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I
-» V ^
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M^w \onK
AJATnEMATlCAL SOCIETY:
Of MAI'MiniAI'Ct Al. Si-IfcXciE.
^ ' " i. .NO, I.
BULLETIN OF THE
NEW YORK
MATHEMATICAL SOCIETY.
A Historical and Critical Review
OF Mathematical Science.
EDITED BT
Thomas S. Fiske and Harold Jacoby
GOMMITTEE OF PUBLICATION.
VOL. I.
October 1891 to July 1892.
NEW YORK:
PUBLISHED BY THE SOCIETY,
41 East 49th Street.
1892.
PreM of J. J. Little A Co.
Afltor Place, New York
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PREFACE.
In presenting this^ the first volame of the Bulletin of
THE New York Mathematical Society, the editors feel
that they should express their deep sense of the obligation
under which the encouragement and active assistance of the
members of the Society have placed them. The interest which
the Journal has excited, both in this country and abroad,
shows how real is the need of a historical and critical journal,
devoted to mathematical science, and published in the Eng-
lish language. It is to be hoped that in the future the Bul-
letin may be able to extend its work, and ever the more
adequately occupy the field which it is its aim to cover.
LIST OF PUBLISHED PAPERS
BEAD BBFOBE THE NEW YOBK MATHEMATICAL SOCIETY,
TOGETHEB WITH PLACE OF PUBLICATION.
B0CHEB9 Maxime. Gollineation as a Mode of Motion. Bul-
letin of the New York Mathematical Society, toI. i.,
pp. 225-231, July, 1892.
Ekgleb, Edmund A. A Geometrical Construction for Find-
ing the Foci of the Sections of a Cone of BeToIution.
Transactions of the Academy of Science of St. Louis,
vol. VI., pp. 49-65, April, 1892.
Fisee, Thomas S. Weierstrass's Elliptic Integral. Annals
of Mathematics, vol. vi., pp. 7-11, June, 1891.
On the Doubly Infinite R^ucts. Bulletin of the New
York Mathematical Society, vol. i., pp. 61-66, Dec,
1891.
Jacobt, Harold. On the Determination of Azimuth by
Elongations of Polaris. Monthly Notices of the Royal
Astronomical Society, jpp. 106-113, Dec., l691.
Macfablakb, Alexander. On Exact Analvsis as the Basis
of Language. Bulletin of the New York Mathemati-
cal Society, vol. i., pp. 189-193, May, 1892.
McOlintock, ifimory. On the Algebraic Proof of a Certain
Series. American Journal of Mathematics, vol. xiv.,
pp. 67-71, Oct., 1891.
On Independent Definitions of the Functions log x
and ^. American Journal of Mathematics, vol. xiv.,
pp. 72-86, Oct., 1891.
Mebbiman, Mansfield. Final Formulas for the Solution of
Quartic Equations. Bulletin of the New York Mathe-
matical Society, vol. i., pp. 20S-205, June, 1892.
Pupisr, M. I. On a Peculiar Family of Complex Harmonics.
Transactions of the American Institute of Electrical
Engineers, vol. viii., pp. 579-585, Dec, 1891.
Steinmetz, Charles P. Multivdent and Univalent Involu-
tory Correspondences in a Plane Determined by a Net
of Curves of the n-ih. Order. American Journal of
Mathematics, vol. xiv., pp. 39-66, Oct., 1891.
On the Curves which are Self-reciprocal in a Linear
Kul-system, and their Configurations in Space. Amer-
ican Journal of Mathematics, vol. xiv., pp. 161-186,
April, 1892.
Stbinoham, Irving. A Classification of Logarithmic Sys-
tems. American Journal of Mathematics, vol. xiv.,
pp. 187-194, April, 1892.
BULLETIl^ OF THE
NEW YORK MATHEMATICAL SOCIETY
OCTONART NUMERATION,
BY PROP. W. WObLSBT JOHNSON.
The comparatively small progress toward universal accept-
ance made by the metric system seems to bo dne not alto-
gether to aversion to a change of units, but also to a sort of
irrepressible conflict between the decimal and binary systems
of subdivision.
Before the introduction of decimal fractions, about 1585,
no connection would be felt to exist between the established
scale of numeration and the method of subdividing physical
units, and it would probably never occur to any one to sub-
divide a unit into tenths. The natural method is to bisect
again and again. The mechanic prefers to divide the inch
into halves, quarters, eighths, and sixteenths. The retailer
of dry goods, whose unit is the yard, divides it into halves,
quarters, and eighths, totally ignoring the inch. The mar-
iner not only divides the horizontal angular space in which
his course is laid down into quarters, thus recognizing the
right angle as the natural unit,* but divides the space be-
tween the cardinal points of the compass into halves, quarters,
and eighths. Where decimal money has been introduced
quarters are insisted on in spite of their inconsistency with
tne decimal system. We are compelled to coin quarter dol-
lars, and prices are very commonly c[uoted in eighths and even
sixteenths of a dollar. Great Britain is compelled to coin
eighths of a pound sterling, though half a crown contains a
fraction of a shilling. The French divide the litre into
quarters. The broker expresses prices in halves, quarters,
and eighths of one per cent.
This irrepressible conflict would, of course, never have
* The uncompromising advocate of the metric sjrstem will not content
himself with the centesimal division of the degree, but insists upon the
centesimal division of the quadrant, although it is difficult to see now the
latter could possess any advantage in the way of facilitating numerical
computations. But why do they not go further and advocate the centesi-
mal division of the whole circumference ?
2 OCTONARY NUMERATION.
existed, but all would have been harmouy, if the radix of our
system of numeration had been a power of two. Mr. Alfred
S. Taylor published in 1887 a very interesting pamphlet on
"Octonary Numeration/' being a paper read before the
American Philosophical Society. After an extended review
of the question, with many interesting historical notes, he
argues in favor of the octonary system, and then proceeds to
** develop the scale of notation thus selected, and to derive
from it an ideal system of weights and measures/'
This is not the place to consider the merits of a system
of weights and measures ; we propose therefore to consider
only the theoretical merits of the octonary system. We regret
that in his ingenious paper Mr. Taylor has caused his system
to wear an outlandish look by employing new names, not only
for his units of weights and measures, out also for the num-
bers from one to eight, and even new characters for the seven
digits. We see no necessity for changing the characters or
the names of the digits, although it would bo necessary, in
order to avoid the use of an old name in a new meaning, to
replace the suffix -ty by a new one to denote the second place
fwhich Mr. Taylor, havingchanged the names of the digits,
aid not find necessary). Wo might use the suffix -ate — thus
the octonary 40 would be read, jourate, that is, four eights ;
66 would bo Teadfivate-six, that is five eights and six.
The only advantage of a large radix, qud large, over a
smaller one is in diminishing the number of figi^^es reauired
on the average to express a given number. The number of
figures is inversely as the logarithm of the radix ; and, in
passing from the mdix ten to the radix eight, it increases only
in the ratio of 10 : 9. The ratio increases rapidly for smaller
radices, until for the binary system it becomes 10 : 3, as com-
pared with the denary, and 3 : 1 as compared with the octonary
system. To set against this we have, m favor of the smaller
rJftdix, the simplicity due to dispensing with superfluous char-
acters ; but of far more importance is the simplifying of the
multiplication table. For example, the octonary multiplica-
tion table stands thus :
1
2
3
4
6
6
7
2
4
6
10
12
14
16
3
6
11
14
17
22
25
4
10
14
20
24
30
34
5
12
17
24
31
30
43
6
14
22
30
36
44
52
7
16
25
34
43
52
61
In. comparing the labor with which this table could be
committed to memory with that required by the denary
OCTONABY NUMERATION'. 8
table^ it would seem fair to disregard in both cases not only
the line and column corresponding to 1 (although our Ger-
man friends insist upon ein mal eins), but also those cor-
responding to 2 and to half the radix, on account of their
simplicity. Thus the difficulty would be about as 6^ to 4* ;
indeed, it seems safe to say that the difficulty experienced by
children in acquiring the multiplication table, and that of
older people in retaining it in a condition fresh enough to
be used without an effort of thought, would be reduced more
than one-half even by this slight decrease in the magnitude
of the radix.
For a further decrease of radix, the difficulty of the mul-
tiplication table decreases rapidly : for the binary system no
multiplication table exists, but even for the radix four the
difficulty has practically disappeared.
But this advantage of a very small radix is, as mentioned
above, attended by a rapid increase in the number of figures
required to express a given number ; and the inconvenience
arising from this source has, we think, been frequently under-
estimated. Binary arithmetic, in which the characters 1 and
alone are used, has even been proposed by some enthusiasts
as a substitute for logarithmic computation. Mr. Taylor, in
commenting upon this system, after mentioning the absence
of tables to be committed to and retained in the memory,
says: ** Every form of calculation would be resolved into
simple numeration and notation. In fact, calculation as an
effort of mathematical thought might be said to be entirely
dispensed with, and the labor of the brain to be all trans-
ferred to the eye and hand. A perfect familiarity with the
notation of the scale, and with the simple rales of position,
would enable the operator to determine in every case by mere
inspection, whether the next figure should be a 1 or a 0. It
follows that the only errors possible in such a work would
be the merely clerical ones of the eye or hand; ♦ * *
and it may well be doubted whether, in all important and
lengthy calculations, the binary system would not be found
to afford a real economy of labor, instead of an increase as
has been generally supposed. ''
Now it is to be remarked that the number of figures used
in calculation would increase at a rate much greater than
that of the number of figures used in expressing results. For
example, in performing a multiplication in the binaiy nota-
tion, the number of figures to be written down (after making
due allowance for the greater proportion saved by the occur-
rence of ciphers in the multiplier) would bo about five times,
instead of three times, the number occurring in the same
operation performed in the octonary notation.
Again^ whenever the columns to be added are of consider-
4 OCTONABY KUMEBATION.
able length their summation, though executed by mere count-
ing and the determination of the numbers " to carry/* would
require fixed attention, and inyolve liability to error ; so
much so, that the words we have itah'cized in the quotation
appear hardly iustified. The numbers to carry would be
inconveniently large, especially if mentally expressed in the
binary system. Indeed, counting in this system would
obyiously be very much more liable to error than in the
denary (or in the octonary) system, which gives highly dis-
tinctive names to all such numbers as have to be carried in
the mind in the course of calculation.
The same objection exists, though to a less degree, to the
(quaternary system, so that the labor of accurate calculation
in this system, although perhaps less than in the denary sys-
tem, would probably exceed that which would be required in
the octonary s;^stem.
The conclusion appears to be inevitable that, considering
only the two features which depend upon the mere size of
the radix, ten is decidedly too large and four too small a
radix, so that the ideal radix in this respect is about six
or eight.
Passing now to the intrinsic character of the ra(}ix, it is
desirable that the radix should be divisible by simple factors.
Thus it is universally admitted that an uneven radix would
be quite out of the question. It is indispensable for a multi-
tude of purposes that even the least instructed persons should
be familiar with the distinction between even and uneven
numbers, and able to recognize at a glance to which class a
given number belongs. It was formerly the custom to extol
twelve as an ideal radix, because of its divisibility by two,
three, four, and six. Divisibility by three, although incom-
parably less important than divisibility by two, would no
doubt be a great convenience, much more so than divisi-
bility by five ; but it is doubtful whether much weight
should be given to divisibility of the first power of the
radix by four, so long as wo do not adopt a purely binary
system (that is, two or a power of two for radix). We
ought rather to consider only the prime factors of the radix,
so that six would possess all the advantages of twelve, and
since on the other score twelve is far too largo a radix, six
would be far preferable to it. (The number oi figures used
to express a given number would for six exceed that for
twelve only in the ratio 7:5, and would exceed that for ten
onlv in the ratio 9:7.)
Against this advantage of divisibility by different i>rime
factors we have to set the advantages of a purely binary
system. Theoretical considerations here point in the same
direction as the practical ones rehearsed in the first part of
OOTONARY KUMBRATION. 5
this paper. Owing to the unique character of the number two,
it mast be admitted that the expression of a given number
in powers of two gives a better notion of its intrinsic character
than expression in powers of any other number. Accordingly
the binary system has always been regarded as theoretically
the ideal system, although for practical purposes the great
number of figures used in expressing numbers is an msu-
perable objection. Now it is to be noticed that if the radix
IS a power of two, we have virtually all the advantages of the
binary system. For example, if we have a number expressed
in the octonary system, we have only to substitute for the
characters 0, 1, 2, .... 7 their binary equivalents 000, 001,
010, 111 to obtain the number in the binaiy system.
The digital expression of a number in the octonary system
would be much more suggestive of its intrinsic nature than
expression in any non-binary system, for the highest power
of two contained as a factor m a number is its most important
characteristic. Again the distinction between numbers of the
form ^n + 1 and those of the form 4;i + 3 is of great import-
ance in the theory of numbers, and in the octonary system it
would be obvious at a glance to which of these classes a given
uneven number belongs. So also with the distinction between
** evenly even '* and ** unevenly even " numbers. It is inter-
esting also to note that the square of every uneven number
would end in 1, the preceding figures expressing a triangular
number. Thus the uneven squares in octonary notation are
1, 11, 31, 61, 121....
We have seen above that, if divisibility by another prime
factor besides two bo regarded as the paramount desideratum,
six would be preferable to ten as a radix. But the tests of
divisibility by small divisors (such as the familiar one for nine
or three) would always to a great extent serve the same purpose
as the divisibility of the radix. These tests depend upon the
lowest value of n for which r" — 1 or r" + 1 (r being the radix)
is divisible by the divisor in question ; and they consist in
reducing the given number to one of n places which will
give the same remainder when divided by the given divisor.
This is done in the firat case by the addition of periods of n
figures each, beginning with the units ; and in the second
case, by the addition oi periods of 2n figures, followed by sub-
traction of the second period of n figures from the first. For
example, with the radix ten we can test for each of the divi-
sors seven, eleven, and thirteen, which are factors of 10* + 1,
by reducing to six places by addition of periods of six, and
then to three places by subtraction of the figures represent-
ing thousands from the first or unit period of three figures.
Let us see how the matter would stand in the octonaiy sys-
tem. For seven we should add all the digits^ and for mne
6 TEACHING OF QEOMETRY IN GERMAN SCHOOLS.
(and therefore for three) we should add by periods of two.
Again since 8' + 1 = 6 x 13, wo should test for five and
thirteen (or oneate-five) by reducing to four figures by addi-
tion, and then to two figures by subtraction. Among small
primes, eleven is the least adapted to the octonary system,
but for this divisor we might convert the given number to the
binary system, then reduce to ten figures by addition, and to
five by subtraction (since 2* + 1 = 3 x 11), and finally recon-
vert into an octonary number of two digits.
As there is no doubt that onr ancestors originated the deci-
mal system by counting on their fingers, we must, in view of
the merits of the octonary system, feel profound regret that
they should have perversely counted their thumbs, although
nature had differentiated them from the fingers sufiScieutly,
she might have thought, to save the race from this error.
THE TEACHING OF ELEMENTARY GEOMETRY
IN GERMAN SCHOOLS.
Inhalt und Methode des planimetriachen Unterrichts. Eine
verglcichende Planimetrie. Voo Dr. Heinrich Schotten. Leip-
zig, B. G. Teubner, 1890. 8vo, pp. iv. + 870.
Whoever has followed the efforts of the Association for
the Improvement of Geometrical Teaching in England in
the course of the last ten years will have been struck by the
slowness of the progress made and the paucity of the practical
results attained. In Germany there exists no such society ;
but a powerful agitation for the reform of geometrical teacn-
ing has been in progress there for at least sixty years, ard
with particular force during the last two decades. And yet,
even from Germany, with its well developed and highly
centralized system of education, comes the complaint that
progress is slow and much remains to bo done.
Recent statistics have shown, in particular, that the most
widely used text-books are far from being the best. Thus,
while Hubert Miiller^s Geometry, which may be regarded as
the best representative of the ** modern school, '' reached its
•third edition in 1889, after a lapse of fifteen years from its
first appearance, Kambly^s very inferior text-book, whose
faults and mistakes have frequently been exposed and com-
plained of, appeared in 1884 in its 74th edition.
This book of Kambly's easily leads in the list of text-books
used in various schools ; it is adopted in 217 schools, the
next in order being another rather inferior book^ by Koppe,
TBACHIKQ OF GEOMETRY IN GERMAN SCHOOLS. 7
introduced in 51 schools ; then follow Mehler's, nsed in 44,
Beidt's in 29, etc., while there are 55 mathematical text-
hooks used in bat one school each. Similar statistics for onr
American schools would be both interesting and instructive.
Still the signs of improvement are not wanting. Some
very good text-books of geometry have been published in
recent years and are making, though slowly, their way to
the front; preparatory CpropsBdeutic '') courses in 'in-
tuitive '' geometry in connection with geometrical drawing
have been introduced in many schools, and are generally
recommended by the school boards ; excellent classified and
graded collections of problems have appeared and are in
actual use ; and, above all, the whole subject of the improve-
ment of geometrical teaching has been ventilated and dis-
cussed with great thoroughness and completeness.
In this last respect the work done by Honmann's Zeitschrift^
cannot be estimated too highly, 'fhe volumes of this jour-
nal, specially devoted to the discussion of scientific instruc-
tion in the secondary schools, are replete with material for
the study of this question, the editor himself being one of
the principal contributors. It is to be hoped that the Bul-
letin of the New York Mathematical Society may, in the
course of time, perform a similar service towards the improve-
ment of mathematical instruction in this country.
On the other hand, the custom of many German schools
of publishing scientific and educational essays in connection
with the school calendar {^^Programm^') has given a wel-
come opportunity to many experienced teachers to express
their ideas on the subject and to propose improvements.
The material that has grown up in this way is somewhat
bewildering in extent, and, moreover, not very readv of access
to American students. A full set of Hoffmann's ieitnchrift
is probably to be found in but very few libraries in this coun-
try; many of the older ^^Frogrdmme" are hard to obtain;
and of the legion of German text-books of geometry that
have appeared during the present century only a very small
number, of course, have found their way into American
libraries.
The attempt made by Dr. Schotten to sum up the results
of the various efforts of reform in geometrical teaching in
the secondary schools and to give a critical survey of the
literature of this subject, will therefore be welcomed by all
interested in elementary geometrical instruction.
The title of Dr. Schotten's work is perhaps somewhat mis-
leading, as it does not indicate that his study is confined
♦ J. C. V. Hoffmann's Zeitachrift f^r rruUhemcUiaclien und naturuns-
senschaftlichen Unterrichi, published by Teubner, Leipzig.
8 TEACHING OF GEOMBTBY IK GEBMAlSr SCHOOLS.
entirely to German books and papers. Nor is there any
indication on the title-page that the present yolume is only a
first instalment of the work ; a second yolume is announced
in the preface and on the last page of the book, but even this
would not seem to exhaust the subject.
There is no table of contents, and no general index, a
serious defect in a work of this kind, which may perhaps be
remedied in the second volume. The book is of course made
up in a large measure of quotations, interspersed with critical
remarks by the author ; unfortunately, tne arrangement is
far from convenient, and in some instances very awkward.
In general, however, the author has well accomplished his
exceedingly laborious task. He shows a thorougn acquaint-
ance with the literature of his subject, as well as good judg-
ment and discrimination in making nse of it.
In an introductory essay. Dr. Schotten briefly states his
views on what is desirable m the way of reform. He sustains
these views, which are not over radical, not so much by argu-
ment as by a large array of quotations from various sources.
They may therefore be taken to fairly represent the better
thought of the day on the subject, at least in Germany.
It may be of interest to give a short account of these fun-
damental principles in teaching geometry which have found
the approval of so large a body of experienced and well-
trained teachers.
The study of mathematics in the Oymnasium should begin
with geometry (in Tertia, i. e., in the fourth and fifth years
of the whole nine-year course), being followed by algebra in
Secunda (also two years), while the lust two years {Prima)
are reserved for trigonometry and a thorough review of the
whole course. There is no urgent demand for increasing the
exte^it of* mathematical instruction ; but what is taught should
be taught well, that is, with thoroughness and accuracy. The
object of mathematical teaching in the Gymnasium is not to
form mathematicians, but to improve the mind, not only by
training in logical thinking, but oy accustoming the student
to precision of language in writing and speaking, by awaken-
ing his self-activity through the solution of problems, and in
the case of geometry in particular, by forming and practising
the power oi mental intuition {^^Anschauuna'*).
These objects, however, cannot be attained by the so-called
Euclidean method of teaching geometry. While Euclid's
arrangement of the propositions has long been abandoned in
German text-books, his synthetic method of proof is still
retained in many books. Here reform is most peremptory.
The *^ genetic " method should pervade the whole course ;
that is to say, the student should be led up in a natural way
to each proposition, so as to see clearly its connection with
TEACHIli^O OF OEOMETBY IK OEBHAK SCHOOLS. 9
what precedes, and finally conceiye of ifc, not as a single
artificial experiment in reasoning, but as an essential member
of an organic whole.
The proof of a proposition should be obtained by what the
Grermans call the "heuristic*' method, i.e., by the process
that would naturally be adopted by any one trying to jind the
proof himself anew. A "synthetic reconstruction of the
proof may finally be added in some cases.
Frequent reyiews are of course required to keep the student
constantly aliye to the conyiction that he is studying a well-^
connected system, and not a mass of detached single tacts.
The introduction of some of the ideas of modern projectiye
geometry (symmetry, dualism, theory of rows and pencils,
correspondences, etc.) will be found a great help in building
up a natural system of geometry. But whereyer used, these
ideas must be closely interwoyen with the whole system ; it is
decidedly objectionable to merely put these matters into an
appendix at the end of the book us is sometimes done.
There is howeyer a fundamental diflBculty in introducing
the ideas of projectiye geometry into elementary teachings
It lies in the fact that the circle is the only curyeS line con-
sidered in ordinary elementary geometry, while in modern
geometry the circle appears as a yery special case of a conic
section.* This circumstance will indicate how far we may
go in applying the methods of modem geometry to an ele-
mentary course, provided the study of the conic sections
be excluded.
Let us now turn to the main body of Dr. Schotten's
work. It is divided into five chapters : (1) Space, (2) Geom-
etry, (3) The Space-Forms (solid, surface, line, point), (4)
The Plane, (5) The Straight Line. In a second yolume the
author promises to treat in a similar way of (1) Direction
and Distance; position of points, lines, and circles in their
mutual relations ; metrical relations ; (2) The Axiom of
Parallel Lines (Eucl. XI.); (3) The Angle; (4) Auxiliary
Geometrical Ideas, such as equality, motion, dimension, con-
cept, definition, proof, explanation, postulate, theorem,
axiom, form, magnitude, position, figure, locus, symmetry,
etc. ; (5) Method.
The author prefaces each chapter by a brief statement of
his own views, and then follow quotations from all those text-
books or other works that express any original ideas on the
subject of the chapter. Comments by the author on these
quotations are usually giyen in foot-notes. But it must be
♦ See 0. Rausenbheoer. Elementargeometrie dea Punktes, der Oera-
den und der Ebene, systemaliach und kritisch hehandeU. Leipzig,
Teubner, 1887, pp. 2-8.
10 TEACHING OF OEOMETBT IK GEBMAK SCHOOLS.
said that the whole is not sufficiently well digested, and it
requires some labor (which the aathor might have spared the
reader) to get at the final results.
It will not be necessary to pass in review here the manifold
and widely different views of the fundamental conceptions of
geometry collated in Dr, Schotten's book.
The tendency in Germany seems to be at present to escape
as far as possible the hidden dangers that await the teacher
at the very threshold of geometry in the definitions of such
ideas as space, geometry, the point, the plane, etc., by two
means : (1) by requiring a preparatory course in geometrical
drawing in which the student shoald become thoroughly
familiar, in a practical way, with the fundamental geomet-
rical ideas ; (2) by a strict adherence to Pascal's rules.
As these rules do not seem to be as widely known as thev
deserve to be,* they are here transcribed in full from Pascal s
essay, ^^ DeVesprit giomiirique.''\
Bules for definitions. — " 1. N'entreprendre de dfefinir au-
cune des choses tellement connues d'elles-m^mes, qu'on n'ait
point de termes plus clairs pour les explicnuer. 2. jS^'omettre
qucun des termes un pen obscurs ou equivoques, sans defini-
tion. 3. N'employer dans la definition des termes que des
mots parfaitement connns, ou d^jd. expliqu6s."
Eules for axioms. — "1. N'omettre aucun des principes
n6cessaires sans avoir demand^ si on I'accorde, quelque clair
et evident qu*il puisse etre. 2. Ne demander, en axiomes,
que des choses parfaitement ^videntes d'elles-m6nies.*'
Bules for demonstrations. — '* 1. N*entreprendre de demon-
trer aucune des choses qui sont tellement 6videntes d'elles-
m^mes qu'on n'ait rien de plus clair pour les prouver. 2.
