J,//?.
2
tf-o'fqS^ -i<3o/
REPORT
OF THE
THIRDTTEETING
OF THE
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.
The 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. Challis, 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 phseno-
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. Lindley, 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. Clark.
With respect to the next part of the Transactions, which
includes the communications made to the Sections, two
a 2
IV PKEFACE.
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 maybe, it is evidently impossible to publish
a safe and satisfactory report of them from any minutes
which can be taken. The second rule is, not to print any
of the miscellaneous communications at length ; but either
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 lengthf ,
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 Rc;
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. Whe-
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.
f " Experiments on the Quantity of Rain which falls at different Heights in
the Atmosphere."
CONTENTS
Page.
Proceedings of the Meeting ix
TRANSACTIONS.
Report on the State of Knowledge respecting Mineral Veins. By
John Taylor, F.R.S., Treasurer of the Geological Society and
of the British Association for the Advancement of Science, &c. 1
On the Principal Questions at present debated in the Philosophy
of Botany. By John Lindley, Ph. D., F.R.S., Professor of
Botany in the University of London 27
Report on the Physiology of the Nervous System. By William
Charles Henry, M.D., Physician to the Manchester Royal In-
firmary .59
Report on the present State of our Knowledge respecting the
Strength of Materials. By Peter Barlow, F.R.S., Corr. Memb.
Inst. France, &c. &c 93
Report on the State of our Knowledge respecting the Magnetism
of the Earth. By S. Hunter Christie, 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 105
Report on the present State of the Analytical Theory of Hydro-
statics and Hydrodynamics. By the Rev. J. Challis, late Fel-
low of Trinity College Cambridge 131
Report on the Progress and present State of our Knowledge of
Hydraulics as a Branch of Engineering. By George Rennie,
F.R.S., &c. &c 153
Report on the recent Progress and present State of certain Branches
of Analysis. By George Peacock, M.A., F.R.S., F.G.S.,
F.Z.S., F.R.A.S., F.C.P.S., FeUow and Tutor of Trinity Col-
lege Cambridge 185
TRANSACTIONS OF THE SECTIONS.
I. Mathematics and Physics.
Professor CErsted on the Compressibility of Water 353
W. R. Hamilton on some Results of the View of a Characteristic
Function in Optics 360
The Rev. H. Lloyd on Conical Refraction 370
VI CONTENTS.
Page.
Sir John F. W. Herschel on the Absorption of Light by coloured
Media, vieM'ed in connexion with the undulatory Theory .... 373
The Rev. Baden Powell on the Dispersive Powers of the Media
of the Eye, in connexion with its Achromatism 374
R. Potter, Jun., on the power of Glass of Antimony to reflect
Light 377
on a Phsenomenon in the Interference of Light
hitherto undescribed 378
Sir John F. W. Herschel's Explanation of the Principle and Con-
struction of the Actinometer 379
M. Melloni's Account of some recent Experiments on Radiant
Heat 381
John Prideaux on Thermo-Electricity 384
W. Snow Harris on some new Phaenomena of Electrical Attrac-
tion 386
The Rev. John G. MacVicar on Electricity 390
The Rev. J. Power's Inquiry into the Cause of Endosmose and
Exosmose 391
Michael Faraday on Electro -chemical Decomposition 393
Dr. Turner's Experiments on Atomic Weights 399
Prof. Johnston's Notice of a Method of analysing Carbonaceous
Iron 400
R. Potter, Jun. A Communication respecting an Arch of the
Aurora Borealis 401
John Phillips's Report of Experiments on the Quantities of Rain
falling at different Elevations above the Surface of the Ground at
York 401
II. Philosophical Instruments and Mechanical Arts.
The Rev. Wm. Scoresby on a peculiar Source of Error in Experi-
ments with the Dipping Needle 412
The Rev. W. H. Miller on the Construction of a new Barometer 414
W. L. Wharton 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 Cumming on an Instrument for measuring the
total heating Effect of the Sun's Rays for a given time 418
on some Electro-magnetic Instruments 418
Andrew Ure on the Thermostat, or Heat-governor 419
Thomas Davison on a Reflecting Telescope 420
W. L. Wharton on a Steam-engine for pumping Water 421
E. J. Dent on the Application of a glass Balance-siDring to Chro-
nometers 421
E. HoDGKiNsoN on the EflFect of Impact on Beams 421
on the direct tensile Strength of Cast Iron 423
J. I. Hawkins's Investigation of the Principle of Mr. Saxton's loco-
motive differential Pulley, &c 424
John Taylor's Account of the Depths of Mines 427
J. Owen on Naval Architecture 430
CONTENTS. VU
Page.
III. Natural History — Anatomy — Physiology.
Professor Agardh on the originary Structure of the Flower, and
the mutual Dependency of its Parts 433
Professor Daubeny's Notice of Researches on the Action of Light
upon Plants 436
Walter Adam on some symmetrical Relations of the Bones of the
Megatherium 437
R Harlan on some new species of Fossil Saurians found in Ame-
rica 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-
mon Toad 441
J. Blackwall's Observations relative to the Structure and Func-
tions of Spiders 444
W. Yarrell on the Reproduction of the Eel 44G
C. WiLLCox on the Naturalization in England of the Mytilus cre-
natus, a native of India, and the Acematichcerus Heros, a native
of Africa 448
J. Macartney's Abstract of Observations on the Structure and
Functions of the Nervous System 449
H. Carlile's Abstract of Observations on the Motions and Sounds
of the Heart 454
H. Earle on the Mechanism and Physiology of the Urethra .... 460
Burt on the Nomenclature of Clouds 460
G. H. Fielding on the peculiar Atmospherical Phaenomena as ob-
served at Hull during April and May 1833, in relation to the
prevalence of Influenza 461
IV. History of Science.
Francis Baily's short Account of some MS. Letters (addressed
to Mr. Abraham Sharp, relative to the Publication of Mr. Flam-
steed's Historia Calestis,) laid on the table for the inspection of
the Members of the Association 462
Recommendations of the British Association for the Advancement
of Science 467
Recommendations of the Committees 469
Appendix 484
Prospectus of the Objects and Plan of the Statistical Society of
London 492
Objects and Rules of the Association 497
Index 501
o to o
Cft^ CO CO »o
=.J
— CO
CM O >r;
C
O
O J3
;. -a
M
OJ
o
u
o
iX!
CCS
CO
<:
w
P5
H
CO
QO
_ O _
X o w
W a; *>»
i" '^ B
^ P S;
G t- 5j
3 3 QJ
c/^ Ph ^
>- S s S
O .
a s
"" 1-3
O
;«■
H
125
a
o
1-a
-pq iS
H Q
c <u t; s
03
° 5 2
5 S S
s
bo
53
,^»-HOOOOO00O
QulC^O^O COCOCO»-i
^ — «o ■^
^
Q
00 f-i
5 00
«i«
< ^
3
►J •-
03
<! 1/3
*J 4J O = *J
CQ
E
o
CJ rt &, o ••-
3
< u
OS Q
C8
fe
H
«o
«o
k,
(U
a>
t3
^
s
ei
QC
■±3
=4-
o
CO
pi3
^
•O
O
<u
■n
4-1
a
;ri
n
H
S
c2
J3
1
s
JJ
X
c
V
C3
!U
^
n
.*
bn
60
c
CS
c
«0
ira
o
"3
>
ca
to
c
Cm
o
lO
^
o
rt
3)
J3
en
T3
c
J3
a
c
3
o
£
*-•
c
n
•rl
QJ
J2
CO
Q)
13
a
3
■*-»
C5
P.
CO
.a
ti-,
o
u
C
0)
'o
<§
o
o
o
^
§
~i
1
-^
o
CO
o
05
w
(M
rt
s
-^•
^
THIRD REPORT.
PROCEEDINGS OF THE MEETING.
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
Rev. 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.
1833. b
X THIRD KEPORT — 1833.
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 still in 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 phsenomena,
would have been of comparatively little value to himself and to
society, had there not been superadded to them a beautiful
moral simpUcity 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 conferi'ed on him, out of
the funds of the Civil List, a substantial mark of his royal
favour."
The Rev. William Whewell, being called vipon by the
Pi'esident, 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 look 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 . THIIID REPORT — 1838.
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 Hght,
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.
" It 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 had 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 laboiu's of the
Meeting now begun. I am well persuaded that if the President
had taken this ofiice 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
PROCEEDINGS OF THE MEETING. Xlll
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 natui-ally 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,
eflPects 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 — 1833.
tion, some furthei- 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 M^as 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. XY
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 Uberty 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 procure 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 phaenomena are to be referred,
the law of universal gravitation ; though we still desiderate a
clear application of the theory to the details. In 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 ai'e aware, that which makes light to consist in
material particles emitted by a luminous body, and that which
makes it to consist in undulations pi'opagated 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 — 1833.
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 dehvered 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 appeai-ed 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 concihated by having a distinct acknow-
PROCEEDINGS OF THE MEETING. XVll
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 phaenomena,
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 I'efer, which it is less necessary to notice in a general
sketch like the present. The recent discoveries in Thermo-
electricity, of which Professor Gumming 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,
thei'e 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-
Xviii THIRD REPORT — 1833.
legist 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 estabhshment of the properties of each species *. In Che-
• Perhaps I shall not have a more favourable occasion than the present of
correcting a statement 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-saUs ; 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.
" I 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 Mr. Haidinger to be
not simple but compound. It seems, however, that the solution of the difficulty
(for no one now will doubt that it lias 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 at the 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.
XX
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 rviles 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. XNl
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 theoi-eticul speculations
XXii THIRD REPORT — 1833.
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, will 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 occui',
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 eni-iched 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 a 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 phaenomena 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 cry-
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.
" I 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 — 1833.
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 phaenomena. But all the rest of the prospect is compara-
tively darkness and chaos ; hmited 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 even if 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 ive, that
we should ape their mental attitudes ? — we, who can but with
pain and eiFort 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 vi'ho 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
do, any perverse conception of the true scale of his aims and
hopes.
" 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.
" I 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 tide^,
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.
XXvi THIRD REPORT— 183,S.
" 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 hav-
ing read the minutes of their proceedings to the Meeting, the
Rev. 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 present 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-
PKOCEEDINGS OF THE MEETING. XXvni
fessoi' Bavlow, was given, in the absence of the Author, by the
-Rev. W. Whewell.
In the evening, Mr. Whewell delivered a Lecture in the
Senate-house, 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 Parish 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 having
read the minutes of their proceedings, the Rev. 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
600/. Mr. Babbage, at the President's request, explained his
views respecting the advantages which wovild 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.
The President, in his concluding Address to the Meeting,
explained an irregularity which had occurred in the formation
of a new Section. In addition to the five Sections into which
the Meeting had been divided by the authority of the General
* For a particular account of these appropriations, see p. xxxvi.
c2
XXviii THIRD REPORT 1833.
Committee, he stated that another had come into operation, the
object of whicli 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 phaenomena capable of being brought under the sem-
blance of a law are legitimate objects of our inquiries. But there
ai'e 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 among
the objects of the Association. The sciences of morals and
politics are elevated far above the speculations of oiu- 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 phaeno-
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
generahzations, 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 by the 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 Daemon 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 and'body, 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 v/hich 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 Nortliampton,) 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-
ratively 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 it is subordinate to some law, — that it
is a step toward the knowledge of some general truth. Without,
at least, a glinnnering 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 phsenomena, 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 oi'der 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 phaenomena 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 to desert 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 pi'esent 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
svu-ely 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
attainntents in science may tend to our moral improvement; and
XXxii 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 BoreaHs, seen on the 21st of March 1833.
Mr. Hopkins gave an abstract of a paper on the Vibration
of Air in Cyhndrical 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 utihty 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 theinequahty 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. Brunei exhibited and explained a Model in illustration
of his method of constructing Bridges without centering.
PROCEEDINGS OF THE MEETING. XXXUI
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-
posits 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. Dufrenoy entered into a consideration of some phaenomenaof
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
Limest*)ne.
JMr. 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 pai't
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.
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 sear, which he dis-
tributed into five new genera.
The Rev. 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 difterence 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 hving 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 respecting
the reflex function of the Medulla oblongat^i 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 Coimnittee 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 the thanks 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, powder 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 : —
To select those points of science, which, on a review of the
former Recommendations of the Committees, or those contained
XXXvi THIRD REPORT — 18,33.
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 200/. 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.
S. 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 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. XXXVU
6. That the effects of Poisons on the Animal Economy should
be investigated and illustrated by graphic representations ; and
that a sum not exceeding 251. 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 251. 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 a representation 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 -j- be
appointed to wait upon the Lords of the Treasuiy 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. It was 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 %. 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 was immediately 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-
lection of Recommendations and Suggestions printed in the latter part of the
volume.
XXXviii 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 Professor of
Astronomy, Cambridge. John Dalton, D.C.L. F.R.S. Instit.
Reg. Sc. Paris. Corresp.
President elect.— lAeni. 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. Rev. J. Robinson, D.D. Astronomer Royal
at Armagh.
Treasurer.— John 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.
L.S. Professor of Botany. Rev. B. Powell, F.R.S.
Professor of Geometry.
Secretaries for Cambridge. — Rev. J. S. Henslow,
G.S. Professor of Botany. Rev. W. Whewell, F.R.S.
Secretaries for Edinburgh. — John Robison, Sec.
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, Edinbvn-gh. 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.
Edinburgh. George Rennie, F.R.S. Rev. W. Ritchie, LL.D.
F.R.S. Professor of Nat. Philosoohy in the University of Lon-
don. J. S. Traill, M.D. W. Yarrell, F.L.S. &c. Ex officio
members, — The Trustees and Officers of the Association.
,S'ecre/fm>.s.— Edward Turner, M.D. F.R.S. Sec. G.S. Rev.
James Yates, F.L.S. G.S.
F.R.S.
Savilian
F.L.S.
&c.-
R.S.E.
PROCEEDINGS OF THE MEETING. XXXIX
COMMITTEES OF SCIENCES.
I. Mathematics and General Physics.
Chairman. — Sir D. Brewster, F.R.S. &c.
Deputy Chairman. — Rev. G. Peacock, F.R.S.
Secretary. — Professor Forbes.
Viscount Aclare, 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,
F.R.S. G. DoUond, 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,
F.R.S. Sir John F. W. Herschel, F.R.S. E. Hodgkinson.
W. Hopkins. John Hymers. Rev. Professor T. 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. Pottei-, jun. Professor Powell. Professor Quetelet.
Professor Rigaud. Rev. Dr. Robinson. Rev. R. Walker,
F.R.S. W. L. Wharton. C. Wheatstone. Rev. W. Whewell,
F.R.S. Rev. R. Willis, F.R.S.
II. Chemistry, Mineralogy, Sj-c.
Chairman.— 3. Dalton, D.C.L. F.R.S.
Deputy Chairman. — ^Rev. Professor Cumming.
Secretary. — Professor Miller.
Professor Daniell. Professor Daubeny. M. Faraday,
D.C.L. Rev. W. Vernon Harcourt, F.R.S. W. Snow Harris,
F.R.S. W. Hatfeild, F.G.S. J. F. W. Johnston, A.M. Rev.
D. Lardner, LL.D. F.R.S. Rev. B. Lloyd, LL.D. T. J.
Pearsall. Dr. Prout, F.R.S. Professor W. Ritchie. Rev. W.
Scoresby, F.R.S. W. Sturgeon. Professor Turner.
III. Geology and Geography.
Chairman. — G. B. Greenough, F.R.S. Pres. G.S.
Deputy Chairmen. — Rev. Dr. Buckland, F.R.S. G.S. R.L
Murchison, F.R.S. V.P.G.S.
Secretaries.— W. Lonsdale, F.G.S. John Phillips, F.R.S.
G.S.
Dr. Boase. James Bryce, jun. F.G.S. Joseph Carn^j
F.R.S. G.S. Major Clerke, C.B. F.R.S. M. Dufrenoy. Sir
Philip Malpas de Grey Egerton, F.R.S. G.S. Dr. Fitton, F.R.S.
G.S. Rev. J. Hailstone, F.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.
Rev. J. Yates, F.G.S.
IV. Natural History.
Chairman.— Kev. 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. £.
Gray, F.R.S. Rev. Professor Henslow. Rev. 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, ^c.
Chairman. — Dr. Haviland.
Deputy Chairman. — Dr. 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.
Rev. 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.
[ 1 ]
TRANSACTIONS.
Report on the State of Knowledge respecting Mineral Veins.
By John Taylor, F.R.S., Treasurer of the Geological So-
ciety and of the British Associatioii for the Advancement of
Science, Sfc. 8fc.
X 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 obsei'ved
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 writer adopts the Wernerian theory,
and mentions cases which he thinks confirmatory of its truth.
In the four volumes of the Transactions of the Royal Geo-
logical Society of Cornwall, we shall find this subject more
1833. 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. K. 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 difiicult 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 phaenomena 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
occur.
. " Veins cross the strata, and have a direction different from
theirs. Other mineral repositories, such as particular strata or
REPORT ON MINERAL VEINS. 9
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. True 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 mentions 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.
b2
4 THIRD REPORT — 1833.
The only dimension we can ascertain is that across from one
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 flookan by the
Cornish miners.
The clearest idea of a vein will be obtained by imagining a
crack or fissiu'e 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 MetaUica.
Agricola being held to be the first who has written anything
certain on the 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 Jissiires which have been since filled up by de-
grees tvith 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 Rosier, 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 Ki?igr
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 oftheHartz,in 1787; and Linnaeus 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 Cormibiensis 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. V
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 ivater, which variety of
causes and circumstances must infalUbly 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 ideaf ."
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.
+ Miiwialoffia Cornubicnsis, ji. 101. ''
8 THIRD REPORT — 1838.
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 have 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 have 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 have 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 observing
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.
b. By particular internal canals.
c. 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 oft" 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, furthei", 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 ai'isen, 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.
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 eftects.
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.
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
IS 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 vai'ious 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 phasnomena of veins, and that bis experience is
therefore principally confined to jhose 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 fi'equent 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 ci'oss direction, they do not exhibit
the heaves and interruptions of true veins, bvit usually unite.
Dr. Boase says that this statement is opposed to his obser-
vations, and that the phaenomenon 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 phaenomena, 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 wiU appear that there are three leading hypo-
theses.
1st. 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 sola-
i'4 THIRD REPORT 1833.
tions, which deposited the substances with which they were
charged in the veins, or by the process of subhmation 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-
sei'ved.
Similar to these, and occurring with them, and therefore well
known, are others, which, though baiTen 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
REPiORT ON MINERAL VEINS. 15
geologists of the present day, the veins of granite, porphyry,
quartz, &c.
Some of these have been examined below the svirface, where
they pass through coal-fields, or other deposits of useful mine-
rals, but containing in themselves nothing to reward the toU 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 be deposits of ore also which it would be difficult to
refer the structure of to any other hypothesis, particulai-ly such
as contain ores so intimately mixed with the rocks as to form a
constituent part of them.
I 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
accovmt ; 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, Rosier, Henkel, HoiFman, 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 attexnpt 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 trovible ; 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 :
— 1. 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.
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 foi'med. 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 metalhferous 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 diff'er
also in their direction, each class being, however, parallel in
18^3. c
18 THIRD REPORT — 1833.
itself; ami how tliese 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 metalHferous veins, suggests whether some analogies
may not be traced between electro-magnetic ciu'rents 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 apphed 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 httle 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
case 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 difterent metals from those in
the former. They pass through or traverse the other veins,
cutting them through, and sviffering 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 Playfjiir.
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 been found to
be subject to exceptions, tlie 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 is
clear and intelligible, it is that the relative age of veins is made
REPORT ON MINERAL VEINS. 19
out by these fiicts, even although we may not yet be able to
apply rules for every case, — a subject which has been con-
sidei'ed 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
cominff 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 afl^ecting the rock itself with visible efifects 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?
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 contrai'y 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
c2
so 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. Weavei' describes the contents of the great vein of Bolanos 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 fluov
or calc 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 metahic 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 metdllic 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. Bovi^, 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 'i
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 I 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 phaenomena.
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, 1832.
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
richer, 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
aff'ect the contents of the veins ; and appearances as to the tex-
ture and formation of the strata are often regarded by miners
with mox'e anxiety ihan 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-
mainder 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 innumerable 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 in 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 20,827 bings.
Little limestone 287
Four-fathom limestone 91
Scar limeston e 90
Tyne bottom limestone 393
■ 21,688
Gritstone Beds. — High slate sill 107
Lower slate sill 289
Firestone 262
Pattinson's sill 259
High coal sill 327
Low coal sill 154
Tuft 306
Quarry hazel 44
Nattrass Gill hazel 21
, Six-fathom hazel 576
Slaty hazel 18
Hazel imder scar limestone 2
2,365
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 inti-icate 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 to a
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 me a source of great gratification;
as the desire of promoting such inquiries must be my apology
for attempting the task which I have undertaken.
[ ~^7 ]
On the Principal Questions at present debated in the Pliiloso-
j)hy of Botany. By John Lindley, Ph.D., F.R.S., 8fc.,
Professor of Botany in the University of London.
If 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 phagnomena 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, mc
28 THIRD REPORT — 1833.
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 axillae of pre-existing tubes ; an observation
that has been confirmed by Mr. Henry Slack *. It has been
distinctly proved by M. Mirbelf , that the same thing occurs in
the case of Marchantia polymorplia. 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.
f " Recherches Anatomiques et Physiologiques sur le Marchantia po/t/mor-
pha," in Nouv. Ann, du Museum, vol. i. p. 93.
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 Hke 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 svich 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 Marchantia f ,
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 phaenomenon 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.
* Ueher die P or en des Pflanzen- Zell'jewehes, p. 11. Tubingen, 1828.
t Archives de Botaniqiie, vol. i.
30 TIITRD 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. Mohlf, 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.
t Ueher die Poren des Pjiemzen-Zellgeivehes.
X Botanische Zeittmg, October 7, 1832.
§ Transactions of the Societij of Arts, vol. xlix.
REPORT ON THE PHILOSOPHY OP BOTANY. 31
It would, however, appear from the researches of Mirhel*,
that the presence of a twisted fibre within a cell is not always
the cause of the spiral or fibrous character so common in tissvie.
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 ai-e 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 elaiers in the Mar-
chantia, which he describes to the following efi^ect. 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 enlai'ge,
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 ownf, 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 pi-eceding 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 Cycadece^ Coniferce, and Tree Ferns.
» Archives de Botantque, vol, i.
t Introduction to Botany, p. 2.
S3 THIRD REPORT — 1833.
The latter considers that the dotted tubes of Cycadece un-
doubtedly pass directly into the vessels called by the Germans
vasa scalarlformia \ 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-
known, than it can be at the present day. Knowing, as we
now do, that a tree is more analogous to a Polype than to a
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 sufficiently 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 phaenomena. The motion was for the first time beheld
by Amici, the Professor at Modena, who discovered it in the
Char a. 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 phee-
nomenon has been determined with considerable precision.
Among other things, it has been ascertained that in Nitella
the currents have always a certain relation to the axis of growth,
the ascending current vmiformly passing along the side of the
cell most remote fi-om 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-Ranae the fluid has been ob-
» Transactions of the Society of Arts, vol. xlix.
REPORT ON THE PHILOSOPHY OF LJOTANY. 3^
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 Tradescmitia 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 Cheliclonium majus,
and in the stipulae of Ficus elastica.
His observations have been repeated by a Commission of the
Institute, 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
])ut the common rotatorymotion imperfectly observed.
Sfri.'cfure of the Axis. — From the period when INl. 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
regard to dicotyledonous plants ; but Dr. Hugo Mohl has
endeavoured to showf that monocotyledonous stems are not
* Aniinlea ih.i Sr.'ienres, vol. xxii. p. 80.
i Molil, "])i> I'aliiiavum Stnictura," in Marlhis's Genrra ct Species Palmarum.
IS;}.']. D
34f 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 trimk. 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 Cycadece — 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 Cycadece are a slight modi-
fication of vasa scalariformia. Dr. Mohl is also of opinion
that Cycadeae 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 Cycadece, in
regard to its anatomical condition, must be considered inter-
mediate between that of Tree Ferns and Coniferce, just as their
leaves and fructification undoubtedly are. He states that in
Cycadece 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 Cycadece and Exogence 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
* Uebcr den Bau des Cycadeen Stammex und sein Verhaltniss xu den Slamni
der Coniferen und Banmfarrn. 4to. Munich, 1832.
REPORT ON THE PHILOSOPHY OF BOTANY. 35
Malabaricus be correct, where the stem of Cycas c'lrcinal'is is
shown to have several concentric zones, precisely as in other
exogenous trees, it must follow that Di\ Mohl's explanation
would be still more inadmissible ; accordingly, this author dis-
credits the fact of the stem of Cycas circinalis having numer-
ous concentric zones. It is, however, certain, from the speci-
mens brought to England by Dr. WalUch, 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 vmdiscovered
peculiarity in the mode of forming their stem by Cycadecs; 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 exogenous trees.
Pi-ofessor 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 Cycadece : 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, Lycopodiacece,
Alarsileacece, and Mosses, in all 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 Monocotyledons from Dicotyledons, upon principles
different from those generally adniitted. 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
* NuiurHchpn Sijstem des Pflanxenreichs nach seiner inncrcn Organization.
8vo. Berlin, 1^32."
36 THIRD REPORT— 18i]o.
in the stem of Monocotyledons there is only one surfoce 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 connected 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 assei'ted 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 downvt^ards by them. In
support of this it is argued : 1 st, That an anatomical examination
of a plant shows that the woody systems of the leaf and stem
are continuous : Sndly, 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 :
ordly. 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
• Achille Richai'd, Nouveaux EUmens de la Botaniqtte, 5me edit. p. 119.
REPOllT 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 Vita, 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 littea is grafted upon the common
Horse-chestnut, — and that these circumstances are inconsistent
» yuufcaiu- E/emcns de la BoiaiiiqKC, 5mc edit. p. 105.
38 THIRD REPORT — 1833.
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 tliat 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 ceUular 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 also
been proved by Mr. Knight* and M. De Candollef 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 pha?nomena 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 ahvays
produce branches like itself, notwithstanding the long super-
position of new wood which has been taking place in it froni
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 hoi-izontal 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 versa, 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 Transaciiuns, 1S05, p. 257.
•{■ Physiohgic Vcnitcde, p. 158.
REPORT ON THE PHILOSOPHY OF BOTANY. SSi
lateral extension, the diameter of each will remain the same ;
but if one grows more rapidly than jthe 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 versa. It is* 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 — 1st, 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. Turpinf ,
as far as can be collected from a long memoir upon the grafting
of plants and animals ; but I must fairly confess that I 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 leaves,
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, Phyaiologie Vegetale, p. 165.
+ Sue Annales des Sciences, vols. xxiv. and xxv., particularly vol. x.w. p. •13.
J De Candollu, Physiologie f'rgelale, 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 leisvire 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 Wiu'zel
root, beginning with the dormant embryo, and concluding with
the perfectly formed plant.
Arrangemeni 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 verticiilate plants we not unfrequently see that the whorls
are dislocated by the praBternatural elongation of their axis, and
then become converted into a spire ; and the same pha3nome-
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
consider that, as M. Adolphe Brongniart has shown*, what we
calLvvhorls in a flower often are not so, strictly speaking, but
only a series of parts placed in close approximation, and at dif-
ferent heights, upon the short branch that forms their axis.
Dr. Alexander Braun has endeavoured f 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 witli a contemplation of the manner in which the scales of
a Pine Cone are placed, to which a long and ingenious method
of analysis v/as applied, he found that several different series of
spires are discoverable, between which there invariably exist
peculiar arithmetical relations, v/hich are the expression of the
various combinations of a certain number of elements disposed
in a regular manner. All these spires depend upon the posi-
* Annales des Sciences, vol. xxiii. p. 226.
+ Verglekhendr Vritersuchunj iibcr die Ordininfj der Schuppen an den Tan-
nen~apfcn. Ito. IPjd.
REPORT ON THE PHILOSOPHY OF BOTANY. 41^
tion of a fundamental series, from wliicli the otliers 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 ^-^ in-
dicates that eight turns are made round the axis before any
scale 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 M'hich 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.
Structtire 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 inteqnal
parenchyma is arranged with beautiful uniformity, in the man-
ner most conducive to the end of exposui'e 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
* Anitales des Sciences, vol. xxv. p. 215.
42 THIRD REPORT 1833.
is divided completely by a number of partitions caused by the
ribs and jirincipal 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 *.
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
w^ater have no cuticle at ail, has not been disproved, unless in
the case of Marchantia f , whose under surface can scarcely be
said to have a distinct cuticle; but this jilant, 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 exti'emely probable. It was especially found to be
the case by M. Mirbel in his so often quoted remarks upon
Marchaniia.
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 by a membrane. On the contrary,
the observations of M. Mirbel on Marchantia, 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 svirrounding 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
* " Reclierclies Anatomiques et Pliysiologiques sur le Marchantia poli/ruor-
pha," in Nouvcaux Annalcs du Museum, vol. i. p. 7.
+ Ibid. p. 93.
J Suppl. prlmttm Pradromi Floros Novce IloUaiidicc, p. 3.
REPORT ON THE PHILOSOPHY OF BOTANY. 45
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 aii'-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 it is through the
stomata that water rushes into the cavernous parenchyma to
supply the loss occasioned by the abstraction of air.
