and the ships which would 
be included in the experiments proposed would have all the 
essential qualities of a man-of-war, except, that by differing in 


- form, some would be superior to others in respect of velocity. 


Naval architecture is a branch of science which depends so 
essentially upon experiment for its advancement, and the ex- 
periments are necessarily upon so large and expensive a scale, 
as to place it out of the power of individuals, or even of socie- 
ties of individuals, to conduct them. It is, therefore, one of 
those enterprises in science which none but a nation can un- 
dertake ; and it is worthy of so great a maritime nation as En- 
gland to endeavour to advance, at whatever reasonable cost, a 
subject so important to its defence. 


HI. NATURAL HISTORY—ANATOMY— 
PHYSIOLOGY. 


On the originary Structure of the Flower, and the mutual De- 
pendency of its Parts. By Professor AGARDH. 


_TuE observations in this paper were generally directed against 


the commonly adopted view, that the flower is formed of several 
verticils independent of each other. The author remarked the 
difference between the appearance of the verticillated parts and 
their real and originary situations. Adopting the view, that 
the flower is nothing more than a branch, which has been re- 
duced to a mere point whilst its subordinate parts have been 
transformed, he concluded that the different parts of each yer- 
ticillus formed an originary spiral, and are really of unequal 
order, age, and situation, which, in many cases, is still evident 
during the inflorescence ; and imagining the branch shortened 
to a point, it will be found that the upper, later, and weaker 
parts must be the inner ones of the apparent verticillus. 

. The second point of Professor Agardh’s view is, that the. 
stamens are not transformations of petals but of buds. This 
view is consistent with the whole theory of the development of 
plants, as laid downinaseparate work (Organography of Plants), 


_and founded on the principle, that the several appendicular 


parts of the plant are not all transformed leaves, but only one 

part of them are transformed leaves, and others are transformed 

buds, so that to every part which is a transformed leaf be- 
1833. QF 


434 THIRD REPORT—1833. 


longs another part, which is a transformed bud. The stamens 
are now, according to Professor Agardh, the buds of the leaves 
of the flower, or of the petals and sepals. ‘They are therefore 
situated in their axillz, and each stamen belongs to a certain 
petal or sepal, and both organs together form a little flosculus 
as part of the whole flower, in the same way as the carpella are 
parts of the fruit. Thus a Decandrous plant, for example a Ce- 
rastium, consists of 10 floscules, each consisting of a leaflet and 
of astamen. A Pentandrous plant, for example a Borago, 
consists also of 10 floscules ; but 5 of them, those of the interior 
verticillus, are incomplete, bearing no stamens in their axille. 

The observations made on the situation of the parts of the 
flower were collected by Professor Agardh into the following 
general laws or views. ' . 

1. The number of stamens, in all cases where this num- 
ber is determinate, depends upon the number of sepals and 
petals ; and when there seems to be a different normal num- 
ber for the leaves of the flower and for the stamens, it is an 
aberration arising from abortion. 

2. The difference between the flowers which have the same 
number of stamens as of leaflets, and those where their num- 
ber is only the half of the leaflets, is caused by the abor- 
tion of the stamens of a whole verticillus of floscules ; and ge- 
nerally of the corolline floscules. The reason is, that the corol- 
line verticillus is constituted of later parts, which do not arrive 
at complete development. 

3. The same reason is to be assigned to the general fact that 
the petaline stamens are generally the weaker, smaller, and 
later. 

4, The determinate stamens are either 1, 3, or 5 in num- 
ber, belonging to each leaflet, because the buds, according to 
Professor Agardh’s theory laid down in his above-mentioned 
work, originate properly in the axille of two deviating fasciculi 
of spiral vessels, which fasciculi in the leaf (being no other 
than the nerve) are always 1, 3, or 5, &c., and the buds must 
therefore have the same symmetry and number. This is the 
reason of the determinate number of stamens in several Polyan- 
drous families, as, for example, in Rosacee, in which the sepals 
have each 3 stamens, and the petals each 1; and in Phila- 
delphie, in which the sepals have 5 stamens, and the petals 
1: whence the former family must have 20 stamens, and the 
latter 24. ny 
- 5. Some Polyandrous plants have not a determined number 
of stamens. In these Professor Agardh distinguished two cases. 


- 


TRANSACTIONS OF THE SECTIONS. 435 


In some, as in the Ranunculaceae, the buds of the flower are in 
a vascillating state between the form of stamens and flower-buds, 
and even the sepals are in a vascillating state between bractez 
and sepals. In Helleborus, Nigella, &c., the inferior buds are 
nearly like flowers, and also in Ranunculus the buds are only 
to be regarded in the same state of transition to petals, as 
the sterile flowers of Synantheree approach the form of 
petals. By this is explained, not only why the nectaria of 
Helleborus are axillary to the sepals, which they could not be 
if they were originary petals or leaves, but also why the nec- 
taria of Berberidee, which are so nearly allied to Ranunculaceae, 
are axillary to the sepals; and finally, why there is an evident 
transition between stamens and petals in Nympheacee, and 
in all the families allied to Ranunculacee, the buds in the 
flower having an equal tendency to form flowers, petals, and 
stamens. a 

» The other case of Polyandrous plants is where no relation at 
all is observable between the flowers, leaves, and the number 
of stamens. This is to be explained by the analogous case, 
wheré the flowers in capitula, as, for example, of some Synan- 
theree, are without distinct bractez to each single flower ; and 
it is not more singular, that the stamens should be in some 
cases without their respective flower-leaves, than that the 
flowers in some cases should be without determinate bractez. 

6. When some floscules in the same verticillus are sterile or 
without stamens, they are frequently those which are younger 
or later than the others. The same reason is to be assigned 
for the inequality of the stamens in the same verticillus. Ex. 
Personate, Labiate,.in which the two younger stamens are 
smaller, and the youngest stamen fails. 

7. The ternary number in Monocotyledonous plants is de- 
rived from a leaf-bud, in which two outer leaves, or squame, 
turn their back to the stem, and form two sepals; and the 
third sepal is the leaf, in the axilla of which the bud is situated. 
This is evident in Carex (in which the two leaves coalesce into 
the utriculus,) and in the Graminee. 

8. Dicotyledonous plants have their flowers formed on two 
different plans. 
“9. One group of them has originally opposite leaves; on 
these the floscules are naturally in pairs, and when a fifth flos- 
cule exists, it-is to be regarded as the last, and the only one of 
the third pair which has found sufficient room to develop. (See 
Calyx of Dianthus.) 

-°10. The other portion of Dicotyledonous plants has alternate 
leaves, and, in consequence, impair and unequal floscules; but: 
ZE2 


436 THIRD REPORT—1833. 


the floscules have a tendency to take a symmetrical form on both 
sides, so that in this case a floscule exists which is especially 
to be regarded as impair, and which is the first or the last in 
the spiral: This impair floscule is commonly placed either near- 
est to the axis of the racemus (axilis), or outermost in the peri- 
phery (periphericus) of the racemus. 

11. ‘The petaline floscules have a contrary progression to the 
sepaline, so that if the odd or impair sepal is placed nearest 
to the axis of the racemus, the odd or impair petal i is placed 
outermost in its periphery. 

12. The situation of the impair floscule is different in dif- 
ferent families, for example, the odd sepal, (the first or last sepal,) 
is axilis in Labiate, Personate, Umbellifere, and periphericus 
in Leguminose, Rutacee, &ce. 

13. By the situation of the carpella two cases are to be di- 
stinguished. 

14. In some cases the carpella are commensurable with the 
number of floscules. They are then placed either parallel to 
the sepals, as in the Liliaceae, Primulacee, Geraniacee,; Cru- 
cifere, or parallel to the petals, as in the Rutacee, Philadelphus, 
Onagrarie. 

15. In other families, and by far the greatest part, the car- 
pella are two, (complete and incomplete,) and thence not com- 
mensurable with the five divided flowers. In this case one 
carpellum is parallel with the impair sepal, and the other 
with the impair petal. The fruit of the Boraginee and Labiate 
is to be referred to this case, two carpella taken together being 
placed parallel to the impair sepal, and the two others parallel 
to the impair petal, the fifth carpellum having vanished. 


Notice of Researches on the Action of Light upon Plants. 
By Professor DavBeny. 


The author communicated a notice of certain researches 
which he is at present pursuing concerning the action of light 
upon plants, and that of plants upon the atmosphere. 

He considers that he has established, by experiments on 
plants immersed sometimes in water impregnated with carbonic 
acid gas, and at others in atmospheric air, containing a notable 
proportion of the same, that the action of light.in promoting 
the discharge of certain of their functions, and especially that 
of the decomposition of carbonic acid, is dependent neither 
upon the heating nor yet upon the chemical energy of the se- 
veral rays, but upon their illuminating power. 


TRANSACTIONS OF THE SECTIONS. 437 


He regards light as operating upon the green parts of plants 
in the character of a specific stimulus, calling into action and 
keeping alive those functions from which the assimilation of 
carbon and the evolution of oxygen result, and that the de- 
scription of rays which are proportionally more abundant in 
solar than in artificial light are those most instrumental to the 
above purposes. 

With regard to the second branch of the inquiry, Professor 
Daubeny has only proceeded in it so far as to have satisfied 
himself, that in fine weather a plant consisting chiefly of leaves 
and stem will, if confined in the same portion of air night and 
day, and duly supplied with carbonic acid during the sunshine, 
go on adding to the proportion of oxygen present, so long as 
it continues healthy, at least up to a certain point, the slight 
diminution of oxygen and increase of carbonic acid which 
takes place during the night bearing no considerable pro- 
portion to the degree in which the contrary effect is observable 
by day. 

He accounts for the discrepancy between his own results 
and those reported by Mr. Ellis in his work on Respiration, by 
his having taken care to remove the plant from the jar imme- 
diately upon its beginning to suffer from the heat or confine- 
ment, and from his having carried on the experiments upon a 
larger and more suitable scale. 

Considering the quantity of oxygen generated by a very 
small portion of a tree or shrub so introduced, he sees no 


reason to doubt that the influence of the vegetable may serve 


as a complete compensation for that of the animal kingdom, 
especially since this same function appears to belong to every 
plant which has come under his review, whatever may be its 
structure or organization. 


On some symmetrical Relations of the Bones of the Mega- 
therium. By Waiter Apam, M.D 


The author, having examined the bones of the megatherium 
which are preserved in the Museum of the College of Surgeons, 
was led to observe their forms according to their symmetrical 
relations. For this purpose, the coronal breadth of the cra- 
nium is taken as a common term of reference. It measures 


8-75 inches, but in the following scale of proportions its breadth 


is denoted by 10, and all the other measures are altered in the 
same ratio, and expressed by the nearest integers. Dr. Bar- 
clay’s nomenclature is employed. 


438 THIRD REPORT—1833. 


Heap. 620%! Coronal breadth of cranium............ 10 
Mesial thickness of the bony plate form- 
ing the palate and the basilar surface of 
the nasal passage). 32. sc) stele mre teed Jord 2 
Mesial height from the surface of the palate 
to the concavity of the coronal surface, 
about the rostral margin of the rostral 
ELS ge ee IR RE 8 
Greatest height of the fragment of the 
head from the palatal surface in the di- 
rection of the socket of the third molar 12 
VeERTEBRA. . . Breadth of the transverse process of the atlas 14 


of the fifth cervical............ 9 
of one of the largest dorsal .... 12 
of a caudal vertebra, supposed to 
be that next the sacrum...... 25 
of the seventh of the 12 caudal 
which remain.......4....%...% 10 
of the smallest caudal.......... 4 
Length of the body of the atlas........ 3— 
of the fifth cervical............ 3— 
of one of the largest of the dorsal . ae 
of one of the lumbar............ 
Of SixvCHnUAlS Gee ee ee ee 
of the smallest caudal .......... 3 


In the first three vertebrz of the tail in which the length is 
diminished, that dimension is greatest on the upper or dorsal 
aspect, indicating that about the middle the tail had a ten- 
dency to curvature downwards and forwards. 


Longest spinal process of a dorsal vertebra............ 16 

Srrnat Cana, . Width in atlas ......0.....0.000.000. 3 
in fifth conyieal: +s! bateieht . w:tes 3+ 
at She mel visi. :.96- sii wnae mains 4 


The ribs of the megatherium were connected to the sternum 
by osseous attachments. 


Length of the longest rib which has been found (without 


its sternal attachment)... . 00+ « ecisinies rieieeesefe 36 
of the shortest rib found (probably the first). . 15+ 
Greatest breadth of the shortest rib ............+4-- 5 
of the longer ribs........... ao ae 3°5 
Thickness of the longer ribs ...-..........seecceees 1°8 
Srernum .... Breadth of the rostral portion.......... 8 


Mesial length of the same ..... SR 40 


et a. eee ee ee 


He 


PELvis 


Scapuba ,... 


TRANSACTIONS OF THE SECTIONS. 


Greatest transverse extent of the iliac bones 75 


Width of pelvic aperture.............. 16 

Depth: of ditto; .. 9 -alkibiqew apreeisei2 zis |- 28 
Breadth at the mesial margins of the ace- 

ST Bie er at uh oxauddhy oicksileiheg och ater she 18 
between the acetabula and the 

ischial tuberosities .......... 28 

at the ischial tuberosities ...... 24 


Thickness of bone at the ischial tuberosities 3 

Symphysis pubis, its rostro-caudal extent 11 

From the sternal surface of the symphysis 
pubis to the sternal surface of the spinal 


canal, the mesial distance is.......... 39 
Length of the thyroid foramina ........ 12 
Breadth of the same...........0...... 6 


From the lateral extremity of. the left iliac 


bone to the right ischial tuberosity.... 58 . 


Breadth from the acromico-glenoid sinuosity 17 
from the caudal margin of the gle- 


ADU RTI i hn ad ati ige hin mm a 20 
Length of the glenoid cavity............ 7 
Extent of dorsal margin of scapula...... 31 
from the dorso-caudal angle to the 
extremity of the acromion .... 33 
From the acromion to the glenoid cavity, 
Kerentest brewdth)y ies fired oes iene 14 
Thickness of scapula at the dorso-caudal 
alien .sneitisds sl th dS GE A. . 3 
Craviciz,... This strikingly resembles that of man. 
Ptsubeastlins inci shaages. diay Bia tote stasis 17 
Harel: Boma tiv eias gues Ws ww retewo 7 
Rapivs...... Dteribeteetdaand cerv rye slenar lS. latsseed i te aiaia 28 
, Its greatest breadth near the os humeri ee) 
; Its ginpllest printer costes wes ches boo me lee 10 
Aernaganwe: > Bread@edags 9, (ewaamnys oi ioe ese 11 
Length be) eaccyc +i. bts steno bagte)-relsdnatigd> 10 
_ Catcanevo .. Digital breadth.........-.0.. 0.0200 ee © 18 
Greatest dene thts. n:..» .-.-. 0am whybsl 20 
PGE TO oases», psalm rere, *' Sys « 12 
Smallest girth ...... CR ene 9 20 
BOATS Fie...» 5.0 Da PON g ooo wy ony a 28a apa hee 13 
Proximal reat’. fcige 80 Poa. Je eee 14 
Fibular length 22 ypask OS PN st al 24 
Tibial length 2 > teapindee. tO) OTT 85 25 
Tibia, smallest’ girth. +. “Miedetk).2 ae. 16 
Femur ...... Breadthe.< wae. terest he teenie... 18 
Smallest pinky 6) HOW -AWL MIUE A... 30 


Greaheoteleme tly, © ci wivoniieiy lek Mads. alee 32 


439 


440 THIRD REPORT—1833. 


Dr. Adam observes, that by the completion of Mr. Clift’s 
labours in adapting and mounting the remains of this animal, 
some peculiarities now visible in the internal structure of the 
bones will be concealed, and on this account he directs the 
attention of zoologists to the following observations : 

‘© Tn the thicker parts of the ribs as well as of the bones of 
the limbs where broken, there are dispersed and conglomerated 
in the reticulated texture, like the spherules in some crypto- 
gamous plants, numerous round bodies from one tenth to two 
tenths of an inch in diameter. These bodies are hard, but of 
a steatomatous appearance: they seem to have resulted from 
the same exuberance of ossification so conspicuous in the ex- 
ternal surface. The external surface of the thicker parts of 
the bones looks as if formed by a conflict of the osseous spi- 
cule, which are of the size of coarse needles.” 

Dr. Adam is of opinion, that probably the nails of the me- 
gatherium might have been doubled under the foot in the same 
manner as those of a living cognate species, the short-tailed 
manis, the feet of which living species had hitherto been incor- 
rectly figured in zoological works. 


On some new Species of Fossil Saurians found in America. 
By R. Harran, M.D., of Philadelphia. 


The species of saurians mentioned in this communication 
had been all examined by Dr. Harlan, and a full account of 
them is preparing for publication. The following extracts will 
make known the names and localities of these fossils. 

1. Ichthyosaurus Missouriensis.—A fragment of the head 
has been found in a hard bluish grey limestone, near the junc- 
tion of the Yellow-stone and Missouri rivers. 

2. A dorsal vertebra, analogous to those of plesiosaurus, 
except that its length is remarkably greater in proportion to its 
breadth. Found in marl on the banks of the Arkansaw river : 
supposed to belong to a very large individual. The marl con- 
tains many bivalve shells. 

Remains of crocodiles, geosauri, &c., were also mentioned 
by Dr. Harlan as occurring in West New Jersey in marls. _ 


Remarks on Genera and Subgenera, and on the Principles on 
which they should be established. By the Rev. Leonard 
Jenyns, 4.M., F.L.S. 


The object of this paper was to make some remarks on the 
great multiplication of genera at the present day, and to show 


-— 


TRANSACTIONS OF THE SECTIONS. 441 


that in constructing them sufficient attention had not always 
been paid to the true principles of classification. It was par- 
ticularly stated that in this country zoologists had very much 
overlooked the principle which determines that all groups bear- 
ing the same title should be groups of the same value; and 
that in raising to the rank of genera the subgenera of the 
French, they put these last on the same footing with groups of 
a higher denomination, to which in strict reality they were 
subordinate. Instances were brought forward from amongst 
the genera of British birds, in which this disregard to a due 
subordination of groups was particularly manifest. It was 
mentioned that in this way Plectrophanes was made a group of 
equal value with Emberiza, Lagopus with Tetrao, Coturnix with 
Perdix,and Botaurus with Ardea, although it might be clearly 
seen, that in each of these instances the first group rested on 
characters far less important and less numerous than those 
which were common to the two considered as one genus. 

Some remarks were then made on the method of ascertaining 
the value of any new group that presents itself. It was ob- 
served, that to fix this with certainty required a previous ac- 
quaintance with all the other existing groups belonging to the 
same family, and that therefore it can only be determined so 
far as the present state of our knowledge of that family will 
allow. If it be found on comparison that its characters are of 
equal value with those of other acknowledged genera in that 
family, the group in question may be considered as a genus 
also ; but if of less, it is clear that the group itself is one of 
less importance, and must occupy a subordinate station. 

The author concluded with pointing out the impropriety of 
splitting up natural genera, as had been done in some cases, 
merely because they contained a large number of species. He 
stated that the value of a group was not affected by such a 
circumstance ; furthermore, that no groups should exist in our 
systems but such as exist in nature; and that for the mere 
purpose of abridging labour in the search after particular spe- 
cies, it was quite sufficient in the case of extensive genera to 
institute sectional divisions, indicating such sections by signs. 


On some parts of the Natural History of the Common Toad. 
¢ By James Macartney, M.D., F.R.S. 


After commenting upon the unfounded prejudices against the 
whole class of reptiles, and the toad in particular, the author 
corrects an error concerning the mode of feeding of the toad— 


into which even Linnzus had fallen—that the flies are attracted 


442 THIRD REPORT—1833. 


into its mouth by a power of fascination. ‘‘ The toad takes 
its prey in the same manner as the chameleon and many 
other lizards, by projecting its tongue, striking the insect, and 
drawing it back into the mouth, and this it does so rapidly that 
the action cannot be seen; but if a fly alights on the outside of 
a glass vessel in which a toad is inclosed, the creature, thinking 
its prey is within its reach, performs the usual act, and the 
stroke of the tongue is very distinctly heard against the inside 
of the glass opposite to the fly.” 

Stories are very frequently published of living toads being 
found encased in solid rocks and in the trunks of trees, and 
these accounts receive very general credit. To ascertain how 
far this is probable, Dr. Macartney made the following experi- 
ments. 

He placed a toad ina glass vessel covered loosely with a 
piece of slate, and buried the vessel containing the toad about a 
foot deep in a garden; on digging it up a fortnight afterwards, 
the animal was in perfect health, and had recovered from a 
wound it had previously received in the thigh. He then took 
the same toad, and having secured the top of the vessel in such 
a way that no air nor moisture could be admitted, he buried it 
in the same place, and on raising it a week afterwards found 
the animal dead and putrid: from hence he concluded that the 
toad cannot live if moisture and atmospheric air be perfectly 
excluded. It is very probable that toads have been often found 
alive in chasms of rocks, or in hollow trees having a small aper- 
ture through which the air and also insects might enter; but 
that any animal possessing lungs should live for an indefinite 
time without some communication with the atmosphere appears 
quite incredible. 

Cuvier has stated that the toad, although not venomous, yet 
when provoked ejects a liquor from two glands placed on its 
head, which is capable of irritating the skin. Dr. Macartney 
has often had toads in his possession, but never observed any- 
thing of the kind; nor does he believe that the animal has any 
disposition to injure others: on the contrary, the toad is very 
gentle, capable of being domesticated, and of becoming attached 
to those who treat it well. 

It is a popular notion that toads cannot live in Ireland, which 
opinion is in some degree countenanced by the fact of there 
being no reptiles in that country except the water-newt and the 
frog, and the latter was introduced within the last century. It 
is also understood that there are no reptiles in the Isle of Man. 
The climate of both these islands being more moist than that 
of England would be particularly suitable to frogs and toads, 


— 


TRANSACTIONS OF THE SECTIONS. 443 


although it would probably be unfavourable to serpents and 
many kinds of lizards. 

Some years ago the author brought eleven toads from this 
country to Ireland, and as he did not wish them to be propa- 
gated, on account of the alarm and disgust which many weak 
people feel towards them, he buried them in a flower-pot co- 
vered loosely with a slate, to prevent the earth falling in upon 
them. In this situation he kept them for two years, occasionally 
digging them up, for the purpose of exhibiting them and making 
them the subject of experiments. They at length all died during 
a very hot summer, the ground in which they were buried 
having become so dry that the animals could no longer receive 
any moisture; for although the toad eats many insects when 
it is at liberty, it will live and increase in size by imbibing 
moisture alone, for which purpose its skin is provided with nu- 
merous pores. 

The toad possesses greater powers of repairing the effects 
of injuries than most other animals. One of the toads which 
has been mentioned as living beneath the surface of the earth 
for two years had been subjected to the experiment of having 
the upper part of the skull removed, and a portion of the brain 
scooped out. The wound rapidly healed, leaving a depression 
corresponding to the quantity of bone and brain taken away. 
The only effect which remained from this injury was that the 
animal afterwards did not walk in a direct line, but in curves 
to the one side, a fact which has been observed in other in- 
stances consequent to injuries of the brain. Dr. Macartney has 
seen one instance of the same kind in the human subject, the 
person being incapable of locomotion, except in circles, as if 
he were waltzing. 