Prouver toutes les propositions un pen obscures, et n'em-
ployer d leur preuve que des axiomes tr^s-6vidents, ou des
propositions deid accord^es ou demontrees. 3. Substituer
toujours mentalement les definitions k la place des definis,
pour ne pas se tromper par Fequivoque des termes que les
definitions ont restremts.'
Thus, conformably to Pascal's first rule on definitions, we
find that some of the best German text-books do not try at all
to define what is space, or what is a point, or even what is a
straight line.
Strange as it may appear to some teachers, these text-
books do not begin with several pages of definitions to be
committed to memory, followed by a page of axioms again to
be committed to memory. Nor are the demonstrations made
* Dr. Schotten, while quoting them somewhat inaccurately in trans-
lation, says that he does not know in what work of Pascars they occur,
t Pascal, Penaies, ed. Havet, Paris, Delagrave, 1888, pp. 655-566.
TEACHING OF GEOMETRY IIST GEBMi^T SCHOOLS. 11
to cover exactly a whole page when they can be expressed in a
line. Some oi these authors, although well acquainted with
synthetic, and even with non-Euclidean geometry, do not at all
aohor the use of the expressions " direction '* and " distance.^*
Indeed, Dr. Schotten regards these two ideas as intuitivelj
given in the mind and as so simple as not to require defini-
tion ; he therefore bases the definition of the straight line
on these two ideas, or rather recommends to elucidate the
intuitive idea of the straight line possessed by any well-
balanced mind by means of the still simpler ideas of direction
and distance.
It is interesting to compare these views deduced by Dr.
Schotten mainly with regard to their pedagogical value, and
as a result of practical experience in teaching, with the con-
clusions arrived at by Prof. G. Peano* from a purely scientific
1)oint of view and based on the principles of mathematical
ogic.
A more philosophical discussion of the foundations of geom-
etry is reserved in the German schools to the review course
in the Prima of the Oymnasium, Then only will the student
be able to appreciate to a certain degree the niceties involved
in a careful treatment of the fundamental definitions and
axioms of geometry.
It is to oe hoped that Dr. Schotten will continue his studies
in German ** comparative planimetry," and that his second
volume will not be deferred ad calendas grcBcas. It would
also seem desirable that somebody should give us a similar
account of what has been done in other countries in the same
direction, in particular in England, France, and Italy.
In conclusion, the following two text-books might be men-
tioned, out of a large number of others, as giving a fair idea
of the reform movement in Germany :
Hubert Miller, Leitfaden der ehenen Geometrie, Leipzig,
Teubner, 1889 ;
Henrici and Treutlein, Lehrhnch der Elementar-Oeo-
metrie, ib., 1881.
For the more scientific study of the questions involved, the
reader is referred to the following works, in which ample
bibliographical references will be found :
Otto Rausenberger, Die Elementargeometrie des Punk-
teSy der Geraden und der Ebene, Leipzig, Teubner, 1887.
Bekno Erdmann, Die Axiome der Geometrie, Leipzig,
1877.
ScHMiTZ - DuMONT, Die mathematischen Elemente der
Erkenntnistheorie, Berlin, 1878.
«See Riviita di MaUmatica, ed. by Peano, Turin, vol. I. (1891).
pp. 24-25.
12 PICARD'S DEMONSTRATIOlSr.
J. C. Becker^ Ahhandlungen aus dem Grenzgehieie der
Mathematik und Philosophie. Zurich, 1870.
Alexander Ziwet.
Akit Abbob, August 1, 1891.
PICARD'S DEMONSTRATION OP THE GENERAL
THEOREM TIPON THE EXISTENCE OP
INTEGRALS OF ORDINARY DIFFERENTIAL
EQUATIONS.
TRANSLATED BT DR. THOMAS S. FISKE.
The cardinal proposition in the theory of algebraic equa-
tionsy that every such equation has a root, holds a place in
mathematical theory no more important than the correspond-
ing proposition in the theory or differential equations, that
every differential equation defines a function expressible by
means of a convergent series. This proposition was originally
established by Cauchy, and was introduced, with a somewhat
simplified demonstration, by Briot and Bouquet in their trea-
tise on doubly periodic functions.* A new demonstration
remarkable for its simplicity and brevity has been published
by M. Emile Picard in the Bulletin de la SociiU MatMmatique
ae France for March,f and reproduced on account of its strik-
ing character in the Nouvelles Annales des Mathimatiques for
May. This demonstration requires no auxiliary propositions,
and depends upon no preceding part of the theory, except the
simple consideration, that any ordinary differential equation
is equivalent to a set of simultaneous equations of the first
order, t The following is a translation of Picard's demon-
stration.
1. Consider the system of n equations of the first order
du . . V
^ = /i (a;, 2*, v, . . . , w)y
dv ^ , V
dw . , .
♦ Thiorie desfonetions cUiptiqueSj p. 825.
Jordan,' Coutb d' analyse, vol. III., p. 87.
! Bulletin de la Soeiiti MatfUmatique de France, Vol. XIX., p. 61.
Jordan. Coun d^analysej vol. Hi., p. 4.
picabd's demokstbatiok. 13
in which the functions / are continuous real functions of the
real quantities x, u, v, . . . , w in the neighborhood of x^, Uo,
r„, . . . , Wo, and have determinate values as long as x, u, v,
. . . , w remain within the respective intervals
{x, — a, a:. + a),
{Uo — hy u. 4- h)y
(v. — J, V, + h),
■>
a and h denoting two positive magnitudes.
Suppose that n positive quantities A, B, , . . , L can be
determined in such a manner that
I / (x, u\ v', . . . , w') - f (x, u, V, . . . y w) \ ^
<A\u'—u \+B\v'—v\ + . . . -hLlw'—w],
in which | a \ denotes as usual the absolute value of a, and
X, u, V, . . . , w are contained in the indicated intervals.
This will evidently be the case when the functions/ have
finite partial derivatives with respect to w, v, . . . , w.
Starting with these very general hypotheses we will demon-
strate that there exist functions u, Vy , , . , to of x, continue
ous in the neighborhood of x^, satisfying the given differential
equations, and reducing, for x = x^y respectively to Uo, v„
> . n w„
9 »*'»•
2. We proceed by successive approximations. Taking first
the system
du^ ^ , ^
dwx ^ I V
d^^f- (^* ^«> ^., • • • J ^^J>
we obtain by quadratures the functions w„ v,, . . . , Wx^
determining them in such a manner that they take for x^ the
values Ucy v„, , . , y w^ Forming then the equations
dut . f V
^- =/i {Xy w„ v,, . . . , Wx)y
dwt ^ f \
= fn (Xy Ui, r„ . . . , w?i).
<
dx
14 PICJLKD's D£XOS3imjLD03C
we determiDe m^ r^ . . . . tr^hT the coodinon tliat tlicT take
U/r z, the Talaei v^ r^ . . . . r^ respccdrclT. We oonthme
tbii proeeai indefiDiteh-^ :he fsnctio:^ b^ r^ . . . , r. being
coDoecud vith the preceding u^^^ r.^^. . . . , r.^ bj tbe
and, for z^=^ x^ Mtiffring the equations
«. = w^ r. = r^ . . . , r. = r^
We will now prove that when m increases indefinitely, v^
9., . . . 9 tr» /«na toward limits which represent the integrals
sought provided that z remains enfficientfy near z^
Let If be the maximnm almoin te Talae of the functions/
when tbe Tariables apon which they depend remain between
tbe indicated limits. Denote by p a quantity at most equal to
a. If now z remains in the interral
(x. — p, X. -r p),
we hare
I Wi — tt. I < Mp, . . . , I f^i — IT. I < Jfp.
Hence, provided Jfp < i, the quantities tit, r^ ... , Wi
remain within the desired limits, and it is evident that the
same is true of all the other sets of values u, v, . . . , w.
Denoting by d a quantity at most equal to p, suppose that x
remains m the interval
(ar. — 6, X. + 6),
and write
«• — w»_i, = U„, . . , , w^ — w^i = W„, ;
we have, placing m = 2, 3, . . . , «,
dIL
dz
= /l {X, U^xy . . . , Wm-l) — /l (iC, tt«_j, . . . , «?m-2).
dW
(/X
picabd's demonstration. 15
Since
\U^\ <M6, ...,\W^\<Md,
the preceding equations, for m = 2, show that | Z7i | , | F2 1 ,
. . . , I ITa I are less than
(-4 + 5+ ... +L)M6\
Continuing step by step it may be shown that | Z7, | , . . . ,
I fT. I are less than
Md{A +B -h ... + L)^-' 6^\
Now since
u^y and also r«^ . . . w^y will tend toward finite limits if
{A + B -{• . . . + L)6<1.
This condition will be fnlfilled by making 6 suflBciently small.
We see then that w^, v., . . . , w^ tend toward determinate
limits, w, v, . . . , w, which are continuous functions of x in
the interval
(Xo -S,Xo-\- 6)y
6 being the smallest of the three quantities
b 1
^' M' A +B'\- . .. + L'
and that w, v, . . . , w are represented by series which coii-
verge after the manner of a aecreasing geometrical progres-
sion.
Moreover we have
w« = /i {Xy w^i, . . . , w^^) dx + u.,
and, since w«, v., . . . , w„ differ from their limits as little
as we please, whatever the value of x in the indicated inter-
val, when m is sufficiently great, we have in the limit
u = \ fi {Xy u, V, . . . y w) dx -{- w„
\
16 OALCUL D£S PBOBABILITES.
Hence
^ =/x (a:, UyV, . . . , w).
Similar results hold for the other functions. TTie functions
u, V, ... y to are consequently the functions sought.
CALCUL DES PROBABILITifiS. Par J. Bertkand, de
TAcad^mie Fran9aise^ Secretaire perp6tnel de l'Acad6mie
des Sciences. Paris, Gauthier-Villars, 1889. 8vo., LVii
+ 332 pp.
There is possibly no branch of mathematics at once so
interesting, so bewildering, and of so great practical impor-
tance as the theory of probability. Its history reveals both
the wonders that can be accomplished and the bounds that
cannot be transcended by mathematical science. It is the
link between rigid deduction and the vast field of inductive
science. A complete theory of probability would be a com-
plete theory of tne formation of belief. It is certainly a pity
then, that, to quote M. Bertrand, ^^one cannot well under-
stand the calculus of probabilities without having read La
Place's work," and that " one cannot read La Placets work
without having prepared himself for it by the most profound
mathematical studies/'
Though not otherwise is thorough knowledge to be sained,
yet an exceedingly useful amount of knowledge is to be had
without such effort. In fact, M. Bertrand's forty odd pages
of preface on **The Laws of Chance" give an insight into
the theory without the use of so much as a single ^gebraio
symbol.
Listen to this reductio ad dbsurdum of Bernoulli's theory
of moral expectation :
" Mf I win,' says poor Peter, proposing a game of cards to
Paul, 'you must pay three francs for my dinner.' * Meal for
meal,' replies Paul, *you should pay twenty francs in case
you lose, for that is the price of my supper.' *If I lose
twenty francs,' cries Peter, frightened out oi his wits, *I can-
not dine to-morrow: without coming to that you might lose
a thousand ; put them up a^inst my twenty. According to
Daniel Bernoulli, you will still have the advantage.'"
Even more complete is the upsetting of Condorcet's calcula-
tion as to the prooability of the sun's rising.
"Paul would wager that the sun rises to-morrow. The
theory fixes the stakes. Paul shall receive a franc if the sun
rises and will give a million if it fails to do so. Peter accepts
OALOUL DES PBOBABIUTES. 17
the wager. Each morning he loses his franc and pays it. As
the sun rises from morning to morning his cnance daily
diminishes. Paal conscientioasly increases his stake ; Peter
as conscientioasly pays his franc. The obligations remain
eoaitable. The bettors travel throngh twenty conntriesfrom
West to Eafit. Peter always loses ; he pnrsues his fortune
however and takes Paul to the North ; they cross the arctio
circle ; the sun stays a month below the horizon ; Paul
loses 30 millions and thinks the order of nature overturned/'
Even La Place does not escape M. Bertrand's pleasant
raillery, and M. Quetelet has his ideal average man dismissed
as follows :
'* In the bodv of the average man the Belgian author places
an average soul. . . . The typical man would be without
passions and without vices, neither foolish nor wise, neither
Ignorant nor learned, forever dozing : this is the mean
between sleeping and waking ; answeiing neither yes nor no ;
mediocre in evervthing. After having eaten for 38 years the
average ration oi a healthjr soldier, he would die, not of old
age, but of some average sickness that statistics would reveal
to him."
But I must hasten on to the main body of the work : suffice
it to say that in these few introductory pages is packed this
variety of topics :
The Petersburg paradox, D'Alembert and Bernoulli's dis-
pute as to the benefits of inoculation for small-pox, Ber-
noulli's theorem, the ruin of players, inverse probabilities ;
Poisson's law of large numbers, the application of the theory
of probabilitj to statistics, the theory of errors of observation,
thejprobabihty of decisions.
Having seen so much done without any algebra at all, we
are prepared to accept M. Bertrand's statement, as to the
entire work, that " very few pages could embarrass a reader
familiar with the elements of mathematics."
Scarce an integration sign, never a generating function,
really it is charmingly simple and direct : and everywhere
illuminated too by the common sense and mother wit that
are so conspicuous];^ displayed in the preface.
It is characteristic of the method of treatment that all
lurking dangers, all insidious snares, are carefully pointed out
by means of numerous concrete examples.
A good instance of this, though merely one of half a dozen
bearing upon the same point, is the following problem to
show the absurdity of trying to reckon probabilities when
the favorable and possible cases are each infinite in number.
In a circle a chord is drawn at random. What is the
probability that it shall be longer than the side of the in-
scribed equilateral triangle ?
18 GALCUL D£S PBOBABIUTES.
You might say : The probability is unchanged by fixing
one end of the chord. The probability that it shall be long
enough is then merely the probability that it does not lie
without the &ngle made by the two chords of 120^ meeting
at the point. This probability is ^.
Or, you might say : The direction matters not, provided the
chord IS not too far from the centre, viz., not more than half
the radius of the circle. The probability is J.
Yet again : To choose a chord at random is to choose its
middle point. In order that the chord shall be long enough
its middle point must not be without a circle concentric witb^
and of hall the radius of, the given circle. This gives the
probability J.
Similarly, in a chapter on total and compound probabilities,
are a number of problems showing how essential it is, in com-
puting the probability of a compound event, that the proba-
oility to be given to the second composing event shall be that
which it has when the first event is known to have happened.
Even Clerk Maxwell has violated this principle in ootaining
the formula
<p (x) -Ge- *•*•
in which x is the a;-component of the velocity of a molecule of
gas and q)(x) its probability. He assumed that the x, y, and
z eomponents had independent probabilities. That they are
not independent is plain enough from this : if ^ is the maxi-
mum velocity of a molecule, the movement is parallel to the
fl:.axis, and y and z are nothing.
In the chapter on expectation the consideration of the
Petersburg paradox gives an opportunity to again ridicule
Daniel Bernoulli's moral expectation. I quote a few pas-
sages :
**You play, that is the hypothesis. Are you foolish or
wise to do so ? The question is not put.''
** Peter, whose whole fortune is 100,000 francs, wishes a
chance to gain 100 millions. * Nothing is easier,' coolly replies
the geometer whom he consults. * If the game is equitable
you will have 999 chances in a thousand of losing your 100,000
francs.' "
" The theory of moral expectation has become classic, never
was the word more exactly nsed : it has been studied, it has
been commented upon, it has been developed in works justly
famous. Its success stops there, no one ever has made or ever
can make any use of it."
Two chapters are devoted to James Bernoulli's famous
theorem that no possible event can be so improbable but that
there is as great a chance as you please of its happening some
time or other, if only you wait long enough.
OALCUL DES PBOBABILITES. 19
Three disfcinct demonstratioDS are riven, of which the first
is the most straightforward and complete. I give a sketch :
The probabilities of two contrary events being p and q, the
most probable combination in /a trials is that in which the first
event happens ^p times and the second /i q times. By Ster-
ling's theorem its probability is
1 / V^TCpipq ;
and the probability that it happens /jp ± h times is
e-^^/^PQ / ^/27rp^pq.
Approximately true for small values of A, it does to take
this formnla trae for large and even infinite values^ because
then both the true values and those given by the formula are
so small as to be negligible ; e. g. :
Putting fji = 1000, A = 100, « = g = 4, we get for the
corresponding probability
e-^V2 I VlOOO n = 0.000 000 000 520 06.
By a simple artifice, we substitute for (1), giving the prob-
ability of an error A, the formula
for the probability of an error between % and z + dz.
To test this formula, notice that the sum of all possible
Erobabilities is certainty and that we should have and do
X
lave
J — 00
Other such tests can be given.
The probability then that an error shall be smaller than a is
2 6-^dt^ &{a /2pipq).
If a has a determinate value, however large, the proba-
bility that it shall not be surpassed approaches zero as pi is
increased* without limit, which proves Bernoulli's theorem.
The relative error grows smaller and smaller as the absolute
error ctows larger and larger.
To nx the meaning of this, consider how many times a coin
would need to be tossed in order that the probability of ob-
20 CALCUL DBS PBOBABILITES.
tainiDg heads at least a million *more times than tails shall
exceed 0.01. We get, putting /^ for the required number,
0.99 = (1 000 000 ^/% / V^u) = (1.83)
and thence
}A = 597 211 millions.
Not only when the probability of an event is known can the
number of its happenings in a given number of trials be pre-
dicted almost with certainty, but when the probability is un-
known, Bernoulli's theorem can be used inversely to find it.
The ratio of the number of happenings of an event to the
number of trials certainly approaches this unknown proba-
bility as the number of trials is increased.
Nevertheless, two conditions are necessary : the probability
must not change during the trials, and it must have a deter-
minate value.
**The King of Siam is forty years old, what is the proba-
bility that he shall live ten years ? It is different for us than
for those who have asked his doctor, different for the doctor
than for those who have received his confidences ; very differ-
ent indeed for the conspirators who have taken steps to stran-
gle him to-morrow.'*
In a word, Bernoulli's theorem applies to ohjectivey not to
subjective, probability.
An immediate consequence of the theorem is the inevit-
able ruin of any gambler who plays long enough at a fair
game. But the number of games before ruin occurs may be
enormous. Thus, in tossing coins at one franc a toss it re-
quires 624 000 tossings to insure a probability of 0.9 that one
player or the other shall lose 100 francs. Truly a courageous
gambler would hardly be frightened at such a prospect.
If in this very game either party had started in with only
a franc and was to receive a franc a game so long as he
played without losing it, his mathematical expectation would
be mfinite.
Let us return to the inverse use of Bernoulli's theorem.
Consider this problem :
An urn contains pi balls, some white, some black, in an
unknown proportion, k drawings are made, the ball being
replaced after each drawing. There are bbtaincd m white
and n black balls. What is the most probable composition of
the urn ?
Before the trial is made, all sorts of hypotheses are possible
as to the composition of the urn. Suppose all compositions
are equally probable. It follows that the probable ratio of
whit-e to black balls is m : n ; and, that there should be a
deviation e from this ratio, the probability is
CALCUL DES PBOBABILIT^S. 21
where G is independent of €.
The hypothesis that all compositions are o ©nor i equally
prohahle is rarely realized. Suppose that the Balls were put
into the um by lot with a probability i for each color.
We then get for the probable proportion of white balls
(/i + 2m)/2 (/i + m + n),
which lies between i and m/im + n).
If /i is very large this will approach i, no matter what
the numbers m and n are ; if, on the contrary, it is m and
m + n that are large, the fraction is very near to m/{m + »),
no matter what /i is.
The probability of causes is always thus affected by a priori
probabilities.
To what is commonly known as the theory of least squares
three chapters are devoted. In fact, a fourth chapter, on
'^ Errors in the Position of a Point," is really an extension of
Oauss's law of errors.
Verv interesting is the criticism of Gauss's reasoning.
To begin with, can it be strictly maintained that the prob-
ability of an error J is (p{^) ?
" Docs it not depend upon the quantity measured ? "
*^ If you take a weight, if you measure an angle, is there
not a greater chance of a correct estimate if the weight is an
exact number of milligrams, if the angle contains an exact
number of seconds, than if it is necessary to add a fraction ?
If this fraction, not given by the instrument, is exactly i,
is there not a less chance of error in evaluating it than if it is
0.27?"
There is a case where the postulate is rigorously demon-
stratable, but the conclusion is nevertheless only approximate.
Suppose, in fact, that the quantity to bo measured is the pro-
portion of white balls in an urn ot unknown composition.
Of pi balls drawn, m are white.
The fraction m//i is a measure of the ratio sought. The
measure is the more precise as the number of balls drawn is
greater. The operation repeated n times gives the n success-
ive measures.
7W,///, m^/^y . . . , m,/;/.
The most probable value of the ratio deduced from the
drawings is
2 m/n/j^ = 2 {m//^)/n,
the arithmetical mean of n equally trustworthy measures.
Now, if in )u drawings from an urn we get nt white balls,
22 CALCUL PES PBOBABIUTBS.
the probability that the ratio of white to the whole number
of balls shall be m//< is indeed approxinuUely
which is of the form
and precisely what (xaoss's law would give; but if the law
were rigorous the formula should be exact.
The nypothesis that the arithmetical mean of several quan-
tities is tde most probable value leads to inconsistencies. It
requires, for example, that the most probable value of the
square of the quantity sh^ be the arithmetical mean of the
squares. Nor can the objection be avoided by making a dis-
tinction between measures directly observed and those result-
inff from calculation. A mechanic could easily attach to* a
balance a needle to indicate the square or the logarithm of
a weight.
The same objection applies to expressing the probability of
an error as a function of the error alone.
The problem is proposed, " If (Jauss had adopted, instead
of the mean, another mode of combination of the measures,
what law of errors would he have deduced ? "
The problem is not solved, but it is shown that if
J [Ziy ar,, • • • ^ ^u)
is the most probable value of a quantity of which Xi, ic,, . . . , a:,
are measurements, then, in oraer that the probability of an
error shall be a function of the error, /must be the arith-
metical mean of the x's increased by some function of their dif-
ferences. In other words, it must be such that if all the x'b
are increased each by or, it shall also be increased by a.
In spite, however, of all theoretical obJ3ctions to Gauss's
law, constantly accumulating experience completely justifies
its adoption. As to the arithmetical mean, Ferrero has shown
(see Charles S. Peirce's review in Am. Journ. Math, I., 59)
that all functions of the measurements that it would not be
absurd to take for the most probable value of the quantity
measured, will, if the measurements are good, agree in their
results ; while, of course, if the measurements are bad, no
treatment can be expected to give good results.
It is gratifying to find these careful definitions of precision
and weight.
"The precision of one measure is said to be a times that
of another measure when the probability that an error is
contained between z and z -{' dz for the one is the same as
GALCUL DES PROBABILIT^S. S8
that it shall be contained between az and a {z -^ dz) for the
other/'
'^ The weight of one obserration is said to be /? times that
of another, when the consequences that can be deduced as to
the value of a magnitude measured by an observation in the
first system are equivalent to those that can be deduced from
/S observations in the second system, all giving the same
result/*
" If /? is a fraction m/n, it is necessary that m concordant
observations of the first system can be replaced by n concor-
dant observations of the second/'
'^ The system of observations that gives to the error s the
probability
has k for its precision and i" for its weight, if we take for
units of precision and weight those of one observation in the
system for which the probaoility of an error z is proportional
to^«\''
M. Bertrand argues at some length for the rejection of
doubtful observations. He gives no criterion, however, as
Peirce has done, to determine when they shall be rejected.
This is left to the judgment of the computer, with the caution
that the number of retained observations must be large.
The probable value of the square of an error smaller than \,
when those larger than \ have been rejected, is
1 e (kX) - 2k\e-^'^yV^
2nk' [0 (k\)Y
As for Gauss's attempt, in his last memoirs, to break away
from all hypotheses of a law of errors in establishing the
method of least squares, it is shown that neglecting the
squares and powers of the errors is equivalent to assuming
the exponential law.
The equating of the probable value of a function to it^
true value is not unobjectionable. The following example
shows this.
Five angles ?i. If, i,, l^, ?«, have been measured. The geo-
metrical conditions require
h + li — k = 0,
?, + i, - /, = 0.
It is fonnd, however, that
24 CALCUL DES PROBABILIT^S.
Designating the errors really committed by Cj, «„ €„ e^y «,,
we have :
^4 "f' ^1 — 6% ^ /*ij
Ci + ^, — ^, = Af.