Anther, Sj-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 f, that the opi-
nion emitted by Mirbel in 1808 J, 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 tjiat this lining is composed of cellular tissue chiefly
of the fibrous kind, which forms an infinite multitude of little
springs, 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.
* Annales des Sciences, vol. xxv. p. 24?,
t De Cellulis Anther arum fihrous. 4to. Wratislaviae, 1830.
X " Observations sur iin Systcnie d'Anatomie Comparee de Vegetaiix, fondes
sur rOrganization de la Fleur," in Mcmoire.s de I'lnstitut, 1808, "p. W.'A.
§ "Complement des ObsciTations sur le Marchantia jwlymorphu," in Ar-
chives de Botanique, vol. i.
44 THIRD REPORT — 1833.
The Origin of the Pollen, connected as it intimately is with
the singular pha;nomena 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. Mirbellias added some observations*, detailed with that
admirable clearness and precision which give so great a value
to all his writings, and wiiich 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. In a 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 phaeno-
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 sepiu'ated
* " Complement des Observations," i^c, as above quoted.
REPORT ON THE PHILOSOPHY OF BOTANY. '4S
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 triangvdar
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.
Wliat 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 rirginica *.
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. Mii'bel admits the exist-
ence of an outer not distensible coat, and of an inner highly
extensible lining. A curious paper upon this point f 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.
Fert/lhation. — 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 vvho could
not discern the tubes that are projected into the style by the
* Observations upon Orchidon? n?id Asclrpiacfea?, p. 21.
f Beitrage ?:ur Kenntnks dcs Pollen. 4to. Berlin, 1S33.
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 phsenomenon ;
it has since been explained by M. Mirbel. Another case, pre-
senting similar apparent difficulties, occurs in Helianthemutn.
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 Brongniartf, 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
AsclepiadecE. 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 Piecherehes sur la Sfnicture dc VOoule Vegetal et sur ses Deve-
loppements. Also Additions aiix 'Nouvelles Rec/irrclies,' ^c.
•f- Annales des Sciences, vol. xxiv. p. 123. X Linntpa, 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 Asclepiadece 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 Brongniartf 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 phaenomenon 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-
dece 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 vmtil 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 Orchideae and
Asclepiadece. London, October 1831.
f Annft/fis des Sciences for October and November 1831 ; from observations
made in July, August and September of that yoar.
48 THIRD REPORT — ISoJ.
ferences whatever in texture between tliat 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.
Orchidece 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.
Brown f, in itn 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 tliere 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 tlie 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 Orchidece is still far from being
determined. I must particularly remark that the very proble-
matical agency of insects, to m hich Dr. Brown has recourse in
order to make out his case, seems to be singvilarly at variance
with his supposition § that the insect forms, which in Oplirys
are so striking, and which he finds resemble the msects of the
countries in which the plants are found, are intended rather to
repel than i^^ 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 OpJinjdecs ; 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
ReiioufJiera Arac/iniies, the whole genus Oncidiian, Teiramicra
rigldn, several species of Epidendrinn, Cipnbidlum tenuijolium,
Vanda peduncidans, and a host of others ?
* Annales des Sciences, vol. xxiv. p. 113.
t Observations upon the Organs and Mode of Fecundation of Orcliidese and
Asclepiadese.
X Illustrations of the Genera and Species of Orchideovs Plants. Part II.
" Fructification," tabb. 5. 12. 13. ] k
§ " Proceedings of tlie Linnean Society," June o, 1832, as given in the Lon-
don and Edinburgh Philosopliical iMagcuiitc 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 Alstromerias, 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~
lymorplia. 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 Avas, 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 originall)'^ 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
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 the 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, or 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 different 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 grovmds ; 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 PIIILOSOPIIV 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
appHcation 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 CompositcB are anatomically almost identical, and that their
difterence 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 vmex-
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 to escape. As soon as this takes place, the corolla can no
longer remain erect, but falls back toAvards 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
H. 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 a leaf 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 bracteas
of certain Marcgraviacece 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.
E 2
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 gi-een 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 in the Transactions of the Philosophical SocietT/ 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 | 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 petals 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. t vol. v. Part I.
I Nova Acta Academice Naturee Curiosorum, vol. xvi. p. 245.
^ De Antholysi Prodromus.
REPORT ON THE PHILOSOPHY OF BOTANY. S$
instances where one flower grows from within another. And
finally, 6. Proliferousness {Ecblastesis), when buds are deve-
loped in the axillae 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. I 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 phaenomena 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 dii'ected 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 Dot any, p. 313, &c.
54 THIRD REPORT — '1833.
The same principle of growth appears to obtain in Confervas,
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 Vasciilares. In the opinion of Dr. Brown and of
Mirbel, the first rudiment of a plant far more comphcated than
Marchaniia, 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.
Irritability. — I3r. Dutrochet has published f the result of
some experiments v,ith 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
irritabihty. 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, lias
mentioned j the result of some experiments upon the action of
the coloured rays upon germination ; and he has fovmd 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. t ■•'innalcs des Sciences, vol.xxv. p. 243.
J Annates 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 chromiile by De CandoUe, 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
bracteae 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 f , 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 CandoUe J
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.
* Mcmoirrs de la Socictc Physique de Geneve, vol. iv. p. 50.
t Ibid. vol. i. p. 284. % Phjsiologie Vegetale, p. 906.
56 THIRD KEPORT — 1833.
Dr. Dutrocliet 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 Tnfolium 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 acquii'es 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 manui'e 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 CandoUe, 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 f that all plants
part with a kind of faecal matter by their roots, that the nature
of such excretions varies with species or large natural orders :
in Cichoracecs and Papaveracece he found that the matter was
analogous to opium, and in Leguminosce to gum ; in Graminece
it consists of alkaline and earthy alkalies and carbonates, and
in Euphorbiacecs 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
* Aiinales des Sciences, vol. xxv. p. 216.
f De CandoUe, Physiologic Vegetale, 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 a few 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 to a
practically useful purpose, we require positive information upon
the following points : —
1. The nature of the faecal 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 thi;?, ; 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.
[ 59 1
Report on the Physiology of the Nervous System. By Wil-
liam Charles Henry, M.D., Physician to the Manchester
Royal Infirmary,
Introduction. — Tme 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 laminae
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 vital 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 I'especting 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 cnume-
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 elementally 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 striae
may be distinctly perceived, transversely marking the fibrillas."
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 ari'angement
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 lav/s of contractility and ' in-
nervation' must precede the description of the several functions,
which all depend on their single or imited 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. Tims 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-
deUing 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 JUdinbure/h Medical and
Surgical Journal.
RKPORT 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 jiermanent 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 grovmd-
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
Foetus of Vertebrated Animals, by Dr, Allen Thomson.
64 THIRD REPORT — IS33.
human foetus, and the existence of one and the same funda-
mental type in the hrain 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, middles, 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 larger 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
haemorrhage, or prevent the extension to contiguous parts, of
REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. G5
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 o^ierating 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 sensibihty 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 all 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 have their
origin. But a power of effecting regular and combined move-
1833. p
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 vievping 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 injui'y, 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 Physiolorjie, torn. x. p. 361
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 comljination of ideas. Thus
the most docile and intelhgent 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 anunal
* Journal de Physiologie, tom. iii. p. 376.
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.
Cerebelhim. — 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 injui-y. 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, Mhnoires de I' Academie, torn. ix.
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 muscidar 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 simvd-
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
cei'ebellum. Magendie has described f 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. f Ibid. toin. iii. p. 157.
70 THIRD REPORT — 1883.
fully established, opposed as they are by those of so skilful an
experimenter as Magendie. Indeed, while Flourens conceives
the cei'ebellum 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 in a 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 diflTerent 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 tie Physiologie, torn. iv. 399.
REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 71
the left side. Hence it appears that section of the crus lias
more influence over the lateral rotation of the hotly than injury
of the cerebellum itself; and that the impulsive force does not
belong (at least exclusively) to the cerebellum. Wlien 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 Me-
moir es de VAcad^mie des Sciences* a description of some
striking 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 iwegular motions in all directions ensued. When,
* torn. ix. p. 454.
12 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 f. In a rabbit
* Memoires de I'Academie des Sciences, torn. ix. p. 478.
t CEuvres de Legallois, torn. i. p. 247.
REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 73
five or six clays 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 suflS-
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
* Memoir cs de V Academie, torn. ix. 1830.
74 THIRD REPORT — 1833.
might be removed, for though the raisjng 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 I'espiratory 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 meduUge 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 agx'eeing 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. It is 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, conjee-
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 pi-obably 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 point 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 ' pai-alysis'. 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 Avitli 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 oflice 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 sufi^ering; 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 lacei*ated, 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 &pinal 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, torn, 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 vertebrae 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 shai'p 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. In a second paperf 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 I'oots. 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 medvdla,
gives birth, along with motive phenomena, to some evidences
of sensibility ; and, vice versa, 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, torn. ii. p. 276. August 1822. f Ibhl torn. ii. p. 366.
78 THIRD REPORT — 1833.
ends not connected with the medulla, produced no apparent
eftects ; 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 thus 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 obsei'ved 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 inferences.
There is, however, a very I'emarkable case described by Pro-
fessor Royer CoUard, to which these objections do not apply.
Sprevale, an invalided soldier, was upwards of seventeen years
the subject of medical observation in the Maison de Sante 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 {ramolUssement) 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, torn. iii. p. 1 73 ; and Dr. KorefTs,
torn. 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 vipon 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 woi'k of Legallois, entitled ^^ Experiences sur le Prin-
cipe de la Vie, notamment sur celni des Mouvemens du Cceur
et si/r le Siege 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
• CEuvres de Legallois, torn. i. pp. 97, 99, &c.
80 THIRD REPORT — 1853.
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 Avas affirmed to be the constant
result of the operation ; but even in some of Legallois's own ex-
periments f, 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 Inquiry
into the Laivs 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 Mem. de lAcademieX, ^^^^ 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 cii'culation. 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 wovdd 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. 1 1 . (and Journal de Physique, tom. iii.) throw much
* p. 259. t 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, the iris 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 sensorivmi, 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 of nerves. 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 f.
"The motion of the nostril of the same side instantly ceased,
* Natural System of Nerves, p. 3.58.
t See Mr. H. Mayo's Anatomical and Pliijsioloffical Commentaries, Part I.;
and Outlines of Human PLy^ioloijij, 2nd edit., p. 3;34,
1 S.Jo. G
82 TIIIUD 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 f." 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 inquirers X- 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 fvmctions. 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 case the
« pp. 106, 107. t Outlines of Human Physiology, 2nd edit., p. 337.
+ (Euvres, p. 1 54 et seq.
REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 83
dyspnoea 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 dataf.
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 lingvial, 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
oflice of conducting motive impressions.
II. 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 joined 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 Structurd 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 Inquirt/, 3rd edit., p. 109.
t 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 tevtium
quinti paris ramum, maxillarem inferiorem dictum, properat,
insalutato ganglio semilunari, ad similitudinem radicum ante-
riorum nervorum spinalium ?" Siimmerring 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
pain ; 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.
IlEPORT 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 smelUng, 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,
and 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 fifth
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
vninjiired 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 Laine, 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
iri'itation. 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 insensible membrane, instead of inducing an aug-
mented mucous discharge, will excite the inflammatory process
described by Magendie. The disorder of the digestive func-
tion f, 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, lom. v.
•f Outlines of Phyniolngy, p. 71.
REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 87
the thirty-one pairs of spinal nerves, becomes, after the vniion
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."
III. The olfactory, auditory and optic nerves are gifted with
a special sensibility to the otajects 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 diflPerent 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 18S3.
IV. The functions of the ganglia, of the great sympatlietic
nerve, and its intricate plexuses and anastomotic connexions, are
matter, at present, of conjectui'e. 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 Sympathetici 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 J "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 counti'y 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 all 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. t 4to, 1830.
J The Nervous System of the Human Body, p. 129. 'Ito, 1830.
§ Op. cit., p. 11;').
II Transacliom of Oie Medko-chirurykal Socidy of Edinburgh, 1826, vol. ii.
p. 165.
TJEPOKT 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. 1 76, 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-
ti'ol 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.
11. 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 aei-ial
undulations.
IV. The system of the great sympathetic nerve, and its as-
sociated plexuses and gangUa.
[ 93 ]
Report on the present State of our Knowledge respecting the
Strength of Materials. By Peter BarloWj Esq., F.R.S.,
Corr. Memb. Inst. France, 8fc. Sfc.
The 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 covmtry 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 sti*ength 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 IS.'JS.
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 length : 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 6f MATERIALS. 95.
multiplied by the distance of the centre of gravity of their sec-
tion from the lowest point ; abovit 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 eqvially 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
I'atio 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 vi^ithin 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, Sfc, in 1817,
I v/as 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 thiiId report — 1833.
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 I have 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
and fir beams of considerable dimensions, carried on at the ex-
1833. H
98 THIRD REPORT — 1883.
pense of the French Government, from which he has drawn the
following empirical formulae, viz.^
1. In oak bea,„s Ul = ■1784451 (/+ -03). «■
bo V'o
2. In fir beams ?/! = 8161 128 a /r^
o
where P = half the weight in kilogrammes, a the less, and h
the greater sides of the section, /half the length of the column,
and b the versed sine of inflection, the dimensions being all in
metres *.
How far these formulae 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 VE'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
* See Traite Analytique de la Rhistance des Solidei.
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 formulae 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 Magaziiie.
H 2
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 excejDtion 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.
Formula; relating to the ultimate Strength of Materials in
cases of Transverse Strain. — Let I, b, d, denote the length,
breadth and depth in inches in any beam, w the experimental
w
breaking weight in pounds, then will j— r-^ = S 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, ,
bcT^-^'
2. When fixed the same, but uniformly loaded,
1 lie _ ^
3. When supported at both ends, and loaded in the middle,
1 Itv _ ^
4 ^ ZTrf^^ ~ ^'
REPORT ON THE STRENGTH OF MATERIALS.
101
4. Supported the same, and uniformly loaded,
1
8
Iw „
^ bd^-^'
Fixed at b
oth ends.
and loaded in
the middle,
1
6
Iw ^
Fixed the
same, bui
. uniformly loaded,
1
12
Iw ^,
^ bd^~
7. Supported at the ends, and loaded at a point not in the
middle. Then, n m being the division of the beam at the point
of application,
11711 I HI _ „
Some authors state the coefficients for cases 5 and 6 as -rj-
1 . ^
and jp, but both theory and practice have shovpn these numbers
to be erroneous.
By means of these formulse, 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.
Formulce 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.
Case 1.
2.
3.
32., Pw _
Case
4.
5 Piv
8 ^ bdn~ ^
1 ^ bdH-^-
12 Pw
1 ^ bdH- ^•
5.
2 Pw
3 ^ bdH"^
1 Ptv „
1 ^ bdH-^-
6.
5 Pio ^
12 ^ bdH-^
Hence, again, from the column marked E in the following
Ttble, the deflection a given load will produce in any cate 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.
73
elasticity, deduce it immediately from the formula . .^ ^ = 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 formulae 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) ■= n R,
where t is the thickness of metal in inches, c the cohesive power
in pounds of a square inch rod of the given material, n 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.
KEPORT ON THE STRENGTH OF MATERIALS.
103
Table of the Mean Strength and Elasticity of various Materials, as
deduced from the most accurate Experiments.
Names of Materials.
Woods.
Acacia
Ash
Beech
Birch, Common
, American Black
Box
Bullet Tree
Cabacully
Deal, Christiana ,
, Memel ....
Elm
Fir, New England
— .Riga
— , Mar Forest ...,
Green heart ,
Larch, Scotch
Locust Tree
Mahogany
Norway Spars
Oak, English | j^°'"
, African
, Adriatic
, Canadian ...
, Dantzic
Pear Tree
Poon
Pine, Pitch
, Red
Teak ,
Tonquin Bean .
Iron.
from.
Iron, Cast < ,
, Malleable
, Wire
Speci.
fie
Gra-
vity.
710
760
700
700
750
1000
1030
900
680
590
540
550
750
700
1000
540
950
637
580
700
900
980
990
872
760
646
600
660
660
750
1050
7200
c.
Mean
strength of
cohesion on
an inch sec-
tion.
lbs.
17000
11500
20000
11000
11000
5780
12000
12600
12000
7000
20580
8000
12000
9000
15000
14400
14000
12000
14500
9800
14000
10500
10000
15000
7760
16300\
36000 ]
60000
80000
Constants
for trans-
verse
strains.
1800
2026
1560
1900
1500
2650
2500
1550
1730
1030
1100
1130
1140
2700
1120
3400
1470
1200
2260
2000
1380
1760
1450
2200
1630
1340
2460
2700
8100
9000
Constants
for deflec-
tions.
4609000
6580000
5417000
6570000
5700000
10512000
7437000
6350000
6420000
2803000
5967000
5314000
3400000
10620000
4200000
767000
5830000
3490000
Modulus of
Elasticity.
3739000
4988000
4457000
5406000
3388000
5878000
4759000
5378000
6268000
3007000
6249000
4080000
2797000
6118000
4480000
4649000
5789000
2872000
700000047020000
9500000
3880000
8590000
4760000
6760000
5000000
7360000
9660000
10620000
55830000
2257000
5674000
3607000
6488000
4364000
6423000
7417000
5826000
69120000 5530000
91440000 6770000
Ilemarks,
of English growth.
ditto.
ditto.
ditto.
American.
Berhice.
ditto.
English.
Scotland.
Berbice.
America, South.
Results very va-
riable.
East Indies.
East Indies.
Berbice.
[Mean of English
f and Foreign.
[ 105 ]
Report on the State of our Knowledge respecting the Magnetism
of the Earth. By S. Hunter Christie, Esq., M.A., F.R.S.
M.C.P.S., Corr. Memh. Philom. Sac. Paris, Hon. Memb.
Yorkshire Phil. Soc.; of the Royal Military Academy ; and
Member of Tritiity College, Cambridge.
Had the discovery of the loadstone's du-ective 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 phgenomena, though less striking,
yet equally interesting, and some even more difficult of expla-
nation.
These phasnomena 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
Ijitensity 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 phaenomena;
* 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 1 800, 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 cousulting 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, 11 1° 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 J, 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 11^° or 11^° 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.
t The Netv Attractive, by Robert Norman, chap. ix. London 1596.
X 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.
II 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 f ; " 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 4^° 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 evening f. Wargentin at Stockholm in 1750,
and Canton in London from 1756 to 1759, moi-e particularly
observed this phsenomenon; 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.
f Philosophical Transactions, 1701, vol. xxii. p. 10-36.
+ Ibid. 1724, vol. xxxiii. p. 96.
§ Ibid. 1759, vol. xli. p. JWS.
REPORT ON THE MAGNETISM OF THE EARTH.
109
morning *, yet his observations and remarks are of great value
as pointing out the annual oscillations of the needle f. Since
this, the diui-nal variation has been very generally observed,
but by no one with greater care and perseverance than by the
late Colonel Beaufoy ;}:.
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 Geant, 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 Geant 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, nuich 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, torn. xl. p. 345. f Ibid. p. 348.
Many of the results of Colonel Beaufoy 's observations are published in the
Edinburgh PhilosophicalJoumal, vols. i. ii. iii. iv. and vii.
§ Saussure, Voyayes dans les Alpes, torn. iv. p. 302 au p. 312.
does not give the mean results, I insert them here.
As Saussure
Time of absolute maximum. Time of second maximum.
Geneva
Cliamouni ...
Col du Geant
East.
h m
7 56 A.M.
7 34
8 09
West.
h m
1 09 P.M.
1 41
2 00
East.
6 26 P.M.
7 44
5 51
West.
h m
11 17 P.M
10 46
10 17
Extent of Elevation
diurnal above the
change. sea.
15 42
17 06
15 43
Feet.
1305
3453
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
I'ude, 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 f . 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 X, 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 phsenomena 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
phsenomenon 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.
t Philosophical Transactions, 1673, vol. viii. p. 6066.
X New Attractive, chap. vii. § Gilbert, De Magtiete, ^c, Lond. 1600..
REPORT ON THE MAGNETISM OF THE EARTH. Ill
sidered that the earth acts upon a magnetized bai% 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 f. 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 J. 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 phaenomena with such an hypothesis, but
conjectures that the period of revolution of these poles is about
700 years li.
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, Sfc, lib. i. cap. iv.
t Philosophical Transaction)!, 1668, vol. iii. p. 789.
X Ibid. 1673, vol. viii. p. 6065. § Ibid. 1683, vol. xiii. p. 208.
II /6irf. 1692, vol. xvii. p. 563.
lis 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 phaenomena 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
phaenomena 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 phsenomenon.
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. 1 Ifi
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, Ampere and Arago, immediately led to the considera-
tion, whether all the phaenomena of terrestrial magnetism were
not due to electric currents ; and the discovery of Seebeck, that
electric currents are excited when metals having different
powers 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 Gumming 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 Edinbui-gh,
Dr. Traill considers "that the disturbance of the equihbrium
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 equihbrium 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 f.
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 nf the Philosophieal Society of Cambridge, vol. ii. p. 64.
+ Pliihtxophtenl Transartions, 1823, p. 392.
18,'Jrj, 1
114 ' THIRD REPORT — 1833.
application of heat, the phaenomena corresponding to magnetic
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 difterent 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 f. 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 pliEenomena, 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
• "Theory of the Diurnal Variation of the Magnetic Needle," Philosoj)Mcal
Tramactions, 1827, pp. 321, 326.
I- Ibid. pp. 327, 328.
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 w^hich 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 phaenomena 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 phae-
nomena in general. If the meridians wei'e 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,
i2
11(> THIRD REPORT — 1833.
direction and extent of chains of movmtains, 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
having 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 hnes 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
aflbrding no points 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 eivina^ 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 fovind 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 f.
All the lines to which I have here referred have been liitherto
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, Traite de Physique.
f Hansteen, Edinhtrgh fhilosophical 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 hues 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 phsenomenon of terrestrial magnetism is not, in a great
measvu-e, 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 aie 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 M'hich 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.
Altiiough it Avould therefore appear that the rotation of the
118 THIRD REPORT — lSo3.
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 phaenomena, — 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 phaenomena of terrestrial magnetism to electrical cur-
rents existing in these veins -f ; but although we should not be
jvarranted 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 J.
II. Intensity of the Terrestrial Magnetic Force.
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,
particvilarly 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. f ^l>'^- 1830, .p 407.
X Mr. Henwood informs me that he has repeated the experiments of Mr.
Fox ill from forty to fifty places not before experimented on, and that he pro-
poses greatly extending them. i\s 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.
IIEI'ORT 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 lO** 30" a.m.,
when it reached its minimum, and increasing from that time,
attained its maximum about 1^ 30™ p.m. f . 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
needle ; but in the communication containing these observations,
I pointed out the necessity of such a correction J; and since
then, in deducing the terrestrial intensity from the times of vi-
• Edinburgh Philosophical 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 T)e 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 eifects 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 pi-oper 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 fractiu-e. 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.
X Philosophical l^ransaclions, 1825.
l20 THIRD REPORT— 18S3.
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, fi'om 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 obsei'vations at Spitzber-
genf show, more decidedly, the diurnal variation of this force :
there, its maximum intensity appears to have occun*ed at about
3^ 30"" A.M., and the minimum at 2^ 47™ p.m. ; its greatest change
amountino; to -J^ 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-r
able irregularity in the changes which it undergoes. It would,
however, appear, from these observations, that the variations hi
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 A'-hich I founded the theory of the di-
urnal variation of the needle :j:, 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 Society of Cambridffe, 1820.
t Phitosophical Transactions, 1828. ' J Ibid. 1827, pp. 345, 349.
REPOKT 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 formulse with which they
have been compared.
On the hypothesis of two magnetic poles not far removed
from the centre of the earth, if 8 represent the dip, A the mag-
netic latitude of the place of observation, 1 the intensity of the
force in the direction of the dip, and m a constant, then
iind therefore.
-v/ (4 - 3 sin2 8)'
tan S = 2 tan K ;
I = |V(3sin^A+l);
or if i is the angular distance from the magnetic pole, or the
complement of the latitude,
I=|.^(3cos2i+l).
By comparing his own observations with the first of these
formulae, 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 phaenomena 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
formvda and at variance with the other, when these formulae 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 RKPOKT — lS3o.
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 formulas, 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 cavise 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 f, to which
I have already referred, I have suggested an experiment which
I think might throw much light on this subject. I have pro-
posed that a large copper sphere, of uniform thickness, should
be filled with bismuth, the two metals being in perfect contact
throvighout, 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 wei'e illustrative of the principal phaenome-
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 1 827.
f Philosophical Transaclions, 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 phaenomena 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 aerostatic 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 Geant, 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 f.
M. Kupffer X 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, Traite de Physique.
f Voyages dans les Alpes, torn. 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. 1 transcribe the passage from the only edition I can consult, pub-
lished at Neufchatel, 1796. " A' Geneve ces vingt oscillations employ erent
5m 2'; 4™ 50'; 5"*; 4™ 40'; dont la moyenne etoit 5™ 0*"4; le thermometre
6tant a 6 degres. A* Chamouni S"" 33' ; 5" 34' ; moyenne 5" 33''5 ; thermometre
12 deg. Au Col du Geant 5™ 30"-3 ; S-" 30'-5 ; 5" 3r-4; 5^ 34'-6, moyenne
5m 32"-45; thermometre 12-4 degres."
" Or les forces magnetiques sont inversement comme les quarres des tems.
Mais, a Geneve, le tems etoit ■')'" 0'-4 ou 300'.4, dont le quarre = 111155-56 ;
a Chamouni 5" 33'-5 = 333'-5, dont le quarre = 111223. Au Geant 5"
32'-45 = 332'-45, dont le quarre = 11523-0025 ; d'ou il suivroit que la plus
grande force etoit dans laplaine, et la plus petite sur la plus haute montagne,
a pen pres d'une cinquieme : observation bien importante, si elle etoit confirmee
par des experiences repetees, et faites a la meme temperature."
The means of the above observations are 4™ 53* = 293', 5'" 33'-5 = 333'-5,
andS"! 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.
t Voyage dans les Environs du Mont Elbronz. Rapport fait a I'Acadimie
Jmpiriale des Sciences de St. Petersbourg, p. 88.
124 THIRD REPORT— 1838.
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, 370 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 tei'restrial 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, I have,
however, shown that Captain Foster's Port Bowen observations
<lo not warrant the conclusions which have been drawn from
REPORT ON THE MAGNETISM OF THE EARTH. 125
them, and have pointed out cu'cumstances which may, in this
case, 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 phgenomena be-
came unquestionable*. As, however, the magnetic influence
of the aurora boreahs 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 vai'y at every instant, — 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 hei'e 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 ohsei vations to which I here particularly refer, see the Journal of
the Royal Inslitufiu/i, vol. ii. p. 272.
126 THIRD REPORT 1833.
cately 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 i?i
equiUbrio 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 dui'ing 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 phaenomena
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 phae-
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 Kverard
Home, however, indicate the same kind of influence. In a 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. I'he
arc 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 thundei'-storms are so frequent, and of
svich 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 phasnomena 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 curx*ents 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 Transaction!!, 1823, p. 364. The arrangements which I have
just described for determining the influence of the aurora borealis arc 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 phaenomena,
must be rejected ; and if we are to refer these phasnomena 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
fects, and are not likely to know more.
If difficulties meet us at every step when we attempt to ex-
plain the general phaenomena 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 phaenomena, and of their causes ; and, consequently, im-
provements in the methods of observation, and in the instrvi-
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 terresti'ial 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 Hanstepn 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.
REPORT ON THE MAGNETISM OP 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 v/hich 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
methods proposed for determining the times of vibration appear
1833. 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 dihgent ob-
servation of all the phaenomena of terrestrial magnetism, as the
surest means of arriving at a knowledge of their causes : it is
with i-eluctance 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 in a national observatory. Such an omission
is the more to be regretted, seeing that no one has, I believe,
cai'ried 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 phaenomenon 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.
[ liJl ]
Report on the present State of the Analytical Theory of Hydro-
statics and Hydrodynamics. By the Rev. J. Challis, late
Fellow of Trinity College Cambridge.
The problems relating to fluids, which have engaged the atten-
tion of mathematicians, may be classed under two heads, — those
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 iabsolute incompressibilityj as the basis of the
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 circum-
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 ai'e,
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 forses, and the intei-ior 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 natui'e 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 Math^ma-
tique 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. I am 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
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 hmitation 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 capillai'y attraction, which the author svibsequently 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 eiFect
of heat, has proved that the explanation of the phasnomenon 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
• Memoircs de I' Academic des Sciences, Paris, torn. ix. 1830.
134 THIRD REPORT — 1833.
a proof of the law 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 f. 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 phasnomena 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 pei'ma-
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 Hydrodynaynics
by Mr. Moseley %, who has derived them from a principle of so
simple a nature, that, as it can be stated in a 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 lie
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 efl^ect of heat be taken into accoimt, if only the
condition of steadiness remains. The equations of motion are
readily derived from it, because it enables us to consider the
.* Cambridge 183 J.
t 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.
t Cambridge 1830.
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 formulae 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 f . 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 Catnbridge,
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 phsenomenon 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 X 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 III. f Memoires de I'/lcadimie de Berlin, 1755, p. 344.
X Tom. ix. 1830.