There is one fact in the natural history of the toad which the 
author believes to be quite unknown,—the utterance by the 
animal of a musical sound, consisting of one note, so clear and 
pure that it perfectly resembles that which is produced by 
striking a piece of glass or some sonorous metal. The season 
of the year in which this was heard was the latter part of au- 
tumn. Dr. Macartney concludes by observing that ‘‘ One ob- 
ject in studying zoology, and that not an unimportant one, is, 
by closely investigating the habits of animals, to remove the 
prejudices and apprehensions which are traditionally handed 
down to us from those ages in which fable took the place of 
knowledge. Many of these errors and prejudices with respect 
to animals exist in the present day, even amongst well informed 
persons, to an extent that would scarcely be believed unless 
our attention had been directed to the subject. In selecting 


444. THIRD REPORT—18398. 


the history of the toad, I have merely employed a remarkable 
example of the fact.” 


Observations relative to the Structure and Functions of 
Spiders. By Joun Buacxwatt, F.L.S. 


During the last three years the author has been engaged 
occasionally in conducting experiments having for their object 
the determination of a highly interesting question in physiology, 
namely, what are the true nature and functions of the remark- 
able organs connected with the fifth or terminal joint of the 
palpi of male spiders? The opinion advanced by M. 'Treviranus, 
and adopted by M. Savigny, that those parts are instruments 
employed for the purpose of excitation merely, preparatory to 
the actual union of the sexes by means of appropriate organs 
situated near the anterior extremity of the inferior region of the 
abdomen, is in direct opposition to the views of Dr. Lister and 
the earlier systematic writers on arachnology, who regarded 
them as strictly sexual; and the results of the author’s investi- 
gations clearly demonstrate the accuracy of the conclusion 
arrived at by our celebrated countryman. 

In the spring of 1831 Mr. Blackwall procured young female 
spiders of the following species, Epeira diadema, Epeira apo- 
clisa, Epeira calophylla, Epeira cucurbitina, Theridion ner- 
vosum, Theridion denticulatum, Agelena labyrinthica, &c., and 
having placed them in glass jars, fed them with insects till they 
had completed their moulting and attained maturity, which is 
easily ascertained in most instances by the perfect development 
of the sexual organs. He then introduced to them adult males, 
taking care to remove the latter as soon as a connexion had 
been consummated in the usual manner, by the application of 
the palpal apparatus to the orifice situated between the plates 
of the spiracles in the females. He never in any instance suf- 
fered the sexes to remain together any longer than he found it 
convenient to continue his observations, and remarks that their 
union, however prolonged and undisturbed, was invariably ac- 
complished in the manner stated above. After a lapse of se- 
veral weeks the females thus impregnated respectively fabricated 
their cocoons, and deposited their eggs in them, all of which 
proved to be prolific; affording a complete refutation of the 
opinion promulgated by M. Treviranus. 

That there might not remain the slightest doubt, however, 
on the mind of the most fastidious inquirer, in the summer of 
1832 the author brought up from the egg young females of the 
species Epeira calophylla and Epeira cucurbitina, which, when 


TRANSACTIONS OF THE SECTIONS. 4A 


they had arrived at maturity, he treated in the manner de- 
scribed in the preceding cases. In the autumn of the same 
year these spiders deposited their eggs in cocoons spun for 
their reception, out of which the young issued in the ensuing 
spring, having undergone their final metamorphosis in the 
cocoons. 

These experiments, besides effecting the purpose for which 
they were instituted, served also to supply collateral evidence 
of the correctness of M. Audebert’s observations relative to 
the capability of the House-spider, Aranea domestica, to pro- 
duce several sets of prolific eggs in succession, without renew- 
ing its intercourse with the male; for three females of the spe- 
cies Agelena labyrinthica deposited each a second set of eggs, 
and a female, Epeira cucurbitina, laid four consecutive sets, 
intervals of fifteen or sixteen days intervening, all of which pro- 
duced young, though these females had not associated with 
males of their species for a considerable period antecedent to 
the deposition of the first set of eggs. 

MM. Lyonnet and Treviranus, with other skilful zootomists, 
have fallen into the error of mistaking the superior spinning 
mammule of spiders, when triarticulate and considerably elon- 
gated, for anal palpi (palpes de l’anus), denying that they per- 
form the office of spinners, in consequence of their having failed 
to detect the papilla from which the silk proceeds; and in this 
opinion they are followed by the most distinguished arachno- 
logists of the present day. The author is inclined to attribute 
this singular oversight to the peculiar disposition and structure 
which the papillae or spinning tubes connected with the su- 
perior mammule, when greatly elongated, frequently exhibit. 
Arranged along the under side of the terminal joint, they pre- 
sent the appearance of fine hairs projecting from it at right 
angles ; but if the spinners when they are in operation be care- 
fully examined with a powerful magnifier, the function of the 
hair-like tubes may be ascertained without difficulty, as the fine 
lines of silk proceeding from them will be distinctly perceived. | 

In conducting this observation Mr. Blackwall usually employs 
the Agelena labyrinthica of M. Walckenaer: its size, the length 
of-its superior mammule, and its habits of industry, afford a 
combination of advantages comprised by no other British spider. 
_ The purpose subserved by the superior mammulz, when very 
prominent and composed of several joints, is the binding down 
with transverse lines, distributed by means of an extensive la- 
teral motion, the threads emitted from the inferior mammule ; 
by which process a compact tissue is speedily fabricated. 

The foregoing facts supply a striking exemplification of the 


446 THIRD REPORT—1833. 


importance of connecting physiological researches with anato- 
mical details. 

In attempting to drown a small spider, new to naturalists, 

(which the author has named Erigone atra,) for the purpose of 
taking its dimensions accurately by measurement, he was asto- 
nished to find that at the expiration of two days, though it had 
remained under water the whole of the time, it was as lively 
and vigorous as ever. This extraordinary circumstance induced 
him to submerge numerous specimens of both sexes in cold 
water contained in a glass vessel with perpendicular sides, on 
the 21st of October 1832, in which situation they continued till 
the 22nd of November, an interval of 768 hours, without having 
their vital energies suspended. 
. He has tried the same experiment with individuals of other 
species, and some of them have preserved an active state of 
existence for six, fourteen, or twenty-eight days, spinning their 
lines and exercising their functions as if in air, while others 
have not survived for a single hour. It is evident, therefore, 
from these curious facts, that some spiders possess the power 
of abstracting respirable air from water; for though in the act 
of submersion the spiracles are generally enveloped in a bubble 
of air, yet so small a supply is speedily exhausted, and, indeed, 
soon disappears. 

The external and internal organization of such species of 
Araneide as can exist for along period of time under water 
deserves to be attentively examined ; but those species which 
the author has observed hitherto are minute, and it would re- 
quire the hand and eye of an accomplished anatomist, assisted 
by the most delicate instruments and powerful magnifiers, to 
effect this desirable object satisfactorily. 


On the Reproductionofthe Eel. By Wiiu1aM Y ARRELL, F.L.S. 


Sir Humphry Davy, in his “ Salmonia,” considered the 
mode in which eels produced their young as a problem in na- 
tural history not then solved, the more general opinion being 
that they were viviparous. 

The paper commences with a recital of the opinions of various 
writers on this subject, from Aristotle and Pliny to the time of 
Bloch and Lacépéde, and the author states his belief that the 
viviparous nature of eels had been inferred from the circum- 
stance of their being subject to numerous intestinal worms, three 
species of which are named and described as of frequent occur- 


rence. 
The sexes are distinct ; the females oviparous. The situation, 


Spee Oe he 


TRANSACTIONS OF THE SECTIONS. 447 


structure, and peculiarities of the sexual organs are described, 

and the author gives a statement, from his own examinations, 

of the dates at which eels in various ponds and riyers in 

different southern counties deposited their ova and _milt, all 

et which occurred between the 15th of April and the 7th of 
ay. 

The migration of adult eels in autumn, in tide rivers, is con- 
sidered as extending to the brackish water only, and believed 
to be induced by the higher degree of temperature there exist- 
ing. The mixed water is shown by experiment to maintain a 
temperature two degrees higher than the pure sea or fresh 
water, from the combination of two fluids of different densities. 
Eels pass the winter imbedded in mud. 

The return of adult eels is shown by the habits and success 
of the basket fishermen in rivers within the tide-way, who place 
the mouths of their eel-pots up stream in autumn, and down 
stream in the spring. 

The ascent of the fry is described as it occurs in the Thames, 
the Dee, the Severn, and the Parret. 

Sea water contains a much larger proportion of earthy mat- 

ter, and in consequence less air, than the water of rivers, and 
fresh water also yields its oxygen much more rapidly than that 
of the sea; the author states his belief that no instance of a 
freshwater fish going to the sea todeposit its spawn will be 
found, while more than twenty species of truly marine fishes 
ascend rivers to deposit their spawn, obtaining thereby, for the 
vivification of the ova, the assistance afforded by a larger quan- 
tity of oxygen. . 
. The restlessness of eels during thunder-storms, when enor- 
mous quantities are taken, is referred by the author to the high 
degree of muscular irritability known to exist in all animals pos- 
sessing a low degree of respiration, with which coexist. the 
power of sustaining privation of air and food, a low animal tem- 
perature, and great tenacity of life, all of which eels are well 
known to possess. ) pow 

Fishes that swim and take their food near the surface die 
soon when taken from the water, having a higher degree of re- 
spiration and less muscular irritability, compared with those that 
swim near the bottom; and vice versd. . : 

» The paper concludes with descriptions of the characters 
— distinguish three different species of British freshwater 
. ; pPNe 


448 THIRD REPORT—1833. 


On the Naturalization in England of the Mytilus crenatus, a 
native of India, and the Acematicherus Heros, a native of 
Africa. By Cuaries Witticox, Curator of the Museum of 
the Portsmouth and Portsea Literary and Philosophical So- 
ciety. 


Mr. Willcox states that when His Majesty’s ship Wellesley 
was docked at Portsmouth in July 1824, he discovered on the 
lead of the cutwater and under the keel a great number of My- 
tili, which, on examination, proved to belong to the species 
named M. crenatus. ‘The Wellesley was launched at Bombay 
about February 1815, and came into this harbour in May 1816, 
where she remained for upwards of eight years previously to her 
being taken into dock. 

The same species of Mytilus has, however, within the last 
twelve months, been found by Mr. Willcox among groups of 
Mytilus edulis, on the fore part of the keel of several ships on 
being taken into dock, which proves their naturalization in a 
climate apparently uncongenial to their nature. 

The Acematicherus Heros has been found in many parts of 
His Majesty’s dockyard at Portsmouth for some years past. 
Several specimens are in the possession of Mr. Willcox, and al- 
though it was generally supposed that they were bred in African 
timber, imported for the building and repairs of the navy, yet 
it was not until the following circumstance occurred that this 
fact was proved. With the intention of determining this ques- 
tion, Mr. Willcox had for a considerable time been in the habit 
of examining every piece of African timber which came under 
his inspection: at length, whilst a piece of this timber was be- 
ing cut, several small holes, the size of a pea, were discovered 
running in a direction more or less oblique to the fibres of the 
wood, and generally increasing to six or seven times their dia- 
meter at the orifice, the inside surface being perfectly smooth. 
Shortly after, a larva was found surrounded by the dust of the 
wood, and it was carefully extracted from one of these holes. 
This circumstance encouraged Mr. Willcox to make further 
search, and he at length succeeded in finding in another hole 
the pupa, which was taken out alive, and he found it to be that 
of the insect before mentioned. 3 

The perfect insect was kept alive by Mr. Willcox nine weeks, 
by feeding it on sopped bread sweetened with sugar. Several 
of these insects have of late been found in an apparent healthy 
state, at different parts of the island of Portsea; two of them 
(male and female) Mr. Willcox has now alive. It may be, per- 


TRANSACTIONS OF THE SECTIONS. 449 


‘haps, concluded from this circumstance that this species of in- 
sect will ultimately become naturalized in this climate. 


Abstract of Observations on the Structure and Functions of the 
Nervous System. By James Macartney, M.D., Professor 
of Anatomy and Surgery in the University of Dublin. 


The author begins by stating the received opinion respecting 
the structure of the brain, as consisting of two substances; the 
one an opake white pulp, which is considered to be the nervous 
matter; the other a coloured substance, in some places inclosing 
the white, and at other places being imbedded in it. 

It has been long known, he adds, that the white substance in 
many parts assumes the shape of bands or bundles of fibres. 
Dr. Spurzheim did not hesitate to call these fibres nerves, and 
was more successful in tracing their course in some parts of the 
brain than his predecessors had been. 

But the author has employed a method of dissecting the brain, 
which has enabled him to discover that all our former ideas with 
respect to the structure of the cerebral organ fall far short of 
the intricacy with which its several parts are combined. 

In order to perceive the real structure of the brain, recent 
specimens are necessary. The sight should be aided by spec- 
tacles of a very high magnifying power; and as the different 
parts are exposed in the dissection, they should be wetted with 
a solution of alum in water, or some other coagulating fluid. B 
these means it will be observed that all the white substance, 
whether appearing in the form of bands, cords, or filaments, or 
simply pulp, are composed of still finer fibres, which have a 
plexiform arrangement, and that all those fibres, to the finest 
that can be seen, are sustained and clothed by a most delicate 
membrane. By the same mode of dissection, also, it is possible 
to make apparent the existence of still finer interwoven white 
fibres in all the coloured substances of the brain, in many of 
which the nervous filaments are so delicate and transparent that 
they are not visible until in some degree coagulated by the so- 
lution of alum or by spirits. 

Dr. Macartney has thus been enabled to see twenty-six 
plexuses not hitherto described in the brain, the fibres com- 
posing which assume two arrangements, the one reticular, the 
other arborescent. 

The membrane mentioned as pervading the entire substance 
of the brain, and supporting its delicate organization in every 
part, has heretofore escaped the observation of anatomists, and 


yet when the fact is declared, we at once perceive that such a 
1833. 26 


450 THIRD REPORT —1833. 


membrane must exist. It cannot be supposed that a mass of 
the magnitude of the brain, and possessing so definite an or- 
ganization, should form an exception to the fabric of all the 
other parts of the body, and be left unprovided with a mem- 
branous support. ‘This membrane is analogous to the cellular 
membrane; and if we admit that the filaments of the brain are 
similar to the fibres in other parts of the nervous system, we 
may consider the membrane which sustains and connects the 
cerebral plexuses as their proper musclema. 

The pia mater, or vascular integument of the brain, is com- 

posed of two layers; the external of which passes over the 
convolutions of the cerebrum and the gyri of the cerebellum, 
and the internal is reflected between these forms, and gives all 
their exterior surface an intimate covering. The blood-vessels 
seen on the brain are inclosed between these layers, and are 
conducted on the inner layer to the substance of the organ. The 
inner portion of the pia mater is continuous with the membrane 
of the substance of the brain, but becomes so delicate on enter- 
ing the structure of the organ that it is readily detached from 
the brain without apparently injuring the integrity of its sur- 
faces. When the inner layer of the pia mater is obtained in 
connexion with a portion of the vessels and membrane which 
penetrate the brain, it has the appearance of tufts or shreds, 
and as such has been described by Ruisch and Alkenus under 
the name of tomentum cerebri. 
- The musclema of the brain appears in the adult to be only 
furnished with colourless vessels, except in those places where 
red vessels are seen to pass into the substance of the organ; 
but in the foetus, the coloured substance of the convolutions 
may be injected so as to appear quite red. ‘This fact is con- 
sistent with the structure of many other organs during feetal 
life, which in that period of existence receives red injection, 
yet only admits afterwards colourless fluids. ‘The great degree 
of vascularity in the foetus is particularly remarkable in the eye, 
the lining of the labyrinth of the ear, the periosteum, &c. 

The author has ascertained that the actual quantity of the 
sentient substance existing in the brain and other parts of the 
nervous system is extremely small. The bulk of these parts is 
not materially diminished by removing their nervous matter, 
provided their membranous structure be left behind; and 
whenever we meet with the sentient substance in connexion with 
a highly attenuated membrane, as in the retina and in several 
of the cerebral plexuses contained in the coloured matter of the 
brain, it is absolutely invisible until it has undergone some de- 
gree of coagulation. It is, perhaps, not assuming too much 


TRANSACTIONS OF THE SECTIONS. 45] 


from these facts, to suppose that the whole nervous system, if suf- 
ficiently expanded, and divested of all coverings, would be found 
too tender to give any resistance to the touch, too transparent 
to be seen, and probably would entirely escape the cognizance 
of all oursenses. Consistently with this view of the matter, the 
author thinks that we can hardly take upon us to say that the 
simplest animals, and even plants, may not have some modifi- 
cation of sentient substances incorporated in their structure, m- 
stead of being collected, as in the higher classes of animals, into 
palpable membranous cords and filaments. 

The term plexus has been generally employed to signify an 
interweaving or crossing of filaments; but Dr. Macartney is 
satisfied that there is an actual union or intermixture of sub- 
stance in both the plexuses of the brain and of the other parts 
of the nervous system. He has discovered that the roots of 
the spinal nerves, instead of being connected with the medulla 
by mere contact or insertion, as hitherto supposed, actually 
enter into the composition of the filaments of the spinal marrow, 
and that these roots of nerves (as they are called) form com- 
munications with each other within the substance of the me- 
dulla. With regard to the cerebral nerves also, it can be 
shown that they are continuous with the cerebral plexuses in 
their immediate neighbourhood. 

Many of the communications formed between the right and 
left sides of the nervous system are well known, such as the 
commissures of the brain, the crossing white filaments of the 
spinal marrow, the decussation of the pyramids, and the inter- 
change of the two optic nerves in fishes. The author has 
found so many communications to exist between the origins of 
the nerves on the right and left sides of the body, that he is 
‘disposed to believe it to be a general fact. The optic nerves 
in the human subject do not decussate, as some have supposed, 
but form a very intricate plexus where they come into contact. 
This mode of conjunction accounts for atrophy of the tractus 
opticus being in some instances found on the same side, and at 
others on the opposite side to that of the eye affected with 
blindness. 

The facts already observed would justify the opinion that 
the sentient substance is in no place distant or isolated; that 
it is essentially one and indivisible; and consequently the ner- 
vous system differs from all the other systematic arrangements 
in nature. oF 

It appears to the author that this view of the sentient system 
will alone serve to explain the numerous sympathies which 
exist in animal bodies, the occurrence of disease in the higher 

262 


452 THIRD REPORT—1833. 


orders of animals from indirect or remote impression, and the 
operation of all remedies which act through the medium of the 
sensibility. 

The mode in which the sentient substance is arranged, its 
more or less minute subdivision, and the degree of arterial 
vascularity, determine the phanomena of sensibility as they 
come under our observation. Hence, we find that the brain, 
even different parts of it, the spinal marrow, the trunks of the 
nerves, and their sentient extremities, are so differently en- 
dowed, that we might be almost led into the error of supposing 
them all composed of different materials. 

It is well known to surgeons and to experimental physiolo- 
gists, that the brain is not endowed with any feeling, in the 
common meaning of the word. It may be wounded without 
any sense of pain to the individual. 

The trunks of the nerves not possessing the arrangement of 
the sentient substance suitable to common sensation can only 
transmit the feeling of pain. Thus, patients after amputation 
often complain of pain in the part that has been removed ; but 
the author believes that in no instance have they experienced 
natural or agreeable sensations, or have expressed a conscious- 
ness of the presence of the removed limb unattended with 

ain. 

3 The sentient extremities of nerves are alone capable of being 
affected by narcotic poisons. Half a tea-spoonful of the es- 
sential oil of almonds introduced into the substance of the 
brain of a rabbit did not produce the least effect on the ani- 
mal, nor was any effect produced by placing the end of the 
sciatic nerve in a spoonful of the essential oil of almonds 
during half an hour, although the animal was afterwards killed 
in the usual manner by a few drops of this liquid on the 
tongue. 

Impressions on the extremities of nerves sent to the organs 
of sense and to the external surfaces of the body are attended 
with consciousness in the individual, whilst those naturally 
made on the interior surfaces cause no perception. ‘These 
surfaces, however, are amply supplied with nerves, and possess 
a high degree of local sensibility, by which they not only 
discern mechanic forms, but qualities in food and medicines 
that the perceptive powers of the individual cannot distinguish. 
These internal and unperceived sensations are continually 
though secretly influencing the condition of the whole nervous 
system, and are often the cause of remote morbid actions. 
Under some circumstances movement follows impression made 
on the external parts of the body after consciousness has be- 


TRANSACTIONS OF THE SECTIONS. 453 


come extinct. It is known that the ordinary actions of the iris 
correspond with the impressions of light on the retina; and 
the author has observed that the iris continues to move under 
the same law after the animal’s head has been cut off, or the 
eye taken out, as long as the retina retains its local sensibility : 
similar effects take place in other parts of the body. 

The mutual influence of the nerves and spinal marrow seems 
to be all that is necessary during foetal life, as the absence of 
the brain in the acephalous foetus does not interfere with any 
of the functions of the creature until the moment of birth. 

The offices which the coloured substance performs in the 
nervous system have been matter of speculation with anatomists. 
One obvious purpose of its existence is to give support and 
security to the finest subdivisions of the sentient substance ; 
we therefore find that it affords such protection in proportion 
to the necessity: hence, in the brain, the coloured substance is 
soft and tender, while in the ganglia of the nerves it is gene- 
rally dense and firm. Besides, however, forming a nidus for 
the ultimate plexuses of the sentient matter, the coloured sub- 
stance would seem to fulfill some other use not yet ascertained, 
as wherever it exists it exhibits the same character with respect 
to colour, varying from yellow to green or brown. Dr. Mac- 
artney considers the yellow spot in the retina of the human 
eye, and in that of the monkey and lemur, as a ganglion, having 
discovered that it contains a more intricate reticulation of the 
nervous filaments than exists in the other parts of the retina. 

The coloured substances of the nervous system in no degree 
derive their peculiar tints from the blood that circulates in 
them, since the colours are palest in the foetus, and grow 
darker as the nervous system approaches its perfect organi- 
zation. ; 

It is a generally received opinion that the ventricles of the 
brain are cavities or hollow spaces containing some liquid. 
This error has arisen from the common modes of dissecting the 
brain, which necessarily separate the surfaces of the ventricles 
from each other. If, however, the dissection be performed 
without disturbing the natural position of the parts, not the 
slightest appearance of cavity or interspace presents itself. 
The sole use of the ventricles, therefore, seems to be, merely 
to gain an extent of surface necessary to the development of 
the peculiar organization of the brain. Apparently there is 
less superficies in proportion to the magnitude of the mass of 
the brain in man than in that of animals; but if we calculate 
the depth of the surfaces between the convolutions of the cere- 
brum and on the branches of the arbor vita in the cerebellum, 


454 THIRD REPORT—1833. 


together with the internal surfaces, we shall find that the su- 
perficies of the human brain is greater in relation to its bulk 
than that of any other animal. In addition to the surfaces 
already known, Dr. Macartney has ascertained the existence 
of ventricles (so called) in the bulb of the olfactory nerves, and 
in the optic thalami of the human adult brain. In the thalami 
the distinction of surface is obscure, but in the olfactory tu- 
bercles it is sufficiently plain. 