No matter what the multipliers A„ A„ ^, may be, the tri-
nomial
is known.
This tiinomial is a homogeneous function of the second
degree in the errors ; and calling wt the probable value of
the square of one of the errors, that of the product of two
of them being nothing, we shall find for the probable value
of the trinomial, calculated before the measures are taken,
Equating this to the true value gives
m* = (A A" + A A' + A AAf)/(3A, + 3A, + A,),
an infinite number of different values for m*.
This does not furnish, however, the most serious reason
why the chances of error cannot be precisely evaluated.
" It is supposed, a priori, that all the measurements of a
system arc equally precise ; it is impossible in the vast ma-
i'ority of cases to believe in such equality : it is from lack of
mowing reasons for preference that the results are accepted
as equivalent. But, known or unknown, these reasons, if they
exist, must have an influence upon the error really committed
and of which it is pretended to give the probability."
" After having, with immense labor, discussed the transit
of Venus observations of 17C1, Encke found for the parallax
of the sun 8". 49 with a probable error 0".06. He could
therefore bet 300000 to one that the error would not reach
0".42. Nevertheless, astronomers have just accepted the
parallax 8". 91 corresponding exactly to the error 0".42."
^' We can simply affirm, and this is the important point,
that if the sum of the squares of the corrections are small,
there is great likelihood that the observations have been well
made.''
The extension of Gauss's law to errors in the situation of a
point gives for the most probable position of a point the centre
of gravity of the observed positions supposed equally weighted.
Restricting ourselves to a variation m two coordinates, ** the
probability that an error shall be comprised between u and
u-^-du for X and between v and v + e/v for y is
OALOUL DES PBOBABIUT^S. 25
Points of equal probability are upon the same ellipse having
for its equation
u and V designating the differences between the coordinates
of the point considered and the true position, the common
centre of all the similar ellipses whose dimensions are pro-
portional to ^H.
A comparison is made between the theory and the results
of firing 1000 shots at a target.
The law has been partially guessed by Galton in his Dis-
cussion on the Data of Stature, and more fully worked out
by Mr. Hamilton Dickson. (See Natural Inheritance, p. 100
et seq.)
It seems a pity that in the chapter on the laws of statistics
some slight reference at least should not have been made to
Galton's investigations.*
A point liable to be overlooked in applying the laws of
probskbility to statistics is well stated.
** There are plenty of ways of consulting chance that will
give the same mean without giving the same probabilities of
error. Instead of drawing balls from an urn of a given com-
position, we could draw in order from many urns of various
compositions. The average results would be the same as for
drawings from an urn of average composition^ the chances of
error would not be.*'
** If, to take an extreme case, instead of drawing 10,000
times from an urn containing one white and one black ball,
we draw alternately from two urns, one Containing a white
the other a black ball, we shall certainly get white 5000
times, the error will be nothing. '*
To represent tables of mortality, the substitution of several
urns for a single one, seems, a priori, verv plausible. Among
individuals of the same age it is impossible not to find classes
in which the chances of life are unequal."
The book ends very pleasantly with a chapter on the
misapplications of the theory of probabilities to judicial
decisions.
Apropos of this matter, does not the great probability that
attaches to the results of concurrent indepenaent judgments
furnish the strongest possible argument for cultivating
independence, for ridding ourselves of the systematic errors
imposed by education and fashion, by part^ and sect ?
I have very inadequately sketched this most admirable
introduction to the science of probability : the life and vigor
of the original cannot bo reproduced in a brief review.
Elleby W. Davis.
Columbia, August 14, 1891.
26 THE NUMBEB-SYSTEM OF ALGEBRA.
THE NUMBER-SYSTEM OF ALGEBRA, Treated Theo-
retieally and Historically. By Professor H. B. Fine.
Boston and New York ; Leach, Shewell & Sanborn,
1891. 8vo, pp. IX. + 131.
At the present time we frequently find mathematical re-
searches preceded by an historical account of the cjuestion
under discussion : and this is but another proof of the increas-
ing importance of the study of mathematical history. On
the other hand, it is necessary that the history of any branch
of the science form part of such books as are intended for
students. For pedagogic reasons the historical part of a trea-
tise ought to be placed at the end of the volume, or at least at
the ends of the various chapters.
Mr. Fine's recent book takes its place among the not very
numerous works combining a systematic treatment w^ith an
historical account. It may be regarded as an introduction to
the theory of functions of one variable. In this short re-
view I shall only refer to the historical part of the work,
which occupies the latter part of the book (pp. 79-131). The
author begins by noticing the symbols and systems of numer-
ation, and then passes to the history of fractions and irrational
quantity among the Ancients. He then summarizes the prog-
ress of algebra, from the earliest times down to Descartes,
and finishes with the development of the fundamental notions
of algebra, from Newton to Weierstrass and G. Cantor. For
the ancient history, that of the Middle Ages and down to
1000, Mr. Fine has chiefly followed the well-known works of
Moritz Cantor and Hankel. For modem history he has gen-
erally had recourse to original sources.
One or two improbable or inexact statements may be no-
ticed. For instance, Regiomontanus is mentioned as tiie
author of the Algorithrmis Demonstratus (1534).* Again,
the year 1030 is given as the date of the introduction of the
sign -r, and it is said to have been first used by Pell.f These
inaccuracies are, however, of slight importance, and Mr.
Fine's book will doubtless be found of much assistance to
students of mathematics. G, Enestrom.
Stockholm.
Translated from Bibliotheca Mathematica, 1891, No. 2,
vrith additions from the author, by Habold Jacoby.
* There exists a copy of this work, antedating the birth of Regiomon-
tanus» and attributea to Jordanus Nemorarius.
fThe sign is really due to Rahm, and the "date is 1650. Compare
Beman, Bm. Math., 1887, p. 96.
WEST AFRICAN LONGITUDES. 27
WEST AFRICAN LONGITUDES.
Telegraphic Determinations of Longitudes on the West Coast
of Africa, From observations by Commander T. F. PuL-
LEN, R N., and W. H. Finlay, Esq., M. A, F. R. A. S.,
made and reduced under the direction of David Gill,
Esq., F. R. S., Her Majesty's Astronomer at the Cape of
Good Hope. London, Hydrographic Department, Ad-
miralty, 1891 ; pp. 82.
In this volume are recorded the last observations of the late
Commander Pullen, who lost his life from malarial fever con-
tracted while making night observations at Bonny, on the
West African coast. The results have been worked out under
the supervision of Dr. Gill, and the book contains not a few
suggestions and remarks of interest to astronomers. The
instrument employed was an altazimuth by Troughton and
Simms, having a 14-inch vertical circle, read by four micro-
scopes. This was selected as the most appropriate instrument
available for the purpose ; for it was decided to determine time
by altitudes, after a careful consideration of the relative merits
of meridian observations, and those in the vertical of the pole
star. Dr. Gill expresses a very favorable opinion of the latter
method, which has so long been strongly advocated by Dollen
of Pulcova. It was abandoned chiefly because there is no
bright star near the Southern pole.
The results afterwards proved the wisdom of not depending
on meridian transits : indeed, the conditions of the climate on
the West African coast are so unfavorable that there would be
an excessive loss of time if meridian observations only were em-
ployed. Throughout all the observations with the altazimuth
a mean time chronometer was used, without a chronograph.
Before the commencement of the campaign, the two observ-
ers, Pullen and Finlay, met at the Royal Observatory, Cape
Town, and their relative personal equations were carefully
determined by simple but accurate methods. As a result of
these determinations, the correction + 0'.085 was afterwards
applied to the differences of longitude obtained for the various
stations. The time observations with the altazimuth were
made with *' circle right " and ^^ circle left,'' and pairs of stars
were taken at nearly equal altitudes near the Eastern and
Western prime verticals. The mean from any such pair was
then regarded as a complete time determiuation; and in this
way the results came out very satisfactorih'. The time determi-
nations at the Cape were made by Mr. Finlay with the large
meridian circle : and in the exchange of signals Thomson dead-
beat galvanometers were used with success.
28 SOUTH AMERICAN LONGITXTDBS,
Several interesting remarks, due to Dr. Gill, occur in the
book. The method of carrying chronometers (much afiFected
by navy quartermasters) by means of a strap passed through
the handles and over the top, is condemned. Indeed, it is
possible to stop a chronometer, temporarily, when so carried,
oy a peculiar twist of the arm. Dr. Gill recommends holding
the chronometer with both hands in front of the bodv, the
elbows being pressed against the sides. The spring of the
arms is then a ^at safeguard.
In another place, having called attention to the very high
accuracy attained by Commander PuUen after comparatively
little practice. Dr. Gill refers to an interesting remark of
Professor Winuecke's to the effect that ^^ the best training for
an astronomical observer is a long course of accurate work on
land with the sextant." ;
The ordinary method of circummeridian altitudes was used
in measuring the latitudes of the stations. Stars were ob-
served both Siorth and South of the zenith, and certain sys-
tematic differences in the resulting latitudes are explained as
the result of a looseness of the web-frame in the tube. The
experience gained is summarized (p. 48) for the benefit of
future observers with the portable altazimuth, and any one
would do well to consult Dr. Gill's remarks before beginning
work with this somewhat difficult instrument.
The positions of the various astronomical stations are care-
fully described, and the bearings of many surrounding per-
manent objects are set down. The places of the stars used
are almost all taken from the Ephemerides and the Cape
Catalogue. The volume concludes with several appendices
containing various details and examples of observations and
reductions. Harold Jacoby.
Columbia College, New York ; 1891, September.
SOUTH AMERICAN LONGITUDES.
Telegraphic Determination of Longitudes in Mexico, Central
America, the West Indies, and on the North Coast of
South America, with the Latitudes of the Several
Stations, By Lieutenants J. A. NoRRis and Charles
Laird, U. S. N., published by order of Commodore P.
M. Ramsay, U. S. N., Chief of Bureau of Navigation,
Navy Department. Washington, Government Printing
Office, 1891 ; pp. 189.
The above volume contains the results of longitude deter-
minations executed by order of the U. S. Navy Department
SOUTH AMBRICAK LOKGirTDES. 29
in the years 1888, 1889, and 1890. As will be seen from the
title, the observations have been made in very unfavorable
locations, so far as the comfort and health of the observers
were concerned. It is therefore an evidenco of great endur-
ance and skill that so much was accomplished during the
short time many of the stations were occupied. We read
how one of the observing parties was compelled to proceed
with its entire observing equipment, including instruments,
a hundred miles in canoes up the Goatzacoalcos River, poling
against the current. And afterwards another hundred miles
by mule-train through the '^ tangled intricacies of a tropical
forest." This trip took fourteen days.
But we must here occupy ourselves chiefly with the methods
and results of the expedition, from a scientific point of view.
Eleven longitude stations were occupied altogether, and the
careful way in which the observation spots have been de-
scribed, and referred by exact measurements to local per-
manent landmarks, is very much to be commended (pp. 16-
19). When possible, the sites occupied by previous observers
were again used. The instruments employed were two pris-
matic transits made in 1874, by Stackpole of New York, for
the Transit of Venus Commission. Six break-circuit chronom-
eters and two chronographs were used. The values of the
instrumental constants were very carefully determined in the
field, and afterwards verified at Washington. Whenever it
was found possible the exchange of longitude signals was
made automatically, the distant observer's clock recording on
the local chronograph. When this could not be done, in con-
sequence of the weak current through long cable lines, a
mirror galvanometer was used at each end. This was found
to work quite satisfactorily. The mirror galvanometers were
set up in the cable offices, and the sending and receiving times
of the signals were recorded chronographically by the observer
at each end of the line. For this purpose wires were run
from the cable offices to the observing huts or tents, which
were usually quite close.
No allowance was made for personal equation, as circum-
stances would not allow of any adequate determination of
that quantity. In the reductions, the method of least
squares was used throughout the longitude work. The obser-
vations were first preliminarily reduced, and normal equations
were then formed for the determination of the minute correc-
tions required by the preliminary values of the instrumental
constants. The polar stars were weighted for declination, but
it is not stated what formula was used in assigning the
weights. The adopted clock-corrections, however, are not
those derived from the least square solution, but the means
of the results from the separate time-stars, the latter being
30 NOTES.
again reduced \nth the azimuth and collimation constants
derived from the least square adjustment. The polar stars
were excluded in this last process. The adopted values of
the clock-correction, however, are always very nearly equal to
those obtained from the least square reduction, the greatest
difference being 0*.035. (Vera Cruz, 1889, January 17 ;
p. 69.)
The latitude work was all done by Talcott's method, the
star- places being derived from the American Bphemeris, the
Jahrbuch, and the Catalogues of Newcomb, Safford, the
Coast Survey, and Greenwich Observatory.
The volume contains excellent maps showing the surround-
ings of the various astronomical stations, and closes with an
appendix giving the results of the many valuable magnetic
observations made by the members of the Expedition.
Harold Jacoby.
CoLUMBU College, New York ; 1891, Septemb&r.
NOTES.
The oflScers of Section A at the Washington Meeting of the
American Association for the Advancement of Science were :
Vice-President, E. W. Hyde of Cincinnati ; Secretary, F. H.
Bigelow of Washington. The following papers were read :
The evolution of algebra, by E. W. Hyde ; On a digest of
the literature of the mathematical sciences, by Alex. S.
Christie ; Latitude of the Sayre Observatory, by C. L. Doo-
little ; The secular variation of terrestrial latitudes, by George
C. Comstock ; Groups of stars, binary and multiple, Dy G. W.
Hollcy ; Description of the great spectroscope and spectro-
graph constructed for the Halsted Observatory, Princeton,
5f. J., and Note on some recent photographs of the reversal
of the hydrogen lines of solar prominences, oy J. A. Brashear ;
Standardizing pliotographic films without the use of a stand-
ard light, by Frank H. Bigelow ; On a modified form of
zenith telescope for determining standard declinations, and
On the application of the ^^ photochronograph '' to the auto-
matic record of stellar occultations, particularly dark-limb
emersions, by David P. Todd ; Principles of the algebra of
physics, by A. Macfarlane ; The zodiacal light as related to
terrestrial temperature variation, by 0. T. Sherman ; On the
long-period terms in the motion of Hyperion, by Ormond
Stone ; Exhibition and description of a new scientific instru-
ment, the aurora-inclinometer, by Frank H. Bigelow ; The
tabulation of light-curves : description, explanation, and illus-
tration of a new method, and Stellar fluctuations: distinguished
KOTES. 31
from variable stars : investigation of their frequency, by Henry
M. Parkhurst ; On certain space and surface integrals, by
Thomas S. Fiske ; The fundamental law of electroma^netism^
by J. Loudon ; Method of controlling a driving clock, by
F. P. Leavenworth ; On the bitangential of the quintic, by
Wm. E. Heal ; Parallax of a Leonis, bv Jefferson E. Kershner.
The officers elected for the Rochester Meeting are : Vice-Pres-
ident, J. R Eastman of Washington ; Secretary, W. Upton
of Providence.
The first volume of a work entitled '' Synopsis der Hoheren
Mathematik/' by the Rev. J. 6. H^en, Director of the Ob-
servatory of Georgetown College, Washington, D. C, has
appeared from the press of Felix L. Dames, Berlin. Its 400
pages treat of Arithmetical and Algebraic Analysis. The
contents are as follows : Part I., Theory of Numbers. — Part
IL, Theory of Complex Quantities. — Part III., Theory of
Combinations. — Part IV., Theory of Series. — Part V., Theory
of Infinite Products and Factorials. — Part VI., Theory of
Continued Fractions. — Part VII., Theory of Finite Differ-
ences. — Part VIIL, Theory of Functions. — Part IX., Theory
of Determinants. — Part X., Theory of Invariants. — Part XL,
Theory of Groups. — Part XII., Theory of Equations. The
subject of the second volume will be Analytical and Synthetic
Geometry. The entire work is to be contained in four
volumes, which are promised at the rate of one a year.
The deaths of a number of distinguished mathematicians
have occurred since the beginning of the present calendar
year. Among them may be recorded John Casey, died Jan-
uary 3 ; Sophie Kowalevski, February 10 ; Maximilien Marie,
May 8; Wilhelm Matzka, June 9; and Wilhelm Eduard Weber,
June 23. On another occasion we hope to give the readers of
the Bulletin some account of their work and lives.
T. s. F.
We learn from Hoffmann's Zeitschrift * that on the 5th and
6th of October a meeting will be held at Braunschweig, Ger-
many, for the purpose of organizing an " association for the
improvement of the teaching of mathematics and the natural
sciences.'* At a preliminary meeting held at Jena, September
28 and 29, 1890, f and attended by about 90 teachers from all
parts of Germany, the desirability of such an organization was
discussed and fully established, and a provisional constitution
* ZeUachrift fUr den maihematischen und naiurwisaenschaftlichen
Unterrtcht. vol. 22 (1891), pp. 816-318 and pp. 397-39S.
ti6., vol 21 (1890), pp. 561-674 and pp. 611-632.
32 NOTES.
was drawn tip. The association is evidently intended to
represent mainly the teachers employed in the Gymnasium
and liealschule, although there is, of course, no class-restrio*
tion of membership, anybody interested in the object of the
society being invited to "join, and university professors in par-
ticular. The term " natural sciences " is understood to em-
brace physics, chemistry, mineralogy, botany, zoology, and
geography.
The formation of this association is a si^ificant fact in
connection with the general movement for tne reform of the
so called higlier schools that has been going on in Germany
for many vears. The sensation created by the Emperor^
opening address to the committee called to consider the reform
of the higher schools was somewhat abated by the rather con-
servative final report made by this committee. But the
strength of the popular movement is not broken by any
means. It is the avowed object of the new association to pro-
mote and strengthen the teaching of the exact sciences in the
schools of Germany. The activity of the society is to bear
mainly on the following points :
(1) The improvement and more ample use of scientific
apparatus and other mechanical aids to instruction (the very
general term " LehrmitUl'^ may be interpreted to include
also text- books).
(2) The better preparation of teachers for their calling, by
the establishment of special courses and "seminaries for
elementary teachers at the universities, lectures on the teach-
ing of elementary mathematics, etc.
(3) The application of the recent advances in science and
the arts to elementary instruction in the exact sciences.
A full account of this years meeting will probably be pub-
lished in Hoffmann's Zeitschrift. A. z.
Professor W. H. Echols, Jr., recently Director of the
Missouri School of Mines, has been called to a chair at the
University of Virginia.
Professor M. A\ . Harrington, for a number of years Pro-
fessor of Astronomy at the University of Michigan, is now
Chief of the Weather Bureau in the Department of Agricul-
ture at Washington, D. C.
Professor A. 8. Hathaway has resigned his post at Cornell,
to become Professor of Mathematics at the Rose Polytechnic
Institute.
Professor A. L. Baker, of the Stevens School, Hoboken,
N. J., has accepted a call to the University of Rochester.
Professors A. S. Hardy, of Dartmouth, and Fabian Frank-
lin, of Johns Hopkins, have gone abroad to remain during
the present academic year. T. s. F.
-, u.lr«. Ilv >l' II
lUILLETIN
NHW YORK
M A'lH I'lMATrCA^L SOCIETY.
VOL. I. No. 2.
MACMILLAN & GO'S
lATHEMATlCAL TEXT-BOOKS.'
uxinilHtf ml*HiUiiiM»>» •••■i«|ti» ••"m 1
.', VWAItr'^ ?f:.llD OEOMETr'V.
t4AC«4(LLAN «. CO .
OATALOOUE OF THE ASTBONOMISCHE GESELLSCHAFT. 33
CATALOGUE OF THE ASTRONOMISCHE
GESELLSCHAFT.
Catalog der Astronomischen OeselUchaft. Erste Abtheilung.
GatalcM; der Sterne bis zur neunten GWSsse zwischen 80° ndrdlicher
nnd 2^8tkdlicher DeclinatioD fQr das Acquinoctium 1875. Drittes
St&ck, Zone + 65° bis + 70°, beobdchtet auf der Stebn waste
CHRisnANiA. Viertes StQck, Zone + 55° bis + 65°, beobachtet auf
den Stebnwabten Uelsinofors und Gotha. Vierzehntes StQck,
Zone + 1° bis + 5°, beobachtet auf der Stebnwa&te Albany.
Leipzig, 1890. 8 toIs., 4to.
AsTBOKOMERS frequently need the positions of so-called
fixed stars. They are wanted when a clock is to be regulated
to true sidereal or true mean time ; when^ again^ the astronomer
is on his travels and desires to fix his latitude and longitude^
and the direction on the earth of his meridian ; or when he
is observing some planet's or comet's course^ and wishes to
settle the various ri^ht ascensions and declinations it occupies
from day to day and hour to hour, in order from them to cal-
culate its orbit, predict its future course, and test the law of
gravitation.
Thus accurate star places are the basis in one sense of all
astronomy of position ; but they have an interest of their own
which is more prominent now than it ever has been, and is
yearly increasing.
For no star is absolutely fixed ; and the small motions of
the stars which have been long detected are slowly accumu-
lating their effects, and giving evidence that will in time
throw much light on the structure of the universe.
The star which has by some been called the *' runaway"
(Groombridge 1830) moves over half a degree, as seen from
the earth, in about two centuries and a half; so that a sharp-
sighted observer in a dry mountain region (where the air is
transparent enough) could readily detect its motion with the
naked eye in about a life-time. Other stars move so slowly,
in appearance, that a hundred years of the closest telescopic
observation are necessary to detect any slight deviation from
their former position. The constellations exhibit to the eye
substantially the same appearance from century to century ;
it is only in very small details that they seem to alter during
fifty years.
The study of these trifling motions (as they seem to us) is
extremely fascinating to those who have undertaken it ; and
a vast amount of human effort has been spent in the acquisi-
tion of this form of knowledge.
^ The Alexandrine Greeks did something in mapping and
listing the stars, by such simple devices as they possessed ;
8
34 CATALOGUE OF THE ASTBONOXISCHB OESELLSCHAFT.
the Tartar prince XJlugh Beigb had an observatory at Samar-
cand devoted to the same object; Tycho Brahe the Dane
and Hevelius of Dantzic added to the stock of ^ood obser-
vations of star-places ; but the earliest work which is now of
mnch scientific valne was done at Greenwich by the English
Astronomers Boyd. Flamsteed, who labored in the latter
part of the seventeenth, and Bradley, who held his office near
the middle of the eighteenth century, were eminent in their
time; and Bradley especiallv did more to make precise obser-
vations than any one who nad preceded him. With instru-
ments of clumsy build, and inferior in power to those which
are now carriea from place to place over our Far West to fix
a basis for our maps, he succeeded in putting this branch of
astronomy upon a solid foundation. It was not till quite
lately, however, that his observations were made fully avail-
able ; for Bessel, whose immortal Fundamenta Astronomim
appeared in 1818, reduced only a part of them, and the results
which he obtained lacked the minute accuracy which is now
needful and possible. With all these drawbacks BesseVs
study of a part only of Bradley's work gave the science a pro-
di^ous impulse.
Since Bradley's time the art of observation has progressed
very greatly. He was hampered by the intractable qualities
of matter ; his telescopes were small and unachromatic, his
divisions roughly made, his means of reading them crude ;
but his capacity for practical astronomy unrivalled ; incam-
parahilis Bessel calls him.
The instrument-makers Graham and Bird were succeeded
in England by Ramsden and Troughton ; in Germany by
Beichcnbach and the eldest Rcpsold. The opticians who
made single lenses were distanced by Dollond with the achro-
matic object-glass, which Fraunhofer improved and nearly
perfected. The telescope-tubes became shorter, retaining the
same power; object-glasses of greater and greater aperture
became practicaole with moderate dimensions in the ma-
chinery. The micrometer-microscope was used both in di-
viding and reading off divisions, so that a circle of 10 inches
in diameter is now as good as or better than a quadrant of 8
feet radius ; and the spirit-level of a few inches in length is
far more accurate and trustworthy than the plumb-line run-
ning through two stories of a house. In fact the mechanical
construction of astronomical instruments has become a time
fine art.
Astronomical observation of the first order now requires a
subtle psychological analysis ; the human brain ana body
have become tools for the mind to investigate and employ ;
just as we search into the minute errors of division which
the finest workmen leave in our instruments — errors thought
CATALOGUE OF THE ASTBOKOHISCHE GESELLSCHAFT. 35
too large if a line is misplaced by a twenty-thousandth of an
inch — 80 must we' inquire most scrupulously and carefully
into the possibility that our senses may lead us astray when
we are under the completest strain of attention.