136 THIRD REPORT— 1883.
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 f. 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 vis 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,
* Mecanique Analytique, P&tt II. § xi. art. 34.
t 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 the application of ana-
lysis to physical questions. Had a diff'erent 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,
estabUsh 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
138 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, Avhich is published in
Vol. iii. Part I. of their Transactions. I am 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 ajjplications of arbitrary ftinctions.
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 abstx*act 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 ?
Euler and Lagrange determined the velocity of propagation
in having regard to the 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
density 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 f. 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, z, and t, 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 aerial 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 J, 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 disturbance
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
* Memoires de V Academic, An 1779. ,
t " Memoire sur la Theorie du Son," Journal de I'Ecole Polytechnique,
torn. vii. cah. xiv.
J Transactions of the Philosophical Society of Cambridge, vol. iii. Part III.
§ In M. Poisson's writings this quantity is udx + v dy + ir dz.
140 THIRD REPORT 1833.
an exact differential of a function of three variables, i-enders 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 axisj-.
But no general method exists of distinguishing the ipstances 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 Paris \ 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.
HI. 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 open end. 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
* Mecanique Analytique, Part II. § xi. art. 16.
t Memoires de I' Academic de Berlin, 1 755, p. 292.
% torn. X. 1831.
REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 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 a loop. 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 Chromatics of the Supplement to the Encyclopcedia
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 f," 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,
* Memoires de V Acad^mie des Sciences, Paris, An 1817.
t Ibid. torn. X. p. 317.
142 THIUD REPORT — 1833.
by combining analysis witb a delicate set of experiments, re-
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 difterential equations of fluid motion assume a very sim-
ple form for the case of oscillations of small velocity and extent,
and seem to oiFer 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 integi'al, 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, i. 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 Mecanique Ana'
li/tiquef the same result is obtained analytically. The princi-
pal 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 formulas required for the complete solution
of the problem, and the theory, derived from these formulae, of
waves propagated with a nniformly 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. ait. 3G.
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 Mdmoires 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 theoryf , M. Poisson answers J 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 formulae, 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 wa,ves produced by the immer-
♦ vol. lii.
f Bulletin de la Societe Philomatiqiie, Septembre 1818, p. 129.
X "Note sur le Probleme des Ondes," torn. viii. of Mimoires de I'Acadhnie
des Sciences, p. 371.
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 a priori
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. Poissoii's works this equation is '-1? -f — = 0,
1833. T.
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 pai--
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 (ip), 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 formulas, and discovering from them all the laws of the
phasnomenon. 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 Societe Philoi7ia-
tique*. The note before spoken of in the eighth volume of the
* An 1817, p. ISO.
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 1 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 VEcole Po-
lytechniqiie. 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 AVaves " 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 " les ondes denteUes." 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
I'apid 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), (3),
(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 aiif Experimente gegrim-
det*. The experiments of M. Weber were made in a manner
not sufficiently agreeing with the conditions supposed in the
theoi'y to be a correct test of it. They, however, manifest a
general accordance with it, and confii'm the existence of the
small accelerated waves near the place of distui'bance, 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, 182,5.
REPOUT ON HYDROSTATICS AND HYDRODYNAMICS. 149
The theory has been also put to the test of expermient 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 a 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 iheiv 3femoires.
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 1^. 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
watei', 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 being
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 f.
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-
* PhHosophlcal Transactions for 1832, p. 399.
t 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
the London and Edinburgh Philosophical Magazine and Journal for Septem-
ber 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.
I
[ 153 ]
Report on the Progress and Present Stale of our Knowledge
of Hydraulics as a Branch of Engifieering. By George
Rennie, Esq., F.R.S., §-c. ^-c.
Part 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 Hydrauhcs 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 — 1833.
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 cohunn 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 particles 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 OP 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 vis 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 j)er quce derivantur Fliiviornm Aquce, 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 2 A H X fTTTjT^j 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 ^rd of its
height. These experiments would have been quite contrary
to expectation, had not Sir Isaac Newton observed that water
issuing from an orifice f ths of an inch in diameter, was contracted
Ijths 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 Q9,^ ; and aug-
menting it in this proportion, the opening should have been
~ A H jTqqq, or l^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, Hydrody namica , seu de
Viribns 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
formulae 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 formulae, 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 PauJ Lecchi, a celebrated Milanese engineer, entitled
Idrostatica esaminata ne suoi Principi e Stabilite nelle suoi
Regole delta Mensura delta Aeque correnti, containing a com-
plete examination of all the different theories which had been
proposed to explain the phgenomena 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
efilux 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 expei'iments 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 deinve from them any satisfactoi'y conclu-
sions. But Michelotti is justly entitled to the merit of having
made the greatest revolution in the science by experimental
* Sec also Memorie Idrostatico-storiclie, 1773.
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 ItaUans, the science owes the greatest obligations.
Accordingly, the Abb^ Bossut, a most zealous and enhght-
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 Traits Theorique et Experimental (THijdro-
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 equilibrivun 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 reservoh", the same quantity of water is always dis-
charged, whatever be the decUvity 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 Abbe Bossut
had opened out a new career of experiments ; but the most dif-
ficult and important problem remaining to be solved related to
I'ivers. 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
and islands, 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 gi'eat
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' Hydraidique verifies par un grand nombre
d' Experiences, faittis par Ordre du Gouvernement, 2 vols. 1786,
(a third volume, entitled Principes dHydrauUque 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 difFei'ent inclinations or angles, of
from 90° to iqMj*^ P^'^* ^^ ^ 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.
g = 16'087, the velocity in inches which a body
acquires in fidling one second of time.
* Edinburgh Encyclopedia, Art. Hydrodynamics, by Brewster.
ON HYDRAULICS AS A BRANCH OF ENGINEERING. 161
n = an abstract number, which was found by ex-
periment to be equal to 243*7.
then . = _^Il{^^I^:}L_ _ 0-3 (^^ _ o-i).
ys — log. V s + I'o
Such are some of the objects of M. Dubuat's work. But his
hypotheses are unfortunately founded upon assumptions which
render the appUcations 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 engineers of his time : he was director of the Ecole des
Fonts et Chauss^es, 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 single 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 = S^-, where g" is = 16"087 feet, the velocity acquired
z s
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.
s = the slope or declivity of the pipe.
« = 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 " Experiences destinees a deter-
miner la Coherence des Fluides et les Lois de leurs Resistances
dans les Mouvemens tres lents," proves, by reasoning and facts,
1st, 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.
3rdly, 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
phasnomena 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 of the 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, Ingenieur en chef
ON HYDRAULICS AS A BRANCH OF ENGINEERING. 163
des Fonts et Chaussees, and Director of the Works of the
Canal I'Ourcq 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 terms
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*88 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. Giraird'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 stir le Mouvement des Eaux courantes : Paris 1804. Recherches
sitr les Eaux publiqiies, ^-c. Devis general du Canal I'Ourcq, ^-c.
+ Memoires dfs Savans Ef rangers, 8fc. 1815.
M 2
164 THIRD REPORT — 1833.
Thus v= - 0-0469734 + a/ 0-00^2065 + 3041-47 x G,
which gives the velocity in metres : or, in English feet,
?;= - 0-1541131 + \/~0W3751 + 32806-6 x G.
When this formula is apphed to pipes, we must take G = :^DK,
rj I 7 rr
which is deduced from the equation K = j . When
it is applied to canals^, we must take G = R I, which is deduced
from the equation I = :p, 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-
mulae, and from those of Dubuat and Girard. In both cases
the coincidence of the observed results with the formulae are
very i-emarkable, but particularly with the formulae of M. Prony.
But the great work of M. Prony is his Noiwelle Architecture
Hydraulique, 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 phaenomena
of the contraction of the fluid vein, and given an account of the ex-
periments of Bossut, M. Prony deduces formulae 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 vi^ith cer-
tainty the correction to be applied to the theoretical expendi-
tures of fluids through orifices and additional tubes. The phag-
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 formulas 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 formulae for the mean velocities of rivers.
When V = velocity at the surface,
and U = mean velocity,
U = 0-816458 V,
which is about f V.
These velocities are determined by two methods. 1st, 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-
/ metres
mined by the formula V= V2gh=\/ 19-606 h = 4-429 -//*.
When water runs in channels, the inclination usually given
amounts to between ^^^q*^ ^^^ ^o o*^ P^'^' ^^ *^^ length, which
will give a velocity of nearly 1^ 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 xzW*-^
and y^o*^' ^^^ some rivers are said to have ^^%o*^ only.
M. Prony gives the following formulas, 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 + V0'005 + 3233. R71;
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
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
\J = - 0-0248829 + -/ 0-000619159 + 717-857 DZ
L
scarcely differs more or less from experiments than ^^^ or gL.
The preceding formulae 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 formulae given for open and close canals,
M. Prony has remarked that these formulae 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 phaenomena with sufficient exactness.
The following formula applies equally to open or close canals :
U = - 0-0469734 + ^Z (^0-0022065 + 3041-47 ^V
But the most useful of the numerous formulae 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 189-4
Ditto by Eytelwein's formula 189'77
Ditto by Girard's formula 188-26
Ditto by Dubuat's formula 188-13
Ditto by Prony "s simple formula 192-32
Ditto by Prony 's tables 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 1st, 0-000436 U + 0-003034 V^- = glR = gl-;
2ndly, U = ^;
3rdly, R w' - 0-0000444499 .wj- 0-000309314 y = 0.
This last equation, containing the quantities
w
QIw and R = — ,
p
shows how to determine one of them, and, knowing the three
others, we shall have the following equations :
4thly, p = 0.000436 Qw + 0-003034 Q^'
rfui T P (0-0000444499 Qw + 0-000 309314 Q^)
6thly, . = 0-000436 ± ^[(0-000^ + 4 |-003034) g RI] Q^
These formulae 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 formulas 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 W) = gl w.
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 lat^-
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-
rior of a whirlpool ; and had affirmed that when motion is pro-
pagated in a fluid, and has passed beyond the aperture, the
• Recherchcs Physico-mathematiques sur la Thcorie des Eaux courantes,
par M. Prony.
168 THIRD REPORT — 1833.
motion diverges from that opening, as from a centre, and is
propagated in right Unes 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 ai'ise 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 :
1st, 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.
Sndly, 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.
3rdly, That the expenditure may be still fiirther increased,
* " Calix devaxus amplius rapit." — Frontinus 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 pubHc 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 Ifaliana, vol. iv.
170 THIRD REPORT — 183.3.
In the year 1801, M. Eytelwein, a gentleman well known to
the pubUc by his ti'anslation 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 Hydraidik, 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.
3. That the contraction of the fluid vein from a simple orifice
in a thin plate is reduced to 0*64.
S. For additional pipes the coefficient is 0*65.
4. For a conical tube similar to the curve of contraction 0*98.
5. For the whole velocity due to the height, the coefficient
by its square must be multiplied by 8*0458.
6. For an orifice the coefficient must be multiplied by 7*8.
7. For wide openings in bridges, sluices, &c., by 6'9.
8. For short pipes 6"6.
9. 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 soUd bodies, no more force 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 f; for Professor
. Robison found that the time of oscillation of the fluid in a bent
* See Nicholson's translation of Eytelwein's work.
t See my "Experiments on the Friction and Resistance of Fluids," Philo-
sopkical 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 -j-^tlis 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 „„ „ ,
or 29*13 inches ; and the fall in two miles being eight inches,
we have v'(8 x 29*13) = 15*26 for the mean proportional of
■j-^ths, or 13*9 inches, which agrees very nearly with Mr. Watt's
velocity.
The Professor has, however, deduced from Dubuat's elabo-
rate theories 12*568 inches. But this simple theorem applies
only to the straight and equable channels of a river. In a
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 ^-^^ for every foot of the whole
depth. Dr. Young thinks y^g*^^ ^^ *^^ 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 f rds 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 Nicholsons Journal for 1802, vol. iii. p. 31.
172 THIRD REPORT — 1833. '
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 Gennete*, 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 Gennete pretends to
prove by experiments, from observing that the Danube absorbs
the Inn, and the Rhine the Mayne I'ivers ; 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 Generate de Wiebeking, Wattmann's M^moires sur
I'Art de construire les Canaux, and Funk Sur F Architecture
Hydraidique generate, which are sufficient to determine the
coefficients under different circumstances, from velocities of
fths to 7| feet, and of transverse sections from J- 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 formulae for the contraction
of fluid veins through orifices f, 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.
* Experiences sur le Cours des Fleuves, ou Lettre a un Magistral Hollandais,
par M. Gennete. Paris 1760.
+ " Recherches sur le Mouvement de I'Eau, en ayant egard a la Contraction
qui a lieu au Passage par divers Orifices, et a la Resistance qui retard le Mouve-
ment, le long des Parois des Vases ; par M. Eytelwein," — Memoires de I'Aca-
dimie de Berlin, 1814 and 1815.
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 watei'-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 fths 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 if.
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 / denote the height due to the friction,
d = the diameter of the pipe,
a = a constant quantity,
we shall have, / = V^ ^^ and V^ = •^—,.
•' d al
But the height employed in overcoming the friction corresponds to the differ-
ence between the actual velocity and the actual height, that is, /= A 5-,
where b is the coefficient for finding the velocity from the height.
Hence we have, V^ = =^-= and V = V — — -.
ab' I ab^l -\- d
Now Dubuat found h to be 6-6, and aV^ was found to be 0-0211, particularly
when the velocity is between six and twenty-four inches per second. Hence
or more accurately, V =: 50 v ( -, ^, |.
•' \/+ 50 d/
+ 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, Me-
moire sur les Roues Hydrauliques, and Aubes Courbes par dessovs, ^fc. 1 827.
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 Institutioti, 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 oncesf, furnished fth 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-
gland 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 phaenomena
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 phaenomena, as we before stated,
had been long noticed by Michelotti and others, but nothing
precise had been established on the forms and remarkable phag-
nomena of the fluid vein itself. Venturi had given three ex-
amples.
M. Hachette, in two memoirs presented to the Academic
Royale des Sciences in 1815 and 1816, also considered the
* Notices Historiques, par M. Mallet. Paris 1830.
t French measure, or 0-03059 French kilolitres.
ON HYDRAULICS AS A BRANCH OF ENGINEERING. 175
form of veins ; and in his Traits des Machines, he states that
he had ah'eady 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 "Experiences sur la Forme et sur la Direction
des Veines et des Courans d'Eau, lances par diverses Ouvertures,"
was read to the Academy of Sciences at Turin by M. Bidone,
giving 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 vai-iable 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 i?i equilibrio, and may be assimilated to
the phsenomena 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
state of permanence, and expended from a very small orifice,
as compared with the sections of the containing vessel, accord-
ing to the theory of Euler; and finds that the magnitude of the
section of the contracted vein does not depend upon the velocity
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 f rds 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
Bossut 0-660
Borda 0-646
Venturi 0-640
Eytelwein 0-640
Hachette 0-690
Newton 0-707
Helsham 0-705
Brindley and Smeaton 0-631
Banks 0-750
Rennief 0-621
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 Mihtary School at Metz, for permission
to undertake a series of experiments on a scale of magnitude
sufficient to estabhsh the principles laid down by those authors,
and serve as valuable practical rules for future calculations.
The apparatus consisted, 1st, of an immense basin, having
an area of 25,000 square metres ; 2nd, of a smaller reservoir,
having a superficial area of 1 500 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'Idraulka: Milano 1818. Recherche Geome-
triche fatte nella Scuola degli Ingegneri pontifici d'Acque e Strode, Fanno
1821. Milano.
t "On the Friction and Resistance of Fluids," Philosophical Transactions of
1831.
ON HYDRAULICS AS A BRANCH OF ENGINEERING. 177
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 m
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 g^odth 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,
1st, 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 1"6901 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 = lo\/¥gli = l{Ji-¥) ^/2g ^^—^ being the theo-
retical expense relative to the velocity ;
or the theoretical expense, having regard to the influence of
the orifice.
• That is, where/ = 020 metre, being thehorizontal breadth of all the orifices;
h = the charge of the fluid on the lower part of the orifice ;
A'= the charge in the upper or variable side of the orifice ;
oz= k — h' the thickness of the vein of water.
18.^3. N
178 THIRD REPORT 1833.
-' The conclusions to be derived from these Tables are,
1st, That for complete orifices of twenty centimetres square
and high charges, the coefficient is 0*000 ; 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 0'61], 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. Lesbroa and Poncelet to
diflferences 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 I h V 2gh, 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 avithors 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.
The authors conclude their memoir by recommending their
experiments for adoption in all cases of plate orifices situated
* 4nnales de Chimie et de Physique for 1830, torn. xliv. p. 225.
ON HYDRAULICS AS A BRANCH OF ENGINEERING. 17i)
at a distance from the sides and bottom of the reservoir, pro-
mising to investigate with similar accuracy in a future memoir
the cases which may occur to the contrary.
A note is appended to the memoir by M. Lesbros, contain-
ing formulae for calculating the effective expenditure of com-
plete orifices ; and also a Table of constants, which gives the
effective expenditure of each orifice as compared with experi-
ment. We have been thus particular in detaihng 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 i-ephed 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 18SI,
N 2
180 THIRD RKPORT — 1833.
On the suliject of lij'drometry ^ve 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, 1st, 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 discs 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 MS
Square plates . . 22'010 in air, . . 1 '36 in water.
Round globes . . 10-627 0-75
3rdly, That with circular orifices made in brass plates of
gJg-th of an inch in thickness, and having apertures of i, ^, f , y
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
4 feet 0-621
For additional tubes of glass the coefficient was,.
for 1 foot 0-817
4 feet 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 expenditui-e 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,
1 : 2-6 8 feet ,
1 : 2-0 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 41 inches, the length 14,930
feet, and the altitude 51 feet, it was proved by Mr. Jardine
that the formulae 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 formulae 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
formulee 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 phaenomena 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 — lS'6o.
Appendix.
Since the foregoing Report was read to the British Associa-
tion a paper, entitled " Memoire sur la Constitution des Veines
Liquides lancees par des Orifices Circulaires en mince paroi,"
has been communicated to the Academy of Sciences at Paris,
by M. FeUx Savart, 26 Aout 1833. The author, after detailing
very minutely the different phsenomena 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 lancee verticalement de haut en bas
par un orifice circulaire pratique dans une parol plane et hori-
zontale est toujours composee de deux parties bien distinctes
par I'aspect et la constitution. La partie qui touche a I'orifice est
un solide de revolution dont toutes les sections horizontales
vont en decroissant graduellement de diametre. Cette premiere
partie de la veine est calme et transparente, et ressemble a un
tige de cristal. La seconde partie, au contraire, est toujours
agitde, et paralt denuee de transparence, quoiqu'elle soit ce-
pendant d'une forme assez reguliere pour qu'on puisse facile-
ment voir quelle est divisee en un certahi nombre de ren-
flemens allonges dont le diametre maximum est toujours plus
grand que celui de I'orifice.
" 2°. Cette seconde partie de la veine est composee de gouttes
bien distinctes les unes des autres, qui subissent pendant leur
chute, des changemens periodiqvies de forme, auxquels sont dues
les apparences de ventres ou renflemens reguherement espaces
que I'inspection directe fait reconnaitre dans cette partie de la
veine, dont la continuite apparente depend de ce que les gouttes
se succedent a des intervalles de temps qvii sont moindres que
la duree de la sensation produite sur la retine par chaque goutte
en particulier.
" 3°. Les gouttes qui forment la partie trouble de la veine
sont produites par des renflemens annulaires qui prennent
naissance tres pres de I'orifice, et qui se propagent a des inter-
valles de temps ^gaux, le long de la partie limpide de la veine,
en augmentant de volume a mesure qu'ils descendent, et qui
enfin se separent de I'extremite inferieure de la partie limpide
et continue a des intervalles de temps egaux a ceux de leur
production et de leur propagation.
ON HYDRAULICS AS A BRANCH OF ENGINEERING, 183
"4°. Ces renflemens annulaires sent engendr^s par une suc-
cession periodique de pulsations qui ont lieu a rorifice meme ;
de sorte que la vitesse de lecouleuient, au lieu d'etre uniforme,
est periodiquement variable.
"5". Le nombre de ces pulsations, m^me pour des charges
foibles, est toujours assez grand, dans un temps donne, pour
qu'elles soient de I'ordre de celles qui, par la frequence de leur
retour, peuvent donner lieu a des sons perceptibles et compa-
rables. Ce nombre ne depend que de la vitesse de I'ecoule-
ment, a laquelle il est directement proportionnel, et du diametre
des orifices, auquel il est inversement proportionnel. II ne pa-
rait alter e ni par la nature du liquide, ni par la temperature.
"6°. L'amplitude de ces pulsations pent etre considerable-
ment augmentee par des vibrations de m^me periode commu-
niquees a la masse entiere du liquide et aux parois du reservoir
qui le contient. Sous cette influence etrangere, les dimensions
et I'etat de la veine peuvent subir des changemens remarqua-
bles : la longueur de la partie limpide et continue pent se
reduire presqu'a rien, tandis que les ventres de la partie trouble
acquierent vine regularite de forme et une transparence qu'ils
ne possedent pas ordinairement. Lorsque le nombre des vibra-
tions communiquees est diiferent de celui des pulsations qui
ont lieu a I'orifice, leur influence peut meme aller jusqu'a
changer le nombre de ces pulsations, mais seulement entre de
certaines limites.
" 7°. La depense ne paralt pas alteree par l'amplitude des
pulsations, ni meme par leur nombre.
" 8°. La resistance de I'air 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 lancees horizontalement ou
m&me obliquement de bas en haut ne difFere pas essentiellement
de celle des veines lancees verticalement de haut en bas ; seule-
ment le nombre des pulsations a I'orifice parait devenir d'autant
moindre que le jet approche plus d'etre lance verticalement de
bas en haut.
" 10°. Quelle que soit la direction de la veine, son diametre
decroit toujours tres rapidement jusqu'a une petite distance de
I'orifice ; mais quand la veine tombe verticalement, le decroisse-
ment continue jusqu'a ce que la partie limpide se perde dans
la partie trouble : il en est encore de m^me quand la veine est
lancee horizontalement, quoiqu'alors le decroissement suive une
loi moins rapide. Lorsque le jet est lance obliquement de
bas en haut, et qu'il forme avec I'horizon un angle de 25° a 45°,
toutes les sections normales a la courbe qu'il decrit deviennent
184 THIRD REPORT — 1833.
sensiblement %ales entre elles, a partir de la partie contractee
que touche a rorifice. Enfin, pour des angles plus grands que
45°, le diametre de la veine va en augmentant depuis la partie
contractee jusqu'a la naissance de la portion trouble ; de sorte
que c'est seulement alors qu'il existe une section qu'on peut
k juste titre appeler section contractee."
185
Report on the Recent Progress and Present State of certain
Branches of Analysis. By George 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 College, Cambridge.
The 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 tmdertaking 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 accurat^^ and logical examination of
the first principles of algebra : in the first place, the proper
186 THIRD REPORT — 18o3.
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 tlie 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 sciences 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 distinc-
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 superstruc-
ture which is raised upon them, form only one or more links in
the great chain of propositions, the tennination 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 Avhich those sci-
ences 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 otlier 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, universal 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 imjjossible 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-
lowing 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 sijmboUcal 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, M^hen 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 connexion 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 per\ising 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 pi-inciples 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, wiiich 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 tvyro 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 requii-es 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
titles 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 indetermination 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-
gnious 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 liis 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 thos6
terms indolence acquiesces ; much time is wasted on a jargon whicli 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 w 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 an}' change
had taken place in the science of algebra between the age of Ferrari, Cardan,
Des Cartes, and Harriot, and the end of the 1 8th century ; and by considering
all algebraical formulae as essentially arithmetical, he is speedily overwhelmed
by the same multiplicity of cases (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 X —x" = 2400 ;
if we subtract both sides of this equation from 2500, we get
2500 — 100 a; + x- = 100,
or X- — 100 X -\- 2500 = 100,
inastnuch 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 — a; = 10,
and in the second,
a? — 50 = 10.
The first of these simple equations gives us « = 40, and the second x = 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 4 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 lohole 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 I'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 CJERTAIN BRANCHES OF ANALYSIS. 198
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 repvesente, says he, les grandeurs qui doivent servir d'ac-
croissements, par des nomhres precedes du signs +. <?' l^s grandeurs qui doivent
servir de diminutions par des nomhres precedes du signe — . Cela pose, les signes
-\- et — places devant les nombres peuvent itre compares, suivani la remarque
qui en a He faite^, a. des adjectifs place's aupres de leurs suhstantifs. 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. He sub-
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 6,
he savs that
4-«=-|-A + h=z — K
— a = — A — b = + A;
and therefore,
-f(+A) = + A +(_A) = -A
- (-h A) = - A - (- A) = -h A ;
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 h be whole numbers, it may be proved that a 6 is identical with
h a : therefore, a b is identical with b 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
less inconsistently with the professed grounds upon which the science is
founded.
' By Buee in the Philosophical Transactions, 1806.
1833. o
194 THIRD KEPOUT — 1833.
that the operations of addition, subtraction, multipUcation 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 b, to its meaning when a is less than b,
or from that of the product a b, when a and b are abstract num-
bers, to its meaning when a and b 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 sypibols and
their combinations, constructed upon its own rules, which may
* The operations in arrithmetical 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 generall_y,
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 intei'preta-
tion : by this means, interpretation -w'tM follow, and woiprecede,
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 supj^osition or assumption that the symbols
in symbolical algebra are perfectly general and unlimited both
in value and rejjresentation, 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 + c from a is denoted
hy a — b — c, or that a — {b + c) = a — b — c, its application
being limited by the necessity of supposing that 6 + e 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 — {b + c) = a — b — c for all values
of the symbols, and therefore, also, when 6 = a, in which case
we have
a — {a + c) = a — a — c=:— c.
In a similar manner, also, we find
a — {a — c) = a — a + c=^ -|-e = cf.
We are thus necessarily led to the assumption of the exist-
ence of such quantities as — c and + c, 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.
t For it appears from arithmetical algebra that a — a^O, and that a — a
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 + c is identical with c, but that — c is a
quantity of a different nature from c : 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 satisfyj-.
In a similar manner, the result of the operation, or rather
the operation itself, of extracting the square root of such a
quantity as « — 6 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 -v/ — I ; so that if we should suppose b =■ a -\- c, we should
get V {a — b) = '/{« — (« + (?)} = V [a — a—c) = V [ — c)
= V' — \ cX- In a similar manner, in order to make the ope-
ration universally applicable, when the ?«"' root of a — 6 is
required, we assume the existence of a sign v^ — 1, for which,
as will afterwards appear, equivalent symbolical forms can al-
ways be found, involving a/ — 1 and numerical quantities.
By assuming, therefore, the independent existence of the
signs +, — , X/\, and v^ — 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 ?iot 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 + c.
\ Amongst these conditions, the principal is, that if — c be subjected to
the operation denoted by the sign — , it will become identical with + c: thus,
a — ( — c) ■:= a-\- c. It does not follow, however, that the sign — thus used,
must necessarily admit of interpretation.
X 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 V — 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.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 197
In the concurrence of the signs + and — , in whatever man-
ner used, if two hke 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 h-, or more frequently by a conven-
tional position of the quantities or symbols with respect to
each other : thus, the product of a and b is denoted by a x b,
a . b, or a b ; the quotient of a divided by b is denoted by
« -r- A, or by -V.
•' b
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 \mth the corresponding restdts in arithmetical alge-
bra, lohen 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 of a 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-
cal algebra conformably to the svmbolical conditions to which they are sub-
ject.
198 THIRD REPORT — 1833.
have been suggested only by the corresponding rules in arith-
metical algebra. They cannot be said to he founded xx^on 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 natiu-e of
the operation for all 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 contifiue to be equivalent,
whatever those sijmbols 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 specif c 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 + ;r)" has a
necessary existence whenever the nature of the operation upon
\ + X which it indicates can be defined ; that is, when « is a
whole or a fractional, a positive or negative, number f ; but for
all other values of n, 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, symbolically 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.
+ The meaning of (1 + «)" 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 equivalent 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-
sti-uction 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 sufficient, for deducing equivalent forms,
without any reference to any subordinate science, we shovdd 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 foi-m, 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 arhitrary as-
sumptions, in as much as they are arhitrar'ily 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
numbers. If the conclusions thus obtained do not involve in
their expression any conditiofi which is essentially connected
v-it/i 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
+ . — I (1)" and (—1)" and their equivalents (for the symbolical form of such
signs is not arbitrary), comprehend all those signs of affection which aie re-
quired by those operations with which we are at present acquainted.
+ Some formula; are essentially arithmetical : of this kind is 1 . 2 . 3 . . . r,
in which r must be a whole number. The formula "K"' — ^) • • • (w — r+ 1)
1.2 . . . r
IS symbolical with respect to m, but arithmetical with respect to r. Such cases,
and their extension to general values of/-, will be more particularly considered
herealter.
202 THIRD REPORT 1833.
expressions as m a 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
= (/« + n) a, and that ma — n a ^= {m — n) a : the same con-
clusions are true likewise for all values of ?n, n and a. In
arithmetical algebra we assume a^, aP, «"*, &c., to represent a a,
a a a, aaaa, &c., and we readily arrive at the conclusion that
«"* X «" = «'"■'" ", when m and n are whole numbers : the same
conclusion must be true also when m and n are any quantities
whatsoever. In a similar manner we pass from the result
{cf) " = «"", when « 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.