The author concludes with stating his belief that every as- 
semblage of the nervous filaments in the form of plexus is 
destined to fulfill an especial purpose, and with the anticipation 
that at no distant period we shall be able to understand many 
of the phznomena of sensation which have been hitherto 
veiled in the utmost obscurity. 


Abstract of Observations on the Motions and Sounds of the 
Heart. By Hveu Care, A.B., Demonstrator in the 
School of Anatomy in the University of Dublin. 


The circumstances in the history of the heart’s action which 
have been most the subject of controversy within late years 
may be enumerated as follow:—Ilst, the expansion and con- 
traction of the auricles and ventricles, commonly called their 
‘systole’ and ‘diastole’; 2nd, the beat of the heart against 
the side of the chest; 3rd, the arterial pulse; and 4th, the 
sounds perceptible during the heart’s motion. With a view to 
the explanation of these phenomena the author has made some 
experiments on living animals, the results of which he was 
desirous of communicating to the Association. 

In experiments of this kind it is desirable, as well for ensuring 
the means of accurate observation as for the sake of humanity, 
to diminish as much as possible the suffering of the animal. 
This can be accomplished by the use of the artificial respiratory 
apparatus, the animal having been suddenly deprived of sensa- 
tion without shedding its blood. But the author has found 
that the application of this apparatus causes the heart to con- 
tinue and terminate its motions in an unusual manner, and is 
therefore liable on some points to mislead the observer. In 
those cases in which the employment of artificial respiration is 
not expedient, there is much advantage in using very young 
animals for experiment. In this stage of life, as well as in ani- 
mals of the inferior classes, the different organs appear to haye 
a comparatively independent existence; and as their functions 
are in many instances performed with little disturbance under 


—— oe 


aC ths 


Se eee rere rr rrr 


TRANSACTIONS OF THE SECTIONS. 455 


serious injury to the individual, they also retain their vitality 
long after their separation from the rest of the system. From 
the same causes very young animals appear to suffer less pain 
during experiment than those of mature age. 

After discussing the methods of experimenting, the author 
proceeds to describe the opinions which have been held by 
other persons on the subjects in question, and to compare them 
with the conclusions to which his experiments have led. 

Ist. It has been asserted by Bichat, and his celebrity has 
induced many to adopt the opinion, that the ventricles possess 
a power of active dilatation, by means of which, when their 
systole has terminated, they are enabled to invite into their 
cavities the blood from the neighbouring auricles. The author, 
however, has ascertained by experiment that there is no such 
dilating power in the ventricles, but that these muscles, when 
their state of contraction has ceased, become perfectly soft and 
flaccid, like all other muscles in their state of repose, and thus 
readily admit the blood from their respective auricles, which 
had become distended during the systole of the ventricles. The 
feeling of resistance which was mistaken by Bichat for a dilating 
power, and was supposed by him to accompany the diastole of 
the ventricles, the author has ascertained to be caused by the 
swelling of their muscular fibres during their systole. 

The auricles contract but little upon their contents in man 
and in the higher classes of animals, the small quantity of blood 
which the ventricles discharge at each contraction being com- 
pensated by the frequency of their movement; while in the 
cold-blooded animals, in which the heart acts with less fre- 
quency, the degree of expansion and contraction of both au- 
ricle and ventricle is much greater than in the former classes, 
and the quantity of blood sent through the heart at each move- 
ment is much larger. sind 

2nd. The impulse of the heart against the side of the chest, 
commonly called its beat, has been explained by different 
writers in various ways. Mr. Hunter supposed it to have been 
caused by the straightening of the curve of the aorta during 
the systole of the ventricles, whereby the apex of the heart was 
thrown forwards. Meckel refers it, in part, to the elongation 
of the arterial tubes during the ventricular contraction, and 
partly to the swollen state of the auricles at that time, by which 
the ventricles are pushed forward against the side of the chest. 
Harvey mentions an opinion held by some in his time, and which 
has been lately revived, namely, that the beat is caused, not by 
any active power in the ventricles, but by the muscular con- 
traction of the auricles during their systole, by which the blood 


456 THIRD REPORT—1833. 


being sent with force into the ventricles, distends their cavities, 
and causes them to strike against the chest. This opinion, 
therefore, supposes the beat of the heart to coincide with the 
ventricular diastole. Various other suppositions have been put 
forward upon this subject by different authors. 

The author’s experiments show that the beat of the heart is 
coincident with the systole of the ventricles, and is caused by 
the peculiar shape which these parts acquire in their contracted 
and hardened state, their middle part becoming globular and 
prominent, and their apex being, as Hunter expressed it, ‘ tilted’ 
forward. During their systole the ventricles, like other muscles 
in a state of contraction, become swollen and hard to the touch, 
as was observed long since by Harvey. The greatest quantity 
of muscular fibre being situated about their middle part, where 
the ‘ musculi papillares’ are placed, this part during the systole 
assumes a globular and prominent form, projecting in front, 
and by its protuberance behind pushing forward the body of 
the ventricles. The apex is ‘ ¢é/ted’ forward for the following 
reason. The author has ascertained, by unravelling the struc- 
ture of hearts prepared by boiling, that the fibres which pass 
from the base to the apex, on the front of the ventricles, are 
considerably longer than those similarly placed behind. In 
some human hearts he has found them in the ratio of five to 
three; the shape of the ventricles being nearly that of an ob- 
lique cone, whose base is applied to the auricles, and whose 
longest side is in front. Now it is a law of muscular action 
that fibres are shortened during their contraction in proportion 
to their length when relaxed. For instance, if a fibre one inch 
long lose by contraction one fourth of its length, or one quarter 
of an inch, a fibre two inches in length will lose one inch by a 
contraction of equal intensity. We have seen that the fibres 
which by their contraction cause the apex to approach the base 
of the ventricles, are much longer on the front than on the back 
part, and, consequently, the former are more shortened during 
their contraction than the latter. The apex, then, does not ap- 
proach the base in the line of the axis of the ventricles, but is 
drawn more to the side of the longer fibres, that is, towards 
the front, thus producing the ‘tilting’ forward. 

This conclusion is strengthened by the fact that the forward 
motion of the apex of the ventricles is always proportioned to 
the obliquity of the form of these cavities in different classes of 
animals. In the heart of some reptiles, the frog for example, 
in which the lengths of the fibres of the ventricle before and 
behind are nearly equal, the tilting of the apex is scarcely dis- 
_ cernible. The obliquity is greater, as far as the author has 


TRANSACTIONS OF THE SECTIONS. 457 


been able to observe, in the human heart than in that of any 
other animal. 

Mr. Carlile has ascertained, also, that the ventricles assume 
this form during their contraction, after they have been sepa- 
rated from the auricles by a ligature, and even after they have 
been removed from the body, and placed in a vessel of tepid 
water, or keld upon the hand, the auricles having been pre- 
viously cut off; in all which cases the peculiar motions which 
accompany their contraction and relaxation were observed to 
recur as long as their power of moving remained ; proving that 
the beat of the heart is produced altogether by the action of 
the ventricles during their systole, and that in these, as in all 
other muscles, the peculiar forms assumed during their con- 
traction depend upon the relation, as to length and position, of 
the fibres of which they are composed. 

ord. The arterial pulse, which is produced by the jet of 
blood sent from the left ventricle into the aorta during its 
systole, has been stated by Bichat and many other writers to 
be synchronous throughout the whole arterial system. But 
the experimenter can ascertain in his own person that the pulse 
is successive at different distances from the heart. If the hand 
be placed over the region of the heart, and the radial artery be 
felt at the same time, an interval will be distinctly perceptible 
between the beat and the pulse ; and if the anterior tibial artery 
be substituted for the radial, the interval will be found still 
greater. Repeated observations of this kind show that the in- 
tervals of time between each beat of the heart and the corre- 
sponding pulse in different parts of the body are proportioned 
to the distances measured along the arteries, from the heart to 
_ the respective parts; and a knowledge of this fact leads, without 

further anatomical inquiry, to the conclusion that the beat of 
the heart is coincident with the ventricular systole. For, as 
the intervals of time between the beat and pulse are propor- 
tioned to the distances from the heart to those parts where the 
pulses are felt, it follows that when the distances become eva- 
nescent the intervals of time will also vanish. Consequently, at 
the origin of the aorta the pulse will coincide as to time with 
the beat of the heart; but the pulse at the origin of the aorta 
isnecessarily synchronous with the ventricular systole, by which 
the blood is driven into that artery; and therefore the beat 
_of the heart will coincide with the ventricular systole, a conclu- 
sion which agrees with that drawn from positive experiment. 
The proportion which exists in the pulse between the in- 


tervals and distances is dependent upon the elasticity of the 
arteries. | 


458 THIRD REPORT—1833. 


4th. An explanation of the sounds of the heart has become 
necessary since the employment of the stethoscope in ascer- 
taining the state of internal parts. Laennec has well described 
these sounds, and properly refers the first to the rush of blood 
from the ventricles during their systole. But, in supposing 
that the second sound is produced by the auricular systole, he 
has fallen into an extraordinary error, as the second sound 
follows immediately after the first one, whereas the auricular 
systole precedes the ventricular. This mistake has been no- 
ticed by different writers since Laennec’s time, who have re- 
jected his explanation, and substituted others in its place. 

From the observations which the author has made, he has 
no doubt that the second sound is caused by the obstacle which 
the semilunar valves present to the passage of the blood from 
the arteries back into the heart, at the termination of the ven- 
tricular systole. 

At each contraction of the ventricles a quantity of blood is 
driven by them into the trunks of the arteries, which, being 
already full, accommodate the addition to their contents by a 
lateral expansion of their parts nearest to the heart. When 
the systole of the ventricles is at an end, the elastic force of the 
arteries, acting upon their contained blood, drives it towards 
the heart, its entrance into which is prevented by the sudden 
closing of the semilunar valves: and thus a shock is communi- 
cated to the front and upper part of the ventricles, and to the 
adjacent trunks of the arteries, which may be heard by the 
ear placed over the region of the heart. ‘The relation, as to 
time, which the second sound has to the first, its abrupt cha- 
racter, and its coincidence with the end of the ventricular systole, 
have led the author to adopt the foregoing opinion. 

Mr. Carlile then described the experiments from which the 
greater number of the preceding conclusions have been drawn, 
and having detailed the circumstances of some made upon living 
subjects, proceeded to relate those which follow. 

1. Artificial respiration having been established in a rabbit 
which had been strangled, and the heart having been exposed, 
the following observations were made. 

The finger being applied successively to the front, back, and 
each side of the ventricles, conveyed the sensation of hardness 
and impulse when the ventricles assumed the globular form, and 
of softness and flaccidity when they became flattened and ex- 
panded. The end of a probe being laid on the front surface 
of the ventricles, was raised nearly a quarter of an inch during 
the former of these states, and sank, causing a slight depression 
on the surface, in the latter. The probe was more elevated 


TRANSACTIONS OF THE SECTIONS. 459 


when‘placed on the middle point of the surface, or on the front 
of the apex, than when placed elsewhere. 

The right wing being held aside, so as to admit of the right 
auricle being’ seen, this was observed to swell during the con- 
tinuance of the ventricles in their hardened state, and to dimi- 
nish its size from the instant in which their flaccidity commenced, 
its degree of contraction being, however, inconsiderable. _ The 
contraction of the appendix was preceded by that of the rest of 
the auricle, and followed by the instantaneous movement and 
hardening of the ventricles... The contraction of the different 
parts of the auricle was successive, commencing at the venz 
cave, and terminating at the appendix, of which last the con- 
traction was much more sudden and distinguishable than that of 
any other part. 

The heart in this subject continued to beat for an hour, when 
the motions in all its parts ceased, and nearly at the same time ; 
both auricles and both ventricles remaining distended, soft, and 
full of blood. The heart, separated from the body, was thrown 
into tepid water, where it remained, soft, and without motion, 
and had lost the power of contracting itself. 

2. A rabbit having been strangled, the heart was exposed 
while still beating. In about 10 minutes the left ventricle 
ceased to move, and had contracted itself firmly. In a minute 
or two afterwards all motion was at an end in the left auricle, 
which was also contracted. The right ventricle continued. its 
movements for 45 minutes, and during its contraction the apex 
of the heart was drawn a little to the right side. The right, 
auricle continued to possess motion for an hour and three quar- 
ters; and for the last 20 minutes its contraction proceeded 
slowly, and with a motion apparently vermicular, over its sur- 
face ; always commencing at the part contiguous to the venz 
cave, and ending at the appendix. The right auricle and ven- 
tricle contained each some blood when their motions ceased ; 
but, the heart having been thrown into tepid water, they gra- 
dually expelled their contents, assuming, as those of the left 
side had done, a firm and contracted state. 

The difference of the states in which the hearts were found, 
after their motions had ceased, in the last two experiments, is 
remarkable, and appears to admit of the following explanation. 
In the last experiment, in which no means were employed to 
continue respiration, the left side of the heart soon ceased to 
move; because a continuance of the functions of the lungs, as 
proved by the experiments of Bichat, is necessary to the main- 
taining of its actions. The firmness of its contraction shows, 
that although its ordinary motions had ceased, it still retained 


460 THIRD REPORT—1833, 


a considerable share of organic life, as it is known that muscles, 
whose vitality is quite extinct, have no power of contraction. 
In the experiment in which respiration was artificially main- 
tained, the left side of the heart continued to beat for an hour, 
the sustained function of the lungs affording to it a motive for 
prolonged action ; but having been deprived of the influence 
which the central parts of the nervous system extend to organs 
in vital connexion with them, its powers of life were exhausted 
by the long continuance of its motions, and when these ceased, 
it was quite dead, and incapable of a vital contraction. The 
right side of the heart in the last experiment seems to have 
participated in the exhausted state of the left side, because its 
motions had been performed with much more energy during 
their continuance than would have been the case had not re- 
spiration been artificially maintained. 


On the Mechanism and Physiology of the Urethra. By 
Henry Earte, F.R.S., Professor of Anatomy and Surgery 
to the Royal College of Surgeons. 


The author, having been lately engaged in delivering a course 
of lectures on the anatomy and diseases of the urinary organs, 
was led to prosecute his inquiries into the minute structure of 
the urethra, and to avail himself of the aid of comparative ana- 
tomy to elucidate the subject. The results of this inquiry he 
related briefly to the Section, with a view of reconciling some 
of the discordances of opinion at present existing, and of ex- 
plaining the double functions of the organ. 


On the Nomenclature of Clouds. By —— Burr. 


In the course of some meteorological observations, Mr. Burt 
found the variations in the forms of clouds to be so numerous, 
that it was difficult, by the use of Mr. Howard’s nomenclature, 
to describe them with sufficient accuracy. 

In consequence, he suggests the propriety of defining the 
leading sections of clouds by peculiarities of their external con- 
stitution, and of characterizing the minor divisions by the ex- 
ternal forms of the masses. 


TRANSACTIONS OF THE SECTIONS. 46] 


On the peculiar Atmospherical Phenomena, as observed at Hull 
during April and May 1833, in relation to the prevalence of 
Influenza. By G. H. Frevpine, M.R.C.S.L. 


. The author observes, that the true causes of epidemic dis- 
eases being for the most part unknown, all the unusual circum- 
stances which occur during their prévalence, especially if these 
be capable of estimation by exact comparative measurements, 
should be carefully recorded. ‘The value of meteorological ob- 
servations, as tending to determine the most important of the 
variable conditions of this interesting problem, is insisted on, 
and the results of his own observations are presented as prov- 
ing that the state of the atmosphere during the period of the 
prevalence of influenza at Hull in 1833 was extraordinary. 
The following are the numerical results. 
- 1832, 1833. 
Gia at We ee Ca 
April. May. Diff. April. | May. Diff. 


Mean pressure of the air. | 30°063 | 29-989 | 0-08 29°799 | 30°177 | 038 
dew point ..... 40°006 40°808 0°80 37°823 | 45°253 | 7:48 
temperature in shade | 46°674 | 49°767 | 3-09 44-706 | 55°393 | 10°69 
temperature in sun. | 68°348 | 72:865 | 4°52 7:251 | 84:122 | 16-97 
max. temp. in shade | 53°933 | 57°809 | 3°87 51°310 | 63-690 | 12°38 
min. temp...... 39°416 41°725 2°31 387103 | 47-096 | 8:99 

Quantity of rain in inches 3°820 2-240 { 1°58 4530 0-600 |} 3:93 


From the columns of differences it will be seen how much 
more sudden and violent in all respects was the transition from 
‘April to May in 1833 than in 1832. The number of hours in 
which the sun thermometer, which has a blackened bulb, 
could be used was, in April, 81; in May, 158. ‘The winds in 
April were easterly at the beginning and end, W., S.W., and 
N.W. in the middle ; in March generally S., varying to the E. 
and W. Rainy days in April, 23; in March, only 2. 

The 16th of May is particularly mentioned as affording a 
remarkable instance of contrast between the years 1832 and 
1833. In 1833, during 14 hours, the thermometer in the shade 
averaged upwards of 70°; during 8 hours nearly 75°; from 2 
tod p.m. 77°. The thermometer in the sun for 19 successive 
hours was upwards of 90°. Minimum temperature of the fol- 
lowing night 49°. Range of temperature in the sun 47°°5; in 
‘the shade 28°. In 1832, during 13 hours, the thermometer in 
the shade averaged rather more than 48°; in the sun at 3 P.M. 
the thermometer reached 62°°8; at 2 and 3 in the shade 50°°8 
and 51°, which was the maximum. Minimum of the following 
night 33°. 


462 THIRD REPORT—1833. 


In conclusion the author states that he does not offer these 
data as affording a complete explanation of the prevalence of 
influenza, but remarks that it is difficult to imagine otherwise 
than that such sudden changes from cold to heat, from wetness 
to dryness, from midday heat to cold evening fogs, must have 
had a very decided and general influence on the health of the 
human body. i 


IV. HISTORY OF SCIENCE. 


A short Account of some MSS. Letters (addressed to Mr. Abra- 
ham Sharp, relative to the Publication of Mr. Flamsteed’s 
Historia Celestis, ) laid on the table, for the inspection of the 
Members of the Association. By Francis Batty, V.P.R.S., 
President of the Royal Astronomical Society. 


Tues letters belonged to the late Mr. Abraham Sharp, and 
were found some years ago in a box deposited in an old lumber. 
room, filled with various books and papers, which had been 
considered as of so little use that they were frequently taken 
out by the servants to light the fire, and were otherwise de- 
stroyed and lost. The present collection of them, which was 
preserved from such destruction, consists of above 120 letters 
from the celebrated Flamsteed, and of about half that number 
from Mr. Crosthwait, his assistant at the Royal Observatory, 
all addressed to Mr. Sharp, who at that time lived at Little 
Horton, in Yorkshire, on an estate of his own. It is probable 
that these are the letters alluded to in the life of Mr. Sharp, 
inserted in Dr. Hutton’s Mathematical Dictionary, the parti- 
culars of which, however, have never yet been made public. 
They are now the property of a relation of the late Mr. Sharp, 
residing in London, by whose permission they are exhibited for 
inspection. 

It is well known that Mr. Sharp divided the mural are that 
was erected at Greenwich for Flamsteed’s use, and that he 
was for some time the assistant there. He afterwards retired 
to his estate at Little Horton, where he lived a very secluded 
life, passing most of his time in astronomical calculations. 
Flamsteed employed him to compute the places of several of 
the stars in his Catalogue, from the original observations ; and 
_an extensive and friendly correspondence was kept up between 
them till the time of Flamsteed’s death, and was afterwards 
continued with Mr. Crosthwait, who superintended the print- 


TRANSACTIONS OF THE SECTIONS. 463 


ing of Flamsteed’s works. ‘This correspondence embraces a 
variety of subjects; but the principal, the most novel, and the 
most interesting is the account of the repeated difficulties and 
impediments which delayed and almost prevented the printing 
of the Historia Celestis. 

The date of the first letter, in the present collection, is 
February 6th, 1701-2, at which time it appears that Flam- 
steed was preparing to publish his work, which was not com- 
pleted till twenty-four years afterwards, being six years after 
his decease. He commenced the publication at his own cost 
and risk; but after he had expended a considerable sum 
of money, the subject was mentioned to Prince George of 
Denmark, who undertook to defray the expense of bringing 
out the work: and here his troubles began; for, in the first 
place, the Prince declined the publication of the maps, which 
Flamsteed considered the most important part, and such as, 
in his opinion, would tend most to the “ glory of the work ;” 
and secondly, the committee of the Royal Society, to whom 
the superintendence of the business was intrusted, appear, from 
the whole tenor of these letters, to have thrown every obstacle 
in the way to prevent the progress of the printing. It is not 
directly stated who were the members that formed that com- 
mittee, but it is evident from the correspondence that Newton 
and Halley formed a part of it ; and Flamsteed can never touch 
on this subject (and it forms a prominent portion of his letters,) 
without expressing his opinion, in no very courteous language, 
of their unfriendly and hostile conduct towards him. 

It was in 1704 that the Prince offered to undertake to defray 
the expenses of the printing ; but so many impediments were 
thrown in the way (oftentimes frivolous and vexatious,) that it 
was not till the end of the year 1707 that the first volume only, 
the least interesting part of the work, was completed. Before 
the second volume was commenced,’ the committee required 
Flamsteed to deposit in their hands a duplicate copy of the 
Observations, as well as of his Catalogue, which he accord- 
ingly did, sealed up. New causes for delay, however, were 
brought forward, and before the second volume was sent to 
the press Prince George died. During the whole of this time 
Flamsteed had received only £125 towards the expenses of the 
work ; and as he saw no prospect of any further support from 
Government, he resolved to wait for better and more favour- 
able times. 

He then demanded from the committee the return of the ma- 
nuscript Observations and Catalogue which he had deposited 
in their hands, which request they appear to have refused. 


464. THIRD REPORT—1833. 


The breach was now complete, and the subsequent letters are 
filled with complaints of the conduct of the committee ; and 
Flamsteed eventually commenced legal proceedings against Sir 
Isaac Newton for the restitution of the MSS. But it is prin- 
cipally on Dr. Halley that the force of his indignation falls ; 
and if the circumstances referred to in the letters be correct, 
(of which there does not seem to be any doubt, although the 
motives of the parties may have been misinterpreted,) Flam- 
steed had just cause for complaint and redress; for he charges 
Halley, in direct terms, with having surreptitiously purloined 
the manuscript Observations and Catalogue deposited with the 
committee, and with having published them in a garbled and 
incorrect manner. It is acknowledged that the seals were 
broken; but it is pretended that this was done by an order 
from the Secretary of State, for what purpose, however, does 
not appear. It is notorious that Halley did publish an edition 
of Flamsteed’s Catalogue, and extracts from his Observations, 
in the year 1712, which is the work alluded to by Flamsteed ; 
and as Flamsteed could never recover back the MSS., there 
is no doubt that these were the documents made use of. In 
fact, the matter is not disguised by Halley, in the preface. 
Flamsteed remonstrated against this conduct; calls Halley ‘a 
malicious thief,” and bestows on him other opprobrious epithets. 
In the year 1716, Flamsteed obtained an order from the King 
to have the remaining (unsold) copies of this work delivered up 
to him, for the purpose of being destroyed: 300 copies were 
consequently sent to the Observatory, which, he says, he ‘‘ sa- 
crificed to truth”; and he appears to have missed no oppor- 
tunity of destroying every copy that came into his possession. 
Such is Flamsteed’s history of the edition of 1712. 