After 1790 efforts were made, in several directions, to cata-
logue the stars. Piazzi at Palermo, and Bradley's successors
at Greenwich, spent most of their labors upon the brighter
stars ; those, some ten thousand in number, which are visible
to the naked eye, or approach such visibility. Lalande at
Paris undertook the gigantic task of observing all that his
telescopes would reach. Fortunately for him, he had but
small mstruments. He succeeded m observing over forty
thousand. Bessel, after completing his reduction of Bradley
in 1818, obtained in 1819 a meridian circle capable of dealing
with all ninth-magnitude stars not below or too near his
horizon ; and did all that was in one man's power towards
cataloguing the stars, nearly 150,000 in number, which come
into this category. He confined himself to less than one-half
the sphere, or about two-thirds of his range of visibility, and
took as many stars into his catalogue as he could observe.
His scholar, Argelandcr, was at first his assistant at Kdnigs-
berg ; but was promoted to the observatory at Abd in Finland,
afterwards removed to Helsingfors, and after excellent ser-
vice there in another direction went to Bonn on the Rhine
about 1840. There he extended BesseFs Zones in substan-
tially Bessel's way to the neighborhood of the North pole
(the immediate vicinity of this point had already been sur-
veyed by Schwerd at Speyer), and afterwards from the south-
ern point reached at Konigsberg towards the southern
horizon. This work has since been continued by our emi-
nent countryman Gould to the South pole, on a truly
gigantic scale, and with a still greater approach to complete-
ness.
But in 1852 Argelander took a new departure. Up to this
time astronomers had generally used their instruments of
observation in searching for the stars. They are somewhat
ill-adapted to this purpose. They bear high powers, and
their fields are small, and illuminated, so as to make visible
the spider-lines on which the bisections are made. All these
circumstances cause the loss of many stars which are missed
in the process of sweeping. So Lalande, with a small meri-
dian instrument, picked up many stars afterwards overlooked
by Bessel and Argelander ; while they found many which
Ijalaude ought 'to have been able to see. Nay more : Arge-
lander had spent his leisure at Bonn, while his observatory
was in building, before he had even a temporary shed for his
transit instrument, in making a catalogue of naked-eye stars.
In this there were some 40 of the fifth and sixth magnitude,
36 CATALOGUE OP THE ASTRONOMISCHE GESELLSCHAFT.
which had never been seen through any telescope, so far as
records indicated.
His new plan was to make a working list of stars to the
ninth magnitude inclusive, by a survey of the heavens (the
celebrated Bonner Durchmusterung), to include stars to
the tenth. In this survey no special pains were taken to set
accurate places ; the aim simplv was to locate every such star
nearly enough to identify it and map it somewhat roughly on
a chart. A small telescope of three inches aperture was used
with a field lighted only by the stars themselves, and a painted
scale visible in this manner to replace the fine spider-lines of
the more accurate instruments. With this apparatus a star
could be roughly observed every four seconds, with a student-
assistant to watch the clock-timing of passage ; while in the
whole work the number of stars averaged seven or eight to
the minute.
This observation was done twice for every part of the
Northern hemisphere, and down to 2° south of the equator,
between the years 1852 and 1859; and gave a catalogue of
324,198 stars, accurate enough to find them. Later Arge-
lander's assistant, Schoenfeld, who did a great share of the
actual work, extended it to the parallel of 23° south: leaving
the continuation to the South pole to be effected by Dr. Gifl
at the Cape of Good Hope.
From this greatest of all star catalogues in size the stars
whose magnitude was 9.0 or brighter were selected for more
deliberate and precise observation. There were more than
100,000 of them ; and the observations have been now almost
entirely completed.
The work was accomplished according to a plan formulated
by Argelander in 1868. It was to be done, as Lalande, Bes-
sel, and Argelander himself had previously worked, in zones
bounded by parallels of declination. The rule was made that
each star was to be twice observed, and with more pains and
less hurry than had been possible in the previous zone obser-
vations. Thus the whole labor was beyond the powers of
one astronomer, and it was divided among a number. The
first year or two saw beginnings in Helsingfors (Finland),
where Krueger, another collaborator and son-in-law of Arge-
lander, labored ; in Kazan and Dorpat (Russia), in Christiania
i Norway), in several German observatories, at Cambridge in
Snglana, Chicago, and Cambridge in Massachusetts. Vari-
ous circumstances interrupted, the Chicago fire and conse-
quent financial ruin of the establishment, and the call upon
some astronomers in Europe for service thought more prac-
tical. The final arrangement of zones has been as fol-
lows :
CATALOGUE OF THE A8TB0N0MISCHE GESELLSCHAFT. 37
80°to 75°, Kazan.
75 to 70, Dorpat.
70 to 65, Chnstiania.
65 to 55, Helsingfors and Ootha.
55 to 50, Cambridge^ Mass.
60 to 40, Bonn.
40 to 35, Lund, Sweden.
35 to 30, Leyden.
30 to 25, Cambridge, England.
25 to 15, Berlin.
15 to 5, Leipsic.
5 to 1, Albany.
1 to — 2, Nicolaief .
The space around the North pole had been, meanwhile,
again surveyed in the most thorough manner by Carrington
and others, so that the limits here given covered the necessity
in the Northern hemisphere. German observatories have
done nearly half the work, the remainder being divided in
somewhat unequal proportions between Russia, America,
Scandinavia, England, and Holland. The great Russian
observatory of Pulkova furnished the indispensable funda-
mental catalogue. Of course with equal breadth the polar
zones are the smaller; thus of the three catalogues already
published, Chnstiania (5° wide) contains 3,949 stars, Hel-
singfors riO° wide) 14,680, and Albany (only 4° wide) 8,241.
This makes, in less than one-fifth of the Northern hemis-
phere, or one-tenth of the whole heavens, a erand total of
26,870 ; but some hundreds of these may be duplicates (be-
tween Helsingfors and Christiania), as the zones are made to
overlap at the edges.
The limits of the present article are too narrow to enter
into the technical details of the observations and reductions.
The admirable introduction by Professor Boss to the Albany
zone (printed in English) can be referred to as the best ac-
count of Argelandcr^s plan in our language ; the original
instructions are given in the Vierteljahrschrift der Asirono-
mischen Oesellschaff, Vol. II. The whole undertaking is in
fact almost the original cause of the formation of the Society,
which has since undertaken many other serious problems, and
has become the leading astronomical society of the world.
The results, reduced to 1875, are extremely accurate. Of
course more time spent on every single observation would
have rendered them still better ; but all indications show
that, in the great majority of cases, each coordinate of a
star's place whI be found accurate within a second of a great
circle; so that if a star changes place but two or three sec-
onds in a century, its motion can be detected before the year
38 CATALOGUE OF THE ASTBONOHISCHE QESELLSCHAFl!.
2000. And as all BesseVs and Lalande's stars^ however faint,
have been reobserved, there is a vast mass of material now
ready for the studv of proper motions. A few years now will
see the printing or the row of stately volumes which will con-
tain the results of several centuries (where all the work is
combined as if done by one astronomer) of human labor.
A continuation to the declination —23° is now in pro^ss
in America, Algeria, and Austria ; Gould's great work, about
the same time, defers the necessity of going farther, although
it does not render it superfluous. Photography will doubtless
be called in to make this problem easier; or, rather, the
Southern zones will be included in the present photographic
survey, and perhaps repeated later by the same method.
The comparison of the three volumes mentioned at the
head of this article is in many respects instructive. The
astronomers were of different nations, employed widely vary-
ing instruments, and in one respect a different method. Fearn-
ley of Cliristiania, and Krueger of Helsingfors and Ootha
were pupils of Argelander, and employed the old "eye and
ear'* method (elaborated by the Greenwich astronomers of the
last century). In this the transits of the stars across the
meridian arc watched by the astronomer, who continually
counts by the ear the beats of his clock. If this makes too
little sound, he can reinforce it by an electro-magnet. He
notes where the star is at the integral second (or half-second)
before it passes the wire, and where at the second or half-
second after ; and estimates the tenths by comparing a second
or two afterwards what psychologists call the "traces" on his
memory. The method is not always the most precise possi-
ble, as it requires long training so to regulate the mental pro-
cesses that uniform results shall bo obtained ; but in high
declinations, where the stars appear to move much slower, it
has certain advantages, and is always free of the annoyance
that the sheets or tapes of the chronographic method must be
read off. Argelander himself never used the more modem
American method, which is, other things being equal, the
more accurate, but is not always the one which produces
equally accurate results with the least labor.
The American is the telegraphic method. The star is seen
approaching the wire, and the observer touches a telegraph
key when he estimates that it has reached it. This instant
is mechanically recorded on a chronograph. In one point
this seems to be less accurate than the other ; a very faint
star is usually misplaced by the fact that the observer lingers
in his judgment that the phenomenon has taken place wnen
the effect is hard to see ; so that the right ascensions of faint
stars are too large when chronographically determined.
The Albany zone was so observed. Professor Boss of course
A PBOBLSM IN LEAST SQCJABES. 39
determined how much each star was delayed in observation
by this process ; nsin^ an ingenious method invented by Bessel
of artificially diminishing the light of the stars as seen
through the telescope without altering the character of the
image^ and so found that his own mental processes delay his
judgment by about a hundredth of a second per magnitude ;
that is, he would observe a star of the eighth magnitude
seven-hundredths of a second later than one of the first in the
same place ; and so put it forward a second of arc and a small
fraction in right ascension.
On the other hand, the Albany observations of right ascen-
sion are rather better, one b^ one, than those made at Hel-
singfors. This was probably m part due to Krueger's anxiety
about his declinations, which gave him more trouble, owing
to the weakness of his instrument in that respect. Fearnley,
on the other hand, had a zone so far north {66"^ to 70'') that
with the old method he was able to equal the equality of Boss'
work in right ascension with the new, while his employment
of verniers instead of reading microscopes has somewhat
impaired his declinations.
But, all told, the uniformity of the three catalogues, due
to the excellent plan formulated by Argelander, is more sensi-
ble and far more important than the trifling discrepancies in
execution. The plan is in fact the quintessence of modern
practical astronomy in the subject witn which it deals. That
It has been so warmly welcomed and so thoroughly executed
by astronomers over the whole civilized globe is at once a
proof of the excellence of their training and of the great
advance which has been made in giving the human mind con-
trol over its own processes and over material objects.
Teuman Henry Safford.
A PROBLEM IN LEAST SQUARES.
BY PKOF. MANSFIELD MESSIMAN.
To determine, by the method of least squares, the most prob-
able values of a and b in the formula y = ax + b when the
observed values of both y and z are liable to error,
I. Let Xx and y„ x^ and y„ x^ and y, be n pairs of ob-
served values of two variables known to be connected by the
relation
y z:z ax + b.
40 A PBOPLEM IK LEAST 8QUASES.
If the observed values of x were free from error, the mofit
probable values of a and h would be deduced by the applica-
tion of the common rules of the method of least squares.
There would then be n observation equations of the form
flx 4- 3 — y = 0,
from which would result two normal equations
[a;'] a + M J - [xy] = 0,
[a:] a 4- »3 — [y] = 0,
whose solution gives for a the value
^ _ ^ [^y] - M [y]
II. If however the observed values of y are free from error
the formula should be written
- y ^ = ;
a^ a '
then by forming the normal equations and solving, there is
found for a the value
^ _ » [y'] - [y] '
n [xy] - [x] [yY
which in general is quite different from that given in I.
III. llow shall the most probable value of a be found when
the observed values of both x and y are subject to error ? The
following is the solution which I made in February, 1891,
when considering the problem at the request of the Director
of the Observatory of Harvard College :
Let the weight of each observed value of y be unitv, and let
the weight of each observed value of x be //. Then let ax and
a, be computed by the formulas in I. and 11. The most prob-
able value of a is then one of the roots of the equation
«• + U,-««) a-g^O.
IV. The demonstration of the last formula will be given in
full in a paper which is to appear in the report of the U. S.
Coast and Geodetic Survey for 1890. Here there is only space
to illustrate its application by one or two numerical examples.
A PBOBLEM IN LEAST SQUABES. 41
When a has been computed the most probable value of h is
directly found from
n
y. An interesting corollary is applicable to the case where
a is known a prion and ax and a, are derived from obserra-
tions. Then from III. the value of g is
1-1
a Ox
VI. As an example of the application of III. and IV. let
the following be simultaneous observations of two thermome-
ters having the same exposure :
No.
y
X
1 23466789
9° 10° 10° 11° 11° 11° 12° 12° 13°
10° 10° 11° 10° 11° 12° 11° 12° 12°
It is required to find the relation between the scales, or the
values of a and b in the formula y = ao; + J, regarding the
weights of the two series as equal.
Here ^ = 1, ti = 9, M = 99, [x'\ = 99, [x"] = 1095, [y'l
= 1101, \xy\ = 1095. These inserted in I. give a, = 1 and
inserted in II. give a, = 2. Then from III. there results :
«• + (1 - 2) a - 1 = 0.
from which a = + 1.618 and a = — 0.618. The former of
these is the value required (since it makes the sum of the
squares of the residual errors a minimum, the latter making
that sum a maximum). From IV. the value of b is now found
to be -6.798. Thus,
y = 1.618a; -6.798
is the most probable relation resulting from the given observa-
tions. The common rules of the method of least squares
would give y = a; if observed values of x be taken without
error and y = 2 x — 11 if observed values of y be without
error.
VII. As an illustration of the use of V. let the following
be estimations of the magnitudes of stars by two observers :
42 A NEW ITALIAN MATHEHATICAL JOURNAL.
No. : 1 2 3 4 6 6
y : 8° 9^ 10° 10° 10° 11°
X : 9° 9° 11° 9° 10° 9°
It is required to find the weight g^ it being known a priori
22
that a = 1. Here, from I. there is found a^ = s^, and from
32
IL fl, = Q^r ; then from V. there results
22
_ 22 - 32 _ 10
^ ■" 22 - 29 ■" 7 '
or the weight of the first series of obserrations is to that of
the second as 7 is to 10.
VIII. If the equation between the variables be of a degree
higher than the first, as 2;' = aw* + by values of a and h may
be deduced by following the above method, regarding «* and
w* as observed values corresponding to y and x. Since, how-
ever, the real observed values are z and w I am not prepared
to say that the results deduced for the parameters a and will
be strictly the most probable ones according to the principles
of the method of least squares.
Lehigh Univebsity, October, 1891.
A NEW ITALIAN MATHEMATICAL JOURNAL.
Rivista di Mateniatica, diretta da G. Peano. Torino, Fratelli
Bocca, 1891.
Almost simultaneously with the Bulletin of the New York
Mathematical Society, a new journal of a somewhat similar
character has been founded in Italy. Like the Bulletin, the
Rivista di Matematica is a monthly of at least sixteen pages
8vo. According to the prospectus "its scope is essentially
didactic, its principal object being the improvement of the
methodsof teaching." The Rivista will contain "articlesand
discussions concerning the fundamental principles of the sci-
ence and also the history of mathematics." ** The review of
text-books and all publications having reference to the teach-
ing of mathematics will form an important feature.'' Ques-
tions and inquiries about mathematical subjects sent to the
editor will be either answered dii'ectly or published in the
A KEW ITALIAN MATHEMATICAL JOUBNAL. 43
journal. Articles intended for the journal may be written in
any of the principal langaages and will be translated if neces-
sary. Subscriptions (7 francs per annum) are to be sent to
the publishers, Fratelli Bocca, Turin.
The editor, Prof. Giuseppe Peano of the University of
Turin, is well known through his original investigations in
Mathematical Lo^c and in Orassmann's Oeometrical Calculus,
as well as through his rigorous and elegant treatment of the
Infinitesimal Calculus. His own contributions to the Rivista
so far (the first number appeared in January, 1891) relate
mainly to the fundamental logical principles of the science of
mathematics.
Among the longer articles by other contributors we find an
interesting paper (pp. 42-66) by Professor Segre, of Turin,
addressed to his students, in which he points out some of the
distinctive features of modem mathematics and gives whole-
some advice to the young mathematician who wishes to engage
in original research. The author is evidently inspired by
what may be called the modern Gottingen school (Riemann,
Clebsch, and in particular Felix Klein), insisting as he does
on the organic unity of the whole of mathematics, warning
against excessive specialization, and recommending that the
young mathematician should make it his object to bring to
Dear as far as possible all branches of mathematical science on
the particular subject of his investigation. It is curious to
note that, in the opinion of Prof. Segre, there exists a very
pronounced preference for the study of pure geometry, to the
injury of analytical studies, among the younger generation of
Italian mathematicians. Some remarks in this paper as to
mathematical rigor and the use of hyperspace gave rise to an
interesting discussion between the author and the editor (pp.
66-69, and pp. 154-169). Other contributors are A. Favaro,
G. M. Testi, E. Novarese, C. Burali-Forti, G. Vivanti, etc.
Among the reviews, the very full account given by Gino
Loria of R de Paolis' theory of geometrical groups * is most
prominent (pp. 105-120). E. W. Hyde's Directional Calculus
tinds a competent and appreciative critic in the editor (pp. 17-
19).
Alexander Ziwet.
Ann Abbob, August 10, 1891. ^
* R. DE Paolis, Teoria dei gruppi geometrici e delle corrts^ondeme che
8ipo88ono stabilire tra i loro elementi. Memorie della Societa Italiana
delle IScienze delta dei XL., voL VII. series III.
44 THE PHOTOCHBOKOORAPH.
THE PHOTOCHRONOGRAPa
Thephofochronograph, and its application to star transits.
By J. G. Hagek, S. J., and G. A. Fasgis, S. J., Geoigetown College
Obfiervatorj. Georgetown, D. C, 1891. 4to, pp. 86.
The authors of the above publication are the first to lay
before the astronomical world a solution^ or at least a partial
solution, of the verjr important problem of meridian transit
photo^phy. The instrument they have employed consists
essentially of an electromagnetic shutter or '* occulting bar,**
which can be attached to the eye-end of a transit instrument
or meridian circle. The apparatus is so arranged that the
current from a break-circuit clock moves the occulting bar
every second in such a way that the image of a star in tran-
sit is impressed for a moment upon a photographic plate
mounted behind the bar. A line of *' star-dots '* can after-
wards be developed upon the plate. In order to refer the
dots to the collimation axis of the instrument, a glass reticle
plate, ruled with one vertical reference line, is permanently
fixed in the tube, directly in front of the sensitized surface,
and in contact with it. After the star transit is over, it is
easy to impress the line upon the sensitized plate, by allowing
the light of a lantern to fall for a moment upon the object-
glass of the telescope. While this is being done, the line of
star dots is shielded from the light by the occulting bar, now
permanently interposed between the dots and the light. This
method of impressing the reference line upon the plate is ex-
cellent, and is further improved by ruling the line with a
break in the middle, so that none or the dots can possibly be
" occulted " by the line itself. The plates are measured with
a micrometric apparatus, by means of which it is easy to
determine the instant of the passage of the star across the
reference line.
The process thus very briefly outlined is given by the
authors with all possible detail ; even the preliminary appara-
tus, subsequently discarded as imperfect, being carefully de-
scribed. Other experimenters in the same field should there-
fore be greatly aided by the present work. In this connection
it is proper to refer to the earlier observations of L. M.
Butherfurd, of New York, who successfully employed an
arrangement essentially similar to the photochronograph
many years ago.* In the collection deposited by Mr. Euther-
*B. A. Gould, Memoirs of the National Academy of Sciences, vol. iv.,
p. 175.
L. M. RurnEBFURD, American Journal of Science and Arts, vol. iv,,
Dec. , 1872.
THE PHOTOCHBOKOORAPH. 45
fnrd at Columbia GoUe^* are many negatiyies showing lines
of star-dots, together with mierometric measures of the same.
We shall now enter into a somewhat more careful examina-
tion of some of the statements contained in the book^ taking
them up in order. It is diflBcult to see why only one yerticai
line has been used on the reticle plate. The authors refer to
their reason for this (p. 12), but without anywhere definitely
stating it. One would think the presence of several vertical
lines would offer a valuable control of possible irregular ex-
pansions of the film during development. Nor would there
De any compensating disadvantage, for the admirable device
of breaking the lines in the middle would prevent any inter-
ference with the star-dots. The effect of irregular distortion
of the film would not be eliminated by the method of meas-
urement (p. 24). It is gratifying to find (p. 13) that no
trouble was experienced from a jarring of the instrument by
the reffular beats of the occulting bar.
Probably the most important difficulty of the method is
touched upon by the authors in speaking of collimation
(p. 17). In fact, it may safely be said that the photographic
transit instrument will not be applicable to the finest funda-
mental work, until it becomes possible to determine the col-
limation and level constants photographically ; without re-
versal in the Y's, and without the use of the hanging level.
In all the observations so far made, the collimation constant
has been determined from reversals alone, and the hanging
level has always been employed. A very interesting remark
occurs (p. 18) in connection with personal equation. By
watching the occulting bar through an eye-piece while a star
is in transit, the existence and effect of the observer's per-
sonal equation become very obvious.
We now come to Part Ii. of the book, which treats of the
reduction of the observations. This part is the work of
J. G. Hagen, S.J. ; the first part, in which the instrument and
methods are described, being by G. A. Fargis, S.J. The
screw of the measuring micrometer has been examined for
both periodic and progressive errors, according to the usual
methods. The author very justly concludes that it is advis-
able to determine the screw value separately from each plate,
though errors due to an oblique mounting of the plate in the
tube would not be eliminated thereby, as seems to be implied
in the text (p. 22, c). The adopted method of measurement,
by which the dots are taken in corresponding pairs at nearly
equal distances on both sides of the central line, has much in
its favor. With regard to the example of a series of transits
* J. K. Rbbs, AnnaU of the New York Academy of Sciences, vol. vi,
June^ 1891.
44 SOMESChATTKE 09 HYCBASICS.
(p, 34) it maj be i^d that the data are not safficient to draw
Cfmcltimfma of a reir defioitire character, berond the fact
that the method gires re$alt« of rerr satisfactory accuracy.
The azimath constants for the evening (two in number) and
the eoUimation constant hare been derired from the fifteen
obserrations themselTea. Their ralnes are stated to be the
^'most probable'' ones. If they have been obtained by a
least-square redaction in which the clock-rate was ignore<fy it
is not remarkable that the final residnals show no eridenoe of
a clock-rate (p. 35).
In Ci^inelusiony we ma^ accord to the authors of this book
the credit of having invented and made public a photo-
graphic method by which meridian transits may be observed
witti high accuracy, and with a complete freedom from per-
sonal equation, Ii there is a weak point,- it will be found in
the determination of the instrumental constants. The many
other important jmrposes for which the photochronograph is
very well a<lapted we shall not touch upon in this place.
Some of them have already been described in print, and
many others will doubtless shortly come into prominence.
Harold Jacoby.
Columbia College, New York, 1891, October.
NOMENCLATURE OF MECHANICS.
IIY T. W. WRIGHT, PH.D.
TiTE nomenclature of mechanics is in a somewhat confused
condition. Tliero is some excuse for this because the science
is one of the oldest, and at the same time one of the most
propfrcHsivo, as it certainly is the most comprehensive. New
tornm arc hoiiipf introduced, others are being suggested to
take the ])luce of old ones ; but the naturally conservative
cling to the old, and hence we have a duplication, and in
Honjo (Mises a trii)lioation of names for the same thing. At
the tliroshold wo are mot by a difficulty. How shall we define
nuHjlianicH P Originally the science of machines, it is by some
defined as the science of matter and motion. By others the
term dynamics is ai)])lied to the science of matter and mo-
tion, and the term mechanics is discarded. The tendency at
pn^sont seems to be in the direction of the latter method.
The science is founded on three principles or laws laid down
by Newton. These laws were originally enunciated in Latin^
uiid the number of translations is very great. Here is a source
NOMENCLAUUBE OF MECHANICS. 47
of confnsion. With a new translation come in new terms or a
change in the meaning of old ones. For example, Newton's
first law is called by some the law of inertia. What is inertia ?
Is it inertness, a mere negative property, or is it a property
admitting of measurement, a Quantitative property ? When
we come to the second law we nave the idea of mass promi-
nently brought forward. Since the second law includes the
first, why introduce the term inertia at all ? Is not mass
suflBcient ? Call the first law the law of mass and the second
the law of mass-acceleration. The reformers who drop inertia
in the first law would have us call centre of gravity centre of
mass, and moment of inertia moment of mass. The first of
these changes, centre of mass for centre of gravity,, is well
under way and will probably prevail. The change from mo-
ment of inertia to moment of mass meets with less favor. In-
deed, the new name seems as objectionable as the old, for the
moment is not a simple moment, but a second moment.
Next in importance to a proper translation of the laws of
motion is the settlement of the question of how weight shall
be defined. One school use it in the sense of mass ; another
in the sense of force, it being the attractive force of the earth
on mass ; while a third contend for its use in both senses.