Thus, Tr~''"0~('o+'2") a, = a, and therefore -^ must
mean one half of a, whatever a may be ; a- x a = a^
= a = a, and therefore a must mean the square root of a,
whatever a may be, whenever such an operation admits of
interpretation. In a similar manner -^ must mean one third
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 general theorems ma -\- na = (m -\- n) a and ma — na= {m — n) a,
m —VI
ar'Xa!' = 0'" + " and \ = a*"" ", (a"')" = o™" and (o*") " =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 ^/a, and a^ to denote ^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 + or
X + «*, and — «^ X — « , will equally have for their result
+ a or a, by the general rule for the concurrence of similar
signs and the general principle of indices : in a similar manner
d^ may be affected with the multiple sign of affection (1)^, if
there are any symbolical values of (1)^ different from + 1 (equi-
valent to the sign +), which will satisfy the requisite symbo-
lical conditions f . 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 resvdts 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 — I 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, Definitions.
f That is, it' there is any symbolical expression different from 1, such as
^ '-, and ^ , the cubes of which are identical with 1.
In 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 — 1833.
general rule for the concurrence of signs. In a similar manner
we may consider (1)* (a)* as equivalent to (r/)* ; (1)^ (o)^ as
equivalent to («)^; (1)" a" as equivalent to «" ; (—1)" («)" 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 n 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 (« + l)th
must become equal to zero. Thus, the series
(1 + £)• = (1)" (l + » 0,- + -f-^' ^ +
+ "i'ra'\"/".T" -'' + ^°-)
indefinitely continued, in which n 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
n is general in value as well as in formf .
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 coefiicients alone undetermined. It is evident, however,
that if there existed a general form of this series, its form could
* 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.
f If such a series should, for any assigned value of n, have more symbolical
values than one, one of them will be the arithmetical value, inasmuch as one
symbolical value of l" is always 1.
X In the Nov. Comm. Petropol. for 1771.
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 ini-
poses upon the binomial, might be very easily shown; for if
m and n be whole numbers, then if the two series
{\ \ xf -V^\\ ^■ mx ■\- -\ ^ x^ -j- &c. J
(1 + a-)« = 1« (l + n X + '--^^^-^^^2 +- &c.)
be multiphed together, according to the rule for that purpose,
we must obtain
(1 4- xy^''^ l™-*-" ( 1 + (m+ n)x + ^ \2 ^^7
a series in which m + n has replaced m or n in its component
factors : and in as much as we must obtain the same symbolical
result of tliis multiplication, whatever be the specific values of
m and n, it follows, that if the same form of these series repre-
sents the development of (1 -|- or)" and (1 + x^, whatever m and
n may be, then, likewise, the series for the product of (1 + a:)"'
and (I + xY, or (1 + x)'"+", would be that which arose from
putting wi + w in the place of m or n in each of the component
factors. If, therefore, we assumed S {uri) and S (ra) to represent
the series for (1 + xj" and (1 + x^, when m and n are any
quantities whatsoever, then (1 + a-)" x (1 + .r)« = (1 + ^)'« + "
= S (w -f- w) = S (;h) X S (w) ; 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*""*"", for all values of ra and n,
which determines the interpretation of «"' 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 + xj", for all values of w,
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
(1-1- J")" 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 + x, and we can also
determine the corresponding series by the processes of arith-
metical algebra ; and we likewise interpret (1 + xf and (1 + xf
to mean the square and the cube root of 1 + x, in confoi-mity
with the general principle of indices. The coincidence of the
series for (1 + xf and (1 + xf, whether produced by the
processes of arithmetical algebra, or deduced by the principle
of the permanence of equivalent forms from the series for
(1 + x)", 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 universally 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 law 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 x = COS {2r i: -\- x),
and — COS x = cos { (2 r -|- 1) tt 4 x}
propositions which are only true when r is a whole number,
the limitation is conveyed (though imperfectly) by the con-
ventional use of 2 r 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 9 + \/^^l sin fl)"
= cos (2 ?• w TT + ?i fl) + -v/ — 1 sin {2 r mr + n 6),
we may suppose w to be any quantity whatsoever *, but r 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 :
thvis, the formula 1 x 2 x 3 r, commencing from 1 , is
essentially arithmetical, and limited by its form ro 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) ('' — 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 — I) ... {n — r + I),
r72 77. r
which is so extensively used in analysis, is unlimited with re-
spect to the symbol w, and essentially limited with respect to
the symbol r : it is under such circumstances that it presents
itself in the development of (1 + a;)".
In the differential calculus we readily find
'^ = n{n-l). ..{n-r + l)x"-',
and in a similar manner also
•^(.r« + C, x'^-' + C, x'-^ + . . C„) = w (w-1) . . (H-r+ 1>«-'-:
dx^ 1
in both these cases the value of n 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 -|- x)
when n 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 n may introduce : for
(cos 6 + V"^ sin tf)" = (1)" (cos n6+ V^-i sin n 6)
= (cos 2 /• M TT + -\/— 1 sin 2rn-!r) (cos n6 + -v/ — 1 sin n 6)
= cos {2 rn TT -{■ n d) -\- V' — 1 sin (2 r n rr -\- n 6)
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 n is concerned only. This arithme-
tical coefficient of differentiation (if such a term may be ap-
pHed to it,) will present itself in the expression of the rth dif-
ferential coefficient (r being a whole positive number,) of all al-
gebi-aical functions *; and it is for this reason that we aveojjpa-
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 formulae.
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
/ d X \ log — )
(adopting Fourier's notation,) is equal to 1 . 2 . . . . 7z, when n is
a whole number ; and that consequently^ under the same cir-
cumstances, the coefficient of differentiation or
/ d X \ log — )
n (rt — 1) . . . (w — r + 1) =
^'^.(logiy '
and in as much as \hQ form of this equivalent expression is not
restricted to integral and positive values of r, we may assume
• IT, -f 1 ^ f^' « (-l)'-.2.3...(r+l)a;' -
* Inus if M = -; — ; — r, we have -. — 5 =: 7^ — \ — srrxi
X (1 - ^ ('^ - ^) , r(r-l)(r-2) (r - 3) _ ^^ ] .
\ 2.3a;2+ 2.3.4.5 a;'' '/*
I , d'u 1.2 r
and if M = -TT. — I — 5;, we have
V (1 + a;7 dxr — (1 — a"- ) »•+ 4
X ( 1+ JL . '• (>• - ^) , 1 ^(,.-l)(r-2) (r-3) , & 1
12= x" "^ 2= .4= a-4 J
+ This definite integral, the second of that class of transcendents to which
Legendre has given the name of Intecjrales Euleriennes, was first considered
by Euler in the fifth volume of the Commentarii PetropoUtani, in a memoir
on the interpolation of the terms of the series
1 + 1x2 + 1x2x3 + 1x2X3X4 + &c.,
which is full of remarkable views upon the generalization of formula; and
their interpretation. The same memoir contains the first solution of a pro-
blem involving fractional indices of differentiation,
1833. 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
resti'icted 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
Ix2x3x X,
I '^ 2 '^ 3 '^ x'
111 1
1 H i. —
1" 2" ' -^"
1 4- " ~ ^ I ^ ~ ^^ a — (x — 1) b
'^ a + b '^ a + 3b '^ a + {x + l)b'
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.
* Instiiutiones 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. 21 1
It is easy to construct formulae 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
« + <S sin (2 r TT + ^) + y sin (g rir + flp + &c. ,.
a + /3sin9 + ysinfl' +,&c. ^ fv)>
which is, or is not, identical with 4) (r), according as r is a whole
or a fractional number : such functions are termed nticlulating
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
X '^ ^ O^s D'
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,
we shall readily derive the fundamental equation
r(l +r) = rr(r) t (1)
which is in a form which admits of all values of r. It appears
* Traits des Fonctions Elliptiques, torn. xi. p. 476.
t In as much as
t^"-" ,„ _ r (1 + n)
-x^ =
X'' = A a;^
d j;"-' r (1 + r)
and
dn-r+l r (1 + W)
(lx«-r + 1 * — r (r) '^ — D X —TAX ,
it follows that r A = B, and therefore also that
"which is the equation (l) : and it is obvious that the transition from
dn-r d»-r+ 1
.x" to
(which is equivalent to the simple differentiation of A .V, when A is a
constant coefficient), will lead to the same relation between T (1 + r) and
r(r), ivhateiJe)-he the value of »•, whether positive or negative, whole or frac-
tional. Legendre has apparently limited this equation to positive values of r.
p2
212 THIRD REPORT — 183J}.
also from this equation that if the values of the transcendent
r (r) can be determined for all values of r which are included
(Fonctions Elliptiques, torn. ii. p. 415,) a restriction which is obvious!)' unne-
cessary.
There are two cases in which the coefficient of .x"-'' in the equation
d'- x" r (1 + n)
dxr ~ r(l + n — 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 F (1 -|- w) will be infinite when n is any negative whole
number ; the denominator T (l +n — r) will become infinite when n — 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
r(l -l-n) ^ . .
is negative, then the coefficient fT/T^; — ;— Tn becomes finite, in as much as
r (— i) (if t be a whole number) = j— ^ 1( — IV ' ^^^ ^ ^^^ disap-
d~r 1 . „ .
pears, therefore, by division : thus all the coefficients of , _^ • — are infinite,
unless r be a negative whole number, such as — m, in which case it becomes
1 . 2 . . TO . ( — l)*", a result which is easily verified. In a similcir manner it
d'' 1
would appear that the coefficients of , _y • — are infinite, when w is a posi-
tive number, unless r be a negative whole number equal to, or greater than, n.
The coefficient „ , , — — ^ will become equal to zero, when I + n — r
is, and when 1 -|- n is not, equal to zero or to any negative whole number ; for,
under such circumstances, the denominator is infinite and the numerator is
finite.
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 coefficients of constant quantities
are equal to zero, whilst the differentials or differential coefficients to general
indices (positive whole numbers being excepted,) are variable.
Thus
da d . a x' a T (l) . , a d~^a
dx^ dx^ I'Ci) " V-^x' ^^-i
a (r 1) , , 2 a/j: d-ifl r(l)
= "rnr • "- * = ^v ■■ d^i = F(2j • «*"■"' = «^'
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
C „ dro C r (1 - «) ,._„_^ .
a = = ^prTTvi x-", we get
r (0) ' ^ - dx^ — r (0) T(l — n — r)
if n be a positive whole number, F (1 — ?^) = oo , and this expression is finite un-
less r (1 — n — »■) = oo, in which case it is zero : if r be also a positive whole
= Ca; + C
REPORT ON CERTAIN BRANCHES OF ANALYSIS. S13
between any two successive whole numbers, they can be der
termined for all other values of r. Euler * first assigned the
number, it is always zero : if r = — 1, it is finite when m = 1 : if r := — 2,
it is finite when >j = 1 or « = 2 : if r = — 3, it is finite when « = 1, or w = 2,
or M = 3 : and generally if r be any negative whole number, there will be
finite values corresponding to every value of n from 1 to — r ; wo thus get
rf-'O _ n
dx-i
dx-^
3 5 = r-S + Ci a; + C2
d x-^ 1 . 2
d-« _ C A-"-i , Ci . ^"-2
d^»- 1.2. .(».-]) + 1.2..(«-2) +U-2.* + U-i.
This is the true theory of the introduction of complementary arbitrary func-
tions in the ordinary processes of integration.
More generally, if r be not a whole number,
d^O _ _C_ Y{\-n ) ^_„_^
dx''- r (0) ■ r(i — M — ?•)
which will be finite when n is a positive whole number and when 1 — m — r is
not a negative whole number : thus if w be any number in the series 1, 2, 3 , . .,
and iir=k, then
d^O
dx-~
C
. X T
r (-i)
or
c
ari
or
r
c
(-4)
X'
,1
and
so on for ever
: consequently.
tFO _
^ + ^ +
C2
■7
-f &c.
in hi)
^?iii
ttiti
2.
d x^ x^ x^ a; -
In a similar manner, we shall find
.fi r\ 3 1 . C2 . C3 ....
? 2 = C a;^ + Ci a^ H r "I T + &c. tw tvjimtum,
a a; -
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 7iot, and whenn — r
is, a negative whole number.
Thus,
d^x „ d3«2_ d^ x^ _ dJ x~^ _^ d~^ ^~ ^-(\.
= 0, — — 0, — 0, u, — _ u .
d x^ dx^
d x^ d x^ dx
_ 5
7
and similarly in other cases.
(4.) The differentials of 00 are not necessarily equal to », but may he finite.
If we represent 00 by C F (0), we shall find
Commetitarii Petrop., vol. v. 1731.
214 THIRD REPORT — 1833. '
value of r ( "H" ) = V'"', by the aid of the very remarkable ex-
pression for TT, which Wallis derived from his theory of inter-
^'(^)^° = Cr(0) . r(l).iZ!::=(_l)r_l.l.2..(r-l).^-r,
d X'' r (1 — r)
whenever / is a positive whole number.
Conversely also,
rf-i . a;-i r (0) T, r (0)
= — -^ . x", where — ■^—' = oo .
dx-^ T{1) r(l)
d-^ . x -i _ r(o)
d a;-2 r (2)
d-^ . a;-i _ r (0)
d a;-3 r (3)
d-T . a-' _ r(0)
x\
d x-r V {r)
the arbitrary complementary functions being omitted.
(5.) The occurrence of infinite values of the coefHcient of differentiation will
generalljf be the indication of some essential change of form in the transition
from the primitive function to its corresponding differential coefficients.
Thus,
<f-i 1 r (0) „ , , ^
. - = — ^-1 . aO = log X + C ;
dx-i X r(l) o -r .
this last result or value of -~— :■ . a:" being obtained by the ordinary process
of integration : and generally,
^'_^^-^ = LM xr-i + c^'-'' + Ci ^'-' + &c
dx-r r (r) re*) r(7-— 1) '
the first term of which is infinite, in all cases in which r is not a negative
whole number, in which case it becomes equal to ( — 1)"'' 1.2... ( — r) x''-^,
the complementary arbitrary functions also disappearing. If we suppose,
however, r to be a positive whole number, and if we replace frjvs • "^ ^Y
its transcendental value already determined, we shall get
- , = Hog « + C !■ + „ + &c.,
dx-r r (r) '■ " ^ -I r (;• — 1)
which may be replaced by
^' =m {'»«'+ (-')'rW.ra-0 + c} +-£l^,',+ . . ..
which is in a form which is true for all values of r whatsoever, and which
coincides, for integral values of r, with the form determined by the ordinary
process of integration.
More generally,
d-r . x-» _ r (1 — r) ^~„+r ; C ■ A"-" , Ci ..r'-"- '
.dx-r r(l—n + r)'^ r(r — n + l)T{r — n) '^ ' ' "
which is finite, whilst r is less than n ; and when r and n ai'e whole numbers,
p ffi f\
becomes = (— 1)'' . — i ^ '' a''-", omitting complementary functions.
1 ()i)
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 215
polations ; and subsequently, by a much more direct process,
which lead to the equation,
r (r) r (1 r) = -r^ — (when r > < 1) :
^ < ^ sin r ir ^
If, under the same circumstances, r be greater than n, the coefficient of dif-
ferentiation becomes infinite, and its value, determined as above, becomes
x'—» ,, , Ci a*"-"-'
= roOVir-n + l) {'°S*' + ^} + r (r - u) + ^'=-
^iWrcT^TTTT) ^^"° " + ^^ '^' "" ^'•-«+i) r («-r) + c}
(r — 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 ivfinite, in consequence of the intro-
duction of log « 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.) Uti = (a X -I- i)«, then
dru r(l-J-w)«'' . .V Ci"--!
mrr = Y\\--\-i^-V) • («* + ^y-" + rT=7) + • • • •
For if V = a X -j- b, then -j— = a and -7— j = ; and therefore
d^ u T (\ ->t- 7i) /dv\- , C«'-> , „
Thus if M = (a; -|- \y, we get
rf«^ ^ ^^' V (- i) x" r (- 4) a;^
8 .„, ,^4 , C , C, , C2
(x+ir + ^ + ^ + ^ -I-&C.
*^ X- x' X-
If we replace (a; -f 1)2 by a'2 -)- 2 -|-a; 1, we shall get
^ _ L(^) ^.^ , 2 £11) J 4. Eil) _L
d„4 - r a) '^ + 2 J, ^^^ * + r (i) • x4
C Ci
r (— \) x^ r (— 4-) «*
It thus appears that the two results may be made to coincide with each
other, when (ar -)- 1)t jn the fi.rst 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 calculus : thus,
if u = (a; -I- 1)2, we find
216 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 F (r) may be calculated to any
required degree of accuracy for all positive values of r, and has
a"* x^ X" a-' 1 „ „
If we replace (« + 1)2 by a;2 + 2 a; + 1, we find
d-^u X* '-^^ «:- r. . r.
I^=r2 + T + T+C^+Ci.
d-^ u
It is obvious that these two values of -, 5 cannot be made identical,
without the aid of the proper arbitrary functions.
dr u
(7.) Let u = f" where v =.f{x) : and let it be required to find TTr'
d'' u
llie general expression for -j-^ , when r 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.
d V , .^d^ V , dr u
li -J— =:p and if -j—^ = c, a constant quantity, then ~r^'=- n (n — 1)
.... {n — r -\- I) v"-'' f
c r (r — I) £_^ _i_ r (r— 1) (r — 2) (?• — 3) c^ v^ 7
V + 1 (ra — r + 1) 'y^l.2.Qi — r+l) {n— r + 2)"^'^ ^H
r (1 +w) „ r r (r — 1) cc , „ )
+ T(^ - --' + r(-;-i) --^ + ^-
which is in a form adapted to any value of r.
Tf _ ' 1 ■
dx^
+
X- x^
Rational functions of x may be resolved into a series of fractions, whose
denominators are of the form {x -f- a)", and whose numerators are constant
quantities, whose rth differential coefficients may be found by the methods
given above. Irrational functions must be 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.
X
- (^- /o
3 . 5
2"'a-4
3.5.7
!) l 2* a;2
2^* x^
+ ^ + ^.;
+ &c.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 217
given tables of its logarithmic values to twelve places of decimals,
with colmiins 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
r (/•) = (r - 1) (r - 2) .... (r - 711) F (r - m),
or r {r — m) = -, ^r-, \^ -, .,
^ ' (r — 1) (r — 2) . . (r — m)
where m is a whole number, will explain the mode of passing-
from the fundamental transcendents, when included between
r = and I, or between r = 1 and 2, to all the other derived
transcendents of their respective classes f. The most simple of
such classes of transcendents, are those which correspond to
(^) = s/-,
which alone require for their determination the aid of no higher
transcendents than circular arcs and logarithms. In all cases,
also, if we consider F (r) as expressing the arithmetical yaiwe of
the corresponding transcendent, its general form would require
the introduction of the factor V, considered as the recipient
of the multiple signs of aftection 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 analysis %, and
* Fonctions Elliptiques, torn. ii. p. 490.
tTh„s,r(-L) = V». r(i-) = i. V^,r(±) = IJ Vx.
^/ Tc, &c.
X The consideration of fractional and general indices of differentiation was
first suggested by Leibnitz, in many passages of his Commercium EpistoKcum
with John Bernouilli, and elsewhere ; but the first definite notice of their
theory was given by Euler in the Petersburcjh Commentaries for 1731 : they
have also been considered by Laplace and other writers, and particularly by
Fourier, ill his great work. La Tkcorie de la Propagation de la Cliakur. The
last of these illustrious authors has considered the general dillcrential 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 /*+ 00 /»+ CO
<?•«= — ^ <p(«)d« / (p{a.)dqcosq{x — oi.),
- 03 - CO
which immediately gives us,
6r (px _ 2 /*+ CO />+ CD (?»•
~d^-lrj <^{«-)dciJ <P{cc)dq-^^r COS q(x-o,);
- CD - 03
which can be determined, therefore, if ^ cos q (,x — a.) can be determined,
and the requisite definite integrations effected. If, indeed, we grant the prac-
ticability of such a conversion of (p («) 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 f<, of the exponential funetion e"""^ to be mf^ e'"'^, and consequently
the ^th differential coefficient of a series of such functions denoted by 2 A^ e*" *
must be represented by 2 A^ rtT 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
hypothetical existence to ^ ^^ for general, as well as for integral values
of ft.. M. Liouville then supposes that all rational functions of x are ex-
pressible by means of series of exponentials, and that they are consequently
reducible to the form 2 A^ e*"^, and are thus brought under the operation of
his definition. Thus, if x be positive, we have,
— = / e-"*^'
X .J
a.
and therefore.
d^\ r^
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 219
and have been chiefly du'ected 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-
Tvhich is easily reducible to the form,
I)
,.1 +M
^1 (-irr(i + ^)
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
/ -1 \ M -r* ^1 I \
developement of the formula — for all values of u, which the
principle of equivalent forms alone can reconcile, will best show how little
progress has been made when the ^ttth differential coefficient of — is reduced
to such a form. ^
M.Liouville adopts an opinion, which has been unfortunately sanctioned by
the authority of the great names 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 formulas with refer-
ence to those values of the symbols involved, upon which the divergencj' 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 formulae 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 /3 be an indefinitely small quantity, that
p/Sx p— /3r pm/Ji _ „— n/3i
iy — e e ^j. e e
2/3 (m + w) /3
and that consequently any integral function A -j- A x -)- . . A„ xP, involving
integral and positive powers of x only, may be expressed by 2 A^ t"*^, where
m is indefinitely small ; and conversely, also, 2 A^j^ e"''' may, under the same
circumstances, be always expressed by a similar integral function oix. M. Liou-
ville, by assuming a particular form,
^^^ 2/3 '
where C is arbitrary, and /3 indefinitely small, to represent zero, and differen-
tiating, according to his definition, gets
_ dx^ '^ 2V/3 2 '
but it is evident that by altering the form of this expression for zero we might
show that was equal cither to sera or to ivjinity ; and that in the latter
220 THIRD REPORT — 1833.
scendental functions, such as e"**, sin m x, and cos m x, 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
<P
of negative or fractional powers of a; in the expression for , which were
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 x 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 + Ci a; + C2 a:2 + Cs x^ + &c.,
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
r ( 1 + >»; . |.jjyg ^fjgy deducing the formula
T{l+n — r)
d'' • 7 VtT^ {-ly.a'-. r (n + r)
("'" + ^) = 1 .2...(«-l) (ax + b)n + r'
dx''
which is only true when n is a whole number, he says that no difficulty pre-
sents itself in its treatment, whilst n + r is > 0, but that T {n + r) be-
comes infinite, when »i + 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, T {n + r) is only infinite
when n -\- r 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,
dr 1
in connexion with the important case ' {ax -\- &)" , nearly all his conclu-
dx'T
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, I observe one remarkable example of the abuse of the first principles of
„, , „ d^ cos mx ... ,
reasoning m algebra. There are two values 01 , one positive and
dx'^
the other negative, considered apart from the sign of m, whether positive or
negative : but if we put cos m x = — cos m x + — cos m x, we get
' ,v li
dr cosmx 1 «■ cos mx "i. d' cos m x _
dx- ^ d^^ ^ dx^
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 221
which is adapted to the immediate apphcation of the general
principle in question.
Thus, ii' u = e'" *■, we get
d x-"^^ ' d x-" "^ "" d X" ~ "* ^ '
when r is a whole number, and therefore, also, when r is any
quantity whatsoever.
it u = sm m X, -J— = m sm {-—■+mx), -z — -„ = 7n^ sin
dx \2 / d X-
d^ U ( T "K \
(w + m x), . . . . y—y = m'' sin ( —^ + m x j when r is a whole
number, and therefore generally. In a similar manner if
u = cos m X, or rather ti — cos m {Vf x, (introducing P 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
fjh' -ft \ T TT I
- — = (m a/ ly COS < —^ + {m \^ I) x >, whatever be the
value of r. If u = e" * cos m x, we get, by very obvious re-
ductions, making o = . - and fl = cos ' — ,
-, — = p'' e"'^ cos (m X + n fl).
dx^ '^ ^ '
It is not necessary to mention the process to be followed in ob-
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 — ^°^ *" ^, instead of
two ; and, in a similar manner, if we should resolve cos m x into any number
of parts, we should get double the number of values of cos m x ^ j^ ^j^.^
dr
principle of arbitrary combinations of algebraical values derived from a com-
aJx
mon operation was admitted, we must consider -^ — as having two values,
and its equivalent series
x^ + x^-\-x- + &c.
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 difteretit 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 tn x x cos n x, &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
starthng 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 .7)1 , . . . (r), where m is a whole number repeated twice,
thrice, or r 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^, ... tm'': in a similar manner we can rea-
dily pass from the factorials * 1.2, 1.2.3,... 1 . 2 ... r, to
their assumed equivalents r(3), r(4), . . . r(l + r), as long as r
is a whole number. The transition from m^ and r (1 + r) when
r is a whole nvimber, to 7n'' and r (1 + r) when r is a general
symbol, is made by the principle of the equivalent forms ; but by
no effort of mind can we connect 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 r(l + r) = 1 . 2 . . . r, the function
gamma. Kramp, who has written largely upon its properties, gave it, in his
Analyse des Refractions Astronomiques, the name oifaculte nimierique; but in his
subsequent memoirs upon it in the earlier volumes of the Annales des Ma-
thematiques of Gergonne he has adopted the name oi' 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-
suaffe.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 223
The law of derivation of the terms in Taylor's series,
, d i( J cP u li^ cP u h^ , p
'' = ^^ + ^ • ^' + ^- O + ^^ • TT^TS + ^^^
is the same as in the more general series
and if we possess the law of derivation of —, — and of —, — ^, we
(JL X (IX
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
d'' u . .
the determination of -. — : but it is the second of them which
dx^
altogether separates the interpretation of-r— ^ from that of -7—,
d
or rather of -7—^ when r is a whole number, unless in the par-
ct oc
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 froni 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-
* 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 kmd may be found in the researches
of Poinsot respecting certain trigonometrical series, which
will be noticed hereafter, and wliich 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
u to ■^—^, when r becomes a whole positive number. The gene-
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
belono- 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 deri-
* Euler, in the Peiersburgh 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 _ am (1 _ a»») (1 — a™-i) (1 — C") (1 — a*"-!) (1 — C'-Z)
T^^ + 1^^2 + 1 - «' "^ '
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 (»n + 1)* and following terms in
.
the first case, and the reduction of every term to the fonn -^ in the second,
would form the proper indications of a change in the constitution of the equi-
valent function corresponding to these values of w 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 p«^itive signs of ope-
ration which are used in arithmetical algebra ; but such signs,
though accurately defined and hmited 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 2rmT-\- \/ — \ sin 2 r w tt and cos {2r -\- \) n w +
-v/ — 1 sin (2 r + 1 ) w T ;
2r?nr-v/~l 1 (2j-+ 1) n!r\/— 1
or e and e^
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 aftected 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 ai'e included in (1)" and ( — 1)", are expressible generally
by cos fl + v^ — 1 sin fl, or by « + /3 V^ — 1 ,where a and |3 may have
any values between 1 and — 1 , zero included, and where «^ -f- /3^
= 1. To all quantities, whether abstract or concrete, expressed
by symbols affected by such signs, the common tei'm impossible
has been applied, in contradistinction to those possible magni-
tudes which are affected by the signs -|- and — only.
If, indeed, the affections symbolized by the signs included
under the form cos 9 -|- -/ — 1 sin fl, admitted in no case of an in-
terpretation which was consistent with their symbolical condi-
tions, then the term impossible would be correctly npplied to
quantities affected by them : but in as much as the signs + and
1 833. Q
226 THIRD REPORT — 1833.
— , when used indepefndently, and the sign cos 6 + V — I sm d,
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
+ a and — a denote two equal lines whose directions are op-
posite to each other, then (cos 9 + v^ — 1 sin fl) a may denote
an equal line, making an angle 9 with the line denoted by -^ a ;
and consequently a ^ — \ will denote a line which is perpen-
dicular to -1- a. This interpretation admits of very extensive
application, and is the foimdation of many important conse-
quences in the application of algebra to geometry.
The signs of operation 4- 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 + {— h) — a — b, and a — {— b) = a + b; or the al-
gebraical sum and difference of a and — b, 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 9 + V —I.
sin 6, and cos 9' + -/ — 1 sin 9', the results will no longer de-
note the arithmetical (or geometrical) su?n 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 parallel to
them. Thus, if we denote the line
A B by a, and the h ne A C at right
angles to it by 6 V — 1 , and if we
complete the parallelograms AB D C
and AB C E, then a + b \/ - i will
denote the diagonal A D, and a — b \/ — I will denote the
other diagonal B C, or the equal and parallel line A E.
It is easily shown that a + b \/ ^^ = V'Ca^ + *^ (cos 9
cos" ^ a
+ »/ —\ sin 9), (where 9 = — r ^ , J , and a — b i/ — I
a: v'(«' + b^) {cos 9 — a/^^I sin 9} ; it follows, therefore, that
* Peacock's Algebra, chap. xii. Art. 437, 447, 448, 449.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 227
a + b V — I and a — b ^Z — I may be considered as repre-
senting respectively a single line, equal in magnitude to
\/{a^ + b^) *, and aiFected by the sign cos 9 + -/ — 1 sin 9 in
one case, and by the sign cos 9 — •i/ — 1 sin Hn 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 b V —\.