During all this time, no further progress had been made in 
printing the Observations. The first volume only was com- 
pleted, but this did not contain any of the observations made 
with the mural arc at Greenwich; the second, which was to 
commence with those observations, was not yet begun. Flam- 
steed, however, had printed, for private circulation only, a cor- 
rect copy of his Catalogue of Stars, to counteract the effect of 
Halley’s spurious edition; but no steps had been taken towards 
forwarding the main work, which had now lain dormant upwards 
of ten years, and which was much increased by the numerous 
observations made during that period. At length, not being 
able to regain possession of the MSS., he was obliged to copy 
them again from the original entries, which was a great trouble 
and expense to him; and towards the end of the year 1717, 
he sent the first sheet of the second volume to the press; re- 


TRANSACTIONS OF THE SECTIONS. 465 


solved to proceed in the work at his own cost. Before 
his decease, which happened on Dec. 31st, 1719, he had 
completed that volume, having been occupied nearly nineteen 
years in the prosecution of the work, struggling with difficul- 
ties of various kinds, and thwarted and opposed in various 
ways. Itis to his perseverance and public spirit, supported 
afterwards by the gratuitous exertions of Mr. Sharp and Mr. 
Crosthwait, that we are indebted for the British Catalogue, 
and for that vast mass of observations made at the Royal 
Observatory, which are still of use in various branches of 
astronomical research, and which will render his name illus- 
trious as long as the science exists. 

The correspondence of Mr. Crosthwait relates principally to 
the difficulties, impediments, and delays which still prevented 
the work from being brought to a final conclusion ; and it may 
be safely stated, that had it not been for the extraordinary 
exertions of Mr. Sharp and Mr. Crosthwait, the whole would 
never have been completed. ‘The Catalogue was reexamined 
and compared with the observations, and afterwards reprinted 
with several amendments. The preface cost him much trouble: 
it was required to be translated into Latin, but no one could 
be found adequate to the task, though repeatedly attempted. 
Mr. Pound undertook it, but eventually declined it; and it 
was at last accomplished by a Dissenting minister. The third 
volume was at length finished, and the whole work published 
in 1725, six years after Flamsteed’s death. 

There remained now only the maps, the construction and en- 
graving of which appear to have cost as much trouble and vex- 
ation as the letter-press. Only one of them was finished (Orion) 
when Flamsteed died. For the rest we are indebted to Mr. 
Sharp, who constructed them anew, according to Flamsteed’s 
principles, from the Catalogue. Sir James Thornhill drew the 
figures of the constellations, and recommended engravers for 
the work; but the charges of the English artists were consi- 
dered so enormous, that Mr. Crosthwait went over to Holland 
for the express purpose of engaging some of the best Dutch 
engravers to complete the work. ‘The vexatious delays which 
necessarily occurred by adopting this method, its increased ex- 
pense, and the constant attention requisite to prevent mistakes, 
dispirited Mrs. Flamsteed, and a temporary stop was conse- 
quently put to the work, although Mr. Sharp (now much ad- 
vanced in years) and Mr. Crosthwait were willing to continue 
their services. Atlength, some English engravers being found 
who offered to execute the maps at a more moderate expense, 


the labours of these gentlemen were renewed, and continued till 
1833. 2u 


466 THIRD REPORT—1833. 


the time of Mrs. Flamsteed’s death, which took place on July 
29th, 1730. 

Here the correspondence ceases, probably on account of the 
circumstances mentioned in the last letter, whereby it appears 
that Mrs. Flamsteed did not leave either Mr. Sharp or Mr. 
Crosthwait a single farthing for all their services; neither had 
they received any remuneration since Mr. Flamsteed’s death 
for their unparalleled exertions in her behalf. 


*.* Since the above statement was written, Mr. Baily has 
discovered amongst Flamsteed’s manuscript papers, deposited 
at the Royal Observatory at Greenwich, all the Answers of Mr. 
Sharp to the above letters of Flamsteed; thus constituting a 
complete correspondence between the parties for nearly eighteen 
years. 


CORRIGENDUM IN PROFESSOR POWELL’S PAPER. 


In the abstract, given in the Proceedings of the Physical Section, of Professor 
Powell’s paper a formula is introduced (p.377) which the author finds, since 
the paper was printed, is incorrect; this however does not affect the rest of the 
paper: but the whole will shortly appear in detail in another form. 


a 


RECOMMENDATIONS 


OF 


THE BRITISH ASSOCIATION 


FOR THE 


ADVANCEMENT OF SCIENCE. 


Tue following Reports on different Branches of Science have 
been drawn up, at the request of the Association. 


Vol. I. 


On the progress of Astronomy during the present century, by 
by G. B. Airy, M.A., Plumian Professor of Astronomy and 
Natural Philosophy, Cambridge. 

On the state of our knowledge respecting Tides, by J. W. 
Lubbock, M.A., Vice-President and Treasurer of the Royal 
Society. 

On the recent progress and present state of Meteorology, 
by James D. Forbes, F.R.S., Professor of Natural Philosophy, 
Edinburgh. — 

On the present state of our knowledge of the Science of 
Radiant Heat, by the Rev. Baden Powell, M.A., F.R.S., Sa- 
vilian Professor of Geometry, Oxford. 

On Thermo-Electricity, by the Rev. James Cumming, M.A., 
F.R.S., Professor of Chemistry, Cambridge. 

On the recent progress of Optics, by Sir David Brewster, 
LL.D., F.R.S., &c. 

On the recent progress and present state of Mineralogy, by 
the Rev. Wm. Whewell, M.A., F.R.S. 

On the progress, actual state, and ulterior prospects of Geo- 


108: by the Rev. Wm. D. Conybeare, M.A.,F.R.S., V.P.G.S., 


Clea, the recent progress and present state of Chemical Sci- 
ence, by James F. W. Johnston, A.M., Professor of Chemistry, 
Durham. 

On the application of Philological and Physical Researches 
to the pea of the Human Species, by J. C. Prichard, M.D., 
F.R.S., 

2H2 


468 THIRD REPORT— 1833. 


Vol. II. 


On the advances which have recently been made in certain 
branches of Analysis, (Part I.,) by the Rev. G. Peacock, M.A., 
F.R.S., &c. 

On the present state of the Analytical Theory of Hydro- 
statics and Hydrodynamics, by the Rev. John Challis, M.A., 
F.R.S., &c. 

On the state of our knowledge of Hydraulics, considered as a 
branch of Engineering, (Part I.,) by George Rennie, F.R.S., &c. 

On the state of our knowledge respecting the Magnetism of 
the Earth, by S. H. Christie, M.A., F.R.S., Professor of Ma- 
thematics, Woolwich. 

On the state of our knowledge of the Strength of Materials, 
by Peter Barlow, F.R.S. 

On the state of our knowledge respecting Mineral Veins, by 
John Taylor, F.R.S., Treas. G.S., &c. 

On the state of the Physiology of the Nervous System, by 
William Charles Henry, M.D., F.R.S. 

On the recent progress of Physiological Botany, by John 
Lindley, F.R.S., Professor of Botany in the University of Lon- 
don. 


The following Reports have been undertaken to be drawn 
up, at the request of the Association. 

On the theories of Capillary Attraction and of the Propagation 
of Sound as affected by the development of Heat, by the Rey. 
John Challis, M.A., F.R.S., &c. 

On the state of our knowledge of Hydraulics, (Part II.,) by 
George Rennie, F.R.S. 

On the present state of our knowledge respecting the con- 
nexion of Electricity and Magnetism, by S. H. Christie, M.A., 
F.R.S., Professor of Mathematics, Woolwich. 

On the state of the science of Physical Optics, by the Rev. 
H. Lloyd, M.A., Professor of Natural Philosophy, Dublin. 

On the state of our knowledge respecting the application of 
Mathematical and Dynamical principles to Magnetism, Electri- 
city, Heat, &c., by the Rev. Wm. Whewell, M.A., F.R.S. 

On the recent additions to our knowledge of the Phenomena 
of Sound, by the Rey. R. Willis, F.R.S., &c. 

On the state of our knowledge respecting the relative level 
of Land and Sea, and the waste and extension of the land on 
the east coast of England, by R. Stevenson, Engineer to the 
Northern Light-houses, Edinburgh’. 


1 Communications of facts relative to this subject are much wanted, and may 
be addressed to Mr. Stevenson, Civil Engineer, Edinburgh. 


RECOMMENDATIONS. 469 


On the state and progress of Zoology, by the Rev. Leonard 
Jenyns, M.A., F.L.S., &c. 

On the state and progress of Systematic Botany, by G. Bent- 
ham. 

On the state of our knowledge respecting the influence of 
Climate upon Vegetation, by the Rev. J. S. Henslow, M.A., 
Professor of Botany, Cambridge. 

On the state of Physiological knowledge, by the Rev. Wil- 
liam Clark, M.D., F.G.S., Professor of Anatomy, Cambridge. 

On the state of Pathological knowledge, by John Yelloly, 
M.D., F.R.S. 


RECOMMENDATIONS 


OF 


THE COMMITTEES, 


WITH NOTICES OF DESIDERATA IN SCIENCE BY THE 
AUTHORS OF REPORTS. 


[The Recommendations adopted at the Cambridge Meeting have an asterisk 
prefixed. | 


ASTRONOMY. 


Tue Committee for Mathematical and Physical Science stated, 
that it would tend much to the advancement of astronomy and 
the art of navigation, if the observations of the sun, moon, and 
planets, made by Bradley, Maskelyne, and Pond, were reduced. 

It was resolved by the General Committee, that a represen- 
tation to this effect from the British Association be submitted 
to Government, in the hope that public provision might be 
made for the accomplishment of this great national object; and 
that a deputation, consisting of Professor Airy, Mr. Baily, Mr. 
Davies Gilbert, and Sir John Herschel, be appointed to confer 
with the Lords of the Treasury on the subject. 

‘The application was immediately complied with by the Go- 
vernment, and an advance of 500/. has been made by the Trea- 
sury towards the reduction of the observations from the year 
1750 to the present day. 


470 THIRD REPORT—1833. 


Desiderata noticed in Professor Airy’s Report, p. 187. 


1. Directions for placing a thermometer so as to indicate cor- 
rectly the Temperature of the Air at the place of observation, 
for Refraction-corrections, the external and internal tempera- 
tures being supposed as nearly as possible equal. 

2, Experimental Data for the Theory of Refraction— 

What is the law of the decrease of temperature, or of 
density, in ascending ? 

How does this vary at different times ? 

Can any means be contrived for indicating practically at 
different times the modulus of variation ? 

Does the refractive power of air depend simply on its 
density, without regard to its temperature ? 

Is it well established that the effects of moisture are al- 
most insensible ? 

Can any rule be given for estimating the effect of the 
difference of refraction in different azimuths, accord- 
ing to the form of the ground? 

When the atmospheric dispersion is considerable, what 
part of the spectrum is it best that astronomers should 
agree to observe ? 

3. An investigation of the coefficient of Nutation from the 
Greenwich circle-observations. ; 

4, The reduction of Bradley’s and Maskelyne’s Observations 
of the Sun and Planets, on a uniform plan. 

5. Remeasurement! of the elongation of Jupiter’s Satellites, 
to correct the estimate of the mass of Jupiter. 

6. Separate investigations, from observations, of the diminu- 
tion of the aphelion distance and perihelion distance of Encke’s 
Comet, for the purpose of testing the truth of Encke’s assumed 
law of density of the resisting medium. 

7. Calculations of the perturbations of Biela’s Comet for the 
interval between 1772 and 1806, and of those of the node and 
inclination from 1806 to 1826, for the purpose of ascertaining 
the identity of the comet of 1772, and examining whether this 
comet gives any indication of a resisting medium. 

8. Verification of Burckhardt’s formule in the Mémoires 
de U Institut for 1808, and extension of them to terms depend- 
ing on the inclination. 

9. Theory of the perturbations of Pallas, and of Encke’s 
Comet. 


1 Professor Airy himself has since made the required measurements, and 
given a determination of the mass of Jupiter. 


RECOMMENDATIONS. ATI 


TIDES. 


* That a sum not exceeding 200/. be devoted to the discussion 
of observations of the Tides, and the formation of Tide Tables, 
under the superintendence of Mr. Baily, Mr. Lubbock, Rev. 
G. Peacock, and Rev. W. Whewell. 

That the Association should endeavour to procure the gene- 
ral establishment of systematic Tide Observations along the 
coasts of Great Britain and Ireland, and that the standing Com- 
mittee on Tides be requested to select such places' as may 
appear to them most important for this purpose; that the di- 
rection, and, if possible, the intensity of the wind should be ob- 
served, as well as its critical changes after having set for some 
time in a particular direction; and that the altitude of the cur- 
rents of air should also be, as far as possible, remarked. 


METEOROLOGY. 


1. That the Committee in India be requested to institute such 
observations as may throw light on the horary oscillations of 
the barometer near the equator. 

_ 2. That the Committee in India be requested to institute a 
series of observations of the thermometer during every hour of 
the day and night. 

3. That a similar hourly register be established at some mili- 
tary or naval station in the South of England?. 

4. That the decrease of temperature at increasing heights in 
the atmosphere should be investigated by continued observa- 
tions at stated hours and known heights. The hours of 93 
A.M. and 84 p.M., as giving nearly the mean temperature of the 
year, are suggested for the purpose. (See Report, p. 218.) 

5. That persons travelling on mountains, or ascending in bal- 
loons, should observe the state of the thermometer, and of the 
‘dew-point hygrometer, below, in, and above the clouds, and 
determine how the different kinds of clouds differ in these re- 
spects. (See Report on Meteorology, vol. i. p. 245.) 

6. That the temperature of springs should be observed at 
different heights above the mean level of the sea, and at dif- 
ferent depths below the surface of the earth, and compared with 


1 Directions for observing the Tides, extracted from Mr. Lubbock’s Report, 
and Mr. Whewell’s Memoranda, are inserted in the Appendix. 

? Observations in agreement with this recommendation have been commenced 
at Plymouth and Devonport, under the directions respectively of Mr. G. Har- 
vey and of Mr. Wm. Snow Harris. 


472 THIRD REPORT—1833. 


the mean temperature of the air and the ground.—Detached 
observations on this subject will be useful, but a continued and 
regular series of results for each locality will be more valuable’. 
(See Report, vol. i. p. 224.) 

7. That series of comparative experiments should be made on 
the temperature of the dew-point, and the indications of the 
wet-bulb hygrometer, and that the theory of this instrument 
should be further investigated. (See Report,vol. i. p.243—246.) 

8. That particular attention be paid to the improvement of 
the instruments of meteorological research. 

9. That Mr. Phillips, and Mr. Wm. Gray, jun., be requested 
to undertake a series of observations on the comparative quan- 
tities of rain falling on the top of the great tower of York Min- 
ster, and on the ground near its base ; and that similar obser- 
vations be instituted at other places”. 

A standing Committee was appointed, consisting of Profes- 
sors Airy, Christie, and Forbes, Dr. Dalton, Dr. Robinson, Mr. 
Potter, and Mr. Scoresby, to draw up instructions? for ob- 
serving Auroras, and to endeavour to establish corresponding 
observations in every part of the kingdom. 


Desiderata noticed in Prof. Forbes’s Report. 


1. Verification of Dr, Dalton’s theory of the constitution of 
the atmosphere, by direct experiment. (Report, vol. i. p. 206 ; 
Phil. Trans. 1826.) E 

2. Experiments in various latitudes upon the temperature of 
the earth at moderate depths, by means of thermometers with 
long tubes; with a view to determine the position of the ‘ in- 
variable stratum,” where external causes cease to produce any 
effect. (Report, vol. i. p. 221.) 


1 The height of the springs may be determined with sufficient accuracy by a 
common portable barometer. 

2 The observations at York have been made at three adjacent stations of 
known height, with gauges made on the same mould, and measured by one gra- 
duated glass vessel: they have been continued from the Ist of February 1832 
to the present time. From the results, it has been inferred by Mr. Phillips 
that the diminution in the quantity of rain, at the higher stations, has a certain 
constant dependence on the height of the station, and on the condition of the 
air as to moisture in the different periods of the year. For the further elucida- 
tion of this subject, it is desirable that experiments upon the same plan should 
be tried in other situations, and especially where the climate is of a different 
character from that of York ; in the humid atmosphere of Cornwall, for exam- 
ple, and in the drier air of the midland counties. Gauges exactly similar to 
those in use at York will be supplied from thence to persons undertaking to try 
these experiments, on application to the Secretaries, 

3 An abstract of the directions which have been drawn up by the Committee, 
is given in the Appendix. 


a a 


RECOMMENDATIONS. 473 


3. Experiments on the solar and terrestrial radiation. (Re- 
port, vol. i. p. 222.) 

4. Observations on the horary oscillations of the barometer, 
at considerable heights above the sea. This more particularly 
applies to places near the equator’. 

5. Additional observations to determine what is the influence 
of the moon on the height of the barometer. (Report, vol. i. 
p- 234. See also Arago, Annuaire for 1833.) 

6. The application of the hygrometric correction to the ba- 

rometric formulz for heights. (Report, vol. i. p. 254.) 
_ 7. Observations on the phenomena of wind at two stations, 
at considerably different elevations, (p. 249.) The direction of 
the wind should be noted in degrees, beginning from the south 
and proceeding by the west. 

8. Magnetical observations, regularly conducted, especially 
with a view to auroral phenomena. 


OPTICS. 


* That a sum not exceeding £50 be appropriated to the 
construction of a telescopic Lens, or Lenses, out of rock-salt, 
under the direction of Sir David Brewster. 


Desiderata noticed in Sir David Brewster's Report. 


The determination of various constants, namely, 

1. The refractive indices of the two pencils in all crystallized 
bodies, measured in reference to definite points of the spectrum. 

2. The angles at which light is polarized by reflection from 
crystallized and uncrystallized surfaces. 

3. The inclination of the resultant axes of crystals having 
double refraction, for different rays of the spectrum. 

4, The dimensions of the ellipse which regulates the polariza- 
tion of metals and their alloys. 

5. The circularly polarizing forces of fluids and solutions. 

6. The refractive and dispersive powers of ordinary solid and 
fluid bodies, measured according to the method of Fraunhofer. 

7. Experimental determination of the effects of the absorp- 
tion of light by gases upon the light of the fixed stars. (p. 322.) 


1 Those who may possess such observations, continued for one or more weeks, 
with observations of the temperatures of the mercury and of the air, and the 
probable corresponding temperatures of the air at the level of the sea, are re- 
quested to transmit them to Professor Forbes, Edinburgh. _ The local position 
of the point of observation should also be noticed. 


474. THIRD REPORT—1833. 


MAGNETISM. 


1. That a series of observations upon the intensity of 'Ter-. 
restrial Magnetism be executed in various parts of the king- 
dom, similar to those which have been carried on in Scotland 
by Mr. Dunlop. (Some experiments, made in consequence of 
this recommendation, by Dr. Traill, are given in the published 
Reports of the Association, page 557.) 

2. That observations should be made in various places with 
the Dipping-needle, in order to reduce the horizontal to the 
true magnetic intensity. 

* A standing Committee, charged with promoting these 
objects, has been appointed, consisting of Professors Christie, 
Forbes, and Lloyd. The latter gentleman has undertaken to 
make observations on the magnetic intensity in Ireland, before 
the next Meeting of the Association. 


Desideratum noticed in Prof. Christie's Report. 


* A regular series of observations conducted in this country 
on the diurnal variation of the needle. 


ELECTRO-MAGNETISM. 


The Committee recommend for further examination the Elec- 
tro-magnetic condition of mineral veins. (Consult on this sub- 


ject the paper of Mr. Fox, Phil. Trans. 1830.) 


RADIANT HEAT. 


Desiderata noticed in Professor Powell's Report. 


1. Improvement of the means of obtaining accurate indica- 
tions of small degrees of radiant heat: the thermo-multiplier 
of MM. Nobili and Melloni to be subjected to examination’. 
(vol. i. p. 297, &c.) 

Determination of the following questions (p. 298.) : 

2. Do the ratios of the conducting powers of substances re- 
main the same for all thicknesses ? 

3. It is alleged that in certain cases simple heat is radiated 
freely and directly through transparent media : Is it meant that 


1 Professor Forbes gave an account of the performance of this instrument 
at the Cambridge Meeting. 


RECOMMENDATIONS. 475 


the manner of its transmission in such cases is strictly analogous 
to that in which light is communicated ; or is it only an ex 
tremely rapid communication by conduction? What circum- 
stances can be fixed upon to determine our view of the matter? 

4. Taking into account the thickness, state of surface, &c., 
of a body exposed to radiant heat, does any portion of time 
elapse before it acquires heat from the source; or before it 
begins to radiate it again, when acquired ? and how soon will 
it commence radiating on the opposite side; or according to what 
law does the heat distribute itself over or through the body? 
These questions are put in reference chiefly to the action of 
the body as a sereen, and to the possibility of accounting for 
an apparently direct transmission of heat without the necessity 
of supposing any other principle than that of conduction. 

5. What are the modifications which radiant heat undergoes 
in passing through small apertures? (p. 299.) 

6. Sir J. Leslie found that the focus for simple heat, in the 
concave reflectors he used, was different from and nearer to 
the reflector than that for light: Is this confirmed by more 
extensive and exact observations? and what is the precise focal 
distance in different cases? (Leslie’s Inquiry, p. 14.) 

7. What is the proportion of heat reflected at different in- 
cidences ? 

8. What radiation takes place in vacuo? (p. 300.) 


CHEMISTRY. 


1. That British Chemists be invited to make experiments 
for removing doubts respecting the proportions of Oxygen, 
Azote, &c., in the atmosphere; for determining the proportions 
of Azote and Oxygen in Nitrous Gas and Nitrous Oxide ; and 
for more accurately investigating the specific gravity of the 
compound gases in general. 

2. That Dr. Dalton and Dr. Prout be requested to institute 
experiments on the specific gravities of Oxygen, Hydrogen, 
and Carbonic Acid, and that a sum not exceeding 50/. be ap- 
propriated to defray the expense of any apparatus which may 
be required. 

3. That Dr. Turner! be requested to extend his researches 
into the atomic weights of the elementary bodies, and to re- 
port on the progress recently made inthis branch of chemical 
science. 


1 Dr. Turner reported the progress of his researches to the Meeting at 
Cambridge. 


476 THIRD REPORT—1838. 


4. That Mr. Johnston! be requested to undertake the expe- 
riments which have been suggested to the Committee, into the 
comparative analysis of Iron in the different stages of its ma- 
nufacture. 

*5, That a series of experiments on the effects of long-con- 
tinued heat be instituted at some iron-furnace, or in any other 
suitable situation; and that a sum not exceeding 50/. be placed 
at the disposal of a Sub-Committee, consisting of Professor 
Daubeny, Rev. W. V. Harcourt, Professor Sedgwick, and 
Professor Turner, to meet any expense which may be in- 
curred *. 