The question was debated by some of the ablest physicists in
England two or three years ago but no definite conclusion
was reached. This and' the relation
W= mg
form probably the center of greatest confusion in elementary
mechanics. The perplexity of a beginner as to whether in a
given problem he shall multiply or divide by g is extreme,
and the mournful thing is that this is not owing to his own
stupidity. The pit has been dug for him and is persistently
kept open waiting for new victims.
The nomenclature is deficient in several respects. We have
no single term for the unit of velocity, the foot per second^
nor for the unit of acceleration, the foot per second per second;
but must use these long phrases where a monosyllable ought
to suffice. The most satisfactory suggestion I have seen is to
use/.5. for unit velocity and/.5.5. for unit acceleration. Nor
have we any name for the absolute unit of force in the British
system. It is true that some recent writers use Prof. James
Thomson's term the poundal for unit force. If we say
poundal shall we say ouncal, tonal, etc.? Consistency would
seem to force us to do so. The terms sound odd enough. Is
the gain in simplicity in the dynamical formulas expressed in
absolute units over that of the gravitation system a sufficient
excuse for introducing terms that will probably never be used
48 A TBEATISE OK UKEAB DIFFEBEKTIAL EQUATIOHS.
outside of the lecture room ? What engineer would use foot-
poundal for example ? The nomenclature is also redundant
A single instance will suffice. Shall we say Tis-yiya, Hying
f orce^ or kinetic energy ? AH three are used to denote the
same thing to the mystification of the beginner. All three
can be found in text books of recent date. To my mind
there is no doubt but that kinetic energy is the propef term.
Now, the confusion, deficiency, and redundancy being
granted, what can be done ?* No one writer can do much to
effect a change. But an association such as the New York
Mathematiccil Society can do much. Expressions of opinion
through the pages of this journal would probably lead to some
more definite understanding than now exists. At least some
of the more g:laring absurdities and contradictions of our pres-
ent system might be abated. Besides, it might tend to curb
the ambition of writers to introduce ill-considered terms such
as " heaviness '^ or "centre of weight" for centre of gnmty
and the like.
Union Ck>LLEOB, 1891, October 10.
A TBEATISE ON LINEAB DIPPEBENTIAL EQUA-
TIONS. Vol. I. Equations with uniform coefficients.
By Thomas Cbaig, Fh.D. New York ; John Wiley &
Sons, 1889. Svo, pp. ix. + 516.
The appearance of Fuchs's two memoirs in 1866 and 1868
respectively, gave an impetus to research on linear differential
equations which has resulted in the development of an enor-
mous literature on the subject, consisting of articles and
memoirs scattered through mathematical journals and the
proceedings of learned societies. The systematization and
presentation in a body of the principal methods and results
developed in these isolated papers, is the work which has
been undertaken by Professor Craig, and which has success-
fully issued in the first volume of the most advanced treatise
on pure mathematics ever published by an American author.
Whilst the presentation of the subject as a whole must prove
of advantage to those few mathematicians who have access to
the memoirs whence it draws, upon the many to whom the
original sources are not open it confers an inestimable boon.
To the English-reading student further it manifests in his
own language the substance of what is for the most part in
the original in French or Gorman. Praise is due the author
for the scrupulous care with which he credits every writer
A TBBATISE ON LINEAB DIFFEBEKTIAL EQUATIONS. 49
qnotedy and for the falness of his references^ which pye an
added value to the volume. A glance at these references
cannot fail to impress upon the reader a sense of the over-
whelming influence which the continental element has had in
shaping the development of modem differential equations.
In lact^ an analysis shows that of the sixty-odd names quoted
in the volume more than three-fourths divide themselves
about equally between the French and Germans^ and of the
remainder some eight majr be claimed by the English speak-
ing peoples : so that if this showing in relation to the pop-
ulations of the countries concerned could be fairly consid-
ered as furnishing a criterion relative to the generality of
interest manifested among the several peoples in the develop-
ment of the subject, such interest in America and England
as compared with that in France and Oermanjr might be
averaged as 1 to 7. The dropping of the average in the com-
parison, it may be frankly owned, would not advantage the
showing of America.
The reader in his progress through this treatise will con-
stantly have to do with the modern theory of functions, and
will meet with some simple applications oi the theory of sub-
stitutions. Both of these departments, with their numerous
applications and possibilities of further development, offer a
field whose successful cultivation on the continent shows a
productive power giving as yet no si^ of exhaustion. Pro-
fessor Craig's book will have accomphshed a useful mission if
it helps to awaken American students to a sense of the
work that is being done in Europe, and, as a consequence,
rouses them to a realization of what is being left undone in
America. There seem, however, at present to be definite
tendencies making for the elevation of mathematics in Amer-
ica, and it may not perhaps be idle to indulge a hope that
America will yet contribute in a fitting proportion to the de-
velopment of the science. The preliminary knowledge of the
theory of functions necessary to the reading of Jrrofessor
Graig's book may be obtained from Hermite's Cours, To
the student who desires an acquaintance with the theory of
substitutions one can recommend Netto^s Substitutionen-
theorie and Serret's Cours d'Alaebre Superieure, though so
far as is necessary for understanding the applications of the
latter theory in the volume under consideration, a very par-
tial reading of its treatment in either of the works mentioned
will prove sufficient, and, in fact, a few words of explanation
from one familiar with the substitution notation would prob-
ably suffice. The American student of mathematics who
acquires a knowledge of these branches will in general do so
by his own unaided efforts, for courses in them are offered by
but a small number of our universities, and further, as re-
60 A TREATISE ON LnTBAR DIFFERENTIAL EQUATIONS
gards nnassisted stnd^r^ it may aD fortunately be said that few
of our colleges and universities give a course in mathematics
whose discipline prepares a man for such study. The fault lies
perhaps not so much with the higher institutions of learning
as with the preparatory and hi^ schools, into whose hands
our potential young mathematicians first fall, and which as a
general rule allot to the study of algebm and geometry a
time utterly inadequate to the laying of a basis on which the
college can satisfactorily build. On tne other hand, almost all
our college professors, among whom we find, of course, the
great majority of our mathematicians, are overworked. Teach-
mg absorbs the energy and spontaneity which should be spent
upon private study and research. For the latter scanty allow-
ance is made, except in a few of our larger universities, con-
spicuous among which is that university in which the author
01 our treatise is a teacher. The lack of stimulus and encour-
agement due to the isolation in which the American mathe-
matical professor has been wont to live, may (it is not an
unreasonable anticipation) be remedied in some degree by the
founding of a mathematical society of national scope with the
publication of a bulletin. Thus may be fostered among
American mathematicians a fellow interest in their science,
to illustrate the advantages of which we might cite the sub-
ject of the work before us, which has been developed since
the publication of Fuchs^s memoirs, only by the cross-working
of scores of European mathematicians.
Before the appearance of these two memoirs the only general
class of linear differential equations for which a solution had
been found was that in which all the coeflBcients are constant,
but with the application of the modern theory of functions a
new field opened up. In this theory the critical points of
a function i)lay an all-important role, and, as can be readily
shown in the case of the equation which constitutes the
theme of the volume under review, the critical points of the
integrals of the equation are included among those of its co-
eflBcients. This property evidently gives us some hold upon
the integrals and is, when combined with the fact that the
general integral is a linear function of the particular integrals,
more fruitful of results than would readily be anticipated, re-
sults of which but a few can here be hinted at.
The work opens with a recapitulation of the genei'al proper-
ties of linear differential equations, followed by an extended
modem treatment of the equation with constant coeflBcients.
It then takes up the theory of the differential equation
A TBEATISE ON LIKEAB DIFFERENTIAL EQUATIONS. 51
where the coefficients jo,, />,,....,/?, are uniform functions of
X, having only poles as critical points. Let y„ y,, y, de-
note a system of fundamental integrals of (1). If now the im-
aginary variable x make the circuit of a critical point in the
plane, returning by any path to its point of departure, the
coefficients, since they are uniform, will return into them-
selves^ and the equation will be unaltered. Any integral of
the original equation, then, necessarily remains such, and can
at most have transformed into a linear function of the n
fundamental integrals. It is now shown that among such
transformed integrals, there will be at least one which will
transform into itself multiplied by some constant s which is
determined as the root of an equation of the nth degree in 8
called in reference to the critical point in question, the char-
acteristic equation for the system of fundamental integrals
y,* y,» . . . . , y-.
There will be as many such integrals as there are solutions
to the characteristic equation ; and, in fact, corresponding
to a A-multiple root s^ of this equation there will be a group
of A. integrals w„ w„ . . . . , ux which, when the variable x
completes the closed circuit, may respectively be shown to
transform into
• • . • •
where the coefficients s are all constants, and the aggregate
of such groups corresponding to the different roots of the
characteristic equation will constitute a system of fundamental
integrals of the differential equation. The theory is given for
the point x = considered as the typical critical point, the
reasoning for any other critical point a being obtained by sub-
stituting (a;— fl) for x wherever it may appear in our formula?.
The group of integrals given above are now shown to be of
the following forms :
(2) <
U, = 3fi{ (p,^ 4- <p,, log x}
u, = afi {^„ + ^„ log x 4- ^„ log'a;}
ux = x^i {^x, + <?>Ajog a; + + ^^ log k-ix}
where the ^'s are uniform in the region of our critical point
52 A TREATISE ON LIJSTEAB DIFFEBENTIAL EQIJATIOKa
2; = 0, and such that anyoDO of them can be expressed in
terms of those whose second subscript is 1; 9>„, ^„ f
<Paa differing from one another only by constant factors and
imr^ being equal to log s ; we find that this group (2) may
be replaced by a number of sub-groups possessing precisely
the properties just enumerated^ and can further show that the
transformation effected by a circuit of the critical point may
be represented thus :
S =
y^> y.> "'I ^y»» «i(y, + yJ^ • • -^ «,(y* + y*-«)
y\y y\y • • • ; ^^y\^ »Xy\ + y'i). • • •> «i(y.' + y.'-O
the interpretation of this notation being that a certain system
of n independent integrals of our equation represented by
the symbols y., y, y i» • • -^i* • • '^^ ^^® \^ii, are by circuit
of the critical point transformed into the expressions corre-
sponding in position on the right, where «j, «„.... are solu-
tions of the characteristic equation. The aggregate of trans-
formations thus effected is indicated by the letter S and is
called the substitution for the point in question. As pre-
sented here it is said to be in its canonical form. There will
be such a substitution corresponding to a single circuit of any
critical point, to a multiple circuit of the same, or to a circuit
including any combination of critical points, all substitutions,
by the way, being reducible to successive applications of the
substitutions for different individual points.
The aggregate of all possible substitutions is defined as the
group of the equation. In a later chapter the author by the
aid of the canonical form just given goes into the investiga-
tion of what are called function-groups, these being groups of
functions which under all possible applications of a substitu-
tion-^roup transform into one another. The integrals of our
equation (1) evidently constitute such a group and include, it
may be, smaller function-groups formed by the linear func-
tions of linearly independent integrals less than n in number.
With the consideration of these tne chapter just referred to
concerns itself.
. Reverting now to formulaB (2), if all the <p^B entering into
any one of the integrals u contain only finite negative powers
of Xy the integral is called regular in the region of the point
a: = 0, and with proper choice of r can be written
(3) i?^E aM 9?o + 9^1 log a; +....+ 9?k log*a; \ ,
A TBEATISE OK LINEAR DIFFEBEKTIAL EQUATIONS. 53
where the ^« are uniform, and or'F becomes infinite for
a; = in the same manner as a -\- fi log x + +
^log^o;^ a. A--** heiDg constants. In order that equation
(1) should have a system of linearly independent regular
integrals in the region of the point a; = 0, it is shown to be
necessary and sufficient that every coefficient pj shall have
a; = as an ordinary point or a pole of multiplicity not greater
than y. Denoting by w< the degree of x in the denominater
of p,, the value of i for which w, + w - i = ^ is a maximum
is called the characteristic index of equation (1), and by sub-
stitution in the differential ^uantic F{y) of xp for y, we will
find that xr-pP{xp) developed m ascending powers of x has as its
first term 0{^p) ar^ where 0{p) is an integral function of p of
de^e w — t = V. G^Tp) = is called the indicial equation j
ana it may be snown tnat the number of linearly independent
regular integrals of (1) is not greater than the degree of this
equation. The conditions that it should be equal to this
degree are also determined, and in particular its degree is
observed to be equal to n when all the integrals are regular.
The exponent r in (3), where F is supposed to be a regular
integral, is given by the indicial equation ; and the coeffi-
cients of the q>^s developed in positive powers of x are deter-
mined by substitution of Fin the differential equation.
An extended application of the general theory is made to
differential equations of the second order, particularly to the
equation which has all its integrals regular and possesses but
three critical points. This equation is shown to be trans-
formable to one in which the critical points are 0, 1, oo, an
equation of which the hypergeometric series F(ay y5, y^ x) is
an integral. A complete translation of Goursat's memoir on
this equation is embodied in the work, filling some 150 pages.
An exhaustive discussion is given of its twenty-four integrals,
which divided into six groups of four each, are connected by
some twenty linear relations between integrals selected from
the six groups taken three at a time. The portions of the
plane in which the several integrals have a meaning are also
indicated. An investigation is made of the transformations
admitted by the series when all three quantities a, fiy y, ^^
not arbitrary ; and an extended list of such transformations,
with formulse derived therefrom, is given. The theory of
irreducible equations is briefly touched upon ; as is also, at
greater length, the theory of the decomposition of a linear
differential equation into prime factors, with its application
in the case of equations possessing regular integrals.
In equation (1) we can by a simple transformation readily
get rid of its second term ; and, as is shown in one of the
later chapters of the book, by a transformation ^ = 9^ (^)>
y = «'-* ^''^i) Uy where the form of z is dependent on a differen-
54 NOTES.
tial equation of second order, we may still further rid our-
selves of its third term, the equation so reduced being said to
be in its canonical form. There are also certain associate
equations {91 — 2) in number, the solutions of each of which
consist in a set of variables dependent upon the integrals of
equation (1) and possessing relative to the transformation
mentioned, the invariantive propertv of returning into them-
selves multiplied by a power of z , among these equations
being found the well-loiown equation of the n'** order on
which depends the determination of an integrating factor
for (1).
The volume concludes with a short chapter on equations
with uniform doubly-periodic coeflBcients, a subject which the
author expresses his intention of resuming m his second
volume. Supposing w and w' to be the periods of our coefiS-
cients, by the substitution oix -^ w or x 4- w' for x, they will
remain unaltered and the integrals will transform into linear
functions of one another. By analogy the general theory
already given suggests that the characteristic equations corre-
spondm^ to these substitutions may give us constants s and s%
by which the respective transformations multiply some in-
tegral u. When the general integral happens to be uniform
such proves to be the case, there being at least one integral u
which by the substitutions x -[• w and x -^ w' for x respect-
ively transforms into s u and s'u, and for the determination
of such integrals, as also of the other integrals of the equation,
methods are given. J. C. Fields.
NOTES.
At the meeting of the New York Mathematical Society
held Saturday afternoon, October 3d, at half-past three
o'clock, the Council announced that Professor Henry B. Fine
had been appointed to fill the vacancy in their body. The
following persons having been duly nominated, and being
recommended by the Council, were elected to membership :
Professor Thomas Craig, Johns Hopkins University ; Dr. A.
V. Lane, Dallas, Texas; Professor L. A. Wait, Cornell Uni-
versity ; Professor George Egbert Fisher, University of
Pennsylvania ; Mr. William H. Metzler, Clark University ;
Professor Ellen Hayes, Wellesley College ; Professor George
A. Miller, Eureka College ; Mr. Charles Nelson Jones, Mil-
waukee, Wisconsin ; Dr. J. Woodbridge Davis, New York ;
Mr. Charles H. Eockwell, Tarrytown, N. Y. ; Professor J,
Burkitt Webb, Stevens Institute of Technology.
KOTES. 55
Tho following original papers were read : The Determina-
tion of Azimuth by Elongations of Polaris, by Mr. Harold
Jacoby ; On Powers of Numbers whose Sum is the Same
Power of Some Number, by Dr. Artemas Martin ; A Classi-
fication of Logarithmic Systems, by Professor Irving String-
ham.
Professor Stringham's paper will be published in the
American Journal of Mathematics, and Mr. Jacoby's has been
communicated to the Royal Astronomical Society of London.
T. 8. F.
Ik the course of his paper mentioned above Dr. Martin
presented the following very remarkable series of numbers re-
cently found by him :
4» + 5*4.6» + 7* + 9* + ir = 12*
5* + 10* + ir + 16* + 19* + 29* = 30*
12' + 13* + 15' + 16* + 17'+....
+ 23* + 25* + 27* + 28*+29*+. . . +35* = 50*
l* + 2' + 4' + 5' + 6'4-7* + 9*4-12* + 13*
+ 15* + 16' + 18* + 20' + 2r + 22« + 23» = 28*
The paper will be published elsewhere in eztenso.
Ik connection with Professor Merriman^s article, it may
be of interest to note that Professor Wright, also a con-
tributor to the present number, gives a different treatment of
the same problem in his *^ Treatise on the Adjustment of
Observations,^' p. 206.
William Febrel, the eminent meteorologist, died on Friday,
September 18, at Maywood, Wyandotte County, Kansas. He
was bom in Bedford County, Pennsylvania, January 29, 1817.
He studied at Franklin and Marshall College, and at Bethany
College, being graduated from the latter in 1844. In 1857 he
became an assistant in the office of the American Ephemeris
and Nautical Almanac, and held that position for ten years.
Thereafter, until 1882, he held a special appointment in the
United States Coast Survey. In that year he was made assist-
ant, with the rank of professor, in the Signal Service Bureau,
where he remained until October, 1886, when he made his
home in Kansas City, Missouri. He invented the maxima
and minima tide predicting machine, which is now used
by the Coast Survey in predicting the tides. Professor Ferrel
received honorary elections to the Austrian, English, and
German meteorological societies, and in 1868 was elected to
membership in the National Academy of Sciences. Some of
his principal works are '^ Motions of Fluids and Solids Selative
56 KOTBS.
to tho Earth's Surface," published in 1869 ; "DetenninatioM
of the Moon's Mass from Tidal Obseryationfl,'' 1871; "Oonyeix-
ing Series Expressing the Batio between the Diameter and tfi»
Circumference of a Circle,'' 1871 ; " Tidal Eesearches," 1874 ;
" Tides of Tahiti," 1874 ; " Meteorological Besearehes," in
three parts, published consecutively, in 1875, 1878, and 1881 ;
"Becent Advances iivMeteoroloCT," 1883, and "Temperature
of the Atmosphere and the Earux's Surface," 1884.
It is with regret that we learn of the death of qnr member,
Asher Benton Evans. He died at Lockport, the'place of his
late residence, September 24, 1891. He was an alamnus of
Madison (now Colgate) University of the class of 1860. He
was widely known as an educator, and had been a contributor
to the Mathematical Monthly.
The Cambridge University Press announces : Catalogue of
Scientific Papers Compiled by the Royal Society of London,
new series for the years 1874r-1883 ; The Collected Mathematical
Papers of Arthur Cayley, Sc, i>., F. R. S., Sadlerian Pro-
fessor of Mathematics in the University of Cambridge, Vol.
*IV. ; A History of the Theory^of Elasticity and of the Strength
of Materials, by the late I. Todhunter, F. B. S., edited and
completed by Carl Pearson, Professor of Applied Mathematics,
University College, London, Vol. II.
The Clarendon Press promises : Mathematical Papers of the
late Henry J. S. Smith, Savilian Professor of Geometry %n the
University of Oxford, with portrait and memoir, 2 vols. ; A
Treatise on Electricity and Magnetism, by G. Clerk Maxwell,
new edition.
The October number of the American Journal of Mathe-
matics begins the fourteenth volume. It contains as a
frontispiece an excellent likeness of Professor Felix Klein of
Gottingen.
John Wiley & Sons have in preparation '^A New Element-
ary Synthetic Geometry, Plane and Solid, especially adapted
to high-school work, with numerous examples,*' by George
Brace Halsted, Professor of Mathematics in the University of
Texas. t. s. f.
The Inland Press (The Begister Publishing Company,
Ann Arbor, Mich.) has just issued : *' Practical astronomy,'*
by W. W. Campbell, a short treatise mainly intended for the
use of surveyors and civil engineers; also ''Logarithmic and
other mathematical tables " (to five places), bv W. J. Hussey.
The same house announces as in preparation iwo translations
KOTES. 67
from the Oerman : 0. Dziobek's '^Mathematical theories of
planetary motions/* translated by Prof. M. W. Harrington ;
and E. Netto's "Theory of snbstitutions and its applications
to algebra/* translated by Dr. F. N. Cole. It is to be noticed
that Dr. Netto has not only authorized the present transla-
tion, bat has famished the translator with a lar^e amount of
new material in the form of corrections and additions, so that
some of the chapters of the original are almost entirely re-
written, and the whole work will be considerably increased.
The work will appear early in 1892.
Professor M. W. Harrington having been appointed
chief of the TJ. S. Weather Bureau, the astronomical observa-
tory of the University of Michigan is temporarily in charge
of the newly appointed instructor in astronomy, Mr. W. J.
Hussey. The former instructor, Mr. W. W. Campbell, has
accepted a position as assistant at the Lick Observatory, Mt.
Hamilton, Gal. A. z.
Professor Clarence A. Waldo, recently of the Rose
Polytechnic Institute, is now at De Pauw University, Green-
castle, Indiana. x. s. F.
58 NEW PUBLICATIONS.
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Battebhann (H.). BeitrSge zur Bestimmung der Mondbewegang und
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Galilei (G.). Untcrsuchungen und mathematische Demonstrationen Uber
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Theil III. fOnfter und sechster Tag. Leipzig 1891. 8. 66 pg. m.
23 Figuren. Leinenb. Mk. 1.20
Qa8c6(L. G.). Tablas de Logaritmos, Colo^ritmos y Antilo^ritmos
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Mt^ER (P. A.). Die Beobacbtungen der Horizon tal-Itensitftt des Erd-
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St. Petersburg 1891. gr. 4. 120 pages m. 1 Curventafel. Mk. 4.50
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BULLETIN
NEW YORK
I ATH KM ATICAL SOCIETr.
vol.. 1. .No. 3.
Hl-i'KMBffB, UOl.
.\B\r TOOK
MACMILLAN <& GO'S
MATHEMATICAL TEXT-BOOKS
pt4li,LAf« *. CAi., PuuJiBhars, 1 12 f^-ourm A'<
ON THE DOUBLY INFINITE PBODUCTS. 61
ON THE DOUBLY INFINITE PBODUCTS.*
BT DR. THOMAS 8. FISKE.
The familiar siogly infinite products for tho sine and cosine,
dae to Enler^t
8ina; = .r(l-J.) (l - ^,) (l-^,)...,
or in another form.
Binx = X n \ 1 ,
-« L mTv J'
cos X
= Tiri-, ^1,
were first generalized by Abel. By a brilliant stroke of genius
he obtained for the elementary doubly periodic functions the
remarkable expressions I
X
sn ti: =
7777 fi ^— T-H
|_ mco + mo? J
en a; =
nnTi -- , -^ T-Tvl
nn[i -7 ^-^, rr-.T
dntr =
nn\i j^, — --,1
r/ 77 fl - 7 rr ^-r-j -^— ,1 '
* Rtoim6 of a Lecture delivered before the Society at tho meeting of
KoTember 7, 1890.
t Introductio in Analysin Inflnitorum (1748), lib. I. cap. IX.
t (Euvn8, Nout*eUe 6dition (1881), t. I., p. 848.
Journal f&r die reine u, angewandte Math. (Cbelle), Bd. II., p.
172.
62 OK THE DOUBLY INFINITE PB0DTJCT8.
in which m and m' are independent of each other and assume
successively all integral values from — oo to + oo , the simul-
taneous system m = fn' = alone being excluded in the nu-
merator of the first fraction. Abel^ however^ did not make a
complete and rigorous investigation as to the convergency of
these products nor as to their identity with the functions of
Jacobi. Cayley made the four doubly infinite products con-
tained in the aoove expressions the starting pomfc of a series
of investigations.* He found for them a complete theory,
based in part upon a geometrical interpretation, and upon it
he built up the whole theory of the elliptic functions. Al-
most immediately afterwards, Eisenstein f discussed in a very
elaborate manner, and by purely analytic methods, the general
doubly infinite product
77 7iri ^^- "I,
and arrived at results which, when supplemented by the more
recent theory of primary factors, due to Weier8trass,J have
given to the subject a permanent and classical form.