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 -\- b \Z — 1, and a — b V' — 1,
the sum and difference of a and b V —\, 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 f , 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- -)- b^) has been called the modulus of
a + 6<^ — 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 V — 1 :
conversely the affected magnitude (cos 6 + V— 1 sin d) ^Ja^ + b"- is reducible
to the equivalent quantity a -f- 6 a/— 1, if cos &=. „ , and therefore
sin ^ =
t The sum of a and — b, or a -\- (— b], is identical with the difference of a
and b, 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 b denote lines, we can
readily conceive the process by which we form the results a + 6 and a — 6, at
least when a is greater than b. But when we interpret a + 6 \/— 1 to mean a
determinate single line with a determinate position, we are incapable of con-
ceiving any process or operation through the medium of which it is obtained.
q2
2^g THIRD REPORT— 1835.
tifies the assertion, which we have made above, that quantities
or their symbols affected by the signs +, — , or cos 9 + \^ — \.
sin 6, are only distinguished from each other by the greater or
less facility of their interpretation.
The geometrical interpretation of the sign V — 1, when
applied to symbols denoting lines, though more than once
suggested by other authors, was first formally maintained by
M. Buee in a paper in the Philosojihical 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 Maniere
de repr^senter les Quanfites Imaginaires, dans les Construc-
tions G^om^triques, written apparently without any knowledge
of M. Buee's paper. In this memoir M. Argand arrives at this
proposition. That the algebraical sum f of two lines |, estimated
both according to magnitude and dii'ection, would be the dia-
gonal of the parallelogram which might be constructed upon
them, considei'ed 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. Fran9ais, a mathematician of no inconsiderable
eminence. It was the inspection of this letter, upon the death
of his brother, which induced M. Franpais to consider this
subject, and he published, in the fourth volume of Gergonne's
Annales des Mathematiqties for 1813, 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, Franpais,
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. f -ta somme dirigee.
I Lignes dirigies.
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 * w as published. In this work Mr. Warren proposes
to give a geometi'ical 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 — I, 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
• A Treatise on the Geometrical Representation of the Square Moots of Ne-
tfntive Quantities, by the Rev. John Warren, M.A., Fellow and Tutor of Jesus
College Cambridge. J 828.
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 ynagnitude , it is not necessary that (cos 3
+ -v/ — 1 sin 6) « should represent a line equal in length to the one
represented by o, and also making , an angle 5 with the line re-
presented by a ; but if (cos 3 + v^ — 1 sin 6) a, may, consistently
with the symbolical conditions, represent such a line, without
any restriction in the value of 9, then, if it does represent such
a line for one value of 9, it must represent such a line for every
value of 9 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
sui:>posed that a meaning could be transmitted through a suc-
cession of merely symbolical operations, and that there would
exist at the conclusion an eqvially 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 — I, \/ — \, cos 9 + -v^ — 1 sin 9, as signs
of impossibility, in those cases in which no consistent meaning
can be assigned to the quantities which are aflPected 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 formulae with a view to their equi-
valence, and also under other circumstances, which will be in-
dicated by such means as Avill destroy all traces of the equiva-
lence which would otherwise exist.
The capacity, therefore, possessed by the signs of affection
involving -v/ — 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. Davies 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-angled triangle. In reply to this last
observa tion, it may be observed, that I am not aware that in any case the
sign /\/— I 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
n(n— I) Q , „ , » (« — 1) (« — 2) „_« „ ,.! , -
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 sero and infinity.
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 terras of the second members of this
equation are equal to oue anotlier (when x is less than 1), because they are
the only terms whicli 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 &, we shall find x = 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 b}^ the aid of imaginary symbols is made to
depend upon the analogy which exists between certain geometrical properties
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 233
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 finite quantities,
whether great or small, and to represent them by the symbols
00 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=^^{\ — sr),
if we suppose x to have
any value between + °°
and — 00 : let a circle he
described with centre C
and radius C A = 1, and
upon the production of
this radius,; let a rectan-
gular hyperbola be de-
scribed whose semiaxis is
1, in a plane at right an-
gles to that of the circle:
iftf denote the angle AC P,
then the circular cosine and sine (C M and P M) are expressed by
and — =
2 2 -/— 1
respectively ; whilst the hyperbolic cosine and sine (to adopt the terms pro-
posed by Lambert) corresponding to the angle 6 -v/— 1 (in a plane at right
angles to the former) are expressed by
-^ and V- 1 y ^ )' o"- by -^2~~ 2 '
if they be considered as determined by the following conditions ; namely,
that (hyp. cosine)^ — (hyp- sine)- = 1, and that hyp. cos 6 = hyp. cos — ^,
and hyp. sine ^ = — hyp. sine — ^. 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 analogovs 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 ofinjinities and of zeros, though
when they are considei*ed without reference to their symbo-
lical connexion, they are necessarily denoted by the same sim-
ple symbols oo and : thus there is a necessary symbolical di-
stinction between (00)2, 00 and (oo ) , and between (0)2, 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 00 and respectively.
Though the fundamental properties of and co , 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 x is a real
quantity, we can show that
«^ = 1 + ^ + rr2 + r:V^ + ^^•'
but that we cannot show in a similar or any other manner that
1 . 2 1 . 2 . 3 ^^ °^*^"'
then the equivalence in the latter case is assumed, by considering c*
as the abridged symbol for the series of terms
1 + ^ V- 1 -Y72- TT2T3 + ^^- '
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 ;
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, x^ -I- a x -\- b, which is equal to
x{v('+Ty-AT-o}
is also equal to
whatever be the value of the quantity /3 ; a conclusion which enables us to
reason upon real quantities and to make /3 = 0, when the primitive factors
are required. Similarly, if mstead of ^ ^ y, we suppose
■ = V — "■ . and if mstead of ^ = x.
2 2a/-1
we suppose = x — R, we shall find, whatever /3 may be,
2 V/3— 1
g*V/3-i — j,_j^f_|_ ^^ — I (^ — R)^ a result which degenerates into
the well known theorem e '^~ ^ = y -\- V — 1 x, if /3 = 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 to have been suggested by
M. Buee'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 Fran9ais, 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.
236 THIRD REPORT — \833.
If we assume a to denote a finite quantity, then
(1.) a + = a, and a + oo = + oo .
Consequently does not affect a quantity with which it is
connected by the sign + or — , whilst co , similarly connected
with such a quantity, altogether absorbs it.
(2.) axO = 0, axoo =oo;--- = co and — = 0.
00
It is this reciprocal relation between siero 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 ~ = \ and = 1.
CO
But if those symbols be considered as the representatives
equally of all orders of zeros and infinities respectively, then
~ and may represent either I or « or or co , its final
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
and CO 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, oo and ~ in the formulae which express the
values of x and y in the simultaneous equations,
a X + b y = c \
a' X + b' y ■= c' J
In this case we find
^ t:- and y =
X
'•4^-!} -'{I'-l.}'
„' „ „'
If -7- = -jT, — = — 7) and therefore -z- — -n, then a: = -^
b a a bo
and y = -Q-
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 237
In this case «' = in a, U = m a, and c' = m c, 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-
rence of -r- in the values of the expressions for x and y is the
sign, or rather the indication of that indetermination.
If 4- be not equal to -,-t, but if — be equal to —f, then x — co
b o, a
and y = 00 . In this case vfe have a! — m a,h' =■ mh, but c' is
not equal to m c; and the conditions furnished are inconsistent,
or more properly speaking impossible. In this case, the occur-
rence of the sign oo in the expressions for x 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 identi-
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 go in this case by the
term impossible.
c e' b' b
If -^ = ^7, but if —7 be not equal to — , then x is zero and y
h V a a
is finite, and therefore possible. It is in this sense that we
should include %ero 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 b, a' and b' respectively, which are
f! J. ^ - ]
„2 -f- ^,2 - i'
^ ^ it - \
and consider them as simultaneous when expressing the co-
ordinates of their points of intersection, then we should find
X = //6^ W\ and y = /f^ ^ .
V la^ ~ ci'^S V 16^ ~ b'^S
If we suppose — = —7, or the ellipses to be similar, and at the
same time b not equal to b', then or = op and y = cc , Avhich
238 THIRD REPORT — 1833.
would properly be interpreted to mean that under no cir-
cumstances whatever, whether in the plane of x 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 x and y exist : or in
other words, the sign or symbol oo would in this case mean
that such intersection was impossible. If we supposed — = — ;;
and also h = b',or the ellipses to be coincident in all their parts,
then we should find o^ = -y- and y = ^, 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 b
greater than b', a greater than a\ and -j- not equal to -77 , then
we should find
X = u and y = ^ \^—U
or x = a. \/ — 1 and y = j3^
according as -j- is less or greater than -77. 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 x and y above given : it wovdd appear, therefore,
that the impossible intersection of the curves would be indi-
cated by the sign or symbol go alone, and not by \/—l.
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.
a;2 2/2 y
* If in the equation — + -Tg- = 1, we suppose y replaced by y v — 1,
and the line which it represents when not afifected by V — 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
— ~~ + T2 + «3 + • • • •
a — b a a'- a"
preserves the same form, whatever be the relation of the values
of a and b, and the operation, which produces it, is equally prac-
ticable in all cases. As long as a 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 -jr 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 oo ;
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 co . It remains to interpret the occurrence
of such a sign under such circumstances.
The first member of this equation — — -r 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 — — -y , 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 ^ in a series of such a form under
such circumstances.
Let us, in the second place, consider the more general series
for {a — by, or
„ f, b ^ n{n- 1) b^
n {n -I) in - 2) 1% & 1
The inverse ratio of the successive coefficients of this sei'ies
S40 THIRD REPORT — 1833.
approximates continually to — 1 as a limit, and tlie 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 — . It follows,
a
therefore, that if a be greater than b, the series will be conver-
gent and finite in all cases ; if « be equal to b, it will be 0, 1,
or 00 , according as n is positive, 0, or negative ; and if a be
less than b, it will be infinite.
The occurrence of tlie 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 — i)", in the transition
from a > 6 to a <C 6, which is of such a kind that the correspond-
ing series is not competent to express it : thus, if w = 4» then
{a — by is affected with the sign V — I when a is less than b,
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 — by would appear to attain to zero or in-
finity, but not to pass through it, and no change would appa-
rently take place in its affection corresponding to the change of
affection of « — b; 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 go 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 incompetence 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.
The integral I x^ d x = ^ + C is said to fail when
n ■= — 1, in as much as it appears that under such circum-
x"' ~ '
stances 7 becomes 00 , which is an indication that the va-
ra + 1
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 241
riable part of ^" d x is no longer expressible by a function
under the form r' but by one which must be determined
n + i
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
A /r" + ' a;" "*" ' a" "*" '
jr-; for if we put — -— r + C = — — ^ + C, (borrowing
from the arbitrary constant,) we shall get an expression
-«" + '
n + 1
which becomes -^ when n = — 1, and whose value, determined
according to the rules which are founded upon the analytical
properties of 0, will be log a; + 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
— — — and of -, — . — (where n is a positive number) for all
dx"" X dx'' x" ^
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 = <f (x) = x + \f x — a, then
du , , d^u W , d^u h^ ^
u' = <p (:c + h) = u + j-^h + j^, Y72 + J^' T7273 ^^- '
ft U
and if we suppose x =i a, all the differential coefficients ^,
-^—2, &c., become infinite, which is an indication that no terms
of such a form exist in its developement, which becomes, under
such circumstances, a + V'/'- 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 Taylor'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 <p (.r + h)
according to powers of h may be supposed to be effected. In
the first and common mode we begin by excluding all those
terms in the developement whose existence would be incon-
1833. R
242 THIRD REPORT — 1833.
sistent with general values of the symbols : m 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
2^' = (p (a; + li), if we make
u' = « + A /i" + B /^* + C h" + &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 iH be considered successively as a function of x and of h,
-7—7 = -rrr. , for all values of r, 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
d'^u' _ r(i + a) ^ r(i + &) B,,_„ . .
omitting the arbitrary complementary functions, which will in-
volve powers of h. In a similar manner we shall get
d'u' d^u d" K j„ d^B ,,
dx" dx" dx" dx"'
If these results be identical with each other, we shall find
r (1 + a) y _ d^
rji) ' dx"'
and, therefore, A = -=-?! — ; — n • -i-~^, since r(l) = 1. It is easy
r{l + a) dx"' ^ ^ •'
to extend the same principle to the determination of the other
coefficients, and we shall thus find
d" u h" # u //* o ,, .
or, in other words, it follows that the coefficient of any power
of h whose index is r will be
1 d' u
r{l+r)'dx'^'
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 ?/ will
present to us.
If we suppose x to possess a general value, then u' and u
will possess the same number of values, and no fractional
power of h can present itself in the developement. In this case
T(l + a) = I . 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,
, d u J cV^ u h^ cPu h^ o
du dx^ 1.2 dx^ 1.2.3
It will also follow that the series for m' can involve no negative
and integral power of h ; for in that case the factorial T (1 + «),
which appears in its denominator, would become oo, and the
term would disappear. If it should appear, also, that for spe-
cific values of x 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
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 or oo.
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 u under those circum-
stances. Thus, if we suppose u ■= x t/a — x, we shall find
1 .J-u (-')"'^(l) [ K«-.)-(|-r).r ]
r{\ + r)r[^-rj 1^ {^--rya-xy-i J
if we make x = a, this expression will be neither zero nor
infinit?/ in two cases only, which are when r =i—, and when
3
»' = -g- : in the first case we get,
rf* u , — -
(1)-'-^
R 'Z
844 THIRD REPORT — 1833.
and in the second we get,
— — (t'u 2 ^ ' ■
(I)
3
\a — a)
= >/•=!;
dx^ Ar(0) X X (a -a)
since r(l) =s 1 = r(0), and the symbol in the denominator
3 3
= — ^, is a simple zero. The corresponding developement
of m' under such circumstances is
^/-«2.A^ + -/-I. A^
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 u' 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 h in the developement. For, in the first
place, /*», and the differential coefficient whose index is — , will
possess the same number of values, and the same signs of affec-
tion. If there be a term in u which = P (ar — «)», where P
neither becomes zero nor infinity, when x — a, and where the
multiple values of P, if any, are independent of those contained
- .„ ,1 j.(J^ .V .{x — «)»
in (a: — «)», then it will appear that the term ot —
m
dx»
m
which is independent of {x — «)» is P . — ^ > and that
d X"
m
fill • fl#
all the other terms of —, being either ssero or infinity when
d x^
X ^= <t, 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 u , 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-
iiients, when h = 0, by p, p\ p", p"', &c., we shall find
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 245
none of which become ssero or injinity, in as much as P does not
vanish when x ■= a.
If there exist other terms in m of a similar kind, such as
m' m"
Q,{x — by, R (ar — cy^ , &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 u must be modified when necessary, so
that such radicals may present themselves in single terms of
the form V {x — «)». The same observations will apply to ne-
gative as well as positive values of — , 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,
d~^ 1
p. 211, on the subject of the values of -3 — zTr • — 7> when « is a
whole number. If we suppose, therefore, u to involve terms
such as P (a; — «)», Q (w — Z>)»', &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 :
m m VI
a — a p, d« {x — o)» h"
(^^ rfj ■r(i + ^)
m' m! m'
h — h „, dn' {x — 6)"' Jin'
d x^'
(> - 5)
or.
, du J d^u h^ d^u .h^ ^
«=« + ^^' + ^^r:2 + rf^ 17273+ ^*^"
m
h — h ( Ifi \ Vi
+ &c.
We have introduced the discontinnoiis sigtis or factors :
a — a
g46 THIRD REPORT — 1833.
^ ~ &c., which become equal to 1 when x = a or x =: b, &c.,
X -^ o
but which are ^ero for all other values of x, to show that the
terms into which they are multiplied disappear from the deve^
lopement in all cases except for such specific values of x.
The existence of the terms of the series for u 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 «/. 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 ?/, which
have no existence except for specific values of x. 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
(2 2\^
— " ^" , when x=sa,we should find it to be,
x^ {x — ay
fly
. (x^ — a-f
dx^
x^ {x — of
or
{x + af.-~-,'{x-nf
d x^
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 x = 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. 247
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 functions 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 diiferentials of the first and
higher orders as infinitesimal values oi differences of the first and higher orders.
Such 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,
ii y=. a^ -\- {x — a)^, Euler makes, when a; = a, d y = (d a;)^, instead of
(5 \ = 3 = (d x)^. Iiy = 2ax — x"-\- a^/ (a^ — a;3), he makes,
y) t ^"^
when X = a, d y = a \/ — 2 a . d x^, instead of
,j^ ff"^ (d «)■?
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, a fortiori, therefore, between different orders of differential coefficients.
M. Cauchy, in his Lecons 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 all 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,
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 conventiofial 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 ^T)a to denote 1, when x was taken between
and a, to denote zero for all other values ; ■^Da+^j, to denote 1,
when X was taken between a and a + b, and zero for all other
values ; and similarly in other cases.
Thus, if 2/ = a X + /3 and y =^ of x + /3' 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 and a, and in the second line between the limits a and b,
then we should have generally,
y = -D; (a ^ + ^) + -D/ {«.' X + /30 (1.)
the couvergency 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 afi^ecting 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 verj' 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 Theorie 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 Ampere, in the
sixth volume of the Journal de I'Ecole Poly technique. 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. It is a
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 modem 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 '\f,,
vertex to the base, and if we suppose
A D = «, A B = 6, A the origin of m,.-'
the coordinates, A B the axis of x,
y =. a. X the equation of the line A C,
><
and y = a' .r + /3' the equation of the ■* ^ ^ ^
line B C, then we should find that the value of y represented
by the equation
y = -^Da" . a ^ + ^Dj" (a' a; + /3') * (2.)
would be confined to the two sides A C and B C of the triangle
ABC, excepting only the point C, which cori-esponds to the
common limit of the discontinuous signs. For if we suppose
'Do° and ■^Dj" to be true up to their limits, we shall find, when
X =■ a, that ■^D«° + '^Dj" = 2. If we replace, however,
^D; by ^D; - ^, and ^D," by ^D*" - |^^,
Developement of the Binomial Theorem, Avhich 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 tlie series for (a -\- x)", which they exhibit
under the following form :
(^ + .,)n = «„ + ,,a„-l^.4.....«(«-_l)---(»->-+l).an-r;er
+ -«'-^'(« + ^)"{(^+V^ +
('• + 1)
(r + 1) (r + 2) . . ■ (n - 1)
T • •
(a + xy + i ' 1 (a + xy + ^
Zl^X.
1 . U . . . (n — r—l) ■ (a + .r)"/ '
the remainder being (a + j;)« jC + 1 multiplied into n — r terms of the deve-
1 1
lopement of -77 — ; — ;^ •>..,, or of — r-;.
^ {(a -f x) — aj'^+i' a»- + i
The method which they have employed for this purpose, which is extremely
ingenious, succeeds for integral values of n, 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 '^Dj'* might be replaced, though not with perfect
1 /»o
propriety, by the definite integral _ ^ / dx.
250 THIRD REPORT — 1833.
and if we make, therefore,
y=^'D."-^«^„+^W-i^*^(»'.+^') (3.)
the equation will be true for the ordinate of every point of the
sides A C and C B of the triangle ABC.
More generally, if we suppose y = f^ x, y ■=. (p^x,y ■=■ (^^x,
y = (p^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 x, included
between the limits a and b, b and e, c and d, &c., would be,
+ ('D/-f-^,)fcx + &c.; (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 ^Dj".
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
would readily show that its value is 1 when x is greater than
a, and that it is when x is equal to or less than a. It will
therefore follow that the product
^(logO) e^^^s")'^- ") ^ ^(logO)e^'^e*'>(*-^)
is equal to 1 between the limits a and b, and is equal to at
those limits, and for all other values. And, in as much as
* Memoires de Mathematique 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
p(iogO) _ Q^ yfQ jyjay replace the preceding product by the equi-
valent expression
This expression, which is equivalent to ^Db" '■ j,
has been applied by Libri to the expression of many important
theorems in the theory of numbers *.
The definite integral / — sin r x has been shown by
It
Eulerf and many other writers, to be equal to -^ when x is
positive, to when x is 0, and to -^ when x is negative. It
follows, therefore, that
2 r^dr . (b-a) C (a + b)l
— / — sm ^^ — Tz — - r cos ■< x — ^^ — 7^ — - r >
vJo r 2 L 2 J
= — / ^ — sin (^ — a) r H / — sin (^ — 6) r
vJo r ^ ' TTi/o »*
is equal to 1 , when x is between the limits a and 6, to ^, when x is
at those limits, and to zero, for all other values. If we denote the
definite integral — / — sin — - — - r cos -s x — ^ — -- — '- > r
^ -K JQ r a L 2J
by Cj", we shall get,
xT\ « n«i ^ ^ 1 " — b
^' -^* +2(a:-a) + 2]F=T)'
and consequently the equation of a polylateral curve, such as
that which is expressed by equation (4.), will be,
y = Cb . <P]X -{■ C/ . 9.2 -^ + Cd . ^3 X +&C.,
in as much as at the limits we have <Pi {b) = p^ (b), (p^ (c) s= a^ (<?),
and consequently for such limits Cj" ^j (b) + C/ f^ (^) = f 1 (^)
= ip^ (b), and not 2 (p^ (b).
All definite integrals which have determinate values within
given limits of a variable not involved in the integral sign, may
be converted into formulae which will be equal to 1 within those
• Cr elk's Journal for 1830, p. 67.
t Inst. Calc. Integ., torn. iv. ; Fourier, Theorie de la Chaleur, p. 442. ; Frul-
loiii, Memorie della Societa Italiana, torn. xx. p. 448. ; Libri, Memoires de Ma-
thematique et de Physique, p. 40.
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 formulfE 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 formulae themselves. We shall now
proceed to notice some examples of such formulas.
The well known series f
X 1 1 . 1 .
r TT + -3- = sin j; — ^ sin 2 j: + -?r sin 3 x — -p sin 4.r + &c. (1.)
is limited to integral values of r, whether positive or negative,
and to such values oirit •\- -^ as are included between -^ and
— -^ : the value of r, therefore, is not arbitrary but condi-
* If a definite integral (C) has n determinate values »i, «2, . . . un, 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
«D « = _ (C — eti) ( C — eta) • • • (C — »n) ^ j.
oil X oCq X • . ■ cin
thus in the case considered in the text, we get
= _ 2 (C — 1) ^C — i-) + 1 = — 2 C2 + 3 C.
t The principle of the introduction of r w in equation (1.) by which it is ge-
neralized, will be sufficiently obvious from the following mode of deducing it :
= log {i + e'^~' } - log {1 + e-' ^^}
log
1 I '*'
1 +e
, -X V-l ; ^ . - , ■ ( S'/^ -JfV-A
= log e =. X a/— I + 2r t V — 1 = Ve — e /
and, therefore, dividing by 2 V* — 1» and replacing the exponential expressions
by their equivalent values, we get
X . 1 1 . 1 .
r IT + — = sin a; sin 2 a; + — sin 3 a; sin 4 a; + &c.,
2 2 3 4
where x upon the second side of the equation may have any value between
•+ 00 and — 00 .
REPORT ON CERTAm BRANCHES OF ANALYSIS. 25S
-re
2
tional. If we successively replace, therefore, a" by -^ + t and
— — x,yvQ shall get
•It X 1 . ^ 1 „
rir + -J- + "o = cos X + -^ &\a'ax — -^ cos 3 x
— :j- sin 4 X + &c.
/«• + -; 7r- = COS a: yr sin2x ^ cos 3 jr
4 2 » 3
H — J- sin 4 d7 + &c.
Adding these two series together and dividing by 2, we get
(f J- f^\ •B' 1 1
^-^ — '- * + -J- = cos ^ IT- cos 3 a: + -^ cos 5 X — &c. {2.)
It It
If X be included between g- and —, then r = and r' = 0,
and we get
-r = COS X ^ cos S X + -p- cos 5 j: — &c. (3.)
4 3 5 ^ •'
■jt 3 iz*
If J? be included between -g- and -^, then r = — 1 and r' = 0,
and we get
w 1 1
— -r = cos j: -^ cos 3 ^ + ^r cos 6x — &c. (4.)
4 3 5 ^ '
If the limits of x be -^ and -^, —5- and -^, -^ and
— -3-, ^ and — '-^, we shall obtain values of the series
nt Tt
(2.), which are alternately -7- and — -r-.
X . 1 .
Again, if in equation (1.), or rir + -tj- = sin x — -^sinS^r
+ -q- sin S X -7- sin 4 t + &c., we replace x by tt — x, we
shall get
»•' ff -I -^ — = sin or + -^ sin 2 x + -q- sin 3 J? + -j- sin 4 j? + &c.
Adding these equations together and dividing by 2, we get
254
THIRD REPORT — 1833.
(r + r') IT 1 1
Tf" + -J- = sm X + -TT sin 3 z + — sin 5 jr + &c. (5.)
2
3
which may be easily shown to be equal to -j- and j- altera
4 4
nately, in the passage of x from to v, from ^ to 2 tt, from 2 it
to 3 It, &c., or from to — tt, from — it to —2it, &c.: its
values at those limits are zero.
The series (2.) and (5.) have been investigated by Fourier, in
his TMorie 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.
The equation y
2
is that of an indefinite straight line,
Q A P, making an angle with the axis of x, whose tangent is
■g-j and which passes through the origin of the coordinates :
whilst the equation
y sz sinx zr- &m2x + -jr- sin 3 a; ;i- sin 4 .r + &c.
» o ^
is that of a series of terminated straight lines, d' c, dC,T> C,
&c., passing through points a, A, A', &c., which are distant
2 It from each other : the portion d C alone coincides with the
primitive line, whose equation is y == -^.
Again, the line whose equation is y = ~^, is parallel to the
1
i
IS
G' d, C
* From page IG/" to ipo ; also 267 and 3')6.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 255
It
T
71"
axis of X at the distance -j- above it : the line whose equation is
TT
— -J-, is also parallel to the axis of x, at the distance -j- below
it : the line whose equation is
y = cos X «- cos S X + -^ cos 5 x — &c.
consists of disco7itinuous portions of the first and second of
those lines, whose lengths are severally equal to tt. The values
of y at the points B and b, corresponding to a- = -^ and — ^,
are equal to srero, since the equidistant points D and C, 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 circiUation of the trigonometrical
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 disco7iti7iuous, and that the coincidence
would take place only throughout one period of circulation,
whether from to tt or from — -^ to -^. It remains therefore
to consider whether such a conversion is generally practicable.
Let us take ti equidistant points in the axis of the curve
whose equation is 7/ = 4> x, between the limits and tt, those
limits being excluded : if we denominate the corresponding
values of the ordinate by i/^, i/c^, . . . . i/„, and if it be proposed
to express the values of these ordinates by means of a series
of sines (of 7i terms) such as
o, sin a; + ag sin 2 x + «3 sin 3 4? -f- . . . . + a„ sin 71 x,
then we shall get the following n equations to determine the n
coefficients a^, %> ^3 • • • • ^«*
TT
yj = ttj sm + «2 sm
^2 = «i sin -^ — ^ + O2 sin
y^ = a^ sin - + flTg sm
277 , . 37r
+
«T
n + \ ^ 71+1
.a„.m^^j,
47r . Gtt
— ; — 7 + ff sm
7fi+ \ ^ n + \
+
. ^nn:
71 + V
+
. 3M7r
. ffnSin — — -,,
n + V
256 THIRD REPORT — 1833.
w TT , . 2 tn: . 3 tin . 1^-K
If any assigned coefficient a^ be required to be determined
from this system of equations, we must multiply * them seve-
rally by
n , m It ^ . 2mTr ^ . Smrr _. nmv
2 sm — —T, 2 sin — --r, 2 sin ^ , ... 2 sin r,
n + r n + l n + I n + I
when all the coefficients except a^ will disappear from the sum
of the resulting equations : and we shall thus find
2 f . OTTT . . 2mTt . nrmr'X
«», = — r~\\ Vi sin — --r + Vo sin — —^ + . • . -/„ sin — -— ;- >.
M + 1 L n + \ ^^ « + 1 "^ n + IJ
It would thus appear that it is always possible to determine a
series of sines of n terms with finite and determinate coeffi-
cients, which shall be the equation of a curve which shall have
n points in common with the curve whose equation is ?/ =: <p x,
within the limits corresponding to values of x between and v ;
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 =s <p X, would be complete within those limits only, producing
a species of contact to which the texxn. finite osculation has been
applied by Fourier f . 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 hy y = <p 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 "Theorie 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 I'Ecole Polytech-
nique.
f TMorie de la Chaleur, page 250.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 257
The hypothesis of n being infinite would convert the series
for «„ into the definite integral *
2 /»0
2 /»0 . ,
— / ax sin mx ax.
COS a;
if we make r = x and r = d x : or otherwise if we
?^ + 1 w + 1
assume the existence of the series
(p X ■=■ a^ sin a: + «2 sin 2 x + . . . a,n sin m « + &c.,
it may be readily shown, by multiplying both sides of the equa-
tion by sin m x d x, that
a,n = — / (p X sin m X d X :
and in a similar manner, if we should assume
<p X = Oq cos .r + «i cos x + . . . a^ cos 7n x -\- &c.,
that
cr^ = — / <p X cos mx a X -f .
Thus, if we should suppose ip jp = cos x, we should find
4 C 2 4 . 6 1
= — \ Tj — ^c sin 2 ^ + H — p sin 4 x -f -p — = sin 6 a; + &c. I •
»r[l.o o.o o./ J'
a very singular result, which is of course only true between the
limits and it, excluding those limits %.