*6, That inquiry be made as to the most perfect method of 
purifying Mercury, and that the true specific gravity of the 
metal be determined. 

*7, That an examination be made into the nature and quan- 
tity of the gases given off from thermal waters, whether there 
be any variation in these respects according to season of the 
year, hours of the day, or condition of the atmosphere ; and 
whether there be any changes of temperature in the same 
waters. 

*8. That the gaseous products which are discharged from 
the chimneys of smelting and other furnaces and fireplaces be 
examined, at various periods of the operations carried on in 
them, with a view of ascertaining the compounds which are 
formed when the processes are most successfully conducted, 
and also of detecting the existence of compounds which may 
perhaps be new or valuable. 


MINERALOGY. 


1. That Professor Miller be requested to undertake an ex- 
amination of the form and optical characters of those Crystal- 
lized Bodies which have not been previously determined, and 
that Chemists be invited to send him specimens of perfect ar- 
tificial Crystals. 

2. That Dr. Turner, Professor Miller, Mr. Brooke, and the 
Rev. Wm. Whewell, be requested to cooperate in prosecuting 
and promoting the following inquiries, with a view to examine 
the theory of Isomorphism, and the connexion between the 
crystalline forms and chemical constitution of Minerals : 


1 Mr. Johnston reported the progress of his researches to the Cambridge 
Meeting. 

2 These experiments have been instituted by Mr. Harcourt, in Yorkshire, at 
the Low Moor Iron Works, the property of Messrs, Hird and Co., and at the 
Elsecar Furnace, belonging to Earl Fitzwilliam. 


RECOMMENDATIONS. ATT 


1.) To determine whether the angles of varieties of the same 
species (in the usual acceptation of identity of species) are 
identically the same, under various circumstances of colour, 
appearance, and locality ; and if not, what are the differences. 

2.) To determine the chemical constitution of such varieties, 
—the specimens, mineralogically and chemically examined, 
being in all cases the same. 

3.) To determine what quantity of extraneous substances 
may be mixed with a crystalline salt, without altering its form. 

4.) To determine the angles of the various species or vari- 
eties of isomorphous or plesiomorphous groups, and their 
respective chemical composition’. 


Desiderata noticed in Mr. Whewell’s Report. 


1. To determine the optical differences on which depend the 
distinctions of the different kinds of lustre, metallic, ada- 
mantine, vitreous, resinous, pearly. : 

2. To determine whether the oblique rhombic prism consti- 
tutes a real system of crystalline forms, or is a hemihedral 
form of the right prism. 

3. To determine the limits of magnitude and simplicity in 
crystallometrical ratios. 

4, 'To determine whether chemical groups are strictly iso- 
morphous, or only plesiomorphous. 

5. To determine whether the angles of plesiomorphous 
crystals are separated by definite or by indefinite steps. 

6. 'To determine what are the differences of chemical com- 
position corresponding to differences of optical structure in re- | 
sembling minerals, as apophyllite, tesselite, leucocyclite. 


GEOLOGY. 


*1, That measurements should be made, and the necessary 
data procured to determine the question of the permanence or 
change of the relative level of Sea and Land on the coasts of 
Great Britain and Ireland; and that for this purpose a sum not 
exceeding £100 be placed at the disposal of a Sub-Committee, 
consisting of Mr. Greenough, Mr. Lubbock, Mr. G. Rennie, 
Professor Sedgwick, Mr. Stevenson, and the Rev. W. Whewell: 
—the measurements to be so executed as to furnish the means 
of reference in future times, not only as to the relative levels 


1 Professor Miller reported the progress of these inquiries to the Cambridge 
Meeting. 


478 THIRD REPORT—1833, 


of the land and sea, but also as to waste or extension of ‘the 
land. 

*2. That Mr. Rogers (Professor of Chemistry in Philadel- 
phia) be requested to furnish an account of the progress which 
has been made in investigating the Geology of the United States 
America. 

3. That Professor Phillips be requested to draw up, with such 
cooperation as he may procure, a Systematic Catalogue of all 
the organized fossils of Great Britain and Ireland, hitherto de- 
scribed, with such new species as he may have an opportunity 
of accurately examining’. 

4. That Mr. John Taylor be requested to collect detailed 
sections of the Carboniferous series of Flintshire, with a view 
to a comparison with the same series in other parts of England ; 
—with a view also of ascertaining the circumstances under which 
the Mountain Limestone is developed, after its suppression in 
certain coal-fields in the central parts of England. 

5. That the attention of Geologists be invited to those coal 
districts in the midland counties of England, where, the Car- 
boniferous Limestone and Old Red Sandstone being deficient, 
the coal measures rest immmediately on the Grauwacke and 
Transition rocks ;—with a view to discover whether any cir- 
cumstances connected with the physical structure of that part 
of the island can be stated, explanatory of the local absence of 
the two great formations above mentioned. 

6. That sections and plans should also be collected of the 
Coal-fields of Worcestershire, Shropshire, Staffordshire, Che- 
shire, Lancashire*, and the south-western part of Yorkshire*. 

7. That the Faults or Dykes in the carboniferous rocks in 
Flintshire should be examined, with a view to ascertain whether 
some remarkable differences in their character may not be 
observed, as compared with that of veins and dykes in other 
districts. 

*8. That collections be made of accurate plans of ‘‘ heaves” 
in the Veins of Cornwall and the North of England, with a 
view to determine how far the apparently horizontal heaves may 
be explained by vertical motion. 

*9, That the direction, intersection, inclination, and breadth 


1 This catalogue is commenced, several monographs are composed, and a 
general basis is arranged. Communications, lists of organic remains, notices 
of localities, and specimens of new or undescribed species may be addressed to 
Mr. Phillips, Museum, York. 

2 Mr. Elias Hall of Castleton has constructed a map and sections of the Lan- 
cashire coal-field. 

3 Mr. Hartop exhibited at the Cambridge Meeting a correct map and de- 
tailed section of the coal strata on the river Dun. 


RECOMMENDATIONS. 479 


of the non-metalliferous fissures which cross the planes of the 
strata, and in some instances divide many contiguous strata, 
should be observed, in relation to the same circumstances in 
the dykes and mineral veins of the vicinity ; with a view to 
ascertain whether there be any and what connexion between 
these phenomena’. 

*10. That the history of ancient vegetation should be further 
examined, by prosecuting the researches into the anatomy of 
fossil wood which have been exemplified in Mr. Witham’s re- 
cent volume. 

11. That the quantity of Mud and Silt contained in the water 
of the principal rivers of Great Britain should be ascertained, 
distinguishing, as far as may be possible, the comparative quan- 
tity of sediment from the water at different depths, in different 
parts of the current, and at different distances from the mouth 
of the river; distinguishing also any differences in the quality 
of the sediment, and estimating it at different periods of the 
year; with a view of explaining the hollowing of valleys, and 
the formation of strata at the mouths of rivers. 

12. That the experiments of the late Mr. Gregory Watt, on 
the fusion and slow cooling of large masses of stony substances, 
should be repeated and extended by those who, from proximity 
to large furnaces, have an opportunity of trying such experi- 
ments on a large scale; and that trial should be made of the 
effect of long-continued high temperature on rocks containing 
petrifactions, in defacing or modifying the traces of organic 
structure, and of the effect cf the continued action of steam or 
of water at a high temperature, in dissolving or altermg mine- 
rals of difficult solution. 

*13, That the dimensions of the bones of extinct animals 
should be expressed numerically in tables, so as to show the 
exact relations of their dimensions to those of animals now 
living; and also to show what combinations of dimensions in 
the same animal no longer exist. 

* 14, That the following Geological queries be proposed : 

1.) Are there any instances of contorted rocks interposed 
between strata not contorted ? 

2.) Is there any instance of secondary rocks being altered in 
texture or quality by contact with gneiss or primary slates? 

3.) Is the occurrence of cannel coal generally connected with 
faults or dislocations of the strata? 


‘ Mr. Phillips has undertaken to state the results of his examination on this 
subject in certain parts of the North of England, and requests to be favoured 
with communications relating thereto. 


480 THIRD REPORT—1833. 


4.) What is the nature of the pebbles in the new red sand- 
stone conglomerate in different districts: do they ever consist 
of granite, gneiss, mica-slate, chert, millstone grit, or any other 
sandstone which can be distinctly traced to the coal series ? 

5.) Is. the Red Sandstone of Kelso contemporaneous with 
that of Salisbury Crags; and what relation do they respectively 
bear to the adjacent coal-fields ? 

6.) What is the exact northern boundary of the coal-field of 
the River Liddle? 

7.) What are the relations as td age of the two series of whin 
rocks, one running north-east along the Liddle in Roxburgh- 
shire, the other south-east in the neighbourhood of Melrose 
and Jedburgh ? 

8.) Can the Limestone of Closeburn in Dumfriesshire be re- 
cognised beyond that valley? 

9.) Does the Wealden formation exist in the midland counties 
of England ? . 

10.) What is the character of the districts in which ores of 
manganese occur? 

11.) What is the history of the Heematite of Dalton in Lan- 
cashire, in relation to the beds in which it occurs? 

12.) What are the mineralogical characters of the several 
beds comprised in Forster’s Section of the Strata in the North 
of England; and what are the fossils contained in each ? 


Desiderata noticed in Mr. Conybeare’s Report. 


1. An accurate examination of the conclusions deducible from 
the known density of the earth, as to the solid structure and 
composition of the interior. 

2. The attention of residents in our remote foreign depen- 
dencies is invited to the two great questions of comparative 
Geology and Palzontology. 1. Is there or is there not such a 
general uniformity of type in the series of rock-formations in 
distant countries, that we must conceive them to have resulted 
from general causes of almost universal prevalence at the same 
geological era? 2. Are the organic remains of the same geo-- 
logical period specifically similar in very remote districts, and 
more especially under climates actually different; or are they 
grouped together within narrower boundaries, and under re- 
strictions as to geographical habitats analogous to those which 
prevail in the actual system of things? (p. 410.) 

3. Anexamination of the geological structure of the countries 
constituting the great basin of the Indus, where, if in any part 
of India, it is supposed a complete series of secondary strata 
may be expected. (p. 396.) 


RECOMMENDATIONS. 481 


Desideratum noticed in Mr. Taylor's Report on Mineral 
Veins. . 


A correct account of the affinity that the contents of a vein 
bear to certain of the rocks in which the fissure may be situated, 


ZOOLOGY. 


The Committee recommend to the consideration of Zoologists 
the following subjects of inquiry : 

*], The use of horns in the class Mammalia; the reason of 
their presence in the females of some, and their absence in those 
of other species ; the connexion between their development and 
sexual periods; the reason of their being deciduous in some 
tribes, and persistent in others, 

*2. 'The use of the lachrymal sinus in certain families of the 
Ruminantia. 

*3. The conditions which regulate the geographical distri- 
bution of Mammalia. 

*4.. 'The changes of colour of hair, feathers, and other ex- 
ternal parts of animals ; how these changes are effected in parts 
usually considered by anatomists as extra-vascular. 

*5. The nature and use of the secretions of certain glands 
immediately under the skin, above the eyes, and over the nos- 
trils, in certain species of the Grallatores and Natatores; the 
_ nature and use of the secretion of the uropygial gland. 

*6. How long and in what manner can the impregnated ova 
of fishes be preserved, for transportation, without preventing 
vivification when the spawn is returned to water. 

*7, Further observations on the supposed metamorphosis of 
Decapod Crustacea, with reference to the views of Thompson 
and Rathke. | 

*8. Further observations on the situation of the sexual or- 
gans in male spiders, and on their supposed connexion with the 

alpi. 
‘ “9. The use of the antenne in insects. Are they organs of 
hearing, of smell, or of a peculiar sénsation ? 

*10. The function of the femoral pores in Lizards, and the 


degree of importance due to them as offering characters for 
classification. 


BOTANY. 


1. That Botanists in all parts of Great Britain and Ireland 


be invited to compose and communicate to the Meeting of the 
1833. 21 


482 THIRD REPORT—1833. 


Association, Catalogues of County or other local Floras, with 
indications of those species which have been recently intro- 
duced, of those which are rare or very local, and of those which 
thrive, or which have become, or are becoming extinct; with 
such remarks as may be useful towards determining the con- 
nexion which there may be between the habitats of particular 
plants, and the nature of the soil and the strata upon which they 
grow; with statements of the mean winter and summer tempe- 
rature of the air and the water, at the highest as well as the 
lowest elevation at which species occur; the hygrometrical 
condition of the air, and any other information of an historical, 
ceconomical, and philosophical nature. 

*2, That Professor Daubeny be requested to institute an ex- 
tended inquiry into the exact nature of the secretions by the 
roots of the principal cultivated plants and weeds of agriculture; 
and that the attention of Botanists and Chemists be invited to 
the degree in which such secretions are poisonous to the plants 
that yield them, or to others; and to the most ready method of 
decomposing these secretions by manure or other means. 

* A Committee was formed to conduct a series of experi- 
ments on the growth of plants from seeds, and to preserve the 
results of their experiments, in order to establish the identity 
or confirm the specific distinctions of certain allied plants, and 
to communicate the results obtained, from year to year, at the 
Meetings of the Association. 

Mr. Don, Librarian to the Linnean Society, has undertaken 
to be the channel of correspondence on this subject. 


Desiderata noticed in Professor Lindley’s Report. 


1. An accurate account of the manner in which the woody 
part of plants is formed. ‘‘ Perhaps there is no mode of pro- 
ceeding to elucidate this point, which would be more likely to 
lead to positive results, than a very careful anatomical examina- 
tion of the progressive development of the mangel wurzel, be- 
ginning with the dormant embryo, and ending with the perfectly 
formed plant.” 

2. An investigation of the comparative anatomy of flowerless 
plants, with a view to discover in them the analogy and origin 
of their organic structure. 

_ 3. The cause of the various colours of plants. 

4, The nature of the fecal excretions of cultivated plants, 
and of common weeds; the degree in which those excretions 
are poisonous to the plants that yield them or to others; the 
most ready means of decomposing such excretions by manures 
or other means. 


RECOMMENDATIONS. 483 


ANATOMY AND PHYSIOLOGY. 


*], That the effects of poison on the animal ceconomy should 
be investigated and illustrated by graphic representations ; and 
that a sum not exceeding 25/. be appropriated for this object. 
Dr. Roupell and Dr. Hodgkin were requested to undertake this 
investigation. 

*2, That an experimental investigation should be made of 
the sensibilities of the Nerves of the Brain; and that a sum not 
exceeding 25/. should be appropriated to this object. Dr. 
Marshall Hall and Mr. 8. D. Broughton were requested to un- 
dertake these experiments. 


ARTS. 


* That Mr. Dent be requested to communicate to the next 
Meeting of the Association, a statement of the performance of 
his chronometer with a glass balance-spring. 


Desideratum noticed by Professor Barlow in his Report on 
the Strength of Materials. 


A set of experiments on the application of a straining force 
on vertical columns (of timber, iron, &c.). 


STATISTICS. 


*1, That Colonel Sykes be requested to prepare for publi- 
cation his valuable statistical returns, collected by himself in 
India, relative to the four Collectorates of the Deccan, subject 
to the Bombay Government. 

*2. That Professor Jones be requested to endeavour to ob- 
tain permission to examine the statistical records understood to 
exist in great number in the archives of the India House, and 
to prepare an account of the nature and extent of them. 

*3. The inquiries of this section are restricted to facts re- 
lating to communities of men which are capable of being ex 
pressed by numbers, and which promise, when sufficiently 
multiplied, to indicate general laws. 

* A permanent Committee of this section was appointed. 
Professor Babbage was requested to act as Chairman, and Mr. 
Drinkwater as Secretary. 

In a Report since addressed to the Council by this Committee, 
it is stated, that the Committee having deemed it expedient to 
promote the formation of a Statistical Society in London, a 
public meeting was held on the 15th of March, 1834, at which 

212 


484 THIRD REPORT— 1833. 


it was resolved to establish such an institution. The Society 
already includes more than three hundred members, and has 
issued a statement of its objects and regulations, which is sub- 
joined in the Appendix. 

The Committee remark, that ‘ though the want of such a 
society has been long felt and acknowledged, the successful 
establishment of it, after every previous attempt had failed, 
has been due altogether to the impulse given by the last Meet- 
ing of the Association. The distinguished foreigner (M. 
Quetelet) who contributed so materially to the formation of the 
Statistical Section, was attracted to England principally with a 
view of attending that meeting; and the Committee hail this 
as a signal instance of the beneficial results to be expected 
from that personal intercourse among the enlightened men of 
al] countries, which it is a principal object of the British As- 
sociation to encourage and facilitate.” 


GENERAL SCIENCE. 


* That a sum not exceeding 100/. be appropriated towards 
the execution of a plan proposed by Professor Babbage, for 
collecting and arranging the Constants of Nature and Art'. 


APPENDIX 


CONTAINING DIRECTIONS FOR OBSERVATIONS ON THE TIDES, 
AURORA BOREALIS, &c. 


TIDES. 


OxsErvaTions of Tides along the coasts of Great Britain 
and Ireland will be valuable, both in the construction of more 
accurate tide tables, and as data towards the perfection of the 
theory of tides. 

Observations of the tides should record particularly, 

The time in hours and minutes, and height of high water 
daily, or if convenient every tide. 

The time and height of low water. 


See Appendix, p. 490, for an Abstract of Mr. Babbage’s Plan. 


he” ex. tea aie ee ake 


f 
RECOMMENDATIONS. 485 


The direction of the wind, and the height of the barometer 
and thermometer. 

The direction and velocity of the stream of flow and ebb. 

At what hour (with respect to the time of high water and 
low water) the slack water after the stream of flood, and after 
the stream of ebb, respectively occur. 

The height of the water must be given from some fixed mark 
or line, which should be described accurately, so that it may 
be easily found again at a future time. The observer ought to 
state the manner in which the height was measured ; the man- 
ner in which the moment of high water was fixed upon; the 
time employed, whether apparent or mean solar time, and how 
it was obtained. 

The height of the water at the end of every minute, for 
half an hour before the expected time of high water, and until 
there can be no doubt that the time of high water is past. 
Machines to dispense with this minute attention are described 
in the Philosophical Transactions, 1831, and in the Nautical 
Magazine for October 1832". 

The uncertainty occasioned by waves may be avoided by 
making the observation in a chamber, to which the water has 
access by a small opening, or by fixing in the water an upright 
tube, (of wood or iron, for instance,) the bottom or sides of the 
tube being perforated ; in either case an upright measuring 
rod, carefully graduated, and connected to a float, will rise and 
fall with the tide, and permit, at any moment, the height of the 
water to be read off against the collar through which it works. 
This rod may be so constructed as to leave a moveable index 
at the highest and lowest points. 

A long series of continued observations can alone be of use 
towards the determination of the dependence of the time, 
height, and other circumstances of high and low water upon 
the places and distances of the sun and moon; but a smaller 
number of observations will often be sufficient to determine the 
establishment of any place, with more or less accuracy, accord- 
ing to the number of observations ; and the best mode of do- 
ing this is by comparative observations with some place of 
which the establishment is accurately known, or where obser- 
vations are continually carried on. A few sets of comparative 
observations of neighbouring places will give the relative time 
of high water at these places with considerable accuracy ; and 
thus the motion of the tide-wave and the arrangement of the 


1, Tide-guages may be seen in operation at St. Katharine’s Docks, London. 
An excellent one has lately been set up near Bristol by the Literary and Phi- 
losophical Institution of that city. 


486 THIRD REPORT—1833. 


cotidal lines (or lines along which it is high water at the same 
instant,) will be discovered. It would be very desirable for 
those who have the opportunity, to combine so as to effect the 
detailed description of the tides through some small extent of 
coast, such as that which has been effected by M. Daussy for 
the west coast of France. 


AURORA BOREALIS. 


Notwithstanding the attention which has been paid to the 
phenomena of the Aurora Borealis, and the various hypotheses 
which have been imagined to explain them, it will be found 
that there is a want of information on the points which are 
most necessary as bases of induction; and the British Asso- 
ciation have therefore been induced to appoint a Committee in 
the express view of directing observers to the really important 
features of this meteor, and of obtaining, by a system of con- 
temporaneous observation, data which experience shows cannot 
be derived from insulated exertion. 

The following are the most important points which demand 
the attention of observers: 

1. The elevation of the auroral arches and streamers above 
the surface of the earth. 

2. The determination of the question whether the auroral 
exhibition is accompanied by sound. 

3. The existence of recurring periods of frequency and bril- 
liancy in the Aurora. 

4. The influence of arches, streamers, and other auroral 
phznomena upon the magnetic needle. 


1.) It is recommended to all who intend to observe Auroras, 
to make themselves well acquainted with the names of all the 
principal stars to the north of the equator, especially those 
which do not set here. This will be most easily done by stu- 
dying a celestial globe. Good maps of the stars may also be 
consulted with advantage. Either the proper names or the 
Greek characters with the name of the constellation will be 
sufficient. 

Persons who may prefer to determine the angular elevation 
and position of the arches and streamers by graduated instru- 
ments, must be supposed well accustomed to the use of them ; 
they may, however, be reminded, that telescopic sights are 
for this purpose useless, and that steady instruments, which 
can be handled with ease and expedition, are much more avail- 


a 


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fliwiee sty af Tle Ree, Cd, a et Male n . ang 
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Aoi Bente 4 tyes, 9000 ; 


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Rev. Prof. Sedgwick. 


Tat, SAS:16! Long, W. 2* 26° 


hem. 
7, 25—Strong illumination to the N. W., which gradually expanded to a polot 
few degrces E. of N., and toa puint nearly W. The places of greatest Hlomina. 
tion variable, but chiefly near the magnetic meridian, After some time ascending 
spires of yellowish light in several directions. No well defined Juminous masses, 


§, 5.—The light assumed the form of an ill-defined arch; the vertex was 
nearly in the magnetic meridian, the western end reached the horizon under 
‘Arcturus, the curve passed under 6 and y Ura Majoris. Its breadth was con- 
inidernble, the lower edge ill-defined, and scemet to blend itself with the common 
ight of the region. a 
‘Sireake of Nght rose occasionally towards ti 
by We, and W. Cloude of extreme blacnest were scattered through the 
Jcq of the Aurora, Stars distinctly, but rather dimly sen through the 
fhtest part of the Aurora. 


onith, expecially from N, by Evy 


§ 40.—The black clouds had disappeared, 


& §5.—The upper part of the arch better defined than before, passed between 
jf Urex Maj. and very scar & Uraw Maj. Its vertex in or near the 
sit sie meridian. Lower edge still very ill-defined. For a few minutes 
ea inous region was divided hy a concentric band of a dark dirty grey colour 
the lower arch was ill-defined and unsteady. 


ie 
fo iwo arches 


o,10-—Arch nearly as before. A great expansion of light where the N. B. 
sehof the arch touched the horizon, spreading out to a place a few deprecs 
er Gspellas the centre of this bright cloudy mass of light was about 35° E. 
he magnetic meridian. A few patches of clouds in this region were faintly 
Miated on the edger, as if by the light of the Aurora transmitted through 
Remithere was also n'slight expansion of cloudy light where the N. Ws limb 
yearth came sown to the mountain tops, (65% W. of the raagnetic north.) 