The path which the student naturally follows in the study
of the periodic functions, leads him directly to the considera-
tion of these products and, at the same time, indicates their
paramount importance. A theorem of Jacobi § shows him that
no more general periodic functions of a single variable are
possible than the doubly periodic or elliptic functions. He
learns that such functions are but the ratios of single valued
functions of another class, the so-called theta-functions ; and
these, it is soon seen, are nothing more or less than doubly
infinite products. There is no doubt that the theory of the
theta-functions of a single variable forms the natural intro-
duction to that of the elliptic functions.
Before taking up the general products, the limit of the
single product \
"=^4t~^]*
♦ Camh. Math. Joum., vol. IV., 1845, pp. 257-277.
Jourfi. des Math. (Liouville^, t. X.. 1846, pp. 885-420.
Collected Math, Papers, vol. I., nos. 24 and 26.
f Maihemaiiache Abhandlungeny Berlin, 1847, pp. 213-334.
Journal fUr die reine u. angewandte Math, (Crellb), Bd. XXXV.,
1847, pp. 153-247.
X Aohandlungen der Kdnigl, Akad. der Wiaaenachaften zu Berlin wm
Jahre 1876.
Abhandlungen aua der Punetionenlehre, von Kaki^ Weiebstbass.
BerUn, 1886, pp. 1-52.
S Gesammette Werke, Bd. I., p. 262.
I Cf . Hebmitb, Coura a la Surbonne, Quatrihne Sdition, p. 89.
OK THB DOUBLY IKFIKITE PRODUCTS. 63
when p and q both become infinitely great, should be con-
sidered. It will be found to be iudeterminate. In fact^ if
we l^i^ye in the limit
a being a given constant^ then
A similar resnlt holds for the infinite prodnct representing
cos a;.
In the investigations of Gayley corresponding results were
developed in connection with the double products/ for ex-
ample
u = xnn\i ^— r-/l,
by the introduction of an auxiliary geometrical construction.
The periods oo and oo' being always assumed respectively real
and ima^inary^ a pair of rectangular axes were drawn^ and cor-
responding to every factor in the product a point was set down,
the coefficient of the real period being the abscissa and that
of the imaginary period the ordinate. The entire finite por-
tion of the plane was thus covered with a series of pomts
forming the vertices of a net-work of squares constructed on
the linear unit. These points were all enclosed within a con-
tour of infinite dimensions, the fprm of which depended upon
the relations between the infinite limits of the products.
The value of the product was shown to depend upon the
form of the contour, and in Gayley's memoirs the bounding
contour is regarded successively as a square, a circle, and an
infinite horizontal ribbon, and an infinite vertical ribbon.
By the application of logarithms one obtains
1 d^ • 1
log tt = - a; 22 , — - 22
moj + fn'co' 2 {moo + m'co'y
x'
....
(mco + m'co'y
taking the contour symmetrical with respect to the origin,
t;he terms containing odd powers vanish, or
qS 1 ' X* 1
For another contour, similarly
log W'= — ~ 1^2 7 ; T-TT, — -7-22". ; ;~7r4 — . . .
^ t {moo + m' go') .4 (mo^+.mV)
64 OK THE DOUBLY IKPINITB PEODITOTS.
Hence
the sums extending to the region enclosed between the two
contours. All the terms except the first being infinitely
small, we have
where
;? — i ss* ^ —iff ^^ ^^'
from which is readily seen the relation between two difFerent
systems of theta-fnnctions. The system of theta-functions
corresponding to the infinite horizontal ribbon is identical
with that given by Jacobi.*
In Eisenstein's researches we have
L ma + np + yj
whence
1 a^ 1
log w = — a; 22 ; — r^-- ^ 22
ma -^np +y 2 {ma -^ n/3 -i- y)*
-t 22
3 {7na-\-7iP -^-y)
I • • • •
The whole theory is thus dependent upon that of the very
general series.
22 ,
{ma -t np -h yY
Eisenstein's elegant investigation as to the convergency of
this series has been recognized as fundamental and has found
its way into the text-books, f He deduced as the necessary
condition for convergence
It follows that in the expansion of log u the coeflBcients of all
the powers of x except the first two, have fixed sums indepen-
♦ Jacobi, Fundamenta Nova (1829), cap. 61,
f Cf. JoRDAK, Coura d^AtuUyse, 1. 1., p. 165.
OH" THE DOUBLY INFINITE PB0DTJ0T8. 66
dent of the arrangement of their elements. Since howerer
the first two coefficients may alter their values with a change
in the arrangement of the factors of u, two functions which
are related to each other in this way will he connected by an
equation of the form
One finds in Eisenstein's memoir a very elaborate investiga-
tion as to the nature and value of the quantities p and q, and
the results are applied to a general theory of the elliptic
functions. In spite of the great interest of these further
developments, it is unnecessary for our present purpose to
enter into details upon them on account of the wonderful
simplification brought about through Weierstrass's theory of
primary factors.*
This theory enables us to express anjr continuous function
which does not become infinite for finite values of the vari-
able in a factorized form. It shows us, however, that the
simplest factors of such a transcendental function, should
differ from the linear factors of a rational entire algebraic
function, in that each should have an exponential associated
with it. Thus we find, according to this theory,
sin 2;
-00 L wJ
an expression from which every element of indetermination
has been eliminated. Kow it is evident, after the investiga-
tions of Eisenstein, that we can remove the indetermination
from the product
u = nn[i ^^773—1
by introducing the exponential factor
1 X* '
- ma + nfi + y^ i "" {ma + n^ + y)*
The result may be exhibited as a product of the form
r n ^ I ^ ^*
L ma -i- n/3 + yj
* Weiebstbass, loe. eU.
Cf. also JonDAK, Coura d'Ancdyae, t. II., pp. 815-817.
66 EABLT HISTORY OF THE POTENTIAL.
This product conseqnentlj denotes a function of unique char-
acter possessing all the essential properties of an ordinary
theta-function.
The special case given hy the formula *
in which
w = 2/107 + 2/i'(»',
has been called by Weierstrass the signia-function iT{x), and
is the basis of his beautiful theory of elliptic functions.
EARLY HISTORY OF THE POTENTIAL.
BT FBOF. A. 8. HATHAWAY.
The object of the present article is to correct an error that
occurs in Todhunter's " History of the Theories of Attraction *'
(vol. IL, arts. 789^ 1007, and 1138)^ and that is repeated,
doubtless on Todhunter's authority, in various encyclopaedias.
This error consists in assigning to Laplace, instead of La-
ffrange, the honor of the introduction of the Potential into
dynamics, an honor that the EncyclopsBdia Britannica makes
the basis of a eulogy to Laplace (art. Laplace) in the woi-ds :
^* The researches of Laplace and Legendre on the subject of
attractions derive additional interest and importance from
having introduced two powerful engines of analysis for the
treatment of physical problems, Laplace's Coefficients and
the Potential function. The expressions for the attraction
of an ellipsoid involved integrations which presented in-
superable difficulties ; it was, therefore, with pardonable
exultation that Laplace announced his discovery that the
attracting force in any direction could be obtained by the
direct process of differentiating a single function. He thereby
translated the forces of nature into the language of analysis
and laid the foundations of the mathematical sciences of heat,
electricity, and magnetism.'^
The announcement here referred to was made by Laplace
♦ BiERMANN, Theorie der analytischen Funetionen, Leipzig, 1887,
p. 834.
ScHWARz. Formeln und Lehrsdtze zum Otbrancht der eUipHtchen
FunciioTien, GOttlDgen, 1885.
EABLY HISTORY OF THE POTENTIAL. 67
in the oonrse of a memoir by Legendre between 1783 and
1785 : EneyclopsBdia Britannica ([art. Laplace), — **♦ * ♦ Le-
gendre in a celebrated paper entitled Recherches aur Tattrao-
turn des aphircHdes IwmogineSy printed in the tenth volume
id th^ Divers Savans, 1783, **♦♦''; Todhunter, Hist. Th.
Attr., vol. II., p. 20, " A very important memoir by Legendre
is contained in the tenth volume of the MSmoires * * * prS-
sentispar divers Savans ♦ * ♦ . The date of publication of
the volume is 1785. The memoir, however, must have been
communicated to the Academy at an earlier period ; for, in
the treatise De la Figure des Planites, which was published in
1784, Laplace refers to the researches of Legendre, which
constitute the present memoir : see p. 96 of Laplace's
treatise.''
Todhunter continues, in art. 789 : *^ In this memoir we
meet for the first time the function V which we now call the
Potential, and which denotes the sum of the elements of a
body divided by their distances from a fixed point. The
introduction of this function Legendre expressly assigns to
Laplace. The following are the circumstances :
A point is situated outside a solid of revolution. Le-
gendre has to determine the attractions of the solid at the
point, along the radius vector which joins the point to the
centre of the solid, and at right angles to this direction. He
has found a series for the former ; and he says the latter'
might he determined by similar investigations ; then he adds :
***** mais on y parvient Men plus facilement d Vaide
d^un Thioreme que M, de la Place a Men voulu me communi-
quer: void en quoi il consiste'
Then follows the theorem, which is enunciated and im-
mediately demonstrated. The theorem is that the attraction
dV
along the radius vector is — ^-, and the attraction at right
dV
angles to the radius vector is rs ; where r is the radius
rdu^
vector and 6 the an^le which it makes with the axis of the
solid : these attractions being estimated towards the centre,
and the pole respectively."
As Fis the notation used by Laplace in this announce-
ment, it is plain, I think, where he found this method of
differentiation to get the forces ; for that is the notation used
by Lagrange in the last of several memoirs previous to 1783
in which he made use of the Potential. On this account it
may be well to notice this last memoir first : Theorie de la
Lioration de la Lune. Mimoires de VAcademie royale des
Sciences et Belles-Lettres de Berlin, Annie 1780. CEuvres,
t. v., p. 5.
68 EARLY HISTOBY OF THE FOTEHTIAL.
"The memoir is divided into five sections. The first is
desired for the exposition of a general analytical method for
resolving all the problems of dynamics. This method^ which
I employed in my first memoir on the libration of the moon^
has the singular advantage of requiring no construction and
no geometrical or dynamical reasonings but onlv analytical
operations subjected to a process that is simple and uni-
form. ♦ ♦ ♦ ♦
Let 7«, m', m", ... be the masses of the bodies^ P, Qy J2,
. . . the accelerative forces that attract the body m towards
centres whose distances are /?, j, r, . . ., P', Q', R\ . . . the
accelerative forces that attract the body m' towards centres
whose distances are p', ^', r', ....*** *
Taking into consideration the mutual disposition of the
bodies, one will have several equations of condition among
the variables x, y, z, etc. All these are expressed in terms
of some one or more variables (p,tl?,. . . that are independent
By substitution and differentiation, one will have the general
ecfuation <^d(p + V^# + . . . = : thus i> = 0, !P'= 0, . . . ,
give as many equations as there are undetermined variables,
by means of which these variables are determined. We shall
show how to abridge the calculations necessary to reduce
* * * * to functions of ^, ^', ....**♦♦ In regard
to the terms Pdp -f Q6q + Ror + . . . . and similar terms^
we note that in the case of nature the forces P, Q, B, . .
are ordinarily functions of the distances /?, j', r, • . . , so
that the terms of which they consist are all integrable. This
also furnishes a means of simplifying very much the calcula-
tion of tliese terms ; for it is only necessary, in the first place,
to integrate the quantity Pdp + QSq + B6r -f . . . m the
ordinary way, and then differentiate it according to the char-
acteristic 6, * * * *
Put for abridgment
T- 1 [^ dx^-^dy^-^dz^ dx'^ ^ dy'^ ^ dz' ^ ) .
F= m [{P6p +Qdq-\-R6r-^ . .) -^ m' [{FSp' + Q'6q'
+ R'Sr' + ..)+...,
and suppose a:, y, z, x'y y', «',... expressed in terms of other
variables <^, ^, . . . ; then substituting these values in T and
V and differentiating according to the characteristic d, re-
garding (p, tpi , dg)y dtp J . . ., as the corresponding vari-
ables (a referring to the time) the above equation becomes
BABLY HISTOBY OF TEE FOTENTIAIk 69
wherein -s— denotes the coefficient of 6<p in the differential
6T
of T, and --rr- the coefficient of ddq) in the same differential,
and so for the rest/'
This investigation appears also in the MSchanigue Analy-
ti^ue; but, as we shall see by another example^ Todhunter
did not recognize that the Mechanigue Analyttque, like the
MicJianique Cileste of Laplace, was largely a compilation from
preceding memoirs. Theories of Attraction, vol. II., p. 153,
art. 994 : **The first edition of a famous work by Lagrange,
appeared in 1788 in one volume, entitled Michanique Anaty-
hque. There is nothing in this edition which beara explicitly
on onr subject. But on his page 474 Lagrange gives, in fact,
an integral in the form of a series of the partial differential
equation
da' "^ db' "^ dc' "" '
and from this integral, as we shall see hereafter, Biot drew
important inferences with respect to the attraction of a body/'
The solution here referred to was given by Lagrange in
1781 : (Euvres, t. IV., p. 695, TMorie du Mouvement des
Fluides.
The idea of differentiating in order to obtain the forces first
appeared in Lagrange's memoir of 1763 : (Euvres, t. VI., p.
6, Becker ches sur la Libration de la Lune, Prix de I Acaai-
mie Royale des Sciences de Paris, t. IX., 1764. The kinetic
energy is differentiated to obtain the accelerations, forming
the first part of Lagrange's celebrated generalized equations
of motion given first in complete form in 1780. The potential
is used to obtain the forces for the first time by Lagi*ange in
the memoir /Swr V Equation, Siculairedela Lune; VAcadimie
Royale des Sciences de Paris, t. VII., 1773; Prix pour Fan-
nie 1774 ; (Euvres, t. VI., p. 335.
" If a point A attract another point B with any force what-
ever F, and if J be the distance between the two bodies and
d^ the increment of this distance in supposing that A attracts
B an infinitely small space da, then — F-j- is that part of the
force F which acts in the direction da ; and if one proposes
to decompose this force in three mutually perpendicular direc-
70 EARLY HISTOBT OF THE POTENTIAL.
tions da, rf/?, dy, — ^j-zy — -^5^ *^ ^^^ remaining compo-
nents. If F is proportional to -^y ^bich is the case of celes-
tial attraction^ then
and conseqnently, the three forces are represented by the
coefficients of da, d/3, dy, in the differential of -^. In short,
1 ^
it suffices to find the yalae of ^ ^^^ differentiate it by ordi-
nary methods.
If the }>oint B is attracted at the same time towards dif-
ferent points A, A', A", . . , whose distances from B are
Af Af' M"
A, A', J", . . . , and if the attractions are ^i* -at%9 2^' • • • >
it is plain that one has only to seek the yalue of the quantity
M M' M'*
'A"^'Z^ + -j7, + . . •
and to differentiate it as a function of a, /3, y, when the co-
efficients of da, dfi, dy, in this differential immediately give
the forces sought.
In general, if the point B is attracted by a body of any
figure whatever, whose mass is M, then^ considermg each
element, dM, of the body as an attracting point, it is only
necessary to find the sum of all the quantities --t-, found by
making the quantities that relate to the positiou of dM vary
and regarding a, /?, y as constant ; then, denoting this sum
by ^2, and making it vary as to the quantities a, fi, y, that
relate to the position of B, one has -r— , -r-^r, -^— for the three
^ da dfi dy
forces in the directions da, dft, dy, to which the total attrao*
tive force of the bodv Mon B reduces/' Lagrange then goes
on to apply this method to his discussion of the moon.
In October, 1777, Lagrange read a paper that is devoted
wholly to the potential and its applications to the dynamics of
a system of bodies : Remarques ginerales sur le Mouvement de
plusieurs corps qui s'attirent mutuellement en raison inverse
aes carris des distances. L Academic royale des Sciences et
Belles- Lettres de Berlin, annSe 1777. (Euvres, t. IV., p. 402.
*' Let M, M', M", . . , be the masses of bodies which com-
BA.BLT HISTOBT OF THB FOTEITTIAL. 7X
pose a given system, x, y, z the rectangalar coordinates of the
body Jfin space, x', y', x' those of the body M', and so on.
Put
_ MM'
^- ^{x-xj + (y-y')'+ {z-t'Y
MM"
■•' V (a; - X")' + (y - y")* + (2 - z")'
M'M"
■•■ V {x' - xy + (y' - y")' + («' - z"y + •"'
and let -^ , . . . , ^-^, . . . denote, as usual, the coefficients
of dx, , . . , dx', • . . in the differential of £1, regarded as a
function of 2;, . . . , 2;', . . . .
^ , 1 dD, 1 d£l 1 dXl • ., . .^i_ , . 1
One has -^ j-^, 17. -j-, -77 —r-, for the forces with which
M dx M dy M dz
the body Jf is attracted by the other bodies M'y M"y in the
directions of the coordinates Xy y, z, and so on. It is easy to
be convinced of this by performing the indicated differentia-
tion : for that will ^ve tne same expressions as the decomposi-
tion of the forces that act upon each body in virtue of the
attraction of each of the other bodies, supposed proportional
to the mass divided by the square of the distance. This man-
ner of representing the forces is, as one sees, extremely con-
venient, both for Its simplicity and for its generality; and it
has the farther advantage that one distinguishes by it, clearly,
the terms due to the different attractions of the bodies, for
each of the attractions gives in the quantity £1 a term consist-
ing of the product of the masses of the two bodies divided by
their distance apart. '^
Lagrange goes on to give the equations of motion
^d'x dn ^d^y dn .^d'z
_ dn
dt' '^ dx' ^ dt^"^ dy' ^ df '^ dz' ' " '
multiplies them by dXy dy, dz, . , . , adds and integrates, find-
ing the equation of conservation of energy,
+ J if -j •^. {•+...+ constant
72 EABLY HISTOBY OF THE FOTENTIAi:..
Since D, does not change when the aHK>ordinates change
by equal increments, he finds
dfl dn dn _
dx dx* di" ■ ' ""
with similar equations in the y- and iiP-coordinates.
Substituting
d£l
^d^x d£l __ ^d^y d£l ^d^%
dz dp ' dy de' dz df
and using
he has
(TX ^ (PF ^ d^Z ^
=: — — =
In words, the centre of gravity moves uniformly in a
straight line. The equations of motion are then shown to be
unchanged when the centre of cavity is taken as the origin.
Since £1 does not change to tne first order in a when *
dy __dy' _ _ dz ^ dz' _^ _
z z y y
Lagrange concludes that
/ dn dn\ ( ,dn ,dn\ .
[^^-'diJ-^VW'-d^'J'^-^^
with similar results for the axes of y, z.
Substituting the accelerations for the forces according to
the equations of motion, and integrating, he finds the equa-
tion of conservation of areas
^ydz-zdy ^ ^, y'dz' - z'dy' ^ ^ ^ . ^eonstant;
dt dt
9
and so on.
The article closes as follows :
** These theorems upon the movement of the centre of
gravity have already been given in part by D'Alembert ; but
the manner in which I have demonstrated them is new, and.
EABLY HI8T0BY OF THE POTENTIAL. 73
it appears to me^ merits the attention of geometers by the
utility with which it can be used. One perceives by the same
principles that these theorems will be eoually true if the bod-
ies act upon each other by forces mutually proportional to any
function whatever of the distance ; for, calling /(a;) the force
of attraction at the distance x, and putting
Fix) = j/{x) dx,
one has only to change the value of £1 above into
/2 = - MM' jP( v/ (a; - a?') '^ + (y - y') " + (« - «T)
- MM'F(^^ (x-ixf') • + {y-y"y + (« - z") •) - . • • >
to easily obtain the same results.'^
The next memoir in which Lagrange uses the potential is
that of 1780, already referred to, in which he completes his
generalized equations of motion, and uses the notation Ffor
the potential, which Lai)lace adopts.
Todhunter was not without warning of these facts, for he
says, vol. II., p. 221 : ''I must cite another sentence from
Blot's memoir ; he says on page 208, after introducing the
function F,
Jf. Lagrange a dimontri que les coefficients diffSrentiels
dV dV dV
da' dh' dc'
pris nSgativement expriment les attractions exercSes par le
sphSrolde sur ce mime point, paralUlement aux trois axes rec-
tangulaires, M. Laplace a fait voir ensuite que la fonction
V est assujetie d VSquation cliff irentielle partielle
d'V d'V erF_
da' "^ db' '^ d(^ "
I do not know on what authority the above expressions for
component attractions are assigned to Lagrange ; to me thev
appear due to Laplace : see art. 789, and also pages 70 an^
133 of Laplace^s Figure des Plandtes/'
Todhunter attempts a defense, vol. II., p. 160 : ^'La-
grange now proceeds to consider the attraction of the ellip-
soid on an external particle. He introduces what we call the
potential function, and denotes it by F. If /, g, h, denote
the coordinates of the attracted particle, the attractions in the
corresponding directions are "37 > "T" > -^ • Lagrange does
74 EABLY HISTOBY OF THE POTBHTIAL.
not claim these expressions for himself ; and we know that they
are really due to Laplace : see art. 789."
The arfi^nmeut is equally good if it he made to refer to
Laplace's ni*8t announcement^ given in art. 789, with " Laplace "
and '^Lagrange" interchanged. Moreover, Lagrange had
claimed these expressions in the Berlin memoir of 1777, twenty
years previous to the memoir Todhunter is describing.
Todhunter knew that Laplace constantly embodied the
work of others in his own witnout credit (see preface vol. L) ;
and he cites a breach of etiquette towards Legendre in, these
very memoirs in the matter of Legendre's CoeflScients, vol.
IL, p. 43:
" We will first reproduce a note bearing on the history of
the subject which occurs at the beginning of the menioir.
* ♦ ♦^ * Legendre says :
^ La proposition qui fait Vobjet de ce mimoiref itant di*
montree d*vne maniere oeaucoup plus savante et plus giniraU
dans un mimoire que M. de la Place a dSjA publii dans le
volume de 1782, je dois faire observer que la date de man
mSmoire est anterieure, et que la proposition qui paroU icif
telle qti'elle a Sti Me en juin et juillet 1784, a donnS lieu i
M, de la Place, d^approfondir cette mature^ et Wen prSsenter
aux OeomHreSy une theorie complite.'^'
This refers to Laplace's memoir Figure des Planites, con-
tained in the Paris MSmoires for 1782, published in 1785, and
is the one of which Todhunter says (p. 56) ** in this article
we have for the first time the partial differential equation
with respect to the coordinates of the attracted particle which
the potential V must satisfy : it is expressed by means of
polar coordinates," etc.
Nor did Todhunter neglect foreign memoirs alone, bearing
on his subject ; for if he had read the valuable Report on.
Dynamics by Cayley, Brit. Ass. Rep,, 1862, p. 184, he would
have found the potential function properly credited to
Lagrange, with a reference to the memoir, 8ur P Equation
Seculaire de la Lurie, of 1773.
In conclusion I ought to say that a sentence in Sir William
Thomson's Baltimore lectures (1884), led me to investigate
this subject, Lectures, p. 112 : '*I took the liberty of asking
Professor Ball two days ago wliether he had a name for this
symbol pr'; and he has mentioned to me nabla^ a humorous
suggestion of Maxwell's. It is the name of an Egyptian harp
which was of that shape. I do not know that it is a bad
name for it. Laplacian I do not like for several reasons both
historical and phonetical.''
Robe Polytechnic iNSTrruTB,
Teiie Haute ; 1891, October 22.
THE THEORY OF UOHT. 7ff
THE THEORY OF LIGHT.
The Theory of Light By Thomas Preston, M. A., Lectnrer
in Mathematics and Mathematical Physics, University College, Dub-
lin. London and New York, Macmillan & Co., 1890. Svo.
XIntil within a very few years it has been a matter of con-
siderable difficulty for American students interested in higher
theoretical optics to pursue this study with advantage, for
want of access to the ori^nal memoirs and the absence of anv
adequate presentation of their contents in any of the Amen-
can text-books. For the students of the English universi-
ties, Air}''8 Undulatory Theory of Optics and Loyd's Wave
Theory of Light have been the chief English helps until
the publishing of Glazebrook's admirable rhjrsical Optics.
It has been a ^at pity that the clear and beautiful presenta-
tion of the subject given by President Barnard in 1862, and
printed in the Smithsonian Report for that year, was not long
since published in separate book form, as it would be to-day
one of the very best books on the subject were it printed in a
form accessible to college students having a fair command of
elementary mathematics. I have been greatly surprised to find
it so little known even amoug American students who have
made a special study of the higher optics in European uni-
versities. Besides the English books referred to above, the
admirable report of Professor Stokes on Double Refraction in
the British Association Report for 1862, and the equally ad-
mirable later one on Optical Theories by Glazebrookip 1866,
together with such books as Beer's Introduction to Higher
Optics, Knochenhauer's Undulatory Theory of Light, Lord
Bayleigh's articles in the EncyclopsBdia Britannica and the
Philosophical Magazine, Verdet's Lemons d'Optique Phy-
sique, Poincar6, Briot, Sir William Thomson, and Pro*
fessor Tait have heretofore supplied the special advanced
student with most of his needs. It has been very desirable,
however, that a treatise should be written for English-speak-
ing students, similar to those which Yerdet and Poincar6
have written for the French, dealing both with profound
theory and experimental facts. This seems to have been
done oy Professor Preston of Dublin.