If we should suppose (p .r = a constant quantity — between
the limits and a, and that it is equal to zero between a. and tt,
we should find
(1 — cos«) . (1 — cos2«) . _ (1 — cosoa)
(p X = ^ ^ sin X + ^ ^ -' sm 2 ^ + ■ ^ ~^ x
sin 3 X + &c.,
excluding the limiting value u, when the value of the series is
only -^§.
If we should suppose <p x =. 'Dj" . a-x + "D,^ . («' x + /3'),
which is the equation of the sides of a triangle (excluding the
• Poisson, Journal dc I'Ecole Polytcchniqve, cahier xix. p. 447.
+ Fourier, Th/'oric de la Chaleiir, pp. 235 & 240.
\ IhUL, p. 233 ; Poisson, Joimial de I'Ecole Polytechmque, cahier xix. p. 418.
§ Fourier, Tlicorie de la Chaleur, p. 244".
loutj. s
258
THIRD REPORT — 1833.
limit X = h), whose base is represented by tt, then we shall
find *
2 , / IX f 1 r . sin 2 .r sin 3 ^ o 1
(p X — — [a-n + [a. — c^) b) ismx ^ \- — g &c. |-
2 , ,s fsinZ>sinar , sin26sin2j: , sin3isin3j; ^ 1
+ — («-«) I [2 + 2^ + 32 +&c.|.
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, \iy = <^ xho. the equation of the
curve P C C" Q, and if we suppose
2/ = (p ^ = Oj sin X + ttg s''" 2x + a^ sin 3 x + &c.,
between the limits and w ; and if we make A B = w, A A'
= 2 TT, A B' = 3 TT, &c., we sliall get a discontinuous curve,
consisting of a series of similar arcs, C D, a C", 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 arc C D
alone coincides with the primitive curve.
If we should suppose the same curve to be expressed be-
tween the limits and tt by a series of cosines or
2/ = ^ ar = «o + ^1 cos X + a^ cos 2 x + &c.,
and if we make A B = tt, A 6 = — tt, A A' = 2 tt, A B' = 3 w,
&c., then the trigonometrical equation will represent a discon-
tinuous curve c? C D C D', of which the portions C D and C d,
• Fourier has given a particular case of this series, p. 246.
UKPORT 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 f , 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 (p {x) be an integral function of x then the coefficient of
— in the developement of — log ^— will represent the least
root of the equation <p x = 0. We thus find that the least
of the two quantities « and (3 will be represented by the coeffi-
cient of — m the series for log ^ ^-^ ~, which is
(«+iy + ot47G- (^-±^)
and if we re])lace « and B by — and -^, the least of the two
^ cc p
quantities — and -^, or the greatest of the tv/o quantities « and
^, will be represented by
• Transactions of the Philosophical Society of Cambridge, vol. w. p. 374.
t Ihid. p. 125.
X If we represent the series (2.) by S, we shall get
d«-> S 1 „
__ — or 0,
(— l)»-i T{n) d«»-i
according a,s « is greater or less than (i : thus — — - would represent the at-
traction within and without a spherical shell, which is or — , where « is
the distance from the centre.
s2
260 THIRD REPORT — 1S33.
Thus, if y — a .r — /3 = and y — «' .r — /3' = be the equa-
A B p p-
tions of two lines B C and D C, 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_«^_^) (2,_«/^-_^')^0. (3.)
Now it is obvious that if common ordinates P M, P M' be
drawn to the two lines, the least of them will belong to the sides
of the triangle BCD; if we denote, therefore, P M and P M'
by'^i and y,^, the equation
y =
y\ ya
yi + ys
+
1 . 1
2.4
• (H^f
1
2 J V 2
"will become the equation of the sides of the triangle BCD,
when yi and y^ are replaced by their values ; for y will denote
P M for one side and 2^ "* for the other.
In order to express a discontinuous function <p, which as-
sumes the successive forms ipj, ip,,, f^, &c., for different values
of a variable which it involves between the limits « and j3, /3
and y, y and 8, &c., Mr. Murphy assumes S («i 2;), S (/Sj z),
. . 1 .
S (yi ss), &c., to denote the coefficient of — in the several series
for
and supposes
log
„ </ S («, «) , /> (/ S (/3, z)
+ /3
d S (y, ^)
+ &c.
doL '-''' d^ '-''■'' dy
If a. be less than z or 2 greater than «, then S (a, ^) = «,
, , t. d S (u, z) 1 ■r■a^ 1 i.u 4-1 '■^ S (^, s)
and therefore r"^— ^ = 1 = up be less than z, then ^-^ — -
d «
rf^
1 : if Y be less than z, then injll = 1 and so on; con-
' d y
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 261
sequeiitly, in the first case we have <p = /i = <Pi :
in the second,
9 =/, +/2 = ?2, and therefore/^ = (^2 - ^, :
in the third,
<P =fi +A+f3 = <P3> and therefore /a = (p^ - ^3.
It appears therefore that
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
4 r ^ 4 6 "I
•'■D/cos^ = —\ T-^ sin 2 a; + -^^— psin4^ + -=— =sm6a: + &c. \
11.0 0.0 0,1 I
is permanent within the limits indicated by the sign '*'D^° 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
.r» 1 1 r.r2 - 1 1 r* - 1 1 ^^ - 1 . )t
-D.« log a: = 2-|^^— I + -J . (^-.qpT)^+ y • (:,qri)a+ &«• \
which is true for all values of x between and co , has been
extended to all values of x between — 00 and + co , 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
oi\\ev finite : the first generally accompanies such changes of
form as are consequent upon the introduction of critical values
* These formulae 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.
t This series is given by M. Bouvier in the 14th volume of Gcrgoniie's
Aimalcs ties Malhemutifjiies. The conclusion referred to in the text assumes
the identity of the logarithms of x" and of ( — .r)^, which is in fact the wholo
(jucstion in dispute.
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)*" = cos 7n {2 r TT + x) + m cos (/« — 2) {2 rv + x)
-\ \ — ^ — cos {m — 4') {2rTr + x) + ike.
+ i/^l { sin 7n {2rTf + x) + m sin {m — 2) (2 r if + x)
-,. '^'1^-^ sin {m - 4) (2 r TT -}- x) + &c. (1.)
If we suppose m to be a whole number, this equation degene-
rates into
{2 cos x)"" = cos mx + m cos {m — 2) x -\ ^j — -^ — -'
cos (w — 4) X -I- &c. (2.)
the series first discovered by Euler, and which he assumed to
be true for all values of m. li, however, we should suppose tn
to be a fraction of the form —> we should have g values of
. 9
the first member of the equation (2.), and only one of the second.
And if we should confine our attention to the arithmetical
value (/} of the first, it would not be equal to the second, un-
REPORT ON CEKTAIN BRANCHES OF ANALYSIS. 2G3
less m was a whole number ; for if we should denote the series
of cosines
cos {m2rir + x) + w cos (m — 2) (2 r * + .r) + &c., by Cr,
and the series of sines
sin m {2rTf + X) + m sin {tn — 2) (2rTr + x) + &c., by Sr,
we should find, when cos x is positive,
Cr ^r
^ "" COS 2 m rir ~ sin 2 m rv'
and when cos x is negative.
and
^ cos m {2 r + \) ir sin w (2 r + 1) **
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 2 mrie = or s'm 2m r if =■ 0, when cos x is
positive, or cos m (2 r + 1) * = 0, or sin ?n {2r + 1) w = 0,
when cos x is negative.
In a similar manner, assuming
1 . 2 ... 6
r («2-P) 3
X = n < cos X — ^:j — g — ^ cos'^ j:*
we shall find
cos 7« (2 »• TT + x) = cos n ( 2 r + Y ) '^ • ^
+ cos (w - 1) (2 r + 4") 'T . X'.
If we suppose r to be equal to zero, this equation will become
nif ^ (n — 1)* ^,
cos n X = cos -^ . X + cos 5 . A',
which is the form which has been erroneously assigned by La-
grange* and Lacroixf as generally true for all values of w.
Many other examples of similar undulating functions, ex-
* Cakttl den Funciio)'s, chap. xi.
t TVaite du Caktil Diff. ct lu/erj., torn. i. p. 261.
204 ^ THIRD REPORT — 1833.
pressing the various relations between the cosines and sines of
multiple arcs and the powers of simple arcs, whether ascending
or descending, have been given by Lagrange * and other writers
as general, which are either degenerate forms of the coi'rect
and more comprehensive eqviations, 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 f 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
yflc/or,y 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 (+ aY is identical with {— a)^, 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 :
Iog(+a)2 = log(l)2a2 = 4r7r \/'~i +2p (1.)
log (- af = log (- ly^ a^- = (2r + 1)2 nf \/'^l + 2p (2.)
log a^ = log 1 . o- = 2 /• Tf V' ^ + 2 p (3.)
It thus appears that the values of log ( + «)^ and log ( — a)- are
included amongst those of log «^, but not conversely; and also
that the values of log (+ «)^ and log(— a)'^, the arithmetical
value being excepted, are not included in each other.
* Correajjottdence sur I'Ecole Fulyirvlin'iqHo, torn. ii. p. 212.
■\ In Gergoniie's-yJ«««/pi' des Matlu'inutiqncs, torn. xiv. xv. xvi. xvii.
I In Creile's Journal fur die rcine uiid aiigewandie Muthemafik. Berlin.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 265
Again, if we consider — a" as originating from (— 1) (+ a)
Me shall set
nt
\og — a"" = (2 r + 2 7)1 r' + I) Tf \/ — I + m p* :
2'
if we suppose /k = — , r = and r' = — 1, we shall get
log - x/a =^ p = ^ log « = log i/ a ;
or the logarithm of a negative quantity will be identical with
the logarithm of the same quantity with a positive sign. In a
similar manner, if we suppose 7)i = i~-~, where p is prime to n,
r' = — n and r = ^^-^^, then 2 r + 2 7n r' + 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 f 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 identical with those of l"", A and (—1) 1 *". A,
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 — «*", they will be found to diflfer from each
other by the logarithms of V" and (— I) 1'" only, which are
2mr'K V — \ and {2 r -\- 2 m r' -{- 1) * -/ — 1 respectively. The
logarithms in qviestion are Napierian logarithms whose base is e.
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 qviantities (or signs)
multiplied by the inodiiliis INl : 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-
t 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 ai = + n, that— ;j
is eijually the logarithm of + n and — h. The same remark aj plies to all in-
dices or luyarilhmv which are rational fractions with even dentrainatorb.
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, in as 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 nvmi-
ber of cases in which such a proposition can be considered to
be true *.
* In the 15th volume of the Annales des Mathematujues 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 will be 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 x z= and a- = 1 to be di-
vided into an even number 2 p of parts, (where p is an odd number,) the
values of* will form a series effractions,
J_ ± 3 2 2p-l
2p p Zp 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 h-anche pointillee. 'J he 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, one positive and the other negative, corre-
sponding to every logarithm. In reply to this objection, it is merely neces-
m mp m mp
sary to observe that the values of a" and o"^ or of 1 " and 1 "^ are in every
respect identical with each other, the n 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 formulee, 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 vi'hich 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 foi'ms
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 ^jrorfwce 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
, 2r v
which is not 2 tt v — 1, but ^z /^^., which, thouarh 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 formulie 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 / {&) ^= cos 6 + -v^l sin 3
:= e and makes the series for/(^) and/"' (&), combined with the equa-
tion/ (x &) — a. value of/ {&f, and therefore/"' / ^ = 2 r 5r + ^, the foun-
dation of his logarithmic developements : in other words, he makes e '^~' a
periodic quantity the base of his system of logarithms, an assumption which
is essential to the truth of the formula/"'/^ = 2 ?• ar -f and to the gene-
ralization of the series for/"' 6 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 obscrvationb upon the general principles of analysis.
268 THIRD REPOrwT — 1833.
being jjaid 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 oftend our popular notions of the values
of series as exhibited in their sums. We speak of series as
having sums 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 n 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 sex'ies, arithmetically speaking, may be considered as di-
vergent, and thei'efore 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 in-
finitely great and infinitely little as very difl^erently 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 be 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\biafyse.
REPORT ON CERTAIN BRANCHES OF ANALYSTS. 2C9
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
a + a x + a x^ + a a^ + &c. (1.)
Let s be taken to represent this function, and therefore
s = a + a X + a •r'^ + a a:'^ + &c.
= a + X {a + a X + a x^ + a x^ + &c. }
= a + X s : consequently
a
s = .
I — x
If the arithmetical values of the terms of this series be con-
sidered, and if x be less than 1, then , is the stim of the
1 — X
series : in all other cases it is its generating function.
We may consider, however, s (whether it expresses a sum or
a generating function) as identical with *,, s^, s^, &c., in the
several expi'essions
s= a + X s^
s = a + a X + x^ s^
s = a + a X + a x^ + x^ Sg
s = a + ax + ax'^+ . . . a x^"-^ + x"' s„, :
for if the number of terms of the series * be expressed by n
and if ?i be infinite, we must consider *,, s^, Sq, a . . . Sm as abso-
lutely identical expressions ; for otherwise we must consider an
infinite as possessing the properties of an absolute number, and
must cease to regard infinities 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 — a-\-a — a-\- &c.
was assigned by Leibnitz, upon very singular metaphysical
considerations, to be — : the principle just stated would allow
us to put
^70 THIRD REPORT — 1833.
s = a — {a — a + a — a + &c.)
= a — * ; and therefore s = -r-*,
2
* The same principle would show that the equation
*• = « +/(« +f(a+f{a + . . .)))
is identical with the equation
x= a +f (X) ;
and that
* = «/(«/ (a/ (a...)))
is identical with
X = af(x).
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 = mp + i, their sum would be identical with that of the i first
terms of the series ; and if we should denote those terms by aj, a.,, . . . o ,
and if we should take the successive values of this sum for all the values of i
between 1 axidp inclusive, their aggregate value would be represented by
pai + (p—1) a<^+ (p _ 2) a, + . . . a^.
of which the average (A) or mean would be represented by
pay + (p — I) a.2 + (p — 2) as + . . . a^
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 Commmtarii
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 + 1 — &c. ...
1+0— 1 + 1 + 0— 1 + 1 + &c. ...
1-t-O + O— 1 + 1+0 + 0— 1+&C.
12 3
the first would be equal to — , the second to — -, and the third to — . In the
A o 4
same manner we should find
1 + 1 — 1 — 1 + 1 + 1 — 1 — 1 +, &c.
equal to 1, and
1+1+0 — 1 — 1 + 1 + 1+0— 1 — l+I + l +&C.
equal to — -. The same observations would apply to the series
1 + cos X + cos 2 X -\- cos 3 .r + cos 4 a; + &c.
and
1 + cos a- + + cos 2 X -\- cos 3 .r + + cos 4 .r + &c.
where x is commensurable with 2 tt.
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
I + X and subsequently make x = 1, we get
1 — 1+1 — 1 + &C. (1.)
the value of which is — . In a similar manner, if we divide 1 + a; by
1 + a; + a;2, we get for the quotient
1 — x" + x3 — x^ + x^ — x^ + &c.,
which becomes the same series (1.), though the value of the generating func-
2
tion under the same circumstances becomes — . The same remark applies
to the result of the division of 1 + x + x^ + . . x"' by I + x + x- -\- . . x",
which produces the same series (1.) when a; =: 1, though under such circum-
stances its generatmg function becomes — .
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 + a; by 1 + .r + a^, if the deficient terms be replaced, becomes
l+O.x — x^ + x^ + O.x'^— a^ + x^ + O.x'^ — x^ + &c.,
which degenerates, when x ■=■ \, into the series
1+0—1 + 1+0—1 + 1 + &c.,
and not into the series (1.). The same remark applies to the series which
arises from the division of 1 + a; + . . a;*" by 1 + a; + . . . . x", n "Z m;
which becomes, when x z= I,
1+0 + + + &c. — 1+0 + + &c. +1+0 + &c„
•which is equal, by Bernoulli's rule, to — .
n
But it is not necessary to resort to this expedient for the purpose of deter-
mining the sums of such series ; for the series
Oi + On X + 03 a- + . . a x^~ + a, a'' + &c.
is a recurring series resulting from the developement of
a, + Oo a; + 03 x- -\- . . a x^"
1 - xP
which becomes — when a: = 1. If we replace x by —, this fraction will
become
272 THIRD REPORT — 1833.
rating functions. Foi* 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
1 =a + ax + ax^+ ixc = *,
we shall get
a = (I — x) s = a,
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 aritlametical
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
eti z^ + a^ z^~ + . . a z
which becomes by the application of the ordinary rule of the differential cal-
culus, when s; = 1 or .r = 1,
p Oi + (p — I) a^ + . . Op
~ f
P
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 I'Ecole Poly technique. 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 gum of the series
sin q + p sin (x + q) + p" sin (2 .r + q) + &c.
is equal to
sin q -{- p sin (x — q)
1 — 2 p cos X -\- p^ '
when|) is less than 1, an expression which degenerates, when^ = 1, into
— sinq + ~ cos q cot -j'
which may be considered, therefore, as the limit of the sum of the series
sin q + sin (.r + ?) -|- sin (2 x + </) + &c. in wfiii.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 273
degree of accuracy, to its value by the aggregation of a finite
number of those temns. Many tests of the siimmability of series
(considered as different from the determination of their gene-
rating functions,) have been proposed, possessing very different
degrees of certaintj' and apphcabiUty. The geometrical series
which we have just been considering is convergent or divergent,
that is, summable or not, according us x 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. 324, and following. Cauchy, Cours d' Analyse
Algebrique, 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'AIembert, will immediately lead to the following:
" If u^ represent the m"^ term of a series, it is co?ivergent (or will become so)
if the superior limit of (ii )" be less than 1, when 7i is infinite; divergent in
the contrary case."
" If the limit of the ratio tr^, j to 7/^^ 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 the
following test of convergencj'. " If the limit of the value of the product n u^
\)B finite or zero when n 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
2 log 2 ^ 3 log 3 ^ 4 log 4 ^ ■ ■ ■ ■ ^ ?j log «
may be shown to be infinite, though the product n n^ is equal to zero when w is
infinite. The same acute and original analyst has shown that there is no func-
tion of n whatever which multiplied into u^ will produce a re.sult which is zero
or finite when « is infinite, according as the series is convergent or divergent.
1833. T
274 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 circum-
stances in which such series originate.
The series
] 1 6 62
+ -2 + 73 + &C-
a — b a a^ a^
is convergent or divergent according as a is greater or less
than b. 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 go , 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
r, b n(n-\) b^
(a-bf = a'' < 1 - rt — + — ; ^ . -2
^ ^ l_ a \ . 2 or
~ 1.2.3 • ^ + ^""'J'
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
If, 2b , 3b^ 4 63 -|
a^ [^ a a'' a'' J
I If, 2b . 3b'' 4*3
(« - bf
and
J _ If 2 a 3a^ 4 a^ \
- af~ b^X '^ b + ~b^ "^ 63 + *'^-/
(b
will be divergent in one case, and convergent in the other,
whatever be the relation of a and b, though they both equally
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 875
braically, as well as arithmetically, equivalent to each other. It
might be contended, therefore, that in this instance the sign oo ,
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 — b or b — a. But
though a^ — 2 a b + b'^is equal to (« — b)^, and b^ — 9, ab -\- c?
to {b — (if ; and though a^ — 2 a 6 + Z>^ is identical in value
and signification with b"^ — 2 ab + a^ when they are considered
without reference to their origin, yet we should not, on that
account, be justified in considering (a — hf and {b — af as
algebraically identical with each other. The first is equal to
(+ 1)^ (a — by, and the second to (— 1)^ (« — hf ; or the first
to (- \f {b — af, and the second to (+ 1)^ {b — af. 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
a symbolical change in the quantity denoted by -; j-^ 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 (b — a)", in all cases in
which w is a negative even number. When w is a negative odd
number, the change of constitution of the generating function
is manifest, and requires no explanation.
The two series
and
If, b b^ b^ b^ „ -1
a L « « a a* J
1 _ 1 /, a a^ a^ a'' . \
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 a denote a line, (+ af and (— «)« can only be considered as
identical in their common result a'. When (+ a)2 and ( — a)' are considered
with reference to each other, they are not identical quantities, though equal to
each other.
T 2
^76 THIRD REPORT 1833. t
any change in the constitution of the generating function. They
may both of them, therefore, be considered as representing the
vahie 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
/ , i\n « r I n b « (w — 1 ) 6^ . o ~1
(a + 5)" = «» |l -f - + -hV-^ + ^^-j
when n 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 -f- or — , and the whole series may be replaced
by the symbol oo ."
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 ai'ithmetical quantitiesj
the operations with the same names represent arithinetical
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.
* The equations s = and « = -r 7- will equally give us
s ;= in one case, and s = r in the other, whatever be the relation
, a + b a +
of and b.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 277
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 formulie, 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 co 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 eflPected, tlie 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
ax — ba:"' + cx^ — dx^-\-ex^— fx^ + &c.,
which Euler has given f, into the equivalent series
f "v** y'^ "j^
a — Ti—. — -To .^a + T^i— — To .A^a — 71:—. — vt • ^^ a + &c.
1 + x"' (l + x)2-""'-(l + ^)3-- " {l + xy
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
1 - 1 -h 1 - 1 -f &c.
may be shown to be equal to -^. The series
1 - 3 + 6 - 10 + 15 - 21 + &c.
♦ The essential character ot arithmetical division is that the quotient should
approximate continual)}' to its true value, and that the terms of the quotient
which are introduced by each successive operation should be less and less con-
tinually. In the formation, therefore, of the quotient of j and ,
the analogy between the arithmetical and algebraical operation would cease to
exist, unless a was greater than b, or unless the several terms in the quotient
went on diminishing continually.
f Jnstitulioncs Calculi Dlfferenlialis, Pars posterior, cap. i.
278 THIRD REPORT— 1833.
of triangular numbers to -5-. The series
1 - 4 + 9 - 16 + 25 - &c.
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
, ,,,(«- \f {a -If {a- \y , 5
log a = (a - 1) - ^ g ^ + ^ 3 ^ - —^ + &C--
is divergent when a is greater than 2, and convertible by Euler's
formula into the convergent series
(«-l) , 1 (« - If ,1 {a-\f \ {a- \f .
or by the method of Lagrange into the series
„ (^a _ 1) _ |. (^a _ 1)2 + |- (^_ 1)3 _ &c.,
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^(1 + a:) is expressed by the
continuous expression
(^) '(2*a:)*R,
where R is a quantity whose Napierian logarithm is expressed
by
A B_ C
\ .2.x 3 . 4 . x2 ''■ 5 . 6 . ^i" - &c->
where A, B, C, &c., are the numbers of Bernoulli. The law of
formation of these numbers, as is well known, is extremely
• Fonetions ElUptiques, torn. ii. 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 s, 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 lavys 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 f ; 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
»+i d X
=/:
{{\-2ax + a^){\ -2bx + b^)Y
{
by means of series.
Assuming K = (1 - 2 « x + «2)- 4 and K' = (1 - 2 5 r + b^ *,
let us suppose K and K' developed according to ascending and
descending powers of a and b respectively ; or,
K = 1 + a X, + «'■' Xg f a^ X3 + &c.
K' = 1 + 6 Xj + 62 Xg + b^ X3 + &c.
K' = l + ^Xi + lx,+ lx3+&c.
* Erchinger in Schrader's Commentatio de Siimmatione Seriei, Src. Weimar
1818.
t Journal de I'Ecole Polylechnique, torn. xii.
280 THIRD REPORT 1833.
The coefficients Xj, Xj, Xg, &c., are reciprocal* functions, pos-
sessing the following remarkable property, that / X„ X„ rf .r
= 0, in all cases, unless n = m, in which case / X„ X„ </ a;
1 ^-'
~2n + r .
The knowledge of this property will readily enable us to de-
termine the following four different values of s :
«1 = 1 + -g- + -^ + -y- + &C.
1 (J/ fll ft
"^2 ^ T + ST^ "^ 5^ "*■ 76^ + ^''•
% = — + Q^2 + K-73 + i?^ + &C.
"4 —
a ' 3 a2 -1^ 5 a3 ^ 7 a^
1 1 _1_ _1_
a 6 ^ 3 a2 62 -T- 5 ^3 ^3 -T^ ,j, ^4 ^4
Whatever be the relation of a and b 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-
tegral / K K' c? a:, that one only of these two convergent
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
/» + '
examine how far the definite integral of / K K' c? a: will jus-
tify such an inference.
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 vohnneof the Transactions of the Philosophical Society
of Cumhridge, 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 Integral. Cauchy has used the term recipro-
ea/ function in a different sense; see Exercices des Mathematiques, torn. ii. p. HI.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 281
p-^d^^ _^=.log {P-P- + 9,p ^Th) + const.;
and if we denote by r and r' the extreme values of p, when a:
= — 1 and X — + 1, we shall find,
r + ^dx _ 1 ^ 2r\/ab + ^ab-{a + h){\ + ah )'}
^~J -\ "y ~ 4 s/ah ^^i2r^ //ah-^ah- (a + 6)(l+a6)y
inasmuch as^-^ is 4 a 6 — (« + A) (1 + « 6) in one case, and
a X
— A^ah — {a + h){\ + « 6) in the other. It will appear like-
wise that r and r' will have the same sign, whether + or — ,
in as much as p will preserve the same sign throughout the whole
course of the integration. If, therefore, r' = + (1 + «) (1 + a),
then r = + (1 - a) (1 - 6) ; and if / = - (1 + a) (1 + h\
then r = — (1 — a) (1 — h). It thus appears that (1 — «) (1 — h)
must have the same sign with (1 + a) (1 + 6), and consequently
if a 7 1, and h 7 1, we shall have,
. = _L^ loff (<?-!) (^> - 1) j/^ + 4a 6 - (a 4- 6) (1 + «6)
4 ./a 6 (a + l)(6 + l) v'"«6-4a6-(« + 6)(l +«*)
= - — =- log . Iv «_ + _j_ (stj.i]jinrr out the common divisor
_ 1 - a/« 6 + 1 ,
2^ah-a-h)^ ^^^ log ^^T^^^^Y = ± -4-
If a 2: 1 and 6 l\, we shall find r = (1 - a) (1 - h), and
« = === log I 7= ) = ± «i.
2 A/a 6 ^ Vl - A/a 6/
If o Z 1 and h 7 1, we shall find r = (1 — a) (1 — h), and
1 , ( Vh + a/«^
X =
1 ( Vh + A/a\ ,
log ( ~T ^ ) = + ^2.
2 -/afi
If a 7 1 and 6 Z 1, we shall find r = (a — 1) (1 — h), and
2 \/
It would thus appear that the definite integral would furnish
erroneous values of z if no attention was paid to those values
of the factors of r and »•', 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 to select the proper devclopement for ^r. Thus, if a 7 I
282 THIRD REPORT— 1833.
and 6 7 1, we have r = {a — 1) (b — I) =
{(a-lf{b-lf}-i-
and the value of s (z^) is determined by the combination of the
two last developements. In a similar manner, if a Z 1 and 6 Z 1,
SI (^i) will be formed by the combination of the two first. If
« Zl and 6 7 1, then r = (1 - «) (6 - 1) = ^~ ^— ^ :
and the value of s (^g) is formed by the combination of the first
and third developement. And if a 7 1 and 6 Z 1, then the value
of s (sg) 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 {{l — af}-i 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 s, of which it appears that one value
alone is correct for any assigned relation of a and b to \, 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 ;£. It would appear to me, there-
fore, that not only was the employment of divergent series
necessary for the determination of all the values of z, but that
when the theory of their origin is perfectly vmderstood 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 formulae 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 woi'k ; 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
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 283
furnish. Great simplicity in the exposition and exempHfication
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 ai'ithmetical 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 compreliensive, renders it extremely difficult to
give to the principles of this science such a form as may bring
them perfectly ^vithin the reach of a student of ordinary powers,
and which have not hitherto been invigorated by the sevei'e 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 difterent 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 undvdy 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, in a 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 pubHcation
of a series of elementary works on analysis, and on the appU-
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 conseqviently 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
^86 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 Calcul
des Fonctions and his Theorie des Fonctions Analytiqiies, 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
MTiters 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 Stances de V Ecole Nor-
male and in the Journal de V Ecole Polytechniqiie ; 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
I' 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 Math^matiques Pures of
M. Francoeur possesses merits of a similar kind, being too
much compressed, however, for the purposes of self-instruction,
though well adapted to foi'm 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, li^he Arithmetic, Algebra-f, 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 Mathematiques Pures et Appliquees
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.
t 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 f in which the combinatoi'ial 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 ai'e 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, avery 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 Hal], Cambridge, in some papers in the
Transactions of the Philosojihical 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.
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 z, cos z, tan ^, &c., to denote the
sine, cosine, tangent, &c., of an arc ^, 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 cn-cle
of a given radius, and their sines and cosines are commonly de-
fined with reference to the determination of these arcs, and not
with refei'ence 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 arcs 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 Vvcre ge-
nerally, at least in later writers, reduced to a common standard,
by assuming the radius of this circle to be 1, yet formulae were
considered as not perfectly general unless they were expressed
with reference to any radius whatsoever-}-. In the application,
likewise, of such formulae 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 Analysirn Injinitoruw, vol. i. cap. viii. " Quemadmodum
logarithmi peculiarem algorithmum vequinint, cujus in universa analyst summiis
extat usus, ita quantitates circulares ad certain quoque algorithmi normam
perduxi : ut in calculo aeqiie commode ac logarithmi et ipsae quantitates alge-
braicae tractari possent." — Extract from Preface.
t 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 formulse, 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 formulae 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
formulae with this element long 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.