M ps-—Arch nearly as before, its lower edge better defined, breadth nearly 
vealto the distance between the pointers of Urea Major. and the upper pointer 
ee marly in the centre of the bright space between the upper and lower curves. 
Teter still nearly in the magnetic meridian. Below the luminous zone, the 
Te own to the horizon, was of a most remarkable dark dirty grey colour, through 
se Arcturus was ecen to shine dimly, as if through afog- The great mass of 
wir itthe N. E. end of the arch broke up into vertical masses, separated from 
haber by a dark colour like that under the arch. ‘The dark grey part under 
{fant pierced by cloudy flashes of light. ‘These gradually increase in number 
ey iniemity, and traverse the dark space in waves or zig-zags, being on the 
ote juniiel to the horizon, During their passage they sent up spires of in- 
sey green light resting on light of a faint reddish tinge. Soon after, the arch 
ve erved to break up in points like the teeth of a comb, and the whole space 
Mich had been occupied by the dark grey colour was broken up into ribs and 
tee of lisht, especially in the western region. ‘This appearance was succeeded 
{rndulation’ propagated towarda a spot S. E. of the zenith, The undulations 
ited to be propagated in interrupted arches of light, being visible only in those 
Fite where the lines of undulation crossed certain vertical streamers, All 
jee ndolations seemed, however, to haye one common relation; the vertex of 
teh arch being nearly in the magnetic meridian. For several seconds there was a 
wilalefined arch passing through a Lyre. 

9,40.—Great streaks of light parallel to the magnetic meridian ; 
the N. W.; white blotches of light in region S, K. of the zenith. 

9, 45 —Soeeession of xtreansera ascending from the W. and N. This illumi- 
scion seems to be propagated by them to the region between Jupiter and. « 
lim. It ecemed ns if certain portions of the heavens in this region reflected the 
ct of the nvcending streamers; as the mame definite portions of space were illu 
cued during each coruscation of the ascending light. 

1. 0.—Hright illumination towards the N. Dark patchy clouds to the N. W., 
ef some dark horizontal etreaks of clouds illuminated ot the edges. At this 
vere many streamers appearing to diverge from a point below the horizon, 
setly below the end of the Bear's Tail (y Ura Major.) those which were ncurly 
Sant N. declining 15° from the perpendicular, Soon afterwards undulations 
ial to Le propagated along these diverging lines, and amorphous blotches of 
ii, extending in one direction ax far as Jupiter, were #cen in the region S. of 

iinith, Some of these were stationary and constantly iHluminated others were 
Gknary but were only illuminated at intervals. AU that distance from tho N. 
witon, which seemed to be the seat of propagution, the luminous blotehes never 
ul the forin of streamers or spires of light. 

10. 10.—Siriw and spires of light nearly in the same direction as before. but 
be tright. No light propagated to, or reflected from, any cloudy spaces S. of 
& renith 
10. 15.—Greatest illumination in the Mog. N. Bright streaks of light and 
ses propagated to the zenith, Stara shone brightly through the brightest light. 
W. 20.—No spires or beams of light, but wndulutions propagated from the 

x N. Phenomena repeated wll 10. 40, nearly. 

ae ere illumination due Nz Brigit space extends to Gemlnl and to 
the W. Fine streamers from the N, The bright space arranges itself into an arch, 
fommencing nearly N.. pasing through » Urece Major. ; about 25° high near 
the inagactes meridian (measured only by a geological clinometer.) Below the arch 
he eame dirty grey colour noticed before. Fora short time here appeared a 
the four stars of the Hears 

followed by streamers and beams of light ascending 


dark clouds 


was 
sceond arch passing throug! 
10. 50.—The arch gone 
from Mag. N- 
11. 5.—Strong illumination towards the N. W. 
towards the zenith, but no streamers. 
. 25.—Few undulations ¢ greatest 
il toavery Juminous to Mag, Ne; brightest part defined above by an arch 
2i* high; the light seemed to extend to the ho ons ‘This was afterwards broken 
up, first into ascending spires of light, ani then Into undulations. 
12, 10._The light was dispersed in fleecy ill-defined masses, extending from 
the N. nearly as far as the W. 


Fine undulations propagated 


iumination in the N. 


12, 40.—A very strong {lumination continued till nearly this time, when the 
observations were discontinued, as the phenomena presented nothing of a definite 
character. 


DENT.—GENERAL REMARKS. 


During nearly the whole time of observation, the wateh-face could be read with 
great ease, and several of the notes were written W ith no otber light than that of 
the Aurora. 

It may. T think. be observed generally that the places of grentest iMlumination 
scern liable to be broken up into parts, separated frnm each other by spaces of the 
Girly grey colour above mentioned, and quite distinet from the inky colour of the 
patehes of clouds. 

Lattempted, several times, to ascertain the variation of the ncedie, but found 
the mounting was not sulllciently delicate to give resulis worth recording. 


DENT, near Sedbergh, Yorkshire, " 


TABULAR CONSPECTUS OF OBSERVATIONS ON THE AURORA BOREALIS 


OF THE 12th OF OCTOBER, 1833, REDUCED TO GREENWICH MEAN TIME. 


MANCHESTER. YORK. GUISBOROUGH. 
Three sets of Obervatiors by R. Potter, jun. P. Clare, and W. Hadfield. J. Phillips. WW. L. Wharton. 
N. Lat. 538 297 Long. W. 2° 13! Nu Lat. 53° 58° Long. W. 1° 4 N. Lat. 54°31" Long. W. 1* 3’ 


6, 5 —Poiot arches ard streamers noticed. (IR. P.) 
6, 41) —Long and large streamer directed towards the zenith. (It, P.) 


7. 0}. —Irregular nebulous masses to 60° alt. and more. 


7. 24—Portion of arch §. of zenith covered the principal stars in Delphinus 


and Aquila, it was 5° broad, passed southwards and vanished. 
7. 40}.—Lightin the north which had heen 20m. stationary, at 15° or 20° alt. 
and about 100° extent on horizon is now disappearing, 


&. 10}.—Bright light in the N. 


§. 39.—An arch ill-defined, about 18° high in the middle. 
ere Me mrch has risen a Title, fe better defined, and a dark: shade appears 
clow ite 
8, 534.—The arch has its vertex under % Urea Maj. and its upper edge touches 
y Urs Maj., altitude about 19° 30. (R. P.) 
8. 54.—Arch about 24° above the horizon at its vertex, 5° broad ; below it is 
1 dark parallel space 5° broad; and below this a second parallel arch of light 
334° broad. ‘The epace below this arch to the horizon is very dark like a dense 
binck cloud. One foot of the exterior arch is nearly west, and the other foot about 
35° E. of N. the vertex is in the magnetic north. (P. C.) Mr. Hadfield says 
20° olt. and horizontal extent 120°. 
_ 9. 9—The two arches remain in the same position. Stars are distinctly seen 
in the upper luminous arch, and also in the space within the lower luminous arch. 
9, 14).—n Ursa: Maj. in the upper edge of the arch, the height of which by 
measure = 21" 10% (It. P.) 


9, 21,—Beam or streamer, the first seen this evening, darted suddenly upwards 
from the higher arch about 30° E, of N. It was visible 30 seconds. (P. C.) 

9, 22.—A beam or streamer in the same position, visible 10 minutes. 

9, 27.—A sinall black cloud visible in the dark space within the Jower arch, 
about 15¢ I of N. and 10° high. 

9, 29.—A beam or streamer, rose from the dark space within the Jower arch 
10° E. of N. and continued visible for 2 minutes. Clouds are rising in N. W. #0 
5 to cover a portion of the light. 

9, 31. —Streamers are shooting up rapidly in the W. and the boundary of the 
arch is much less distinct. 

9, 33. —Streamers and gleams or broad flashes of light are rising from various 
parts of the arch, they soon covered almost the whole of the northera part of the 
hemisphere, and passed to the south a little beyond the magnetic zenith. The 
archi has disappeared, but a strong light remains in the north. 


9. 41-—For the Inst 10 minutes the #treamers have been very active, and broad 
flashes of light have suceceded each other with scarcely any intermission, in the 
northern part of the hemisphere: the clouds have increased 40 as to obscure most 
of the aky. 


10. 9, —Cloudla disappearing, streamers active. 


10. 39.— Since the last observation the display of treamers and coruscations 
has been fine. Arch observed at this time, its lower edge had its vertex in Benet 
nasch (16° alt.) and passed a little below A in Bootes, and alittle above > in the 
foot of Urs. Maj. 


11. 9.—Since the last observations the streamers and coruscations have 
diminished in number and brilliancy. Now very few are to be seen, but a bright 
light continues in the magnetic Ns 

11, 39.—The light in the north increasing. 


12, 9—IJt has continued to increase. Another arch is visible in magnetic N- 
the highest point of which és about 15° above the horizon, Vertex in Mizar, 

12 24 —Streamers begin to arise from various parts of the arch. 

12, 39.—Streamers much increased in number and activity + also coruscations 
and large masses of light like white clouds, that almost cover the northern part of 
the hemisphere, 

12. 49.—The light is more brilliant, the arch has disappeared ; streamers very 
bright and active, particularly inthe N. K, ‘The sky is. covered with streamers 
and gleams of light from nearly the northern horizon to a little beyond the 
magnetic zenith, ‘The streamers form a corona at the magnetic zenith, from 
which point the Pleiades are about 1° south. 

12. 54—Many bunds or fillets of light have been {instantly formed, and then 
immediately rose toward the zenith ; their first appearance was in the direction 
id form of the luminous arch of 8. 54,5 then they suddenly tose in the centre 
towards the zenith and disappeared. ‘The east and west extremities of these bands 
acemed to move through a very sinall space and gave the appearance as if the 
extreme points were stationary, whilst the highest point moved round a centre 
with a velocity that earried it through 60° in about one-third of a second of time 5 
the other portions of the bands passed through a less space us they approximated 
to the K. and W. cade, For about ten minutes these bands of light succeeded 
each other with gnat rapidity. ‘he streamers and coruscations of light continued 
very active. 

Ty. 9 The comscations, streamers, and bands of light have been diminishiog 
in activity and brijlianey for 10 minutes, but during the last half hour the bril- 
Hiancy, extent, and geuvity of the aurora far exceeded the appearances in any 

previous part of right. 
m3. Poth eer ‘and coruscations bave ceased, but thereis & broad light 
dn the north. 

13. 39,—Th 


light diminishes in brilliancy. 


hem. 

6. 4-—Mean time. The appearance of the northern sky led me to suspect that an aurora had commenced. : 

G. 44.—By this time the doubr was entirely removed ; beams had been sceo playing, and bright fluctuating light was at in= 
tervals excited at detached points along an arch, rectangled to the magnetic meridian, as if along this line thin vaporous clouds Or 
amoke-wreaths were illuminated from behind. One continual beam was tcen N. 20 W. The phenomena were chiefly on the 
eastern side of the magnetic meridian. 

6. 52—A falling sur was scen, passing nearly horizontally from E. to W., through an arch of 10% east of the Pole star, It 
was bright. Another faint falling star seen through a shorter arch above the Great Bear. 

7. 14.—A falling star from W- to E., declining 40°, colour, red. 

7. 14.—A beam of light N. by W- 

7. 39-—Falling star, directly downwards in the line of the magnetic meridian, seemed to fall through 15° from the arch which 
was now forming. ove 

7. 44.—An arch of light was now distinct (above a dark horizon band), its lower edge one-third of the height of & Ursa: Majoris 

7. 52.—The arch had risen higher, its lower edge was now half the hcight of 6 Ursa Muj., and its upper edge two-thirds of 
that height. No streamers were visible at this time, nor had many been seen before. 

7. 56.—The summit of the arch was now 3* below the stars 8 and y Ursa Maj. It had been rather bright and of a 
yellowish hue, not unlike some distant clouds, a resemblance augmented by its great breadth, which was twice as great as that of 
the black space beneath. A singular notch was noticed in the upper edge of this arch, bearing N. by E. It now began to fade 
and to sink a little. 

7. 57-—Suddenly It appeared double, in consequence of the production of a very narrow faint arch above that secn before, and 
separated from it by a dark band. 

7. 58.—This upper arch rose, so as to include 6 and y Ursa: Maj., in its middle. 
8. 2.—It had vanished away, after rising still higher. 5 

8. 4—A beam of light, slightly arched, and directed obliquely upwards, from about 4 degrees (by estimation) to the right 
(south) of the Pleiades, between q Arietis and Jupiter. This singular beam, brightest in the middle, was so unlike the others, that 
T hesitated for some time to record its appearance. It lasted, with hardly a perceptible change of place towards the south, for 
half-an-hour, but its brightness was variable. 

8. 6.—The arch appeared divided by @ transverse black clouds 


hem 

$, 20.—Well defined arch, passing between « and PB Ursce 
Majoris its summit somewhat above ¢ Ursa Majoris, no radia- 
tions. 

8, $5.—Arch 2¢ or 3¢ higher; faint radiations. 

$, 50.—Nadiations and coruscations, all converging to the 
head of Andromeda. 


8. 16.—The broad belt of light in the north was divided 
by several (ransverse arched bands. 
8. 24—It had beeome higher, but was more confused. 


& 29,—Falling star from just under Jupiter, its course N. 
declining 20°,—rapid. 


8 S4—Arch irregular, banded, the upper edge between 
and f Urs Moj. 
9. 19 —Falling star from Polaris direct through Dubbe. 


9, 0,—Brilliant radiations with prismatic colours, con- 
verging to 2 Andromedm, accompanied with sheets of light 
moving with vast rapidity to the same part of the heavens; 
summit of arch moved considerably eastward. 

9. 10,—Continuous sheets of light and. radiations from the 
whole horizon, between W.N. W. and E. N.E,, all converging 
to the same part of Andromeda, but upon reaching that point 
apparently checked in their further progress, and (those from. 
the N. W, especially) twisted as it were for a short distance 
toward the S. £. and then dissipated in space, 


9. 29.—At this time the aurora assumed a totally new and 
far more imposing character. Beams of greater brilliancy shot 
tip in the west, and the dark horizon band was traversed, in 
arched lines, from W. to E,, by roundish masses of strong 
light, orange-coloured beneath, from which sprung, irregularly, 
many short beams of paler colour: then at intervals a wild 
Fluctuation of almost. connected light from W- to KE, with beams suddenly arising from a general arched basis of light, below the 
more regular arch previously noticed, which became less and Jess distinct, as the undulations grew more frequent, nd the bears 
rose higher. ‘The undulations of light became every moment more rapid and extensive, lost their transverse movement and rose 
higher into the sky, till they covered with their pale and restless illumination the whole of the northern hemisphere, flashed over 
the zenith stars, and even to a cousiderable distance south. As soon us the strangeness of the spectacle permitted me to examine 

with calmness, I found that these flashes seemed all to spring from an arched luminous base (probably the perspective union of 
their images) with a rapid but limited undulation; the higher above this base the appearances) were exami the more 
decided was the evidence of intermitting action, the more disjoincd became the upward waves of light, and at length it was evident, 
that all these waves rose towards the zenith with a motion continually accelerated, and an individuality always growing more 
complete, It was weldom that any two waves flashed at the same instant across the culminating point; when they did so, the 
light was sometimes augmented there. * (Note,) > 

9. 40.—At this time, the Hashes frequently extended much 
beyond the zenith stars 

9, 50.—The arch in the N. N.W, became again distinct, 
including « Urew Maj,, its centre being in line with « Ursce 
Maj. 


9, 45.—Similar appearances ; arch ill defined. 


9. 57.—Now, and also several times before, the flashes cx- 
tended over Jupiter. A beam was seen passing through « Uru 
Maj. 

‘The flashes now became lower and less extensive. 


10. 10,—Beams of light travelling to the westward. 


10, 13,—The zenith has been clear for some minutes. 


10, 34.—Arch noticed again, its centre under ¢ Ursm Maj., 
is height in the middle of the light equal to one-half that 
of) Ursce Maj. 

10, 44—And later, till near twelve o'clock. ‘The arch 
north waa subject to much variation, 
rose at intervals, 


the 
Beams and faint flashes 


10, 45.—Arch vory distinct, its summit considerably lower 
than when Just observed. 


11. O.—Sheets of light still fying toward the head of An- 


dromeda, 
11. 15.—Some clouds (cirro-strati) slowly approaching from 
the N. Sheets of Jight and radiations belng distinctly above the 


sgion of these clouds, 

11. 35.—Splendid sheets of light; ill defined arch. 

11. 50-—Arch ili defined, Its summit near 5 Ursce Majoris, 
more cirro-strati, evidently floating below the radiations. 
12, 4,—There remained only a vague Illumination in the N, W. K = 


Towards 13. the phenomena were repeated with greater illumination; the waves again passed the Zenith, and at Ri 
(43 in. N. W. by N. of York) were seen to pass over Jupiter. z ers 


YORK.—MAGNETIC OBSERVATIONS. 


7. We—Till this time, the needle closely watched wns certninly unaffected by the aurora. 


§. 9.—The southward end of the needle deviated six and seven minutes K. 5 beams travelling east. 
9, 34.—From this time it deviated very rapidly to cast 
9, 4—It had reached its maximum deviation to the east. 


It was at this time that the flashes began to 


cross their culmina- 


ting point. Deviation i. 50 minutes 
9.43. 3 45 10S remicen a 
9. cmprety 13 1h (8 ereresiic conormnencion 
9. viation W. 1 10. 10, Beams travelling weat 
cn seeenenen) 3 10. 13, ae 
10. 0. a0 10. 3h ; 
10, FE Soto sees 


shail make a few remarks. 


the eastward deviation of the south end of the needle, the flashes were much more abundant on the castern side of the magnetic 
meridian, 3. It appeared to me protable, that the deviation of the needle was aflected by the movement of the beams, in such a 
way that its southward end deviated to the same side as the beams moved. 5 


YORK.—GENERAL REMARKS. 


For eleven days before the grand aurora of the 12th of October, the barometer had been very high, and the weather remarkably 
dry; the sun powerful, the nights chill and dewy; on the 11th of Oct, the barometer began to fall, and continued to fall till mid- 


day on the 12th. There was rain on the 11th, ond on the 12th till 2p, m. Irom midday on the 12th, the bara i 
{and continued to rise until 9a. m. on the 1th), the air became colder and drier. i ie ae cen at 


1 call them weres, nor only because of thet 
ion of hight and darkness, by whieh. 


pparent origin in distant undulations of uminons mars, bot also. because of a xery sin 
feov nMfected In is upward cau througu the shy. The ine along which any one wave pasved 
3Conce, orin succession, but only in cer realy rect sich succer 
spolsy=fesembling nothiog so cruel as 5 ca ol tansparent white tmove,—were 
ely Highcand dark. Tesppeared to 


iy was irregularly che torso mate my Tmagin 
Tro anc, I2eemed, on a gigantic scale, not unilie the appearance tou cloud amumer, wien IL callea 


"The culiminating point of these waits was at timesina continual Bott eat une vari 

+ continual Batter, In consequence of the ¥arious arrival 6! 
ferent quarters; bot Ht perbape never bappened thet any thing like radiatiog Ii yy s ‘The i\lomli mating? of all this af 
Beat Seales peo ar UN erent es RUE ner pad 
the magucti¢ needle, chart was obliged Lo aeglect some Boe beams which appeased in various directions between N, W. and N. by Ex Zt 


er Lowards the zenith. 
" ‘one ware followed anoth 
inagjacent™ 


CAMBRIDGE. 


Professor Airy. 
N, Lat. 52° 13° Long. EB. 0° 6 


bm. 

6. 32.—Several faint streamers directed to 9 point S. of « 
Cygni, the elevation of their tases being perhaps 15°, and 
of their tops 60% ; moving slowly towards the W. ; a faint bank 
of light in the N. 


6. 53,—No streamers ; generally a faint light in the north. 


7. 27.-—Much the same. 


7, 52.—In the north a banked yellow cloud, its height half that of 


streamer about as high as the Pleiades, directed toward = Cygni. 


HERON COURT, 4m. N. W. of Christchurch, Hants, 


Hon. Charles Harris. 
Ny Lat. 509 43" ‘Long. W. 1* 0° 


y Ursa: Majoris. One degree to the right of the Pleiades a 


‘This streamer remained stead 
7, Sk—The upper boundary of ihe bright loud was extomely sharps it Begamin te ek chaos 


above Arcturus, below y Ursa: Maj. at exactly half the elevation of y Ursa Maj 


to the Jeft of Arcturus, passed a very little 


the N. nt about half the azimuth O€ B Auriges ‘The Slash back belgw Feached to hee Wott ive eet eae of 


7. 59.—A black line was discoverable very near the upper bounda . 
alittle, thus widening the black line, About Arcturus Healer ee 


8. 2.—The upper part after 
0 that its elevation was perhaps 


} of that of y Ursw Maj. 


1g considerably hnd wholly disappeared, and the lawer had 


‘boat half the height ; it was very black, 


The upper part rose and the lower fell 


wunk a little under y Ursa Majoris 


a The high ; 
* Aritis, parallel to that above mentioned (7. 58.) but hitched, e Pavs Nos mew a bitle further west. A streamer through 


8 7A streamer exactly below Polaris, which pluny 
downwards into the blacks a _— 

8. 12.—More diffuse to the west, the dark part more illu 
minated. 


8. 44.—No remarkable alteration, 


9, 38—The eastern boundary of the arch as before + the 
western had advanced nearly to the W. ‘The lower edge of the 
bright cloud sharply defined, and its height not quite half the 
height of y Urs@ Moj.; the upper edge extended upwards 
about two-third of the remainder of the space towards y Ursue 
Maj. 

9, 42.—A gentle appearance of waving was seen, confined 
first entirely within the limits of the bright cloud, and looking 
like very long horizontal waves, very narrow in the vertical di- 
rection, running vertically upwards. In a few minutes the 
waves became considerable, reached as high as » Ursa Majors 
and still increasing. ‘Throughout the appearance of the wavings 
it was evident that the waves or pulses and the streamers were 
wholly tineonnected; the pulses fashed over the streamers 
without altering them. 

10. 0.—In the N. a very bright patch from which vivid 
streamers broke. ‘Towards the W. it was more lumpy. Pulses 
continued. 

10, 12.—The bright bank had almost entirely lost its smooth 
outline, and was broken into irregular toothed lumps. Pulses 
continued, The pulses when highest extended up to 6 U. Minoriss 

10, 27.—Little light the black below much diminished in 
breadth, No pulses (I believe.) 


11. 47.—Rather lighter. 


13. 0.—Pulses were seen. 


\ 


\ 


hem. 
2/23. —Brights irregular arch, ike a | 
ABav ariboyetia tera tf ania ae SRE EC 


8. 37—Its greatest intensity in N.N, W,, was a litt 
a in Canes Venaticl, which was scen, but faintly, oeO 
‘The lower edge of the arch was about 3° below this star, ‘The 
base of the arch about this time became irregular, aiid. coruscae 
tions were shot up along the whole linc, 


9. 57.—Large bodies of light traversing from B. to Wy 
across the N,, and shooting up brilliant coruscations in 
N.N. E., Neand N. W., some of them reaching an altitude 
of nearly 60°; they were accompanicd by faint flashes of 
ight, at right angles to the coruscatious or beams, and appa 
rently bebind them, resembling narrow Juminous bands of 
smoke, perpetually rolling up and disappearing. 