The work in some respects resembles Verdet^s Optique
Physique. It begins, however, from a more elementary basis,
ana includes the more recent work of Lord Rayleigh and Pro-
fessors Rowland and Hertz. In this latter respect as well as
in some others, it is better than Glazebrook's Physical Op-
tics. It lacks the splendid bibliography of Veraet, and is
on tSie whole much more elemontaiy* . . _
76
THE THEOBT OF LIGHT.
It begins with the simplest principles of wave motion,
treated both with and without mathematics* The difference
between wave and group Telocity is explained at the end of
chap. II. after the manner of Lord Bayleigh's Sound. Hnygenff'B
!)rincipal and secondary waves are well explained^ and Stokes's
aw of the intensity of the light at any point of a secondary
wave is giyen together with a reference to his celebratea
paper on the Dynamical Theory of Diffraction, 1849« At
the end of chap. lY.^ explaining reflection upon the waye
theory^ we haye this admirable remark: ^^In dealing with
problems in the reflection of li^ht we may therefore consider
the light propagated in rays if tt facilitates the solution. Tlei
toe must carefully bear in mind that rays have no physical
existence, for it is waves that are propagated and not rays.'*
(The italics are ours.) In chap. Y., in discussing the energy
equation y the density of the ether is referred to as 'Hhat
property of it which corresponds to the density of ordinary
matter^ and by which it possesses energy when in motion.
We have here a hint of ether inertia that may be due to other
causes than mere mass^ e. g. rotatory inertia. The chapter on
determination of refractive indices^ gives a very full account
of the usual methods^ with references to the original memoirs.
The section on gases is particularly good.
Chap. IX. is on diffraction, ana it is at this point that the
really mathematical part of the book may be said to begin.
The treatment is clear and thorough, and nothing is slurred
over that can possibly help a student to a competent under-
standing of the subject, although the treatment sometimes
seems a little too concise. So vast is the ground covered,
however, that this seems almost a necessity.
This is the onlv book so far as we know which gives a pretty
full discussion of the theory and use of Bowland's concave
Sfratings and an account of" the new determination of wave
engths by Rowland and Bell.
The entire treatment of diffraction is very full and satisfac*
tory, although the remarkable results of Lord Bayleigh and
Professor llichelson on the distribution of the fight from
sources other than points are not given. Comu^s graphical
methods of dealing with diffraction problems however are
quite fully given. The recent applications of these researches
of Bayleigh and Michelson to spectroscopy could hardly have
had a place in the book, which has been out of press for more
than a year.
An admirable but brief presentation of the views of Mac*
CuUagh and Neumann on the relation of the plane of vibra*
tion to the plane of polarization in polarized lights is given^
together with the different suppositions involved as to the
changes of density and rigidity of the ether in crystals ou
THE THEOEY OF LIGHT. 77
which diverse views on this point are founded. One might
wish that some of these more purely physical or dynamical
questions had been discussed at greater length. Keference
is however made to the admirable report of Glazebrook on
optical theories in the British Association Report for 1885.
The papers of Sir William Thomson, Willard Gibbs, Kette-
ler, and Glazebrook, in the Philosophical Magazine, American
Journal of Science, and Wiedemann's Annalen, since that
time, however, are particalarly important, especially the ex-
traordinary speculation of Sir William Thomson on a *Ma-
bile '' ether and Willard Gibbs' able discussion of the same.
The notion of a medium capable of transmitting transverse
vibrations in virtue of a quasi-rigidity imparted to it by
motion, especially rotatory motion, is evidently becoming
more and more important and Sir William Thomson, al-
though apparently still an adherent of the elastic solid
theory, has himself shown how the rotation of the plane of
polarization in a magnetic field can be explained by the
assumption of such an ether. In the last chapter (AXI.)
the author of this treatise has given a sketch, clear and as
simple as the nature of the subject will allow, of the present
state of the electro-magnetic theory of light and the experi-
mental researches of Dr. Hertz on electro-magnetic waves.
Oliver Heaviside has done so much work in this direqtion that
it seems as if some mention might have been made of it. So
also Professor Rowland.
The whole treatment of the subject of polarized light is full
and satisfactory, while, on the whole, also very concise.
There are a large number of examples added to each chap-
ter. These are well selected and the sources indicated from
whence they are derived. The advanced student has thus
pointed out to him what authorities on each part of the sub-
ect may be best for him to consult. One of the very best
eatnres of the book, in our opinion, is the impression it
leaves on one's mind of its being, above all, an able, clear, and
accurate presentation of the subject as it was left by Fresnel
— the Newton of the uudulatory theory of liffht. As Fresnel
left it, so, except that Maxwell and Lorenz have shown that
the vibrations are probably electro- magnetic in character, it
essentially is to-day. Questions as to the ultimate structure
and constitution of the ether are related to the undulatory
theory of light, just as questions as to the mechanism of
gravitation are to the theory of gravitation, as ordinarily
treated. The ordinary mathematical theory and its confirma-
tion by observation and experiment, rest intact, no matter
what may prove to be the physical mechanism by which their
results are brought about. The famous Baltimore lectures
of Sir William Thomson^ the British Association reports
78 NOTES.
(1862 and 1885) of Stokes and Olazebrook, and the later
papers allnded to before^ are the natural sources of informa-
tion for those who wish to ^o into these matters.
For advanced students m colleges and all who wish to
acquire a thorough knowledge of the existing state of the
undulatory theory of lights we recommend this admirable
treatise. The type and illustrations are aLso models of clear-
ness and elegance and reflect credit upon the publishers as
well as the author. JoHBT E. Daties.
UNiYERsmr OF Wisconsin,
MadiaoD, October 12, 1891.
NOTES.
A REGULAE meeting of the New Yoek Mathematical
Society was held Saturday afternoon, November 7, at half-
past three o'clock, the vice-president in the chair. The fol-
lowing persons having been auly nominated, and beinj? recom-
mended by the Council, were elected to membership: Professor
Simon Newcorab, Navy Department and Johns Hopkins Uni-
versity ; Dr. Oskar Bolza, Clark University ; Mr. Charles
Riborg Mann, Columbia College ; Professor Ludovic Estes^
University of North Dakota ; Mr. Herbert Armistead Sayre,
Montgomery, Alabama; Professor James Harrington Boyd,
Macalester College ; Dr. Asaph Hall, Jr., U. S. Naval Obser-
vatory ; Dr. Percy F. Smith, Yale University ; Mr. Edwin
H. Lock wood, i ale University ; Professor Kobert Judson
Aley, Indiana University ; Professor Joseph V. Collins,
Miami University ; Dr. Charles II. Chapman, Johns Hop-
kins University; Professor Albert Munroe Sawin, Univer-
sity of Wyoming ; Mr. Frank Oilman, Lowell, Massachu-
setts ; Professor Henry Parker Manning, Brown University ;
Mr. Charles S. Peirce, Milford, Pennsylvania.
Mr. Charles P. Steinmetz read an original paper entitled
" On the curves which are self -reciprocal in a linear nul-sys-
tem, and their configurations in space. '^
Dr. Edward L. Stabler made some remarks upon the
theory of errors which are equally probable between given
limits.
The nul'System in space, which formed the subject of Mr.
Steinmetz's paper, is a one-to-one correspondence between
points and planes such that any point lies in its conjugate
plane, and conversely. A linear nul-system is one in which
all the planes conjugate to the points of any straight line
NOTES. 79
ifitersecfc one another in a second straight line^ so that there
exists a one-to-one correspondence between the lines in space.
T. s. F.
The 64th meeting of the OeseUschaft deutscker Natur-
/orscher und Arzte (a German association corresponding to
the American Association for the Advancement of Science)
was held this year at Halle a. S., September 21 to 25. If
the list of papers announced in advance as to be read in the
different sections can be taken as an indication of what was
actually done, it appears that the section for mathematics
and astronomy is by far the strongest of all sections, not only
numerically — 25 papers, the next in order being the sec-
tion of physics with 13 papers, then the section for instru-
ments of precision {InstrumentenJcunde) with 8" papers, etc.
— ^but in particular considering the weight of the names repre-
sented, it is worthy of notice that astronomy has hardly any
share in this programme, the subjects belonging almost ex-
clusively to pure higher mathematics. The association has
a special section (only recently organized) for elementary
mathematics and natural sciences and the allied educational
questions. Professor Georg Cantor, of the University of
Halle, was president of the section of mathematics and
astronomy ; Dr. H. Wiener, of the same university, was
secretary.
The following is a list of the papers announced to be read
in this section :
L. Kronecker of Berlin, Opening address ; K. Neumann
of Leipzig, On a question in electrodynamics ; L. Koenigs-
berger of Heidelberg, On the theory of systems of partial
differential eauations ; F. Klein of GSttingen, Account of
recent English investigations in mechanics ; F. Me^er of
Clausthal, Review of the present state of the theory of invari-
ants ; M. Noether of Erlaugen, The fundamental proposition
on the intersection of three surfaces ; Rohn of Dresden, On
rational twisted quartics ; E. Papperitz of Dresden, The gen-
ei'al system of the mathematical sciences ; Worpitzky of Ber-
lin, On the axioms of geometry ; H. Wiener of Halle, On
the foundations and the system of geometry; F. Kraft of
Zurich, The meaning and value of Grassmann's Ausdeh-
nungslehre for the whole domain of mathematics and me-
chanics ; V. Eberhard of Konigsberg, Elements of a syste-
matic exposition of the forms of polyhedra ; F. Miiller of
Berlin, On literary enterprises adapted to facilitate the study
of mathematics ; A. Pringsheim of Munich (subject not
announced); Finsterwalder of Munich, The images in diop-
tric systems of larger aperture and larger field of vision ; W.
Dyck" of Munich (subject not announced); H. Schubert of
80 NOTES.
Hamburg, On a question in cnumerafcive (abzdhUnde) geom-
etry ; M. Simon of Strasburg, On a question in absolute
geometry ; G. Beuschle of Stuttgart, A fundamental system
of identities of the algebraic runctions ; R. Mehmke of
Darmstadt, Description of mechanisms for the mechanical
solution of equations ; Hilbert of Konigsberg, On complete
{voile) systems of invariants ; Stackel, Wangerin, G. Cantor,
of Halle (subjects not announced).
A more detailed account of the meeting maybe given later.
On October 23 the Mathematical Society of the XTniver-
sity of Michigan held its first meeting this falL Professor
F. C. Wagner read a paper on the mathematical principles of
thermodynamics. The society was founded in November,
1890, and has held seven meetings in the course of the last
academic year. Professor W. W. Beman is president ; Dr.
F. N. Cole is secretary. A. z.
At the meeting of the National Academy of Sciences
held at Columbia College, November 10 to 12, the following
papers of a mathematical nature were read : Certain new
methods and results in optics, by Professor Charles S. Hast-
ings ; New pendulum apparatus, by Professor T. C. Menden-
hidl ; Astronomical metnods of determining the curvature of
space, by Professor C. S. Peirce ; Variation of latitude, by
Professor S. C. Chandler; Color system, by Professor 0. N.
Eood ; Reduction of Rutherfurd's photogra])hs, by Professor
J. K. Rees ; Measurement of Jupiter^s satellites by interfer-
ence, by Professor A. A. Micbelson.
Professor Hastings's paper contained some new and very
simple demonstrations of optical formulae already known, as
well as certain important formulae altogether new, including
a general expression for magnifying power applicable to both
telescopes and microscopes. Professor Peirce presented astro-
nomical evidence tending to show that space possesses a nega-
tive curvature, and called attention to various methods of con-
ducting an investigation of this property of space. Professor
Chandler exhibited curves showing that the recently discovered
variation of latitude could be made to explain certain hith-
erto unaccountable discordances in older observations. His
paper was followed by considerable discussion among the
astronomers present ; Professors Young, E. C. Pickering, C.
S. Peirce, Abbe, and Dr. Gould taking part. The chief gues-
tion debated was whether the variation has a terrestrial or
celestial origin. The investigations are being published in
the Astronomical Journal, Professor Michelson described his
recent measurements of Jupiter's satellites at the Lick Obser-
vatory^ and thought that we may hope to measure the angular
NOTES. 81
diameters of some of the brighter stars^ if they be as great as
the hundredth part of a second of arc. His paper was perhaps
the most important one of the session. In it was presented a
new method of measuring the angular diameters of luminous
discs by means of the interference phenomena produced by
them. The experiments made at the Lick Observatory have
been described in the Pvhlications of the Astronomical /Society
of the Pacific. The 12-inch telescope was used, but a telescope
is by no means indispensable for these observations, the chief
requisite being a very favorable condition of atmosphere. It
is to be hoped that these very promising researches will be
continued. h. j.
The series of lectures given last winter at Johns Hopkins
University to teachers and those intending to become teachers
was so successful that a similar series is to be given this
winter. Among the lectures promised we note one on the
teaching of mathematics by Professor Simon Newcomb.
La Nature announces the death of !^douard Lucas, Professor
of Mathematics at the Lyc6e Saint-Louis. His death was due
to injuries received from a mishap at Marseilles during the
meeting of the French Association for the Advancement of
Science, at which he presided over the section of mathematics.
He was the author of many papers, but was most widely
known through his RScrSations Mathimatiques. The second
volume of his more recent work Theorie des Nombres is still
in press.
In an article entitled " Twelve versus Ten,*' which appears
in the November number of the Educational Review, Pro-
fessor W.B. Smith strongly advocates duodenary numeration.
John Wiley & Sons have in preparation a new work on
**The Theory of Errors and Method of Least Squares, '* by
Professor W. Woolsey Johnson. t. s. f.
82 NEW PUBLICATIONS.
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Pfkh, (L. Graf v.). Kometische Stremungon auf der Erdoberfllche u.
daa Gesctz der Analogie im Wfjtgcbuude. Berlin, F, DQmmler'B
Verl. $2.Ba
Baschi (L ). G«>inetriB analitica alk Coordinate (Cartcoiane pioiettiTO
euc). rarma IBUl. 8. iJ4u pg.
a dc la Theoris gjnfraie des Fooctiont. Pull
u. a.ao
RovcBi (E.) ct CoMBEBOUSSE (C. de). Traits dp Qfomitrle. «. MfHcw.
revue ct augment^. l>iirt[a U ; Qeomctrie drn I'eapace. Paris 18B1.
e. lOtt G;i pj;. av. figuiva. H. 7.B0
ScaiJLbR (\V. F,). Ivehrbuch der unbestimmten Gleichnngen deal.
Grades I Dinphnnlische Gleichungen). Ncbst den AhhaDdinnnn dM
llBchel de Mbziriw! in franzJIsiNchcm Originale u. deutsoher ijebsr-
eetzuDg. Rcorbeitct zum 'I'heil noch System Klejrer. Bach I. Stutt-
gart 189 1. gr. 8. 8u. 176 pg. M. 4.80
Sxrrn (Cti.). Elementary treatise on Solid Qeometrj. 8 ed. LondiHi
1801. a 15 a. 342 pg. w. flgiires. cloth. H. 18.00
Stahpfer (S.1. IiOgBrithmisch-trigonomelrisehe Tafeln. nebst vencbie-
dfncn andcrcn iiDtzlichen Tufclii und Fornielp und ciner Anwriauns,
niit Ililfe dersvlben logarithmische Kcchnungcn auszutDhrcD. 14.
Aufl. WienlSfll. gr. a. 24 u. 122 pg. gcbunden. M. S.40
Stewart (S. T,). I'Ibob und Solid Geometry. New Tork 1891. 8. 10
a. 400 pg. w. riguTOs. half leatber. ' M. 4.00
Vkuv (P. yy.). On the distribution of the Moon's Heat and iO variation
with the Pbasi!. Published by the Utrvcht Society of Arts and
Sciences. The Hague 18SI. 4. 45pg. w. It plates. Pmc-essar.
M. S.OO
ig a tabid of
_._ ,, _ _ numben "
the stundard of Itowland's photographic map of thcwlar Hpeelni ,
a. tablL-» of errata a, additions to the Index. Manchester 1691 . 8.
:)Opg. H. !.70
WiLLiAMBOS (H.). An Rlementary Treatise on the Integral Calculus. Bth
ed., revised and enlarged. Cr. 8vo. pp. 478. liongman's. 10s. Bd.
WnoKsiir (11,1. Knnonv I>o)!urytm6w,wj-d, S. Uickstein. WaizsairalSOO.
8. 3U l<g. mit 6 Tufeln. M. 8.80
Zech IP.). Aufgal>en aus dcr thforetiscben Mechanik, nebst AuflSsun-
geu. S. Auflaire. utitcr Milwirkung von C. Cninz, Stuttgart 1891.
gr. 8. 8 u. an j)g. m. ITS FiguroD, M. 4.80
CONTENTS.
Ui" IMil.lv liiliiKi. IV.Iwl. llv p. I.i...»4. ■^,
}few PutiliGuiiDUi
Anieliw IMI Uwarllwi (baiild be aadnsBQii to llx-
U Kui iWL Sttet-t, ?(bw Totb n\j^
ON LISTS OF OOYABIANTS. 85
ON LISTS OF COVARIANTS.
BT DB. BMOBT MCCLINTOCK.
I USE the term coTariants to inclndc invariants, and I write
particalarly concerning lists of covariants (^roundforms) of
the binary quintic and sextic, those of qnantics of lower de-
gree being few and well known. When the weight of a
covariant is spoken of in this article, it must bo understood
to mean the weight of its first term or '^ source/' The symbol
5. will denote that coyariant of the quintic whose weight is n,
and 6« that covariant of the sextic whose weight is m. Thus,
for example, 5, and 6, represent the hessians (weight 2) of the
quintic and sextic respectively. The only case oi ambiguity
18 C,p for which weight there are two covariants : one of these
may be denoted by 6,.., the other by 6,5*.
The table printed on the next page exhibits the terminology
of different writers. Professor Cavley's* superb collection of
the covariants of the quintic, in whictif each is designated by
a letter of the alphabet, is arranged, as will be observed, first
according to the degree in the coefficients, and secondly ac-
cording to the order in the variables. Thus 5^, of the first
degree, is called A, and 5, and 5^, of the second degree, come
next ; but 5. being of order 2 while 5, is of order 0, the letter
B is assigned to 5^, and so on. The small italics contained in
the column headed by the name of Dr. Salmon f are the sym-
bols used in a table at the end of his work, illustrative of
transvection, and denote the scmin variants which form the
sources of covariants of the quintic and of higher quantics as
well. The letters a, g, h, i, j, Jcy are therefore used by him
a)so for the sextic, together with /, m, n, q^ representing
respectively 6., 6,^, G„, 6 . Clebsch t and Gonian § differ but
slightly in their nomenclature. Faa de Bruno | designates
invariants by the letter 1, with subscripts indicating degree,
and other covariants by the letter C, with subscripts indicat-
ing order and degree. The column headed by the name of
Professor Sylvester contains his table IT of germs for the
quintic, each source having its distinguishing germ, i. e., the
coefficient in it of the highest power of the final coefficient of
the quintic. Thus, the quintic being
• Mathematical Papers, II, 273-309 ; Cambridge. 1889.
+ Modern Higher Algebra, 4th Edition.
Ji Theorie der Bindren Algebraisehen Formen, Leipzig, 1872.
) Invariantentheorie, herausgcgcben von Eerschcnsteincr, Leipzig,
7.
I ThSarie des Formes Binaires, Turin, 1876.
1 American Journal of Mathematics, V, 89.
86
ON LISTS OF OOYXRIAKTS.
COVARIANTS OF THE QUINTIC.
Weiffht.
Deg- Order.
A
Salmon.
CUitch.
Gordan.
Fadde
Bruno.
SybMiter,
6.
1-6
U, a
f
f
•
a
6.
2-6
C
H,h
H
9*
c„
ic)
6.
3-9
F
9
T
/
t'...
id)
5,
2-2
B
Six
•
i
•
^«
(e)
6.
3-5
E
k
c„
ay
6.
3-3
D
T,j
•
•
J
c..
(«')
6,
4-6
I
►
c«
a(c)f
6.
4-4
H
e
c,..
W
6.
5-7
L
0,..
ic)V
6..
4-0
J
A
A
/,
ar
6..
5-3
K
c^.
Sa{e')/-2{e)ie)f
5..
5-1
J
a
a
a
c.,.
a{c)r
5.,
6-4
N
c,..
a{d)r
5„
6-2
M
r
T
T
o,„
(cyr
5..
7-5
P
c.„
(c){d)r
5„
7-1
/?
/?
<?.„
a\c)f'
5,.
8-2
R
5
5
5
c,..
<c)T
5..
8-0
Q
K
B
B
/,
{c){d]f'
5„
9-3
s
0,..
(c)T
5„
11-1
T
Y
r
^1.11
icYidr
5„
12-0
U
L
C
c
I„
{cyjf'
5..
13-1
V
•
d
6
c..,.
{cYie-r
5»
18-0
*
w
R,I
R
R
/..
a(c)r
OK LISTS OP C0VABIAKT8. 87
the germ of any covariant is the coeflScient of the highest
I)Ower of/ appearing in its source. In the column in qnes-
tion.
\c\ = oo — i',
f) = a^d - UU + 2*',
[e\ = fl« — A3)d + 3c',
[6^) = ace - aef + 2fcd — c' — J'e,*
J = a'ef + 4ac' + 4rfJ* - 3JV - 6o5c?rf.
This germ-theory of Professor Sylvester will doubtless lead
in fnture to important results. We may even now make
some practical use of it as an aid in reducing coyariants to
their simplest forms.
The collection of covariants of the quintic lately made by
Professor Oayley from his past publications is not likely to
be superseded lor many years. It appears in that great
series of volumes, not yet complete, which will endure as the
noblest monument of their illustrious author. It gives each
covariant in the fullest detail, with all the terms arranged in
the most complete order, and with the numerical coefficients
verified, in every instance, as perfectly as that mode of veri-
fication can accomplish it, by calculations printed at the foot
of the columns. The covanants as published are free from
any inaccuracy which I have been able to discover,! with the
single exception of the one (5 J called I. In this the third
and fifth columns should each be multiplied throughout by
6, and in the second column dbcf — 10 should read a(?e — 10.
Yet, perfect as this collection is, it does not profess to give,
and in fact does not always give, each covariant in its simplest
form. An instance in point may be seen as the result of an
examination of the germs. The germ of Fas printed is the co-
efficient of/ ',namely, in Professor Sylvester's notation, a(cY{d).
If we suppose that note has been taken, as in our column
headed "Sylvester," of the germs of the preceding co-
variants tabulated by Professor Cayley, we see that the
germ of Fis the prodfuct of the respective germs of J and Q.
In fact, the addition of 2 JQ to F as printea would simplify it
Yet it does not follow necessarily that the simplest ground-
* Printed erroneously (u2* in the paper cited. The germ of 630 is also
printed incorrectly.
\ As regards the quintic. The last column of is, Ko. 9 of the quartio,
is incorrect.
88
0!!^ LISTS OF COYABIAITEB.
form may not have a compound germ. The case of 6, ^ is on
instance to the contrary.
The synoptical tables of Clebsch and Qordan do not g^ve
the covariautc^ but merely symbolic expressions indicating
how the CO variants may be compated. Let no one, however,
undertake to compute covariants as directed by the symbolic
analysis. The expressions resulting from the application of
the Glebsch-Oordan formul® are often highly complicated.
For instance, their formula for the covanant of weight 15
gives the complicated function S6BK + 7E0 — 252^, and
that for weight 21 gives 252^+29 OiT— 6950, where ^ means
a form of 5 which I think simpler than P as tabulated, and
tff means a lorm of 5 in some respects simpler than 8. Yet
of course these complicated expressions are true covariants, of
the right weights, degrees, and orders. I mention them
merely to illustrate the necessity, for those engaged in com-
puting and tabulating covariants, of a simple method.
I am unable to prove that the method which I prefer will
in every case produce the simplest form of covanant, and it
will not apply to all covariants, but I have not yet known it
to fail when applied, and so I ^ve it for what it may be worth.