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 BAG (5), and P M to be
drawn perpendicular to the other
line (A B), then we may define the
. PM
sine of fl to be the ratio . p , and the cosine of 9 to be the
AM
ratio -J— Ti. By such definitions we shall make the sine and
A 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 formulae 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 arc 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 9 to be not only measured, but also repre-
sented by this ratio, we shall be enabled to compare sin & and
cos Q directly with 9, 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-j-.
* 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 formulae of trigonometi-y are deduced in conformity with these definitions.
t If we should attempt to deduce the exponential expressions for sin 6 and
cos 6 from the system of fundamental equations,
cos" d -\- sin^ ^ = 1 (1.)
cos 6 = cos (— 6) (2.)
sin ^ = — sin (— 6) (3.)
we should find,
aoaAH , g-A('\/^ gA^A/irT_g-A^v'irT
COS 6 = ^^^— ' and sin ^ = ;;==
2 2 V — 1
in which the quantity A, in the absence of any determinate measure of the
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 formulas 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 subraultiples. 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»x d X f*y d y
If we should denote the integrals / . and / — (com-
l/O VI— X' Jq ^/\—yi
raencing from respectively) by d and d' respectively, then the integral of
the equation
dx dy
would furnish us with the fundamental equation
sin {& 4- ff) = sin 6 cos 6' + cos & sin &', (/3.)
if we should replace x by sin 6, •v/l — x^ by cos 6, y by sin 6', and Vl — y*
by cos i'. If the formulae 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 6 and 6' to represent the integrals
/*J" dx Py dy
of the transcendents / —yr- — ; — ^r and / —,7t- — 9^» then the integral
Jo V(i+*-) Jo Vii—y^) ^
of the equation
d X , d y „ / ^
aAi + ■■') ^Ai + r)
would be expressed by the equation
h sin (i + 6') = h sin & X h cos &' + h cos d X h sin 6', (3.)
if we should make x ■=: h sin 6 (the hyperbolic sine of 6), and V (1 4- «*)
= h cos 6 (the hyperbolic cosine of 6), y ^ h sin $', and Vl -\- y^ =: h cos &',
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 (S.), combined with the assumptions made in deducing it, to frame a
svstem of hvperbolic trigonometry (having reference to the sectors, and not
u 2
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 formulae 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 formulae and greatly to embarrass all their applications.
to the arcs of the equilateral hyperbola), whose formulse would bear a very
striking analogy to the formulae of trigonometry, properly so called.
Abel, in the second volume of Crelle's Jotirnal, 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
X
'^ ^/(l-c2sin2,^)
by 6, and replace sin \p by jc, we shall get
-/
d X
or more generally
-X
^{(l-a'2){l-t2a'2)}
d X
V{(1 +e"*=) {l-<?x^)y
If we now suppose a; = <p ^, V (1 — c^ «") =/ ^ and V (1 + e^-r") = F tf, it
may be demonstrated that
<p{i) + e) - r+e2^2-^2-^-7^2-^» .
, fd.f6'—c-(p6.(pd'.Y6. Fd'
f {0 + 6) — 1 + e2 c2 (p2 tf . (p2 tf'
_ ¥d.¥6'-\-e^(p6.<pd' .fd.fd \
or if, for the sake of more distinct and immediate reference to these peculiar
transcendents, we denote
(p S by sin 6 (elliptic sine of ff),
f dhy cos 6 (elliptic cosine of ff), and
F ^ by sur 6 (elliptic sursine of 6),
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 293
The primitive signs 4- 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 fl + ■v^— i sin 3,
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
«o, (cos 9i + -/ — 1 sin flj) a^, {cos (di + 62)+ V — 1 sin (9i-l- ^2) }«2»
. . . {cos (fli + ^2 + • • • ^»-i) + ^^^\ sin (51+ 9.2 + . . . 9„_,)} «„_„
will be competent to form a closed figure, if the following equa-
tions be satisfied :
then these fundamental equations will become
sin & cos ^' surs ^' + sin ^' cos & surs 6
T ^^'^^'- 1 + e2 c2 sin2 6 sin^ $' ~'
e e
cos 8 COS 6' — c2 sin & sin 6' surs 8 surs i'
f A \^ AW e e e e e e_
COS (» + ff; _ 1 +e2c2^2 ^-^ina ^' •
e e
surs d surs 6' + e'^ sin i sin & cos & cos d'
surs C« + tf ) _ 1 + e2 c2 sin2 ^ sin2 d'
e e
If we add, subtract and multiply, the elliptic sines, cosines and sursines of
the sum and difference of 6 and 6' 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 formulae, some of which are very remark-
able, and which degenerate into the ordinary formulae of trigonometry, when
c =: and c = 1 : we shall thus likewise be enabled to express sin n 6, cos n 6,
e e
surs n 8, in terms of sin 8, cos 6, surs 6. The inverse problem, however, to express
« e e e
sin 6, cos i, surs 6, in terms of sin n 6, cos n d, surs n 6, is one of much greater
e e e e 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 (w -|- 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.
g94 THIRD REPORT 1833.
Co + flfi cosfii + a2COs(9, +S^) + , . «„-, cos (3, + fi2 + . • fi«-i) = (1.)
aisinfli + ff2sin(fii + fl2) + . . a„_, sin (9i + 92 + . . 9„-i) = (2.)
91 + ^2+ ... 9„_, . = (« - 2 r) TT (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 flj,
^2 ~ ^1' ^3 ~ ^2 • • ^«-i "" ^n-2 ^^ ^11 positive, aud 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 *^t?Z/a^e(/ figures were
first noticed by Poinsot in the fourth volume of the Journal
de FEcole 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 aflfected 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 fixed 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 counti'y, many of which are
remarkable for the great simplicity of form to which they have
reduced the investigation of the fundamental formulae. Such
works are admirably calculated to promote the extension of
• See also Peacock's Algebra, 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 formulae, 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 formulae 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 formulas, 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 hlaclc 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 otlier
woi'ks 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 hy
Newton in the greatest of his works, and which it became us,
therefore, from a regard to the national honour and our own,
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 membei', 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 a; — a, we presume
that a is either a real magnitude, or of the form a + (3 v' — 1,
where « and jS 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 unknown
sign different from +, — , or cos fi + V — 1 sin 9, 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 x in the equa-
tion
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 — , unreal; and quantities denoted by symbols affecttd
with the" sign cos + V— l sin 6, as imaginary.
^^S THIRD REPORT — 1833.
following problem, and examine all the consequences to which
its solution will lead.
" To find n quantities x, x^, x^, . . . x„_i, such that their sum
shall be equal to ^9,, 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 p„."
The quantities x, a-j, . . . x„^i, 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 6 + -/^ sin 9, 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^, p^, . . . jo„ :
for if we suppose the first of these quantities x to be omitted,
and Pj, Pg, . . . P„_i to be the quantities corresponding to p^,
pc^, . . . pn when there are (w — 1) quantities instead of n, then
we shall get
a; + Pj =/?„
a: Pi + Pa = p<i,
XV^ + P3=^3,
ea^
^ P»-2 + P»-l — pn-l>
^ Pn-1 = Pn-
If we multiply these equations from the first downwards by the
terms of the series x"-', x"-^, . . . 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
x"~23i a:"-' + j)^ x"-^ -... + (- 1)«^„ = 0. (1.)
In as much as />,, p^, . . . p„ 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 x may represent any
one of the n quantities involved in the problem, we must equally
obtain the same equation for all those n 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 spnbolical composition of the coefficients of
everi/ equation as known, though the ultimate symbolical form
of the roots is not knoivn ; 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 Icnown
signs of aiFection 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 Pj . P^ . P«-i, 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 unities
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 = X 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^, Qg, &;c., to represent the coefficients of
the reduced equation, we should find,
.r + 0-1 + Qi = pi,
xx^ + (x + x{) Qi + Q2 = P2,
^ ^1 Ql + (a- + X^) Q2 + Qg = jOg,
X a-1 Q„_4 + {x + x^) Q„_3 + Q„_2 = p„_i,
X X^ (:)J„_2 = Pnf
from which equations we can determine successively rational
values of Qj, Q^, . . . Q„_2. 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, x^, . . . .r„, be odd, it is very
easy to p)rove that there is always a real value of one of them, x,
which will satisfy the conditions of the general equation (I.) *,
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 n may be either 2 m,
2^ m, 2^ m, 2" ?«, 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 — \,) 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 ' Equations' in
the Supplement to the Encyclopaedia Britannica, written by Mr. Ivory.
300 THIRD REPORT — 1833.
combinations may be either the sums of every two of the quanti-
ties, X, x^,... T„_i, such as X + x^, x + x^^, &c., or their products,
such as X Xi, or other rational linear functions of those quanti-
ties, involving two of them only, such as, x + x^^ + x x^, x + x^
•^ 2 X x^, ov X + x-^ + k 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^, p.^, . . . 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 {2 m — 1) dimensions which will have rational and
known coefficients. Such equations being of odd dimensions
must have at least one real root ; or, in other words, there must
exist at least one real value of one of the sums of two roots,
such as a; + x^, of one of the products, such as x x^, of one
of the functions, x + x^ + x x^, or a: + a;] + k x Xy If the
symbols which form the real sum x -\- x^ 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 f . We shall now pro-
ceed to show that this must be the case.
If we form the equations successively whose roots are x -\- x^
+ k X x-^, corresponding to different values of k, we shall have
one real root at least in each of them. If we form more than
m (2 m — 1), such equations for different values of k, we must
at least have amongst them the same combination of x and x^^
forming the real root, in as much as there are only m {2 m — \)
such combinations which are different from each other. Let k
and X'l be the values of k which give such combinations, and
let «' and (6' be the values of the real roots corresponding ; then
we must have
X + x^ -\- k X x^ ^= u.'
X + .r, + k^ X x^ = /3'
* The formation of symmetrical combinations of any number oi symbolical
quantities x, x^, . . . a-n-i, and the determination of their symbolical values
in terms of their sums {px), their products two and two (pq), three and three
(JO3), 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 Algebraicce, in Lagrange's Traite sur la Resolution des Equations
Ni/meriques, chap. i. and notes 3 and 10, and with more or less detail in
nearly all treatises on Algebra.
t If J + •'^1 = a and xx^ = /i . , where « and /3 are real magnitudes, then
j; = -^- + s'' s " — /3 ? the values of which are either real or of the form
(cos (I + \^ — 1 sin 6) ^^/3, where the modulus \//3 is real.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 301
and therefore
X J-,
k - X-i
"*" + "^1 ~ ' k^-k '
There are therefore necessarily two roots of the equation or two
values of the symbols .r, a:{, x^, . . . Xn-\, such that x + x^ and
X Xj 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
2^ m, then the number of their binary combinations must be
9, 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 form a -\- ^ V — \ \ 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 £'* m, or
any one in an ascending series of orders oi parity, it may be re-
duced down to the next order of parity in a similar manner : and
under all circumstances it may be shown that there must be two
roots which are reducible to the form « + /3 -/ — 1 , where « and /3
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 Normals in 1795f , in the form in which the ques-
tion is proposed. A certain number of symbols, representing
magnitudes with unknoum affections, are required to satisfy
• I-et X + x' = r (cos tf + V — 1 sin ^)
X x' = J (cos <p + V — 1 sin (p)
X -\- x\- — A xx' ^= R-(cos 2;/' + \^ — 1 sin 2 ^l>)
or *■ — x' =i R (cos ^Z' + -v^ — 1 sirn^)
r cos tf + R cos ip (jr sin ^ + R sin ip) i
X — - • H 2 V — 1
= r' (cos ic + -v^ — 1 sin x)
x' = r' (cos X — V — 1 sin x)-
t Lemons de I'Ecole Normale, torn. 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 woi'ds,
the root in an equation of n dimensions, is required to satisfy.
The object of the proof above given is to show that it is always
possible to find n real magnitudes with known aiFections which
ai*e 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 fl in the expression f (cos 9 + V^ — 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 Resolution des
Equations Niim^riques ; the first of those given by Gauss in the
Gottingen Transactions ior 1816*; and by Mr. Ivory in his
article on Equations in the Supplement to the Encyclopcedia
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 Theorie des Nomhres ; by Cauchy in the 18th
cahier of the Journal de VEcole Polytechniqiie ; and subse-
quently under a slightly different form in his Cours d' Analyse
Alg4hrique.
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.
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 moi*e 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
y (x) is a rational function o£ x ; if for different values a and b
of the last term of this equation, where a ^b,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
•v/ — 1, or \/ — l, 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 re nde rs them imaginary), will change in form from a to
7W -f- « -/ — 1 . He subsequently shoAvs that the same result will
follow for any values of V between a and b, and consequently,
* I do not venture 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 + ?? V^— 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 M'ould be difficult to justify by any a priori considerations.
The illustrious author hmiself 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 coeffi-
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 + 'S/ — 1 sin 9), the equation is reduced to the form
P + Q ^/^-i, or ^(P2 + Q2) . (cos <p + V^l sin (f); and the
object of the demonstration is to show that there exist neces-
sarily real values of p and 9, 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 g and &,
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 g and S, 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 § and & ; 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 the 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 minimtim 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 estabhsh 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-
nates des Mathe??iatigt(es*, 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
* 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.
1833. 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 mvist 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 such a
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 S3rmbols 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 n 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 Traits sur la
Resolution des Equations Numiriques ; 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 vdll be quite sufficient for
our purpose.
Thus, the ordinary solution of the cubic equation
a^-Sqx + 2r-Q*
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 q."
If we represent the required numbers by u and v, we readily
obtain the equation of reduction
u^ — 2r u^ + ^"^ = 0,
• This equation may be considered as equally general with
a^_Aa;2 + B«— 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 — 183^.
which gives, when solved as a quadratic equation,
ti'
= r + >^ if — (f),
and consequently,
and therefore
^^q_^ 9'
u {r + v/ {r^ — (f')Y
If we call 1, «, o?, the three cube roots of 1, or the roots of
the equation s^ — 1 = 0, and if we assvime a to represent the
arithmetical value of u, we shall obtain the following three
values of ?< + v, which are
a + -^s a a ■] 2_ « «^ -| 2-.
a a u aw'
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
x^ — 3qx + 2r=0:
for it may readily be shown, in the first place, that their sum
= ; that the sum of their products two and two =: — 3 q; and
that their continued product = 2 r; 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 six values of u, in as much as the values of u and v are inter-
changeable, from the form in which the problem was proposed ; but there are
only three values of u + v.
t Since q ^ i
it is usual to express the roots of the equation x^ — 3q x -\- 2r = 0, by the
formula
x={r+ V(r^-q^)}^ + {'•- V(»-=-?^)}i (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
x^ — 3ctqx + 2r = 0,
x^ — 3a'^qx + 2r=0,
and the formula (1.) expresses the complete solution of the equation
(«« — 2 rf — 27 </3 a;3 = 0,
which is of 9 dimensions. It is the formula
« = M + -|-, where u= {r + V (r^ — q^) }i
and has the same value in both terms of the expression, which corresponds to
the equation x^ — 3qx + 2r — 0.
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 tex-ms :
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 appliccible to general equations of a higher order. Thus,
if it was proposed to find two quantities, u and r, the sum of
whose «"^ powers was equal to 2 r, and whose product was equal
to q, we should find
u = {r + ^(r^ - ?")}«;
1 ^
where m + ^? is the root of the equation
x^-nq ^"- + '^^^ f a;«- - n {n - ^A)^{n- ^) ^ ^„_,^
+ &c. = 2 r*.
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 f 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 concei'ned,) 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 x^, x^, oTg, x^, 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 be true, by a process, however, not al-
together general, by Euler, in the sixth volume of the Comment, ^cad. Petrop.,
p. 226. See also Abel's " Me'moire sur une Classe particuliere d'Equations
r^solubles algebriquement," in Crelle's Journal, vol. iv.
t 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.
x'* — p x^ + q x^ — r X + s = 0, (1.)
sixch functions would be x^ x^ + x^ x^ and (ic, + x^ — x^ — x^^,
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 {x-^ + x^^, if we should
suppose p or the coefficient of the second term of equation (1.)
to be zero *. The function (Xj + ^2) (■'^3 + ^4) would give
three values only under all circumstances. The functions x^
+ ^2 + ^3 ^^^ ^1 ^2 •''^3 ^^^ 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 x^ Xo 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 -j-.
Innumerable functions may be formed which admit of 12 and of
24 values, and one alternate function which admits of two values
only J.
The success of such transformations in 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 irrationality/ 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 Sammhmg von Aiifgaben aus der Theorie der algehraischen Gleickuyigen,
which contains the most complete collection of formulae 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. 6I9.
■f The form of its roots being u and — , they are reducible by the same me-
thods as are applied to recurring equations.
I See Cauchy, Cokvs d' Analyse, chap. iii. and noteiv. The use of such al-
ternate functions in the elimination of the several unknown quantities from n
simultaneous equations of the first order, involving « unknov/n quantities,
will be noticed hereafter.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 311
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 eqviation 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 he, whether they are discovei'able in higher equations
than the fourth.
This inquii'y 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 delta 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 ofw dimensions must be either
equal to 1 , 2 . 3 . , . ??, or to some submultiple of it ; and that
when n = 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 Unmbglich-
keit algebraische Gleichungen von hoheren Graden als dem vierten allgemein
Aufzidosen, 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 1883.
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 VEcole Pohjtechnique , 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 n
quantities, is a submultiple of 1 . 2 . 3 . , . n, and cannot be re-
duced below the greatest prime number contained in n, without
becoming equal to 2 or to 1." If we 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 liis 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 of
a reducing equation, whose root is
Xi + u jTg + «^ .Tg + 0,^X4+ &c.
where x\, x^, x^, &c., are the roots of the primitive equation,
and where « is a root of the equation
a"-' + «»-2 ^ ^«-3 -f . . . a + 1 = 0,
where n 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 f that it is
quite unnecessary to describe it in this place ; and the result,
* Mcmnrie della Sor. Hal., torn. xi.
t Resolution des Eqiiafions Ntimeriques, Note xiii.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 313
as might be expected, perfectly agrees with the conclusions
which are derived from more direct, and, perhaps, more ge-
neral considerations. If n, or the number of roots ^j, .Tg, x.^,
&c., be a prime number, then the dimensions of the final re-
ducing equation will be 1 . 2 ... (w — 2) ; and if w be a compo-
site number := mp, then the dimensions of the final reducing
equation will be
1.2. ..w 1.2...ra
or
{m- 1) m . (1 . 2 . . . J})'" {p -\)p.{\.2... m)P'
according as we arrive at it, by grouping the terms of the ex-
pression
A'l + U X^-\- 0^ Xq + &c.
into m periods of^ terms, or into jo periods of »« 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^ -\- a x
+ 6 = 0, we must employ the auxiliary equation y -\- K ■\- x
= 0, and then eliminate x. To exterminate simultaneously
the second and third terms of the cubic equation a;^ -J- « x'^
+ b X + c = 0, we must employ the auxiliary equation y + A
+ B X + x'^ = 0, and then eliminate x ; and more generally, to
destroy all the intermediate terms of an equation of « dimen-
sions,
x" + «j .I'"-' + a^ x"-' -{- . . . a„ = 0,
314 THIRD REPORT — 1833.
we must employ the auxiliary equation
7/ + A + AjX + Ac^x^ + . . . ;r«-' = 0,
whose dimensions are less by 1 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 will be found that the equations upon which
the determination of A, Aj, Ag, 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 tlie 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 Hoene de Wronski, in a short pamphlet pub-
lished in 181 1, announced a method for the general resolution of
equations. He assumes hypothetical expressions for the roots of
the given equation in terms of the n roots of 1, and of the {m — V)
* In the 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 (w — 1) dimensions, and employs
in the determination of the coefficients of this reduced equation
if ~^ fundamental eqaaiions, designated by the Hebrew letter {^,
and ?«"~^ others designated by the Greek letter i2. 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 Mathematiques, 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 Hoene de Wronski were received with extraordinary favour
in Portugal, where the Baron Stockier, 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
formulae 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 (« — 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 -Q which arc 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, howevei*, had been
limited to values of n 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 x" — 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 al^arren fact, which
in no way contributed to the advancement of our analytical
knowledge.
The appearance of the Disquisitiones Arithmeticee of the
* Memoires de I'Academie 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 Traiie szir la
Resolution des Equations Nameriqties. Poinsot, in a memoir on the solution
of the congruence x" — 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.
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-
x"—l
tions. It was well known that the roots of the equation — ; — r- =0,
where w is a prime number, could be expressed by the terms of
the series
r + r'^ -\- r^ + . . . r"~^,
where r represented any root whatever of the equation, and
where, consequently, the first term r 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 ascei'taining 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* oin,
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-
X
- 1
ferent roots of the equation j = 0, but which also made
manifest the cyclical periods which existed amongst them.
Thus, if a was a primitive root of n, and n — I — mk, then in
the series
**, r , r , r , . , , r , . , . r ,
the m successive series which are formed by the selection of
every k^^ term, beginning with the first, the second, the third,
and so on successively, or the k successive series which are
formed in a similar manner by the selection of every m^^ term,
are periodical ; and if the number m or k 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 n 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 n as there are numbers less than
n — 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 Encyclopccdia Britannka, 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 w — 1.
It follows, therefore, that if the highest prime factor of » — 1
be 2, the resolution of the binomial equation a;" — 1 = 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 n is equal to 2* + 1 and is also
a prime number. Thus, if A' = 4 we have « = 17, a prime
number, and therefore the solution of the equation a:'^ — 1 =
will be reducible to that of four quadratic equations. Similar
observations apply to the equations
^2V 1 _ 1 = and x'^^^+ 1-1=0.
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 f has
stated that the principles of his theory were applicable to func-
tions dependent upon the transcendent /^yj\ 4\> 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 X 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 Resolution des Equations Num.e-
riques. It follows from thence that the coefficients of the reducing equations
will be whole numbers.
■f Disqiiisitiones Arithmetic(B, pp. 595, 645.
I " Sur une Classe particuliere d'Equations resolubles algebriquement," —
Crelle's Journal, vol. iv. p. 131.
§ Crelle's Journal, vol. iv. p. 314, and elsewhere.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 319
algebraical resolution of an equation whose roots can be repre-
sented by
X, 6 X, &^ X, . . . . 6^-' X,
where &^ x = x, and where 6 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 9 ^ and d^ x express any other two of the roots, we have like-
wise
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.
o -J.
If we suppose a = — , the equation whose roots are cos a,
/*■
cos 2 a, cos 3 «, . . . cos /x a is
^^_|.^^-2 + ^./i^ii:_l)^^-4. .. =0 (1.)
which may be easily shown to possess the required form and
properties ; — for, in the first place, cos m a =■ ^ (cos or), where 9
is, as is well known, a rational function of cos « or a:* ; and,
in the second place, if 9 a' = cos m a and 9, a; = cos m^ a, then
likewise 9 9i a: = cos mm-^a =■ cos ^Wj m a = 9i 9 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
Stt 47r 4<nit ^
of which the last is 1, and the ti first of the remainder equal to
the n last. The equation (1.) may be depressed, therefore, to
one o{ n dimensions, which is
x" + ^ x"-' - -T- (« — 1) x"-2 — — (n — 2) x"-3
1 (n-2){n-3) l_ (»-3)(/»-4) _
+ 16' 1.2 "" ^ 32' 1.2 "" i^c.-U{4.)
whose roots are
2 TT 4 TT 2 M
cos ;z r, cos p: — ^, .... COS
TT
2n + I 2/2 + 1 2« + 1
320 THIRD REPORT 1833.
•r/. 2v , 2mn /.
If cos 7i T = X = cos a, and cos ^ -^ = 8 j; = cos »? a,
then these roots are reducible to the form
xjxj'^x, . . . 6"-' X,
or,
cos a, cos m a, cos m^ a, . . . cos m" ' « :
and if we suppose m to be a primitive root to the modulus
2n + I, then all the roots
cos a, cos m a, cos m^ a, . . . cos m"~' «
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
x,6a:,&^a:, . . . 6""' x,
where $^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 n equal parts, the
division of an assigned or assignable arc into 2 n equal parts,
and the extraction of the square root of 2 w + 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 2n = 2", we shall get the case of regvilar polygons of
2"+i 4- 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.
Poinsotf has given a very remarkable extension to the theory
of the solution of the binomial equation x" — 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
^« - 1 = M ip),
whose modulus (a prime number) is p: thus, the imaginary
cube roots of 1, or the imaginary roots of ^ — 1 = 0, are
— 1 + V— 3 — 1 — V — 3 ^^ ^jjg whole numbers 4 and 2,
2 ' , 2 '
* Disquisitiones Arithinetic<E, p. 651.
t Journal de I'Ecole Polytechnique, cahier 18.
1
REPORT ON CERTAIN BRANCHES OF ANALYSIS, 321
which satisfy the congruence
,r3 _ 1 = M X 7,
- 1 + 7 + ^ — 3 + 7
whose modukis is 7; are expressed by
2
and ~ — — , 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 x" — 1 = as resulting from the expression for
those of the congruence or" — 1 = M (/>), when M = ; and
we thus are enabled to infer, in as much as M (/>), its multiples
and powers, are involved in those formulse, 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 (/?) is restored, or when 1 is
replaced by 1 + M (p). When the congruence admits of in-
tegral values of jr, 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,
^p-' — 1 = M (^),
where /> is a prime number, will be satisfied by all the natural
numbers 1, 2, 3, . . as far as (^ — 1) : and it follows, therefore,
that all the rational roots of the equation
a7» — 1 = M (j9)
will be common to the equation
x^-i - 1 = M (^),
the number of them being equal to {d), the greatest common
divisor of w and oi p — \. If <^ be 1, then all the roots except
1 are irrational. If we suppose the equation to be
tp - 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
connecting the theory of indeterminate with that of ordinary
1833. 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 del Nmneri, and in his Memoir es de
Mathdmatiqtie 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 symbohcally expressed. He has shown that
problems which have been usually considered as indeter-
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 jKiust 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. 323
of the progress and existing state of the diiFerent 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 dilFer 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 equar
* 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 diflScult 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.
y2
324 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 opei'ations upon
which their determination depends.
If we svqjpose, therefore, the equal roots to be thus separated
from the equation to be solved, and if we assume a quantity
J which is less than the least difference of the unequal roots,
then the svibstitution of the terms of the series
k A,{Ji — \) A, . . . , 2 A, A, 0, - A, -2 A, ... . - k^ A,
where ^ J is greater than the gi*eatest root, and — /'i A 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 v/here
the pairs of consecutive terms of the series of multiples of A
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 J.
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 I greater
than the greatest root of this transformed equation f, then— ;^
* A negative root is always considered as less than a positive root, unless
the consideration of the signs of affection is expressly excluded.
t Newton proposed for this purpose the formation of the equation whose
roots are x — e, and v/here e is determined by trial of such a magnitude that
all the coeflScients of the equation may become positive. In such a case e is the
limit required. Maclaurin proved that the same property would belong to the
greatest negative coeflBcient of the equation increased by 1. Cauchy, in his
Cours d' Analyse, Note iii., and in his Exercices ties Mathematiques, has shown
that if the coefficients of the equation, without reference to their sign, be
Ai A2, . . Am, and \f n be the number of such coefficients which are different
from zero, then that the greatest of the quantities
\_
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 A 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 in a 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 A 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, k to be the superior limit
required, and a and b to represent any two roots of the equa-
tion, whether real or imaginary, then he has shown that their
difference a — b, 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 Avnales des Mathematiques, 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 vnity 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 x^ + 11 x^ — 10 aS _ 26 a;-! + 31 a.3 + 72 a;2 — 230 x — 348 = 0,
the number 4, which is equal to the greatest of the quantities
1+2^,1 + ^.1 + ^,1 + ^.
13 13 11(3' 116
is a superior limit required ; and if we change the signs of the alternate terms,
we shall have 1 + -— , or 7, a superior limit of the roots of the resulting
equation : 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 with
large numerical coefficients.
326 THIRD REPORT — 1833.
greater than »(»-i) :, if n denote the dimensions of the
equation ; and in as much as H is necessarily, when the coeffi-
cients are whole numbers, either equal to or greater than 1, it
1
will follow that «(n-i) , will furnish a proper value of J,
where k has been determined by the methods described above,
or in any other manner. The chief objection to the use of a
value of J 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 J, 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 J 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 Cartes J gave a limit to the
number of positive and negative roots, but failed in deter-
* Resolution des Equations Numeriques, Note iv.
t Jhid., Introduction.