10. 7.-—A bright coruscation shot through the stor & in 
Ursa Major. ‘The main body of light now faded away, but 
the flashes still continued. 

10. 22.—Flashes more brilliant, and accorpanied by other 
flashes or rapid passages of light from to W. and W. to Es 
alorg apparently a line of vapour, as if they were electric dis 
charges conducted along it: three or four successive Mashes 
seemed to light up the same irregularities in the form of the 
conducting medium, as if a band of vapour existed, invisible 
except when the clectric current passed through ite 

10, 37.—A low arch again formed, its base scarcely 6” above 
the horizon, extending to about 7°; between this and 17° the 
flashing continued, but never rose above Denetoasch Jo Ursa 

ajor. 

10, 52—A brond coruscution remained some time immedi- 
ately under Mizar und Alioth in Urs Major, extending 
further to the horizon, the ashing being vivid across its upper 
portion, 

‘Whe coruscation moved slowly westward and faded away. 


CAMBRIDGE.—GENERAL REMARKS. 


In the pulses I noticed clearly what I had noticed very well on a former occasion, (I think in 1827, certainly before 1626) that 


it was not like clouds of light passing upwards and continuing luminous in their course till they vanished 5 
definite spots only of the sky had the faculty of receiving illumination in the order of vertically upwards 


but it was as if certain 
From the odd way in 


Which these spots were distributed, it sometimes appeared ay if the pulses flashed obliquely upwards. 


In the morning there had been a squall from the N. W. with heavy rain: 


very clears 


she aftermoon dull till near sun-set, when it became 


FIFESHIRE. 


Mr, Lawrence Buchan observed at about 9 r, »f the auroral arch passing near the zenith. (/R; 1’) 


ARMAGH. 


Rev. Dr. Robinson. 
N,Lat.54°21' Long.W.6°30. 


be ~ 

8. 11.—Above a luminous 
mass on the horizon are four 
parallel arches, 3° asunder. 
The upper edge of the lower 
arch pasted below 1 and 9 Ursa 
‘Majoris, and as much below ¢ 


9. 1—Three le 
arches, the prineipal one has 
its upper edge on Polaris, and 
midway between Capella and 
A Aurigw; its lower a tittle 
above £ and y Urs Minoria, 

9. 6.—Archinsameplace, 
but interrupted by what seem. 
like black streamers, directed 
one towanl 6 Aurigw, tho 
rites a above Capella, 

ielr edges 
las semen 

9. 11,—Upper ede of 
has risen to pr ant Copel, 
and seer rch haw risen 
neath it, its upper edge o 
Zand 2 Urwo Mifrts, fn the 
first are now three of the black 
streamers, the upper cromed 
by stripes of light Ba N. Be 

9. 3).—Sheet aurora rolling 
in muvee from all points of the 
northern horizon, to a part 
Aye W. and 1° N, of @ Ans 
dromedan, 


10. 14—Streamers cone 
verging to the samo point. 
Tn that region distinet portions 


of sky become repeatedly 
Wuminous when — streamers 
reach then 


retaining the same 
outline ike invisible clouds 
suddenly lighted up by thems 
10. 20.—The mass of light 
In the Ne W, (below whieh no 
dark space wan visible at any 
Lime this evening, perhaps on 
account of have) nearly exe 
Uinet, and a fow fuint streamers 
‘lone shew themselves 
11. G.—A_ renewal of 
streamers low down on the 
N. W.5 above Lyra and y 
Draconis, below Una there ts 
sheet aurora with o tendency 
to condense itself in tho 
direction of arches. The 
motions are much less rapid 
than those of the streamers, 
and its waves, even when they 
reach the zenith, much fainter. 
‘Whe light of the aurora waa 
sufficient to allow of counting 
seconds on a chronometer. 


AURORA BOREALIS 


TAN TIME. 


CAMBRIDGE. 
Professor Airy. 


N. Lat. 52¢ 13’ Long. E. 0° 6’ 


8. 
8. | 
vertex [{ 


h. 1 
| 
{ 


. 8. 25.—The aurora appeared in the form of a large bright cloud, bounded on 
shooting ower side by the horizon, and on the upper side by an arch of a small circle 
norther} giffering much from a great circle). The extremities of the arch were in 
60° alten, E. and W. N. W. or nearly W. The upper boundary was lower than 6 
ge Majoris by § x distance from « Urse Majoris to 6 Urs Majoris. Several 
ll black clouds were scattered over the aurora-cloud, and above it were several 
ly illuminated, whose light appeared to originate simply in the illumination 
e aurora. 

3. 35.—No change, except that the whole appeared to have moved a little to 

* “west. 

movem, 58.—The form and brightness of the arch had not sensibly altered; but a 
the UPP, black cloud on its face attracted particular attention. The western ex- 
it VanIShity of this cloud was below y Urs Majoris, its horizontal length fully three 
rose hig.; the distance from f Ursze Majoris to y Urs Majoris, its vertical breadth 
S. of thi than one-fourth of its length, the eastern end being somewhat broader than 


at theit western, The aurora-cloud suddenly formed itself into streamers, (or 
was n€,mers were formed in front of it) some perhaps 30° or 40° high, but lasting in 
traces ¢ state only for an instant, and two streamers of sensible breadth shot up either 
Were V@ ont of the black cloud or through it, so as to illuminate it, near its western 
inclined mity in two nearly vertical lines, corresponding to the course of the 
connectamers, whose upper and lower parts were visible above and below the cloud. 
* ,markable change in the zonstitution of the cloud followed immediately ; the 

10. tern half became curdly, the upper edge of its small pottions being luminous 3 
western half began to disappear ; at 9. 15. no trace of the cloud discoverable. 

11. 9. 10.—A shooting star from E. to W. very nearly through 3 Urse Majoris. 


light of the aurora-cloud gradually diminished ; the part which remained 
est was a little E. of N. where some light was still visible at 10, 30. 


und the dipping-needle to be unaffected by this aurora at Armagh. It was 
the dark segment was formed at about 123 h.; the phenomena ceased 
The greatest intensity of the aurora was in the direction of the magnetic 


| 


18th, at 8. 34. a low arch was seen there passing below the feet of U. 
(in the middle) above the horizon; at 8. 50, the lower edge of the lower 
arick. é 


TABULAR CONSPECTUS OF OBSERVATIONS ON THE AURORA BOREALIS 


OF THE 17th OF SEPTEMBER, 1833, REDUCED TO GREENWICH MEAN TIME. 


YORK. 
J. Phillips. 


Lat. 53° 58’ Long. W. 1° 4, 
h. m. 


8. 0.—Auroral arches and beams in the N. N. W. 

8. 9.—Arch 3° or 4° broad, including in its middle and 
vertex § Urse Majoris. It gradually and constantly increased 
in breadth and rose in position. 

8. 14.—Arch includes, nearly in the middle of its breadth, 
Dubhe, Arcturus, and Capella. From this time it grew fainter 
and rose higher. 

8. 34.—Beams or streamers in great number and brilliancy, 
shooting upwards in narrow distinct lines athwart the whole 
northern sky, in front of the arch, from the horizon to about 
60° alt. 


8. 44,.—The arch (which after passing Polaris in its upward 
movement shewed itself double) was now in two distinct parts 5 
the upper rose most rapidly to within 15° of the zenith, when 
it vanished. The lower arch became indistinct, and the beams 
rose higher and more frequently, directing themselves to a point 
S. of the zenith. Many of these beams were at one time joined 
at their bases into a singular reversed arch, of which the centre 
was near the Pole star. These streamers shewed momentary 
traces of colour; in the line of the magnetic meridian, they 
were vertical, towards the horizon E. and W. their tops were 
inclined probably 2U° to the South. They appeared wholly un- 
connected with the arch, 

9. 4,.—No arch visible, 

10. 49. ) A low faint arch stationary, its upper edge nearly 
to reaching to » and y Urse Majoris; its vertex under 
11. 19. ) Mizar (alt. about 18° in the middle.) 


seen at Brussels by M. Quetelet, who in a letter to Professor Airy gives the following description of it. 


MANCHESTER. 
P. Clare, W. Hadleigh, and R. Potter. 


Lat. 53° 29’ 


h. m. 


Long. W. 2° 13° 


8. 9.—Arch at its summit 32° high, very brilliant. 


8. 18,—Arch almost exactly includes a and # and y and 3 
Ursa Majoris. ( R. P.) 


8. 24.—The arch 7° broad, includes Dubhe, Arcturus, and 
Capella, so that Capella is on the extreme upper edge; Dubhe 
rather above the middle of the breadth, ahd Arcturus rather 
below the middle. Centre of the arch a little ER. of § Urse 
Majoris, Extent of the arch 130°. (P. C.) 

8. 27.The upper edge of the arch coincides with » and g 
Ursa Majoris ; the lower edge with 3 Urse Majoris. 

8. 40},—Arch 38° or 39° high, and extending about 160° 
on the horizon. (R. P.) 

8. 44,—-Many streamers in the N. directed towards the 
magnetic zenith. 

8, 44.—Arch passed over Arcturus, S. of Polaris, 3° or 4° 
north of Algol, ending obscurely near the Pleiades (alt. about 
60°, vertex in the magnetic meridian.) /(W. H.} 

8. 49.—Half the hemisphere illuminated; many bright 
Streamers and flashes of light rose from the magnetic N. 

8, 54.—Coruscations frequently ending in an arch 30° or 33° 
S. of the zenith ; the southern edge of the arch passing 1° N. 
of the Pleiades. 1° N, of Scheat, 2° N. of the highest star in 
Delphinus, and just touching y Aquila, and » Serpentis. (W7. H.) 
From this time the streams and light diminished, and were 
subject to slight changes till 

11. 0., when the sky became cloudy. 


1m. W.N.W.ofGOSPORT. 
Hon. C. Harris. 

Lat. 50° 48) Long. W. about 1° 9’ 

Cirro-strati clouds ob- 


scured the auroral arch, which ap- 
peared soon after sun-set. 


h, m. 

9. 524. —A beam in the W. be- 
tween £ and y Ophiuchi. It seemed 
to swerve off gradually to the west- 
ward. 

10. 44.—It had faded away. 

10. 494. ? Arch from N. W. to N. 

to N. E.. Its vertex under 

il. 44. N ¢ Urs Majoris, and the 
edge of its base half-way between 
that star and the horizon. 


GENERAL REMARKS, 


This aurora was seen in many parts of Ireland from 9 to 11, and at later hours of the night of the 17th ; as at Adare, Limerick, Armagh, and Dublin. Professor Lloyd found the dipping-necdle to be unaffected by this aurora at Armagh. It was 


CAMBRIDGE. 
Professor Airy. 


N. Lat. 52¢ 13’ Long. E. 0° 6! 


he m. 

8. 25.—The aurora appeared in the form of a large bright cloud, bounded on 
the lower side by the horizon, and on the upper side by an arch of a small circle 
(not differing much from a great circle). The extremities of the arch were in 
the N. E. and W. N. W. or nearly W. The upper boundary was lower than £ 
Urse Majoris by 3 x distance from « Ursa Majoris to & Ursa Majoris. Several 
small black clouds were scattered over the aurora-cloud, and above it were several 
faintly illuminated, whose light appeared to originate simply in the illumination 
of the aurora, 

8. 35.—No change, except that the whole appeared to have moved a little to 
the west. 

8. 58.—The form and brightness of the arch had not sensibly altered; but a 
long black cloud on its face attracted particular attention. The western ex- 
tremity of this cloud was below y Urs@ Majoris, its horizontal length fully three 
times the distance from £ Urse Majoris to y Urs Majoris, its vertical breadth 
less than one-fourth of its length, the eastern end being somewhat broader than 
the western. The aurora-cloud suddenly formed itselt’ into Streamers, (or 
Streamers were formed in front of it) some perhaps 30° or 40° high, but lasting in 
this state only tor an instant, and two streamers of sensible breadth shot up either 
in front of the black cloud or through it, so as to illuminate it, near its western 
extremity in two nearly vertical lines, corresponding to the course of the 
streamers, whose upper and lower parts were visible above and below the cloud, 
A remarkable change in the constitution of the cloud followed immediately ; the 
western half became curdly, the upper edge of its small-portions being luminous ; 
the western half began to disappear ; at 9. 15. no trace of the cloud discoverable. 

9, 10.—A shooting star from E. to W. very nearly through § Urs Majoris, 
The light of the aurora-cloud gradually diminished 3 the part which remained 
longest was a little KE, of N. where some light was still visible at 10, 30. 


Towards 10 p. m, (Brussels time) an aurora borealis was visible ; the dark segment was formed at about 124 h.; the phenomena ceased 


about 3 in the morning. There were no streamers, (jets lumincux,/ and the light of a yellowish white colour, did not rise above the horizon more than 20 to 30 degrees. The greatest intensity of the aurora was in the direction of the magnetic 


meridian to the north. 
State of the atmosphere—York. Temp. 50. 


Barom, 29.682 rising. 


On the 12th and 16th of September, auroral beams had been seen at York and Greta Bridge; on the 18th, at 8. 34. a low arch was seen there passing below the feet of U. 


Maj., its upper edge very near : and 4 of that constellation. At Durham, about 8. 0., two distinct arches were seen, the upper one 16°, and the lower and brighter one 7° (in the middle) above the horizon; at 8. 50, the lower edge of the lower 


arch was well defined, and 4° above the horizon. 


Auroral phenomena were also seen on this evening at Lymington, and a low arch was noticed by Lord Adare, near Limerick. 


RECOMMENDATIONS. 487 


able for observations of these faint and often fluctuating me- 
teors than others of a more refined construction. 

2.) It is recommended that a magnetic needle be kept in a 
proper place, suspended by a silk fibre or slender hair, (a 
point-support not being delicate enough,) and so mounted that 
deviations can be observed to the accuracy of 1’. It has been 
found convenient to fix in a garden a stone pedestal, on which, 
at three invariable points, the frame of the magnetic needle 
rests under a glass cover. The needle, 9 inches long, and of 
such a weight as to perform about 10 vibrations in a minute, is 
suspended by one slender hair. There are simple contrivances 
to steady the needle when required, and to adjust the length 
of the suspending hair. The scale is divided in degrees for 
30° on each side of the centre, and in 10’ for 1° on each side. 
There is no vernier, but the place of the needle on the scale 
is read off with great ease by looking through a fixed magni- 
fying glass, from an opening at some height above, so as to. 
avoid sensible parallax. Professor Christie has described more 
complete apparatus for this purpose, in the Journal of the 
Royal Institution, New Series, vol. ii. p. 278. The observer 
must leave his watch with the assistant, very carefully remove 
all keys, knives, and other things containing iron, from his 
dress, and all loose iron tools and utensils, to at least 20 feet 
distance from the needle. If these precautions are not scru- 
pulously attended to, the results will be fallacious. It is proper 
to caution the observer that there is a regular daily variation 
of the needle, independent of the Aurora. 

Dipping-needles, unless constructed with the utmost care, 
cannot be considered very satisfactory instruments; yet, if 
' their suspension be sufficiently delicate, they may probably 
very well answer for observations during Aurora, of which the 
object is to determine not the absolute dip of the needle, but 
the change of dip occasioned by the Aurora. The same pre- 
cautions of one certain position, removal of iron, &c., are ne- 
cessary, as in the use of the horizontal needle. 

3.) It is recommended that arrangements be made for ascer- 
taining the error of a watch. If near an observatory of any 
kind, the watch should be compared with the transit clock 
there, immediately after an Aurora: if there is a good meridian 
line or good dial, the error of the watch on mean time should 
be found as soon as possible. If a watchmaker in the neigh- 
bourhood has a good regulator, the watch should be adjusted 
by it, and the mode of keeping the regulator should be ascer- 
tained: if a mail-coach from London passes near, the guard’s 
watch may be consulted. The longitude of the place of ob- 


488 THIRD REPORT—1838. 


servation should be ascertained from a map or otherwise. 'The 
attention of observers is especially called to the point of ascer- 
taining the time correctly, as it is one of the most important 
points, and the one which probably will require the longest 
forethought. 

4.) In default of intelligence of an Aurora, the observer 
should go out of doors to some station where the horizon is 
pretty clear, and look about every evening at 10, Greenwich 
mean solar time, as near as may be. He should keep a journal, 
noting for this time every evening whether there was an Aurora: 
a single word will be sufficient. 

5.) As soon as the observer perceives or receives notice of 
an Aurora, he should, if accustomed to magnetic observations, 
observe the magnetic needle, and should go to some command- 
ing situation with his watch in his hand, and a note-book. A 
person so prepared will have little difficulty in fixing on the ap- 
pearances most worthy of notice. We may, however, point 
out the following : 

I. If there is an arch, the positions of its two boundaries, 
its terminations, &c., should be noted by the way in 
which they pass among the stars, (the proportion of 
distances between the stars admitting of very accu- 
rate estimation by the eye). If, as rarely happens, 
the sky is cloudy, the observer may notice the ele- 
vation and extent of the arch, by moving till it appears 
to touch the top of some terrestrial object, noting his 
situation as well as he can, and the next day observing 
with a theodolite the angular elevation and azimuth of 
the object; or ascertaining the height and horizontal 
distance, and thence computing the angular elevation, 
and observing the azimuth by a common compass: but 
it is recommended not to adopt this method when the 
observation of stars is practicable. Notice should be 
taken whether one edge is better defined than the 
other; whether there is a clear sky or dark cloud 
above or below; whether it terminates at the end in 
sky or in cloud; whether there is any dark band in 
it; whether in its general composition it is uniform or 
striated ; whether stars can be seen through it, &c. 

II. If any change takes place in the situation or appear- 
ance of the arch, the observer should instantly look 
at his watch and set down the time, and then proceed 
to note the change. 

Ill. If there are beams or streamers, the time should be 
noted ; then their position among the stars; then their 


a 


‘RECOMMENDATIONS. 489 


height among the stars; their motion (whether vertical 
or horizontal); the velocity of motion (by the time of 
passing from one star to another); their changes ; their 
permanency ; whether they appear to affect the arch, 
or to be entirely in front of it. 

IV. If there are any black clouds in the luminous region, 
notice should be taken whether the streamers seem to 
have any relation to them; whether the arch seems 
to have any‘relation to them; whether and in what 
manner they increase or disappear. 

V. If there are waves or flashes of light, the observer 
should notice the time of beginning and of finishing ; 
the general extent of the flashes (up and down, as 
well as right and left); whether the flash is a real 
progress of light or successive illumination of different 
places ; and anything else that strikes him. 

VI. The existence and change of colours will, of course, 
be noticed. 

VII. From time to time, the needle should be observed. 
If there are two persons capable of accurate obser- 
vation, it is most desirable that one should steadily 
watch the needle, and the other the sky. 

6.) When all is over, the observer should immediately put 
his rough notes in form, and as soon as possible should com- 
pare his watch with the regulator or other authority for his 
time. 

7.) The next day he should, from a celestial globe, take the 

altitudes and azimuths by means of the stars; he should reduce 
his observed time to Greenwich mean solar time, and he should 
append these reductions to his rough observations. In this 
state the observations are fit for publication, and adapted for 
immediate use. It is desirable that they should be transmitted 
without delay to the Assistant Secretary of the British Asso- 
ciation, Museum, York. 


FALLING STARS. 


M. Quetelet’s mode of observing and recording the charac- 
teristic circumstances of these meteors is contained in the fol- 
lowing extract of a letter from him: 

“‘T take my station out of doors, in a situation which com- 
mands a good view of the sky, with a good map of the heavens 
spread out before me. When a falling star appears, I mark on 
the map the point of its commencement, the line of its course 


490 THIRD REPORT—1833. 


amongst the nearest stars, and the point where it vanished. 
This is done by an arrow-line, which marks the apparent di- 
rection and extent of the course of the meteor. ‘The time is 
carefully noted ; a number of reference is placed on the line, 
and the principal circumstances of the meteor are then regi- 
stered in tables of the following form: 


Magnitude Duration of Time of 
Epoch. No. relative to Stars. the Appearances. Apearannes Remarks. 
& 


Aug. 29 1 2 Q"5 105 6! 4", 

It is important to remark whether the falling star leaves, or 
not, any trace of its course, as sometimes happens, in the form 
of reddish scintillations. The condition of the atmosphere, as 
determined by the usual instruments, should be noted: the 
time must be accurately ascertained. More than one observer 
should be engaged at each station, because the meteors some- 
times succeed one another very quickly, and the duration of 
the phenomenon is too short to permit one person to note the 
position, time, and circumstances of each, with sufficient pre- 


cision }.” 


CONSTANTS OF NATURE AND ART. 


«* Amongst those works of science which are too large and 
too laborious for individual efforts, and are therefore fit objects 
to be undertaken by united academies, I wish to point out one 
which seems eminently necessary at the present time, and which 
would be of the greatest advantage to all classes of the sci- 
entific world. 

*‘ T would propose that its title should be The Constants of 
Nature and of Art. It ought to contain all those facts which 
can be expressed by numbers in the various sciences and arts.” 
(Babbage, Edinburgh Journal of Science, N.S., No. 12.) 

The following extracts from Mr. Babbage’s general plan of 
contents will exemplify the objects and arrangement of the 
proposed work. 

These contents should consist of: 

1. All the constant quantities belonging to our system ;—as, 
distance of each planet,—period of revolution,—inclination of 
orbit, &c.,—proportion of light received from the sun,—force 
of gravity on the surface of each, &c. 


1 Contemporaneous observations are especially desirable on this subject. 
Persons desirous of undertaking the investigation are therefore requested to 
apply to a member of the Auroral Committee, or to the Assistant Secretary 
at York, for information of the evenings and hours appointed for this purpose. 


a 


a, ee 


RECOMMENDATIONS. 491 


2. The atomic weight of bodies. 

3. List of the metals, with columns for specific gravity,— 
electricity,—tenacity,—specific heat,—conducting power for 
heat,—conducting power for electricity,—melting point,—re- 
fractive power,—proportion of rays reflected out of 1000,—at 
an incidence of 90°. 

4, Specific gravities of all bodies. 

5. List of mammalia, with columns for height,—length,— 
weight,—-weight of skeleton,—weight of each bone,—its great- 
est length,—its smallest circumference,—its specific gravity,— 
number of young at a birth,—number of pulsations per minute, 
—number of inspirations per minute,—period of blindness after 
birth,—of sucking,—of maturity,—temperature,—average du- 
ration of life,—proportion of males to females produced, &c. &c. 

After enumerating twenty such general heads of Constants, 
Mr. Babbage observes, that ‘most of them already exist, and 
that the difficulty of collecting them consists chiefly in a judi- 
cious selection of those which deserve the greatest confidence. 
It would be desirable, however,” he adds, “ to insert the heads 
of many columns, although not a single number could be placed 
in them; for they would thus point out many an unreaped field 
within our reach, which requires but the arm of the labourer 
to gather its produce into the granary of science.” Mr. Bab- 
bage expresses his opinion, that if any scientific body of men 
would undertake to form such a collection, and to revise it from 
time to time, it would be a work fraught with advantages to 
knowledge, by continually leading to the more accurate deter- 
mination of established facts, and to the discovery and mea- 
surement of new ones. , 


Persons desirous of undertaking or cooperating in the exe- 
ecution of any of the foregoing Recommendations, are requested - 
to make known their intention to the Secretaries of the British. 
Association, Museum, York. 