X The proper enunciation of this theorem is the following : "Every equa-
tion has at least as many changes of sign from + to — and from — to +
as it has real and positive roots, and at least 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 ofx + e for x would show the number
of roots which are included between 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 rio 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 = to be the equation, and X', X", X''', X'% X\ &c., to
denote the successive differential coefiicients of X, then, if all
the roots of X = be real, the roots of the several derivative
equations X' = 0, X" = 0, X"' = 0, &c., must be real like-
wise ; and if the roots of any one of these equations X^*"' =
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
t/ = XW . XC-^' and X'»-'' = 0,
1/ - X(»->^ . X(» 3) and XC- ) = 0,
y = Xi X"' and X" = 0, y = X X'' 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 Me-
moires de I'Academie 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 Resolution des Equations Nu-
meriques. 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 Crelle'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 — 1833.
a collection of conditions of the reality of the roots of an equa-
tion of n dimensions which are „ 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 a: + « and
X + b, gave no certain indications of the nature and number of
the roots included between a and b, 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 pi'ofound and philosophical writers on analysis
and physical science in modern times, to the examination of the
* Resolution des Equations Numeriques, Note viii. The equation of the
squares of the differences of the roots of an equation will indicate the reality
71 \7l ^^ 1 \
of all the roots, if its coefficients have ^ changes of sign, or be alter-
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 I'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
«" + p^ .t«-i + . . . . pn)y 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.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 329
succession of signs of the function X and its derivatives, upon
the substitution of different values of x. 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 = .r™ + a I a:""-' + a^ x"'-^ + . . .a^ = 0,
and if we write X and its derivatives in the following order,
XW X^'"-'\ x(™-2), . . . X", X', X,
then the substitution of yr and ^, will give two series of re-
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 -^, we put any
limit («) greater than the greatest root of the equation X = 0, and
if in the place of — ^ we substitute any negative value of
a' (— /3) (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 intei'mediate to those extreme values — /3 and a, all
the ?« changes of sign of X and its derivatives, from + to —
and from — to + , will disappear, in conformity wath the fol-
lowing theorems, which are capable of strict demonstration.
1st. If, upon the substitution of any value o{ x, one or more
changes of signs disappear, those changes are not recoverable
by the substitution of any greater value of x.
2nd. If upon the substitution of two values a and b of .r,
one change of signs disappears, there is one real root and no
more included between a and b. If under the same circum-
stances an odd number 2 p + \ of changes of sign have disap-
peared, there must be at least one, and there may he 2 p' + I
(where p' is not greater than 2^) I'eal roots between a and b ;
but if an even number 2 p of signs have disappeared in the in-
terval, there ma?/ 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 b, then no root whatever of the equation X =
can be found between the limits a and b.
3rd. If the substitution of a value a o{ x makes X = 0, then
a is a root of the equation. If the substitution of the same
value of X makes at the same time X = 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 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 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. If the substitution of a makes any
number of consecutive derivative functions equal to 0, then, if
there be an even number 2/> of consecutive zeros, there will be^
or (^ — 1) pairs of imaginary roots corresponding, according as
they are placed between the same or different signs ; and if there
be an odd number 9,p -\- \ of consecutive zeros, then there will
be^ + \ o\ ]i 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 for or 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 — /3
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 a. and cannot ex-
ceed the number of changes of sign corresponding to o^ = 0, or
amongst the successive coefficients of the equation ; and that the
number of real and therefore negative roots between — /3 and
cannot exceed the number of permanences corresponding to
a: = 0, or of changes between and — /3, which is also identical
with the number of successive permanences of sign amongst the
coefficients of the equation.
* I 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 X', X", X'",
&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 \\\s Analyse des Equations Deter-
minees, 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 = x'* — 4<r^ — 3x+ 23 = 0, and underneath X'",
X'", X", X', 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 :
X', X,
- +
- +
X-,
X"',
X"
(0)
+
—
(1)
+
—
(^)
+
+
(3)
+
+
+
(10)
+
+
+
+ +
For X = 0, there is a result placed between two similar
signs ; there is therefore a pair of imaginary roots correspond-
ing to it. Every value of x less than will give results alter-
nately + and — , and there is therefore no real negative root.
For X = I, there is a result 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 to 1, there is no real root between those values.
For X = 2, there is a result placed between two dissimilar
signs ; there is therefore no pair of imaginary roots correspond-
ing, and there is no root between 1 and 2.
For X = 3, there is a loss of one change of sign, and there is
therefore one real root between 2 and 3.
For jr = 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 = ^6 _ 12 > + eOx'^ + 123 .r^ + 4567 x - 89012 =
x^S
x^
x^
X"S
X",
x\
X,
(-10)
+
—
+
—
+
~
+
(-1)
+
—
+
—
+
—
—
(0)
+
—
+
+
+
—
0)
+
—
+
+
+
+
—
(10)
+
+
+
+
+
+
+
All tlie 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 laetween two similar signs ;
there is therefore a pair of imaginary roots cori'esponding, and
consequently a loss of two changes of sign.
There is no root of the equation between 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 —ly^ and {m — r — S)* order from X'*"^, as well
as the wj"' and {tn — 1)* derivatives from X, and that the same
rules may be applied to the separation of the roots of tliese 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 b 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 b, 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, 1, 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,
i{ X. = x'^ — x^ + 4< x^ + X — 4) — 0, then from the scheme
X'% X"', X", X', X,
(-10) + - + _ +
1
(-1)
+
—
+
—
—
1
(0)
+
—
-1-
+
—
1
2
2
3
(I)
+
+
+
+
+
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 j
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 333
that there is one root of X' = 0, and no root of X =: 0, between
— 1 and ; that there is one root of X"' = 0, two roots of
X" = 0, two roots of X' = 0, and"three roots of X = 0, situated
between 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 v/hich 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 X('- + ^), X('^ X^'--'),
will be
(a) + - + or - + -
12 12
(b) + + + - _ _
There will be one real root between a and b in the equation
X' = or X^*"^ = 0, and two roots, whether real or imaginary,
corresponding to this interval, in X = or X^'"'^ = 0.
In the first case, if there be two real roots between a and b,
then the curve whose equation
is y = X =y (a;), where o a = a,
o b = b, a n =f(a), b m =
f (b), will cut the axis at the
points a and ^ between a and b.
The curve will have no point
of inflection between a and b, ^ ^
since X" preserves the same sign, whether + or — ; and there
will be a point t, where the tangent is parallel to the axis,
since X', in the same interval, changes from + to — , or con-
versely, and therefore becomes equal to ^ero between those
limits. In this case, the sum of the subtangents (considered
without regard to algebraical signs) will be necessarily less than
a b ; and if the interval a 6 be subdivided sufficiently, so as to
furnish new limits a' and b', then one or both of these points
will sooner or later be found between the points of intersection
« and /3, and therefore/ («') and/ (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
roots, will be, that the sum of the subtangents or quotients *4-7^
./ («)
* We seek for the limits of the real roots only ; we have no concern with
those of the imaginary roots or of their moduli.
834 THIRD REPORT — 1833.
f (h\
+ fufi ^b — a, when no regard is paid to the sign of/' (a)
and f (b). In this case new Kmits must be taken successively,
intermediate to a and b, until f («') and J" (b') one or both of
them change their sign.
In the second case, if there be two imaginary roots cor-
responding to the interval p- „
between a and b, then the ^
curve whose equation is 2/=X
though similar in its other ge-
ometrical properties to fig. 1,
will not cut the axis between
a and b. In this case the sum of the subtangents a n' and
h nil will either exceed the interval a b, or will ultimately ex-
ceed it, when the interval a 6 is sufficiently diminished. The
corresponding analytical character will be that ^^r^. + ^irrr:
is either greater than 6 — a, or that it may ultimately be made
to exceed it *.
Thus, in the example refen*ed to above, p. 332, write down
the following scheme :
X", X-, X", X', X,
6 8
(0) + _ + + _
13 2 3
(1) + + + + +
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 a: = and x ^=\; then
8 V . . 8 14
we shall find -^ 7 1 and, a fortiori, therefore -pr + rn, 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 and 1.
If we suppose
X=a;5 + a;4 + a^-2x2-|-2jr-l = 0,
* The new values a' and 6' of a and h may be made a 4- ., \~{ and b — \^ ,J .
' f'(a) f{b)'
which are n' and m' respectively : a second trial will generally succeed.
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 335
the corresponding scheme will be as follows :
X, X,
x\
x\
96
x"s
42
X",
X-
(-1)
+
36
+
9
—
+
(-4)
+
—
+
—
+
1
2
2
2
(0)
+
+
+
—
+
24
6
4
2
1
2
a)
+
+
+
+
36
+
10
(1)
+
+
+
+
+
+
If we take the interval from (— 1) to 0, we find two roots in-
42 6
eluded within it ; but since q^ + ht, 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 H" to 0, which includes the same roots, we shall find
9 6 1.
^ + ^ = -^, a quantity equal to the whole interval, and we
are consequently authorized in concluding that the correspond-
ing roots are imaginary. In a similar manner, we find the in-
dication of the existence of two roots between and -^ ; and
2 1.
in as much as -j- = -^ = 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 KEPORT — 183^.
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 pi'eparation, 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 Resolution des Equa-
tions Numeriques, has shown in what manner the equation of
the squares of the differences may be apjilied 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 then' 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 Hmits 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
neai'est 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 f. 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 J. We shall now
proceed to give a short notice of these researches.
* Nouvelle Methode pour la Resolution des Equations Numiriques. It con-
tains the exposition of exceedingly simple and rapid rules for the formation
of the transformed equation whose unknown quantity is a; — 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.
t Resolution des Equations Numeriques, Note v.
X Analyse des Equations determinees, livr. ii., Calcul des Racines.
1833. z
338
THIRD REPORT — 1833.
1. If /(^) = 0, or X = be the equation, /' (x), f" {x\ or
X', X'' its first and second derivatives, then the limits a and h
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 h
must be further subdivided until this last condition is satisfied.
Under such circumstances there will be no root of the equa-
tions/' {x) — and/^' {x) = 0, included between a and h : and
if we suppose y — f ix) to be the equation of a paraboHc curve
CAB, where O a=- a,Oh =^ h,am =f{a), b n ^f{b), then
there will be no point of inflection between a and b, and no tan-
gent parallel to the axis. The analytical conditions above men-
tioned would show that /(«) and f{b) must necessarily have
difi'erent signs.
S. If we suppose b to represent the superior limit of the root
(a), then the Newtonian approximation gives us the new su-
perior limit b' = b — jrjj\ ; a new inferior limit will be found
to be a = « frnl '• these limits are still superior and inferior
limits of the root a, and are both of them nearer to it than the
primitive limits b and a.
If the same operation be repeated by replacing b and a in
f{b) and/(«) by b' 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 n b' to be a tangent to the curve at n, and a' m to be
drawn parallel to n b', then b b' = f'{by ^^^ « «' = f^Ty
since/' {b) = tan w 6' 6 = tan m a a. It follows, therefore, that
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 339
O h' and O a' are the new limits h' and d : and if ordinates h' n'
and a' m' be drawn to the curve, and n' b" be drawn a tangent,
and m' a" parallel to n' b", then O b" and O a" will be the new
values b" 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
f (x), f (x), f{x), in the transition from the inferior limit a to
the superior limit b, they will be found to be the following, it
being kept in mind that the sign oi f{x) alone changes from
+ to — , or conversely.
fix) fix) fix)
+ + -
(2){
(3){
(4){
+ + +
a — — +
b _ _ _
+ - +
+ - -
a - + —
— + +
In the first two cases, the formulae of approximation are
b — >, ,,, and a — ^4-Wj and commence therefore with the su-
perior limit. In the last two cases, the formulas of approxima-
tion are a — 4rW and b — ^-7-Kjandcommence therefore with
the inferior limit. In other words, that limit must in all cases
be selected which gives the same sign to /" (x) andy (^), whe-
ther + or — . The construction of the portions of the corre-
sponding parabolic curves included between a and b 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*.
• If, in the figure p. 338, we join the extremities m and n of the ordinates a m
and 6 « by the chord m N n, which cuts the axis of x in the point N, we shall
cjr^TvT f{a){b — a) , f{h){h—a) ... .
find O N= a — •' .)' ^ ., { = 6 — %)' „/ ( » "^^^^^ g'ves a new ap-
proximate inferior limit in the first two cases considered in the text, and a new
superior limit in the last two. Other constructions are noticed by Fourier,
which give similar results.
In the M^moires de I'Academie Royale de Bruxelles for 1 826, there is a
memoir on the resolution of numerical equations by Dandelin, in which the
analytical conditions which must be satisfied bvthe limit, towards which the
z 2
•1 'Q
340 THIRD REPORT — 1833.
4. In tlie application of these rules some precautions are oc-
casionally necessary. Thus, if/" (x) a.ndf{x) have a common
measure 9 (or), and if a root (a) of ip (x) = be included between
a and b, then there is a point of inflection of the parabolic arc
between a and b at the point of its intersection with the axis.
Under such circumstances, the method of approximation must
be applied to the equation f (x) = 0, and not to tlie primitive
equationy(a:') = 0, for the purpose of determining the value of «.
Again, if there exists a common measure of^"' {x) andf{x), which
becomes equal to zero, for a value of x between a and b, then
there ai-e two or more equal roots of f{x) = 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 — a and b' — «' by i and i', it may be very easily proved that
2f'{b) '
wherey" {a. . .b) denotes some value whichy" (x) assumes when
we substitute for x a quantity between a and b : and if we form
the quotient (C) which arises from dividing the greatest value
of/" («) and/" (6)* by the least value of 2f' («) and 2/' (6),
and suppose k 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 h' 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. Tlie first is the
subtraction of the subtangent b b' or ^ from b, as involved in the or-
dinary Newtonian approximation when the proper limit is selected. The
second is the determination of the value of O N, or a — f\V. ~f (\ > °^
f (b\ ~ f( \ ' ^^ ^^^ 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.
This 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/'" {x) = is included between a and b, it follows that
either/" (a) or /" (b) will be the greatest value of /" (n . . .h): the same
remark applies likewise to/' (a) and/' {l).
REPORT ON CERTAIN BRANCHES OF ANALYSIS. 341
late number * immediately greater than this quotient, and n the
order of the articulate or subarticulate number which is not less
than the difference of the limits b — a, then if we divide f{b)
by/' {b), and continue the operation as far as the {2n + kf^
decimal, and increase the last digit by 1, the quotient which
arises being subtracted from or added to, b, according as f{b)
and/' (b) have the same or different signs, will give a result
which will differ from the true value of the root by a quantity
less than ( tt^ ) • 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 {4>n + 3 kj^, the {8n + 7 kf^, &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
ordonn^e,) 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| 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,
t The course of the approximation, in order to be perfectly regular and rapid,
would require that 2 n + h should be greater than n, or that n should be
greater than — ^, a circumstance which might occur if A: or w was negative.
In such a case it will be necessary, or rather expedient, to subdivide the in-
terval b — a, until the difference of the two limits does not exceed ( — - ) ,
where « is equal to, or greater than, 1 — k.
I Analyse des Equations determinecs, livr. ii. p. 188.
343 THIRD UEPOIIT 1833.
Similar processes, also, have been investigated and applied
with remarkable ingenuity and success by Mr. Holdred *, Mr.
Horner f, and Mr. Nicholson J. The first of these writers, a
mathematician in humble life, vpho 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 aflPected 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 linear, 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 instrvictive 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.
f Philosophical Transactions for 1819.
I Essay oti Involution and Evolution. 1820.
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
determin^es, 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
2/ = 1 - ^ + 22 - WTS'^ "*■ ^~7WT¥ ~ ^^■'
which presents itself in the expression of the law of propagation
of heat in a solid cylinder f of infinite length, Fourier ventured
to apply the principles in question to show that all its roots were
real ; but M. Poisson J has disputed the propriety of such an
application, both in this case and in others : thus, if we suppose
X = 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 knovi^ledge of the contents of the other five books, which were
left unfinished, is derived from an Expose 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.
t Theorie de la Chaleiir, p. 372.
t Journal de I' Ecole Poly technique, cahier xix. p. 381; Memoires de I'ln-
st'Uut, torn. ix. p. 92.
344' THIRD RErORT — 1833
d'+'X
d''+^ X
dx
= e' — b a" e"*,
= e*' — 6a''+' e"'.
.71+2
= e'' — b a"+2 e"'^,
where w is any whole number, or zero. If we now suppose
</"+^X _
and eliminate, by means of this equation, e", we shall get
</»X
dx"" ~
d»+^X
and therefore
d'^X d"+^X
dx" ■ c?a;"+2
: - 6 (1 - a) a» e"^,
= 6 (1 -a)a»+'e«*,
= - J2(l _^)3^2»+lg2«^^
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"'' =
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 6 a" + 2 e TT '/^A
X — ^ ■
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 b, in a function f (x) and its derivatives, gives results
which present the same succession of signs between y*("+'') (x)
andy^"' (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 T/ieorie de la Chaleur.
REPOKT ON CERTAIN BKANCHES OF ANALYSIS. 345
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 betw'een the given limits: thus, \i 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 + 2 ■n-, although it is manifest that there are two
real roots of sin ^ = 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 a; = includes between the limits a and « + 2 tt, 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
the roots of the derivative functions , „ , ^ — -p, -r — r^ , he
has not included those of the factor e'^, which those functions
1 + — ) = 0, it follows that
there are an infinite number of equal roots (where a: = — oo )
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 appHcable. It would follow,
therefore, that the existence of imaginary roots in the equation
X = is no longer contradictory to Fourier's proposition, even
• If the transcendental function denoted by / (j;) be a determinate function,
it will always be possible to assign an interval 3, such that the derivative
function/" {x) = contains no root, or a determinate number of roots, be-
tween a and a + S. If such an interval or succession of intervals can be de-
termined for any one derivative function, such as /(") (.i), 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
of algebraical functions, the point of departure is that derivative function which
ie a constant quantity.
346 irilKi) niijfoRT — luutj.
admitting the correctness of that form of it which Poisson ha^
assigned*.
* If we transform e*' by replacing a; by pi, we shall get the expression
c •* , which may be easily shown as above, and also by other means, to be
equal to zero when x' 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
A a;" + B a^ + C ;r'' + &c. (1.)
where ct, /3, 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
y.-«. e~^^ = A + B xP'-"' + C xy-''+ &c„
supposing 86 the least of the several indices a, /3, y, &c., then if x =: 0, we
1
-~2 . 1
shall find x~'^ e *'=OorA equal to zero ; for if we replace — by y, we
shall get
1 i.
_i. =a;«e^' = y-«ey^
4— a 6— a
= 3,-« + /-« + y + _! + &c.,
^ " ^1.2 1.2.3
all whose terms are positive, and which, when a;=: or y = oo , will necessarily
become equal to infijiity : it follows, therefore, that the function e * 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 Lemons snr 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 ivfinity, 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 x 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-
nouUi *, and very extensively illustrated and applied by Euler f .
If we write down m ai-bitrary 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 Resolution des Equations
Numeriques, 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 /' {x) by / {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 e^'"' '^"^ = 1,
when ra is a whole number, we get g^ + ^ " '^ '^^ = e and therefore
consequently e""*" '^' = 1, whenever w is a whole number, — a conclusion
which M. Clausen characterizes as absurd. Its explanation involves no other
2»i sr \/ —\ . ,
difficulty than that which is included in the equation e =1, 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
arc 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.
t IntroducHo in Analysim Infinitorum, vol. i. cap. xvii.
348 THIRD REPORT — 1833.
+ &C.,
when a, /3, 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 hmited 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, b, 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 AD — BC, BE — CD,
C F — D E, &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 + b. Again, form a series whose terms are
A C - B2, B D - C2, C E - W, &c., which is also a recurring
series, whose successive quotients will approximate to the value
of the greatest product a b. In a 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 + c of the
three first roots ; in the second, the sum of their products two
or two, ox ab + a c -\- be; and in the third their continued
product a b c : 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
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
^3 _ 3 ^ _,. 2-0000001 = 0,
one of the roots is very nearly equal to 1, if we divide the
equation by a; — 1, and neglect the small remainder which re-
sults from the division, we shall get the quotient
x^ -X -2= {x -\){x + 2) = 0,
whose roots are 1 and — 2 ; or we may suppose one of the
roots to be 1*000 1, the second -9999, and the third —2; or
we may suppose two of the roots to be imaginary, namely,
1 + -0001 V — \. 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 ^c = 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*
«" ,^ .o «"'
a-a'Ya+ ^ (F «)^ - ^--^ (F a)^ + &c.
• Lagrange, li^sQlution des Equations Numeriques, Note xi.
350 THIRD REPORT — 1833.
where
«' =^,
a" = -
1_
Fa
a! F" a F"
(F' of ~ (F' a)3
„, _ _ cH F" a 3 a' (F" of
Wa 3 (F'^ af
~ (F «)4 + (F a)^ •
This series was first assigned by Euler, and the observations
which we have had occasion to make in the preceding pages
upon hnear 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 least 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 ip {x) — 0,
divide the equation by x, and take the Napierian logarithm of
the quotient which arises ; then the coefficient of — 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
a;2 4. a a; + 6 = 0,
find the coefficient, with its sign changed, of — in log ^^
(b\
1 + X \, which is
\ b b^ 4 63 Q.5 b^ 8.7.6 ^ , o \
KEPORT ON CERTAIN BRANCHES OF ANALYSIS, 351
and therefoi-e identical with that which arises from the deve-
lopement oi — -^ + V\-r ~ ^)- If^be greater than -^, 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 <p (a;) =
may be found " by subtracting from / (0) the coefficient of
— m f (x) log ?-LJ," This more general theorem evidently
includes the former.
" The sum of any assigned number {ni) of roots of the equa-
tion <p {x) = is equal to the coefficient, with its sign changed,
of — m log ^-^.
X ^ x""
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^^, Xci, ^3, . . ^TO be the m least roots of the equation
(x - a)"' - hF (x) = 0,
d--^{f(a)F{a)}
= "^f^""^ + ^' l.2..{m-\)da--^
Jl_ d^"^-^ {f {a) {¥ {a)Y}
*■ 1 .2' 1.2 {2m-\)da^'"-' "^ ^'
If in this very general theorem we make w = 1, it becomes
the theorem of Lagrange ; and if we make m equal to the di-
mensions of the equation, or greater than ^y power of x 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 oi f{x), 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,
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 ai-e 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, /o»-r(r) r(l»-) reorfr(r) r(l — r)
— 215, — 11 from the bottom,*. jfor a;^:^- 2 + a; 1 read a:^ + 2 a; + 1
71 "' 71
— 221, — 15, /or cos i — read cos- 1 —
e e
cos-i a , , a
— 226, — 2 from the bottom, for / , , ,, read cos-i , „ , ,„
Va^ + o2 Va^ + 62
— 234, — 7, for (0) read (0)»
— 240, — 18, for In this last case (a — 6)" read If we suppose n 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 diventkal read diacritical
— 251, — 11, for cos -{x — i — ^ — '- r r read cos -j a; — ^—^ — - > r
— 259, — 23, dele or the greatest of the two quantities ct and /3
4 4
— 261, — 14, /or — read —
— 263, — 15. for X' read X
— 316. — 29, /or*"— 1 =Oreorfa;" — 1 =0.
Ji,-ri"t ,'/ the ni-it. Assoc. VoIJ. p.3o3 .
'SiJ^/^or' 0€<^1^^
^ppOyraALdfor tkc Conyore^s'ion o.
D^ ^ C
E^A ^-^B
H
WBrown^/cuJ/D .
' [ 353 ]
TRANSACTIONS OF THE SECTIONS.
1. MATHEMATICS AND PHYSICS.
On the CotnpressihiUty of Water. By Professor CErstf.d.
{From a Letter to the iiet;. WilliamWhewell, dated Copen-
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-
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
illusti'ious 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 pubHshed 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
oj 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 pubhshed in 1822, can give very
accurate 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-
surements, however, one of the most important requisites oi
such an apparatus is, that the experiments be performed with
the greatest celerity possible. When the experiment is pro-
tracted, the change of temperature produces great variations in
the volume of the water, -rlw of the thermical measure (1° cenr
1833. 2 a
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 B C D is a strong
glass cylinder, having at the top a cylinder E F G H, contain-
ing a piston I m n o, moved by a screw K K, as in the first
apparatus ; but the handle 1 1 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 c c c, 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 efg h is divided into parts of Vu 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 a a is
covered with a small diving-bell, or rather diving-cap, ppp,
whose conical shape has the advantage of preventing the water
from reaching the top of the tube a a, 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, c di& 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. ^ m s is a siphon ; P, a vessel containing water ;
i i are two buoys of cork for lifting up the bottle and the glass
tube cd; s r 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 -^ 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 1^ 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.
T 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.
355
has already observed that this is not the case. When the large
cyHnder is filled with water, the bottle is to be immersed in it.
If the tube a « 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 c 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 G H. If the large cylinder is full of water, part of
it will be expelled through the siphon t u s. Now the piston is
to be lifted up by means of the screw, whereby the pumping
cylinder is filled with w^ater. When this is done, the siphon is
taken away, and the tube r * 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 c d : 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 c 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.
Before the pres-
sure.
Mean.
When the pressure
has ceased.
Length of descent
in the capillary
tube.
Point to which the
pressure has driven
the water down.
248-9
248-95
249-0
50-35
198-6
249-0
249-2
249-4
50-20
199-0
249-4
249-7
250-0
49-90
199-8
250-0
250-5
251-0
50-10
200-4
251-0
251-5
252-0
49-90
201-6
252-0
252-65
253-3
49-85
202-8
253-3
254-1
254-9
49-90
204-2
254-9
255-5
256-1
49-70
205-8
256-1
256-95
257-8
49-95
207-0
257-8
258-4
259-0
49-80
208-6
259.0
259-5
260-0
49-70
209-8
Mean of descents =
49-96.
2 A 2
356 THIUD RfiPORT — 1833,
" The height of the mercury in the barometer, reduced to the
freezing point, was at the same time 332'36 French hnes. 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-
sphei-es. The pressure reduced to hnes of mercury is thus,
332-36 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 en(,l 0-2025 Th., by the thermometer. The
water stood 10*1 parts, about 35 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 0-0005 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 c 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 measui-ed by weights of mercury. WHien
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 measui'e, 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 mercury
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 miUionths. At 0*09125
Th. the volume of the water augments 71*75 miUionths, having
its temperature augmented 0*01 Th. The heat developed by the
71*75
pressure thus augments its volume 0*00025 . ^^, = 1 "79 mil
lionth, or the recession is 46*77 — 1
0*01
79 = 44*98 miUionths.
Actual experiments have given it 44*89, or 0*09 millionth
* PART OF STAMPFER'S TABLE.
Temperatures ac-
cording to the
centigrade ther-
Volumes of the
water.
Differences.
mometer.
-3
1-000373
2
1*000269
104
1
1*000182
87
1*000113
69
+ 1
1-000061
52
2
1-000025
36
3
1-000005
20
375
1-000000
5
4
1*000001
1.
5
1-000012
11
6
1*000038
26
7
1*000079
41
8
1*000135
56
9
1-000205
70
10
1-000289
84
11
1-000387
98
12
1-000497
110
13
1-000620
123
14
1-000757
137
15
1000906
149
16
1-001066
160
17
1-001239
173
18
1001422
183
19
1-001617
195
20
1-001822
205
21
100^'039
217
22
1-002265
226
23
1-002502
237
24
1-002749
2v7
25
1 003005
256
26
1-003271
266
27
1-003545
274
28
1-003828 1
283
29
1004119
291
30
1'0044]8
299
358 THIHU RKJPORT 1833.
greater. The coincidence is often less perfect. At 0'1775 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 = ] *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. CoUadon 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 •05 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 mei*-
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 toTredgold, 20*45 milUonths 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 I'o 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 v^ater in the capillary tube.
If the dilatation of a rod of glass by 1 Th. is 0-0009, its cubical
dilatation is 0*0027, and the dilatation by an increase of 0-00025
is 0-000000675, or nearly 7 ten miUionths. 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 a little more, yet
the quantity is not great. After all this, I think that the true
compressibility of water is about 46*1 miUionths, 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. I do 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 lai'ge 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 FGH I 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 A B C D, 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
SQd 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.
" I 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, ^y William R. Hamilton, 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, 1st, 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 laic, 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 bf propagation of the light, for
any proposed combination.
For example, in the case of a single plane mirror, supposed to
coincide with the plane of ,r y, we may propose to determine the
laws of the two extreme directions of the hnear path by which
light goes to an eye {x y %) from an object {x^ y' z'), or (ex-
pressing the same thing more fully,) to determine the final co-
sines a /3 y, and the initial cosines «' /3' y', of the inclinations of
TRANSACTIONS OF THE SECTIONS. 361
this bent path to the positive semiaxes of coordinates, as func-
tions oi X y ss, a^ y' z', 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 foUoM^ing
equations ;
l\ „ 8 V 8 V 1
« = "s — , p = "5^ — , y = ^^ — 5
ox ^ y OS I
,__^ _8V , rv\ ^^'^
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
V = \^(x- x'f + (y - y'f + (^ + %'f, (2.)
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 directions-
functions or direction-cosines, a^y, a^ /3' 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 fnd 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 estabHsh, for such using, a system of general formulae, and has
deduced 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
S62 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 in vacuo, symmetric in all re-
spects, about one axis of revolution, we may take this for the
axis of ^, and we shall still have the equations (1.), the charac-
teristic fmiction V being now a function of the five quantities,
x^ + y^) X x' + y y', x''^ + y'^, s:, z', 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,
T = ao;' + /S?/ + y« - «'^' -|3'y - y's' — 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? + ^^, a a' + ^ |3', u'^ + /3'^; 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^o) + T('> + T^"), (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(2) = P («' + /3^) -h P, (a a' + /3 /3') + P' (a'2 + ^'«) ; (5.)
and T^*^ is small of the fourth order, and is of the form
T^") = Q{u' + ^r + Q; (a^ + /3^) (« «' + ^ /3') + Q'(a^ + /3^) (a'2 + ^'2) 1
+ Q„(«a'+/3^')'+Q'/(««'+/3/3')(«''+/3") + Q"(«'' + /3T; r ''
the nine coefficients, P P, F 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. .16