PROSPECTUS 


OF THE 


OBJECTS AND PLAN 


OF THE 


STATISTICAL SOCIETY OF LONDON, 
Founded on the 15th March, 1834. 


Tue SratisticaL Society or Lonpon has been established 
for the purposes of procuring, arranging, and publishing “ Facts 
calculated to illustrate the Condition and Prospects of Society.” 
_* The Sratisticat Soctery will consider it to be the first and 
most essential rule of its conduct to exclude carefully all opé- 
nions from its transactions and publications,—to confine its 
attention rigorously to facts,—and, as far as it may be found 
possible, to facts which can be stated numerically and ar- 
ranged in tables. 

The first operation of the Society will probably be to sub- 
divide and organize its general council in such a manner as may 
enable that council to deal conveniently with all the subdivisions 
of the subject-matter before it. Those subdivisions will ne- 
cessarily be numerous. 

The whole subject was considered, by the Statistical Section 
of the British Association at Cambridge, as admitting a division 
into four great classes : 

1. Economicat Statistics. 

2. PoxiticaL STATISTICs. 

3. Mepicat STATISTICS. 

4. Morat anbD INTELLECTUAL STaTISTICs. 

If these four classes are taken as the basis of a further ana- 
lysis, it will be found that the class of 
Economical Statistics comprehend, 1st, the statistics of the 

natural productions and the agriculture of nations ; 2ndly, of 

manufactures ; 3rdly, of commerce and currency; 4thly, of 
the distribution of wealth, or all facts relating to rent, wages, 

profits, &c. 

Political Statistics furnish three subdivisions: 1st, the facts re- 
lating to the elements of political institutions, the number of 
electors, jurors, &c. ; 2ndly, legal statistics ; 3rdly, the statis- 


STATISTICAL SOCIETY. 493 


tics of finance and of national expenditure, and of civil and 

military establishments. 

Medical Statistics, strictly so called, will require at least two 
subdivisions; and the great subject of population, although 
it might be classed elsewhere, yet touches medical statistics 
on so many points, that it would be placed most conveniently, 
perhaps, in this division, and would constitute a third sub- 
division. 

Moral and Intellectual Statistics comprehend, |st, the statistics 
of literature ; 2ndly, of education ; 3rdly, of religious instruc- 
tion and ecclesiastical institutions; 4thly, of crime. Although 
fourteen subdivisions have now been enumerated, it is pro- 
bable that more will be required. 

It will not of course be necessary to have a distinct Sub-com- 
mittee of the Council for each of these subdivisions; but a 
convenient division of the Council, and an arrangement of the 
individuals composing it, so as best to deal with all the different 
portions of the common subject, will be a necessary preliminary 
to any systematic course of inquiry. 

When these subdivisions are established, it will be for them, 
subject to the approbation of the Council, to sketch the outline 
of their own operations. A few observations on the more 
general efforts and objects of the Society are all that need be 
presented here. 

It will be desirable that the Society should, as soon as possi- 
ble, endeavour to open a communication with the statistical de- 
Ae he established by Government at the Board of Trade. 

ithout such a communication, constantly kept up, the Society 

can never be assured that it is not doing unnecessarily what the 
Government is doing at the same time and better. ‘The result 
of such a communication would probably be that the Society 
would abandon to the care of the Government some part of this 
very extensive field of inquiry altogether, and more of it par- 
tially, which would still leave a very sufficient, though a aes 
overwhelming task to the Society. 

The Society, having its own work thus somewhat limited and 
defined, may next proceed to consider the best means, Ist, of 
collecting fresh statistical information ; and, 2ndly, of arranging, 
condensing, and publishing much that already exists. Towards 
collecting fresh statistical information, the first step in order, 
both of time and importance, would be the arrangement of a 
good set of interrogatories, to be drawn up under the superin- 
tendence of the Sub-committees, and afterwards examined, 
sanctioned, and circulated by the Council. The careful execu- 
tion of this task is essential both to afford guidance and aid to 


494. THIRD REPORT—1838. 


individual inquirers, and to protect the Society against the in- 
flux of imperfect or irrelevant statements. Willing agents of 
inquiry exist in abundance quite ready to aid in collecting ma- 
terials ; but few of these agents take a very wide view of all the 
objects of statistical inquiry, and indeed few have very distinct 
notions about the precise information the Society may wish to 
collect, even as to any one object. To sketch, therefore, di- 
stinctly by means of interrogatories, carefully and succinctly 
drawn, the whole outline which it is wished to fill up, is the 
only way to secure to the Society the full benefits to be expected 
from their zeal. _ It is difficult to overrate the importance of the 
step which will be made towards the accumulation of statistical 
knowledge from all quarters of the globe, by the publication of 
such a set of questions; but the operation will be as laborious 
as it is important. It properly may, and probably will, form the 
chief object of the exertions of the Council during the first year 
of the Society’s existence. 

Obvious advantages may be drawn from communication with 
intelligent Englishmen about to travel abroad, with residents in 
the Colonies, and with colonial gentlemen resident in England. 
The Society has already the satisfaction of knowing, that it will 
have friends and assistants equally zealous and able in our 
western colonial possessions. Various societies, foreign and do- 
mestic, abound both in our own country and on the Continent, 
some of them already devoted to this subject, and others very 
willing to take it up. In addition to those already in existence, 
the Society may hope to see other local societies springing up 
in every part of the British dominions, in direct and constant 
connexion with the London Society, circulating its queries in 
their immediate neighbourhood, and collecting and authenti- 
cating the answers. A body of facts can be thus most conve- 
niently collected, which may properly enter into a common pub- 
lication, and will afford safe grounds for comparing the present 
condition and future progress of different parts of the empire. 
The London Society, therefore, will carefully cultivate a con- 
nexion with, and. attend to the wishes and suggestions of, such 
local societies, and will look forward to their multiplication 
and correspondence as among the best supports of its own 
continued efficiency. 

The collection, by such means and agents, of new statistical 
materials will form, it will be remembered, only one part of the 
Society’s work. To condense, arrange, and publish those al- 
ready existing, but either unpublished, or published only in an 
expensive or diffused form, or in foreign languages, would be a 
task of equal usefulness. Authentic statistical accounts, even 


STATISTICAL SOCIETY. 495 


of an old date, may perhaps advantageously receive some atten- 
tion. Our Oriental dominions alone present a field of statistical 
research as interesting as it is immense. Many materials, col- 
lected from that field by the meritorious exertions of the East 
India Company, are known to be in existence, and it is to be 
hoped that, sooner or later, they will be brought through some 
channel before the public. 

To point out such existing collections, old and new, their 
character, value, and the degree of interest attached to them, 
will form an appropriate part of the duties of the Sub-commit- 
tees of the Council, and will itself be a considerable step in 
statistical knowledge. The extent to which the Society shall 
deal with the existing materials so pointed out to it, can only 
be considered when the means and resources it is to possess 
are better ascertained. 

It will of course be one prominent object of the Society to 
form a complete Statistical Library as rapidly as its funds may 
admit. 

The proposed annual subscription to the Society is two 
guineas, which may be compounded for by one payment of 
twenty guineas. 


COUNCIL AND OFFICERS 
Elected at the General Meeting, 3rd May, 1834. 


President.—Marquis of Lansdowne. 

Treasurer.—Henry Hallam, Esq. 

Secretaries.— Woronzow Greig, Esq. Charles Hope Mac- 
lean, Esq. E. Carleton Tufnell, Esq. 

Charles Babbage. Esq. William Burge, Esq. Rev. Geo. 
D’Oyley, D.D. John Elliot Drinkwater, Esq. Howard El- 
phinstone, Esq. Earl Fitzwilliam. Rt. Hon. H. Goulburn, M.P. 
Joseph Henry Green, Esq. Edmund Halswell, Esq. Dr. 
Bisset Hawkins, M.D. Rt. Hon. Fr. Jeffrey, M.P. Rev. 
Richard Jones. John Lefevre, Esq. Sir Charles Lemon, 
Bart., M.P. Rt. Rev. Lord Bishop of London. S. Jones 
Loyd, Esq. Rev. T. R. Malthus. G. R. Porter, Esq. Vis- 
count Sandon, M.P. G. Poulett Scrope, Esq., M.P. N.W. 
Senior, Esq. Dr. John Sims, M.D.  Lieutenant-Colonel 


Sykes. Thomas Tooke, Esq. T. Vardon, Esq. Rev. W. 
Whewell. 


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497 


OBJECTS OF THE ASSOCIATION. 


Tue 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 sci- 
entific inquiry,—to promote the intercourse of those who culti- 
vate Science in different parts of the British Empire, with one 
another, and with foreign philosophers,—to obtain a more gene- 
ral attention to the objects of Science, and a removal of any dis- 
advantages of a public kind, which impede its progress. 


RULES. 


MEMBERS. 


All Persons who have attended the first Meeting shall be en- 
titled to become Members of the Association, upon subscribing 
an obligation to conform to its Rules. 

The Fellows and Members of Chartered Societies in the Bri- 
tish Empire shall be entitled, in like manner, to become Mem- 
bers of the Association. 

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

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

Persons not belonging to such institutions, shall be eligible, 
upon recommendation of the General Committee, to become 
Members of the Association. 


SUBSCRIPTIONS. 


The amount of the Annual Subscription shall be One Pound, 
to be paid in advance upon Admission ; and the amount of the 
composition in lieu thereof, Five Pounds. 


MEETINGS. 


The Association shall meet annually, for one week, or longer. 
1833. 2K 


498 RULES OF THE ASSOCIATION. 


The place of each Meeting shall be appointed by the General 
Committee at the previous Meeting ; and the Arrangements for 
it shall be entrusted to the Officers of the Association. 


GENERAL COMMITTEE. 


The General Committee shall sit during the time of the Meet- 
ing, or longer, to transact the Business of the Association. It 
shall consist of all Members present, who have communicated 
any scientific Paper to a Philosophical Society, which Paper has 
been printed in its Transactions, or with its concurrence. 

Members of Philosophical Institutions, being Members of 
this Association, who may be sent as Deputies to any Meeting 


of the Association, shall be Members of the General Committee 
for that Meeting. 


COMMITTEES OF SCIENCES. 


The General Committee shall appoint, at each Meeting, Com- 
mittees, consisting 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 Meet- 
ing. They shall engage their own Members, or others, to un- 
dertake such investigations ; and where the object admits of 
being assisted by the exertions of scientific bodies, they shall 
state the particulars in which it might be desirable for the Ge- 
neral Committee to solicit the cooperation of such bodies. 

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


LOCAL COMMITTEES. 


Local Committees shall be appointed, where necessary, by the 
General Committee, or by the Officers of the Association, to 
assist in promoting its objects. 

Committees shall have the power of adding to their numbers 


those Members of the Association whose assistance they may 
desire. 


OFFICERS. 


A President, two Vice-Presidents, two or more Secretaries, 


and a Treasurer, shall be annually appointed by the General 
Committee. © 


RULES OF THE ASSOCIATION. 499 


COUNCIL. 

In the intervals of the Meetings the affairs of the Association 
shall be managed by a Council, appointed by the General Com- 
mittee. 

PAPERS AND COMMUNICATIONS. 

The General Committee shall appoint, at each Meeting, a 
Sub-Committee, to examine the papers which have been read, 
and the register of communications ; to report what ought to be 
published, and to recommend the manner of publication. The 
Author of any paper or communication shall be at liberty to re- 
serve his right of property therein. 


ACCOUNTS. 


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


TREASURER. 
Joun Taytor, Esq., 14, Chatham Place, London. 


LOCAL TREASURERS. 


Dr. Dauseny, Oxford. Rev. Tuomas Lusy, Dublin. 

Prof. Forsrs, Edinburgh. Dr. Pricuarp, Bristol. 

JonatHan Gray, Esq., York. Grorce Parsons, Esq., Birming- 

Prof. Henstow, Cambridge. ham. 

Witt1am Horton, Esq., Newcas- | Rev. Joun J. Tayter, Manchester. 
tle-on-Tyne. H. Wootcomsz, Esq., Plymouth. 


2x2 


at 
> 


INDEX. 


—= 


Oxszcts and Rules of the Associa- 
tion, 497. 

Officers, Council, Committees, &c. 
XXXviil. , 

Proceedings of the General Meeting, 

ix. 

of the Sectional Meetings, 

XXxil. 

of the Committees, xxxv. 

Recommendations of the Commit- 
tees, 469. 


Absorption of light by coloured me- 
dia, on, 373. 

Achromatism of the eye, on the, 374. 

Actinometer, the principle and con- 
struction of the, 379. 

Adam (Dr.) on some symmetrical 
relations of the bones of the me- 
gatherium, 437. 

Agardh (Prof.) on the originary 
structure of the flower, 433. 

Algebra, on the science of, 185; 
signs of transition, 232; signs of 
discontinuity, 248; convergency 
and divergency of series, 267. 

Analysis, on certain branches of, 185. 

Antimony, glass of, its power to re- 
flect light, 377. 

Architecture, naval, on, 430. 

Atomic weights, experiments on, 399. 

Attraction, electrical, some new phe- 
nomena of, 386. 

Aurora borealis, on an arch of the, 
401 ; directions for observations 
of the, 486. 


Baily (F.), account of some MS. 

- Letters relative to Flamsteed’s 
Historia Celestis, 462. 

Barlow (P.), report on the strength 
of materials, 93. 

Barometer, new, on the construction 
of, 414. 


, portable, new method ofcon- 
structing, 417. 


Barometer, wheel, on the construc- 
tion of a new, 414. 

with an enlarged scale, 414. 

Beams, on the effect of impact on, 
421. 

Blackwall (J.) on the structure and 
functions of spiders, 444. 

Botany, on the philosophy of, 27. 

Brain, on the physiology of the, 63. 

Buckland (Rev. Dr.), address on re- 
signing the President’s Chair, ix. 


Carlile (H.) on the motions and 
sounds of the heart, 454. 

Challis (Rev. J.), report on hydro- 
statics and hydrodynamics, 131. 
Christie (S. H.), report on the mag- 

netism of the earth, 105. 

Chronometers, application of a glass 
balance-spring to, 421. 

Circulation in plants, on, 32. 

Colours of plants, on the, 55. 

Compressibility of water, on, 131, 
353, 

Cumming (Rev. J.) on some electro- 
magnetic instruments, 418 ; on an 
instrument for measuring the heat- 
ing effect of the sun’s rays, 418. 


Decomposition, on electro-chemical, 
393. 

Dent (E. J.) on the application of a 
glass balance-spring to chronome- 
ters, 421. 

Daubeny (Dr.) on the action of 
light upon plants, 436. 


Earth, on the magnetism of the, 
105. 

Eddies in rivers, on the causes of, 
169. 

Eel, on the reproduction of the, 446. 

Electrical attraction, some new phz- 
nomena of, 386. 

Electricity, on, 390. 

Electro-chemical decomposition, on, 
393. 


502 


Electro-magnetic instruments, on, 
418. 

Endosmose and Exosmose, on the 
cause of, 391. oe 

Equations, on the theory of, 296; 
composition of, 296; general so- 
lution of, 305. 

Eye, on the achromatism of the, 374. 


Faraday (M.) on electro-chemical 
decomposition, 393. 

Fielding (G. -H.) on the peculiar at- 
mospherical phznomena during 
the prevalence of influenzain 1833, 
461. 

Flamsteed’s Historia Celestis, ac- 
count of some MS. Letters rela- 
tive to, 462. 

Flowers, on the structure of, 433. 

Fluid motion, review of the theory of, 
131. 

Fluids, on the motion of in pipes and 
vessels, 135, 153; on the resistance 
of, 149, 153; on the velocity of 
propagation in, 136, 153. 


Genera and subgenera, on, 440. 

Glass, its colouring matter dimi- 
nishes its power of transmitting 
heat, 382. 

of antimony, on its power to 
reflect light, 377. 

Gray (W.) experiments on the quan- 
tities of rain falling at different 
elevations, 401. 


Hamilton (W. R.) on the character- 
istic function in optics, 360. 

Harlan (Dr.) on some new species of 
fossil saurians, 440. 

Harris (W. S.) on some new phzeno- 
mena of electrical attraction, 386; 
on the construction of anew wheel- 
barometer, 414. 

Hawkins (J. I.) on the locomotive 
differential pulley, 424. 

Heart, on its motions and sounds, 
454. 

Heat, radiant, experiments on, 381. 

Henry (Dr. W. C.), report on the 
physiology of the nervous system, 


59. 

Herschel (Sir J. F. W.) on the ab- 
sorption of light by coloured me- 
dia, 373; on the principle and 


INDEX. 


construction of the actinometer, 


Hodgkinson (E.) on the effect of im- 
pact on beams, 421; on the strength 
of cast iron, 423. 

Hydraulics as a branch of engineer- 
ing, on, 153. 

Hydrostatics and Hydrodynamics, 
report on, 131. 


Influenza of 1833, peculiar atmo- 
spherical phenomena during the 
prevalence of, 461. 

Iron, mean strength and elasticity 
of, 103. 

, carbonaceous, method of ana- 

lysing, 400. 

, cast, on the strength of, 423. 


Jenyns (Rev. L.) on genera and sub- 
genera, 440. : 

Johnston (Prof.) on a method of ana- 
lysing carbonaceous iron, 400. 


Leaves, on the theory of wood being 
generated by the action of, 36; on 
the arrangement of, 40; on the 
structure of, 41. 

Life, on the term, 59. 

Light, on its absorption by coloured 
media, 373 ; on the power of glass 
of antimony to reflect, 377 ; on a 
phznomenon in the interference 

- of, 378. 

Lindley (Prof.), report on the philo- 
sophy of botany, 27. 


Lloyd (Rev. H.) on conical refrac- 


tion, 370. 
Locomotion, on the function of, 68. 
Locomotive differential pulley, in- 
vestigation of the principle of the 
424, 


Macartney (Dr.) on the natural hi- 
story of the common toad, 441; on 
the structure and functions of the 
nervous system, 449. 

MacVicar (Rev. J. G.) on electricity, 
390. 

Magnetism of the earth, on, 105. 

Materials, on the strength of, 93, 
103, 421, 423, 

Medulla oblongata, on the, 72. 

Megatherium, on some symmetrical. 
relations of the bones of, 437. 


INDEX. 


Melloni(M.), experiments on radiant 
heat, 381. 

Miller (Rev. W. H.) on the con- 
struction of a new barometer, 
414. 

Mineral veins, on the state of know- 
ledge respecting, 1. 

Mines, on the depths of, 427. 

Morphology, on the theory of, 50. 

Mytilus crenatus, on the naturaliza- 
tion in England of, 448. 


Naval architecture, on, 430. 

Needle, on its variation, 106; on the 
change in its direction, J07; on 
the diurnal change in the varia- 
tion, 108. 

, dipping, on a peculiar source 

of error in experiments with, 

412. 


, magnetic, on the dip of, 
109 ; on the variation of the dip, 
110. 

Nerves, on the, 80. 

Nervous system, on the physiology 
of the, 59; on the structure and 
functions of the, 449. 

Newman (J.) on a new method of 
constructing a portable barometer, 
417. 


Cirsted (Prof.) on the compressibi- 
lity of water, 353. 


Optics, on the characteristic function” 


in, 360. 
Owen (J.) on naval architecture, 
430. 


Peacock (Rev. G.), report on certain 
branches of analysis, 185. 

Phillips (J.), experiments on the 
quantities of rain falling at differ- 
ent elevations, 401. 

Physiology of the nervous system, 
59; of the brain, 63. 

Plants, on the circulation in, 32; 
on the structure of the axis, 33; 
on the cause of the formation of 
wood, 36; on the arrangement of 
leaves, 40; on the structure of 
leaves, 41; on the anther, &c., 
43; on the origin of the pollen, 
44 ; on the fertilization, 45 ; on 
the origin of organs, 49; on the 
theory of morphology, 50; on the 


503 


theory of gradual development, 
53; on their irritability, 54; on 
the action of coloured light on, 54; 
on the various colours of, 55; on 
excretions, 56; on the structure of 
the flower, 433 ; on the action of 
light upon, 436. 

Pollen, on the origin of, 44. 

Potter (R. jun.) on the power of glass 
of antimony to reflect light, 377; 
on a phenomenon in the interfer- 
ence of light, 378; on an arch of 
the aurora borealis, 401. 

Powell (Rev. B.) on the dispersive 
powers of the media of the eye, in 
connexion with its achromatism, 
374, 

Power (Rev. J.), inquiry into the 
cause of endosmose and exosmose, 
391. 

Prideaux (J.) on thermo-electricity, 
384. 


Rain, experiments on the quantities 
of falling at different elevations, 
401. 

Refraction, conical, on, 370. 

Rennie (G.), report on hydraulics as 
a branch of engineering, 153. 

Respiration, action of the medulla 
spinalis and oblongata on, 73. 


Saurians, fossil, new species of, 440. 

Scoresby (Rev. W.) on a peculiar 
source of error in experiments 
with the dipping-needle, 412. 

Sedgwick (Prof.), his addresses, x. 
XXVii. 

Solar rays, on an instrument for 
measuring their heating power, 
379; on the diminution of their 
intensity in traversing the atmo- 
sphere, 380; on an instrument for 
measuring the heating effect of, 
418. 

Spiders, on the structure and func- 
tions of, 444. 

Spinal marrow, on the, 74. 

Stars, falling, mode of observing, 490. 

Statistical Society of London, objects 
and plan of, 493. 

Steam-engine for pumping water, 
421. 

Strength of materials, on, 93, 103, 
421, 423. 


504: 


Taylor (J.), report on the state of 
knowledge respecting mineral 
veins, 1; on the depths of mines, 
427. 

Telescope, reflecting, 420. 

Terrestrial magnetic force, on the 
direction of, 106; on the intensity 
of, 118. 

Thermo-electricity, on, 384. 

Thermostat, or heat-governor, on the, 
419. 

Tides, directions for observations of 
the, 485. 

Timber, table of the mean strength 
and elasticity of, 103. 

Toad, on the natural history of, 441. 

Trigonometry, on the science of, 288. 

Turner (Dr.), experiments on atomic 
weights, 399. 


Ure (Dr.) on the thermostat, or heat- 
governor, 419. 


Vegetable anatomy, on, 27; vege- 


INDEX. 


table fertilization, 45 ; vegetable 
physiology, 49. 

Veins, mineral, on our knowledge 
respecting, 1. 

Vibrations, musical, in tubes, on, 
140. 


Water, on the compressibility of, 
131, 353. 
a steam-engine for pumping, 


421. 

Waves, on the problem of, 142. 

Wharton (W. L.) on a barometer 
with an enlarged scale, 414. 

on asteam-engine for pump- 
ing water, 421. 

Whewell (Rev. W.), his address, xi. 

Willcox (C.) on the naturalization in 
England of the Mytilus crenatus, 
448. 


Yarrell (W.) on the reproduction of 
the eel, 446. 


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