If we call by the name of '^ simple transvection that foml
of transvection {Ueberschiebung) in which one of the two
covariants concerned is the quantic itself, my plan is to pro-
duce any desired covariant, when possible, by simple trans-
vection from the nearest available covariant of lower wei^ht^
Simple transvection increases the degree by 1 and the weight
(in the case of the quintic) by from 1 to 5, and it cannot be
performed when the desired increase of weight exceeds the
order of the covariant operated upon. Observing these limi-
tations, it is not difficult to pick out a succession of available
operations, for instance for the quintic, by referring to the
table of weights, degrees, and orders of possible independent
covariants. Representing by [n] the operation of simple
transvection which is to increase the weight by n, we shall
have, successively.
21 5, = 5„
1
4
1
4
1
2
1
5
5, — 5„
5. = 5..
5, — 5^
5. = 5„
5. = 5.»
5. = 5„
5, — "id*
[31 5, = 5,.,
4] 5. = 5
1
3
1
'4
4
5
m
Although, as I have said, I cannot nrove that simple trans-
vection, applied to the nearest, will always produce the
ON LISTS OF COYABIAKTS. 89
simplest possible result, it seems not unreasonable that this
should be the case, since the oj)erand is {presumably in the
simplest form, and the operation is of the simplest character.
The operation just indicated for producing 5, yields an ex-
pression simpler than H, that for 5,. an expression simpler
than P, that for 5,^, an expression simpler than 8* ; the
others produce the corresponding covanants tabulated by
Professor Gayley, which are therefore, in all probability, the
simplest attamaole forms.
Another principle appears to be even more imiK>rtant than
that of simple transvection from the nearest. It is that if for
anjr auantic a ^oundform is wanted for any degree-weight for
whicn one exists for a lower quantic, the same ^'source''
should be employed. This principle enables us to use for 6„
6„ 6^, 6^, 6,, 6„ 6j„ and 6„, the sources of corresponding
degree-weight for the quintic. Yet for some reason unknown
to me this principle appears uniformly to be disregarded in
the formation of 6,,. Even the "germ-table for the sextic''
of Professor Sylvester assigns for 6,, a less simple form than
5,,. That it is less simple may be seen from an examination
of the numerical coefficients :
6,. byFaAde Bruno's table,t ±186, ±330, ±549, ±330, ±186
6„from5.„ ±142, ±168, ±263, ±168, ±142
In fact, Fa4 de Bruno's 6,, is really 2 • 6. . 6, — 3 • 6,..
Tables for the sextic are needed, as complete, correct, and
well printed as those of Professor Cayley for the quintic. If
anjr member of the Society, undeterred by the great labor
which the task will involve, will undertake to compute such
a set of tables for the sextic, to be published, say, in the
American Journal of Matliematica, I shall be glad to con-
tribute towards it my own compntatious of the first seventeen
of the twenty-six groundforms, complete, with those of the
simple forms of 5„ 5 , and 5,^ already mentioned, which
might usefully be published with the sextic tables. The
utility of such printed tables consists largely in their availa-
bility for reference in case of need, and for this purpose they
should be published, not singly or in small numbers as com-
puted from time to time, but in masses. It is for this reason
that I have not thought of publishing the computations just
mentioned. I have made them, indeed, not intending publi-
cation, but in order to verify to the greatest extent my idea
that the easiest way to find the simplest forms is, wherever
practicable, to apply simple transvection as already explained.
t Corrected. As Professor Sylvester points out {loc. cit.\ the tables
printed by Fa& de Bmno, useful as they are, contain many errors. The
last column of this 6i o table is nearly all wrong, and only one colamn of
the five is quite right.
90 OK LISTS OP OOYABIANTS.
«
Gonfining my attention to this question of simplicity, I have
not even made search among mathematical jonmals to see
what has already been done towards the computation of the
more difficult coyariants of the sextic, bnt will do so at the
instance of any member of the Society willing to undertake
the work of completing the series, who may not himself have
access to a large library.
The class of cases to which I have referred as unsuitable
for the application of simple transvection are those in which
there is no groundform near enough upon which to operate.
For instance, to produce by simple transvection the invariant
6j,, of the sixth degree in the coefficients, we should need as
a basis of operation a covariant of degree 5 whose weight
should not be less than 12, and whose weight and order com-
bined should exceed 17. The only groundforms of degree
6, are, however, 6„ and C , the former of order 4, the latter
of order 2, and neither of them can be used to produce 6„.
It is of course possible in such cases to apply simple transyec-
tionto a complex coyariant — as, for instance, to 6^. 6^, of order
(), for producing 6,^ — ^but that will not usually produce the
best results, and it is doubtless preferable to employ trans-
vection (no longer "simple") of groundforms other than
the quantic itself, in accordance with the recommendations
of the text-books. Of the four text-books already cited
which supply formulsd for computing the groundforms of the
quintic and sextic, the formulae collected by Salmon are
apparently the best. So far as my observation has gone, the
application of Salmon^s formulsB has given simple resalts in
most cases. Among the exceptions to this remark are 5,,,
After once applying simple transvection to produce 6,„
for which weight there are two groundforms of the same
degree in the coefficients and order in the variables, we can-
not again employ satisfactorily the rule of the nearest for
producing the other form. Thus, [Ij 6,^ gives 6,,^, and we
cannot again use [1] 6„ for producing 6y^ ; nor can we prof-
itably use [2] 6,„ perhaps because it is not only not so
"near" as [l] 6,^, but even not so "near" as any combina-
tion of [2] 6,, and [1] 6,,. In this case the usual symbolic
formulaX-Jacobian of 6. and 6, — is the best for practical
application.*
The nine groundforms of the sextic which remain to be
computed or collected (if in simplest form) from other pub-
lications — 6,3, for instance, is well known — are, as to weight,
degree, and order, as follows, the weight being denoted by
the subscript : 6,„ 6, ; 6,., 7, 4 ; 6„, 7, 2 ; 6„, 8, 2 ; 6,^
* I have DOt tested [4] 6|,.
OK LI8XS OV OOVABIANTS.
91
9, 4 ; 6,^ 10, 2 ; 6,„ 10, ; 6,^ 12, 2 ; 6„, 15, 0. Of these
nine, five may be deriTed by simple transvection, viz., [4j
6.. = 6.., [5] 6.. = 6„. [3] 6,. = 6 [2] 6.. = 6.., [4] 6 = ft„.
I nave wntten m two places 6„ for 6^^ or 6^^, not known
which is to be preferred, a matter to be settled in either case
most easily by compating a few terms npon each basis. The
three of nigher weights, to which simple transvection will
not apply, may probably be deriyed most simply by means of
the formalsd giyen by Salmon.
To illustrate the process of simple transyection^ which,
althongh sufficiently implied, is not nsnally illustrated in the
books, 1 giye [4] 6^ = 6^ in detail in the form of a table :
OPKBAITD.
MUI^TIPLISBS.
6«.
For (1).
For (2).
For (8).
ae — 4W + 3c*
%af — ebe + ^d
an — 9ce + Sd*
2bg —ecf-\- Me
eg Mf-\- 3e*
-^2
e
— d
c
— b
a
/
— e
d
— e
b
-7-2
9
-f
e
d
c
1 ) = acg—Sadf-h 2<w'— ^ V 4- Sbcf— bde-^c^e^ 2cd? )
2) = ^cg—Sbdf-h9be* + 9cY—l7cde-hadg + Sd*—aef [ 6.
[3 ) = aeg—dbdg 4- 2c*g —af-^- Sbef—cdf—Sce* + 2(f e )
The multipliers in this instance are extremely simple. The
coefficient of a is always 1, as in this case, but in general
those of the other multipliers are other integers. The rule
which I find best for determining the integers forming the
coefficients of the multipliers for simple transvection is given
in another paper, as a special case of a broader rule for trans-
vection in general. The paper in question, '* On the Com-
putation of Covariants by Transvection,'' to be read before the
Society on January 2, 1892, will be printed elsewhere, the
Eages of the Bulletin being intended rather for critical and
istorical notes than for original inyestigations.
92 A FBBKCH ANALYTIOAL GEOHETBT.
A FRENCH ANALYTICAL GEOMETEY.
Lefons de Oiomitrie Analytiqxie. Par MM. Briot et Bouquet.
BeTue et annoUe par M. Appell. professeur & 1a Faoalt^ des
Sciences. Paris, Ch. Delagrave, 1890. 8vo, pp. iii. + 722.
This popular French text-book reached its fourteenth
edition in 1890. At that time^ as we learn from the preface,
changes in the programmes of the schools and improved
methods of teaching had made a revision of the book advis-
able. This pieoe of work was done by M. Appell, a mathema-
tician, whose name is as familiar to American students as to
Frenchmen. The bare list of the articles in the book which
he has touched covers a page and a half, and it is safe enongh
to say that *' nihil quoa tetigit non ornavit" A treatise of
this Kind is of course more interesting to teachers of element-
ary mathematics than to any one else ; to them even a slight
account of a school book which has achieved great and lasting
popularity in a nation where pure mathematics has flourished
80 splendidly and so long^ can not fail to prove interesting by
virtue of its subject.
The book opens with a concise notice of the different sys-
tems of plane coordinates, beginning with rectilinear co-
ordinates in general and the particular case of rectangular
axes ; then passing rapidly over polar and bi-polar systems,
and finally giving a notion of coordinates in general. These
notions are all simple enough when presented in the trans-
parent style of the authors ; in fact plane coordinates are so
much simpler than cuitcs drawn on a sphere that it is a
wonder that school books on geography should not give an
account of them before taking up the subject of latitude and
longitude which almost always proves difficult to young
pupils. The writer was once explaining rectangular co-
orainates at a teachers' institute wnen one of the members
rose and thanked him for inventing them ; he had been try-
ing to teach latitude and longitude without anv of the pre-
liminary ideas necessary to an understanding of the matter.
At the close of the first chapter we read, *' The representation
of figures by equations is the basis of analytic geometry ; it
allows us to apply the processes of algebra to the stuay of
figures. In analytic geometry we are concerned with three
fundamental questions : when a figure is defined geometri-
cally, to find its equation ; reciprocally, when the equation is
given, to construct the figure ; finally, to study the relations
which exist between the geometrical properties of the figures
and the analytical properties of the equations."
Chapter II. takes up the first problem ; various loci, in-
JL FBENCH ANALTTICAL OEOHETBT. 93
dnding nearly all the simple onrres whose names are famil-
iar, are defined geometrically and their equations written
down directly, as a mere statement of the definition in the
language of algebra. The curves are drawn and sufiSciently
de8cril]Ked« In this way the student not only gets a notion
of what a locus is, but, what is far from easy, he comes to
see how an equation, so different in its nature and belonging
to quite anotner realm of thought, can represent a geometric^
figure, and to look upon equations in x and y as orief state-
ments of the truths of geometry. The mind is put in a con-
dition to understand why the manipulation of an equation
may lead to new facts. Without some such preparation it
can hardly be very profitable to try to prove geometrical
theorems with equations ; there can be nothing in the
student's mind corresponding to them, and a gulf which he can
not bridge will exist between the proof and the conclusion.
That the radius of a circle has a constant length is expressed
algebraically by the equation
that the sum of the distances of any point on the ellipse
from the foci is constant, bv
tt 4- 1; = 2a,
and so on ; the polar equations from their simplicity being
first written down. The chapter closes with a list of exercises
of which this is a specimen : ^^ To construct the curve whose
equation in bi-polar coordinates is uv=^ a'; the distance be-
tween the poles being 2a." Of course this lemniscate would
be constructed by drawing circles with their centers at the
poles. The student has been told previously how to find
points on the ellipse in the same way. The problems are
mostly too difficult for a beginner.
Chapter III. treats of the fundamental idea of homogeneity.
A function/ (a, i, c, . . .), we are told, is homogeneous and
of degree m when / (ia, hhy . . .) = A*/ (a, ^, . . .) ; the
sum, difference, product or quotient of any two homogeneous
functions is liomogeneous ; and the same is true of any
Eower, root or transcendental function of / (a, {, t?, . . .) ;
ut the transcendental function must be of degree 0. Thus
sin (—5 — Ti) is homogeneous, while sin (a + V^^) ^® °^^-
All this is sufficiently clear, and what follows shows its vital
importance at the threshold of analytic geometry.
'•When we seek the relations which exist between the
lengths of the various lines A^ By 0, ... of a figure, we
imagine these lines referred to a unit of length which is
94 A. FBBirCH AKALTTIOAL eSOlOXBT.
nsaally not spocifiod and remains qnite arbitnuy.'' Henoe
the reuflonin^ which leads to a relation among the lengths of
these lines is independent of any particular unit, and the
relation must subsist whateyer be the unit Jn particular it
subsists if the unit be divided by h^ that is if the number
expressing each length is multiplied by I;; henoe the relation
is nomogenoousy or at any rate, if not, then it must break up
into several relations which are each homogeneous. An
apparent exception occurs when some line of the figure is
taken as the unit of length, but the exception is explained
and the homogeneity reestablished. '^ The equations which
the theorems of elementary geometry lead to directly, are
homogeneous. . . . The principle of homogeneity can be
used at every step to verify the algebraic tnmsformations
which have been effected.''
The above is a too brief account of a part of this yenr
elementary and most interesting chapter. No part of it is
difllcult even for a youDg student, while it opens up to him a
lino of tlioucrlit which he must follow throughout his scientific
studios in wliatever direction he turns. This is the kind of
work which makes mathematicians and scientists, while the
Htufleiit whoso analytic geometry consists only in manipulating
a few o(|uati()tis of which the meaning is but dimly seen, finds
it a barren and useless subject.
Homo considerations follow, still of the simplest kind,
which lead to the conclusion that ''all rational expressions,
and all irrational expressions containing only square roots, can
bo cc)nHtru(;to(l by moans of a limited number of right lines
and (rirolcH." It is added, but not proved, that no others can.
A littlo reading of this character would turn the attention of
a goodly nuUibiT of bright young men who are still at work
upon Honio of the impossible problems of antiquity, to subjects
more worthy of their abilities.
Hook II. ojKins the study of the right line and circle. The
trcjatment of these loci is similar to that in our familiar text
books and is limited to the needs of beginners. The next
(diapter troats of geometric loci in general. In Book I. the
equations of many loci were obtained simply by writing dovm
the definition in the lanj^ua^e of algebra ; here we obtain the
equations of loci by eliminating one or more variable param-
eters. The coordinates of a point may be explicitly given as
functions of the parameter a, but more usually they are only
implicitly given in two equations
(1) /, (^,y,a) = o,
(2) /, {X, y, a) = 0.
Each value of a gives a pair of curves intersecting in a point
A FBEKCH AKALYTICAL GEOXBTBY. 95
of the loooB, and *^ the equation of the locus is obtained by
eliminating the parameter a between the equations (1) and (2). '
Why this should give the equation of the locus is a difficult
thing for students to see ; but it is less difficult when properly
stated. The result is not a single equation in z and y, but a
pair of equations
(3) / {x, y, a) = 0,
(4) F {X, y) = 0.
which are equivalent to (1) and f2^. Any system of values of
Xf y, a, which satisfies (1) and (2) must satisfy (3) and (4) ;
hence equation (4) is satisfied by the coordinates of every
point on the locus. Conversely, every system of values of x,
y, a which satisfies (3) and (4) must satisfy (1) and (2) ; it
loUows that (4) is the equation of the locus. "
The teacher who reads this book will be everywhere de-
lighted by the careful way in which the raw edges of thought
are hemmed down ; no loose threads are left to ravel out
and destroy the fabric. It can hardly be doubted that this
brave honesty in their elementary school books has much to
do with that precision of thought and clearness of expression
which makes the works of French mathematicians a perpet-
ual refreshment to the reader. It would be an inquiry worth
making, whether students in hi^h schools and normal schools
who never intend to enter college do not get more benefit
from the study of geometry than any others. With them it
is not merely a thing to be crammed for an entrance exam-
ination, but a subject to be studied for its educational value.
It seems especially disastrous to make analytic geometry, a
subject where the preliminary notions are so delicate and
beautiful, a thing to be asked questions about when entering
college ; inasmuch as the amount of knowledge required can
at best be but small, and will almost certainly be acquired
under conditions likely to blight future results.
The remainder of the book, while deeply interesting, is
more advanced ; and it is not the purpose of this sketcn to
do more than call attention to those parts whose study may
possibly be useful to teachers of classes which are beginning
the subject.
C. H. Chapman.
Jomrs HoPKnrs UMivKBsrrT, December 14, 1891.
96 ANNUAL KEVrVSQ OF GERMAN MATHEXATIGIAN8.
THE ANNUAL MEETING OP GERMAN MATHE-
MATICIANS.
Through the courtesy of the secretary^ Dr. H. Wiener, of
the Uniyersity of Halle^ who kindly sent ns adyance sheets of
the Proceedings, we are now enabled to giye a more detailed
account of the papers read before the section for mathematics
and astronomy of the German Naturforscher- Versammlunff
held at Halle, September 21 to 25, 1891. The meetings of this
section constitute at the same time the annual meeting of the
German Mathematical Union (Deutsche Mathemafiker- Verei"
nigung). The section had seyen meetings; the total number
of members registered as present was 70.
1. The first paper read was a report by Prof. Felix Klein,
of Gdttingen, On recent Enalish investigations in mechanics.
The following abstract of tnis paper is translated from the
ProceedifMs.
^* The distinguishing characteristic of the English work in
mechanics in comparison with that of continental writers lies
in its being based on a thorough jgrasp of physical reality and
in the resulting graphical lucidity (durchgdngige Anschau-
lichkeit) of the inyestigations. For this yery reason the Eng-
lish work in mechanics proyes particularly interesting and
instructiye to the mathematician accustomed to a purely
abstract train of reasoning. The usual lack of that method-
ical treatment and mathematical rigor which the continental
mathematician is wont to expect cannot be regarded as a
serious objection ; in fact, it adds to the interest.
Among the matters of detail discussed by the speaker, his
remarks on the history of the discoyery of Hamilton's method
of integrating the equations of dynamics may be of general
interest. The matter seems to be entirely unknown, although
Hamilton distinctly states the facts at yarious places in his
writings, in particular in his first paper on systems of rays
(1824). At the time when Hamilton beean writing, the
emission theory was still preyalent so that the determination
of a ray of light passing through any non- homogeneous (but
isotropic) medium was considered as a special case of the ordi-
nary mechanical problem as to the motion of a material parti-
cle. It may be noticed in passing that the distinction between
this special case and th^ general problem is not an essential
one : by proceeding to higher spaces, any mechanical problem
may be reduced to the determination of a ray of light trayers-
ing a properly selected medium. Now Hamilton's discovery,
according to which the integration of the differential equations
of dynamics is made to depend upon the integration of a certain
AKKUAL MEETIlirQ OF QEBMAK XATHEXATICIANS. 97
partial differential equation of the first order, was simply the
result of the fact that Hamilton, following the CTeat move-
ment just then taking place in physics, undertook to derive,
from the point of view of the nndulatory theory, the results
in geometrical optics already known in the form of the cor-
puscalar theory. Hamilton's method for integrating the dif-
ferential equations of dynamics is, primarily, nothing but the
general analytical expression for the relation between ray and
wave, a distinction which in its physical form was well known
at the time. Considered in this new light it is readily under-
stood why Hamilton gave to his investigations that unneces-
sarily specialized form in which he published them and which
was removed only later by Jacobi. In his investigations on
systems of rays Hamilton had originally in view certain
entirely practical questions relating to the construction of
optical instruments. This is the reason why he operates
throughout with waves of light issuing from single points.
The i^eal meaning of Jacobi's generalization is that any other
waves of light may be used to determine a ray. The general
wave is constructed in optics from the special waves by means
of the so-called principle of Huygens. This construction is
an exact eouivalent to the analytical process bjr which we
ascend in tne theory of partial differential equations of the
first order from any * complete ' solution to the ' general '
solution."
2. The paper of Mr. Papperitz, of Dresden, On the system
of the mathematical sciences, it is announced, will be pub-
lished elsewhere.
3. Mr. Max Simon, of Strassburg, read a paper On the
axiom of parallels.
4. Mr. Franz Meter, of Clausthal, presented an elaborate
report On the progress of the projective theory of invariants
during the last twentv-Jive years, which will probably be pub-
lished in extenso by the German Mathematical Union.
5. Mr. FiNSTERWALDER, of Munich, read a paper On the
images of dioptric systems of somstohat large aperture and
field which is published in the Transactions of the Bavarian
Academy of Sciences {Abhandlungen, Class II, Vol. 17,
Abth. 3, pp. 517-588).
6. Mr. KOHN, of Dresden, spoke On rational twisted guar-
tics, illustrating his remarks with the aid of models.
7. Dr. H. Wiener, of Halle, read a paper On the foundations
and the systematic development of geometry. The following
abstract is ^ven in the Proceedings,
** To be rigorous we may demand that the proof of a mathe-
matical proposition should make use of those assumptions
only on which the proposition really depends. The simplest
ooDceivable assumptions are the existence of certain objects
98 AiSfKUAL XBBTIKO OF OEBXAK XATHEMATICIAITB.
and the possibility of certain operations by which said objects
may be connected. If it be possible^ without further assnmp-
tionsy to connect such objects and operations so as to produce
propositions, these propositions will form a self-suataininp
\in sich begrUndet) domain of science. Such, for instance, is
algebra.
In geometry it is of interest to go back to the simplest
objects and operationB, since starting from these it is pos-
sible to build up an abstract science which will be independ-
ent of the axioms of geometry while its propositions run
parallel to those of geometry.
The projective geometry of the plane ofFers an example.
Let the objects he points and lines, the operations those of
joining and cutting, and let obiects as well as operations be
restricted to a finite number. Throwing off the geometrical
dress we shall have elements of two kinds and two kinds of
operations such that the connection of any two elements of
the same kind produces an element of the other kind. The
feometrical propositions obtained on these assumptions (apart
rom combinatory propositions inyoMng the number of ele-
ments) are closing propositions (Schliessungssdize), if this
term be taken to mean propositions about certain lines and
points such that every one of the lines contains at least three
of the points and every one of the points lies at least on three
of the lines. Such are for instance : (1) Desargues's theorem
of perspective triangles, and (2) Pascal s hexagram theorem
applied to two lines.
The proof of such propositions cannot be obtained from the
given objects and operations ; in other words, this domain of
geometry is not self-sustaining. If however the proof for any
one sucn proposition (or for several) be taken from some
other domain, then, by its repeated application a closed do-
main of plane geometry may be obtamed. Thus, proving
Desargues's theorem by means of solid geometry, we obtain
the domain embracing all propositions usually derived by
means ot geometrical addition of vectors or points. The at-
tempts at deriving proposition (2) above from (1) have not
been successful. Another possibility would be its derivation
bv projection from a space of three or more dimensions, or
else (which is easily done) by introducing the idea of con-
tinuity. These two ^* closing propositions/^ however, are
sufficient to prove, without further considerations of con-
tinuity or infinite processes, the fundamental proposition of
projective aeometryy and thus to develop the whole domain of
linear projective plane geometry.
Similarly it is possible to build up a solid geometry resting
on the point, line, and plane as fundamental elements, or ol^
jects. JBut in this case we obtain a self-sustaining domain.
AKKUAL MBBTINO OF OEBMAN ICATHEMATICIANS. 99
These considerations can be extended to higher spaces. It
will however be more important to descend from the plane to
the geometry of the line. The onl^ element we here have is
the point ; there can be neither joining nor cutting. It thns
becomes necessary to borrow an operation from another do-
main ; as snch we may nse constructions executed in the
plane but concerning only points lying on our line^ especially
constructions of projective, involutory, and harmonic groups
of points. It appears that the construction of harmonic
groups is sufficient as the following proposition can be proved :
If in a line two pairs of points of an involution^ or three pairs
of corresponding points of a projective system be given, it is
possible to construct the corresponding point to any other given
point by a finite number of constructions of harmonic points.
Other domains are obtained by introducing other assump-
tions. Thus the geometry of order presupposes the pi^oposi-
tion that on a closed line four points can be divided in a defi-
nite way into two pairs that separate each other. Still other
domains depend on the assumption of the continuity of the
elements, which may be either the analytical continuity of
the method of limits, or the geometrical continuity that nnds
its expression in the necessary meeting of points moving in a
certain way in a line.'*
8. Mr. ScHUBEBT, of Hamburg, read a paper On the enu-
merative geometry of p-dmensional spaces of the first and
second degrees,
9. Mr. Eberhabd, of Konigsberg : Elements of the theory
of forms of polyhedra. An elaborate work by tne author on
tnis subject has just been published {Zur Morphologic der
Folyedery Leipzig, Teubner, 1891).
10. Mr. BoLTZMANN, of Munich : On some points in Max-
welVs tJieory of electricity ; will be published in the Fro-
ceedings of the section for physics.
11. Mr. Hensel, of Berlin : On the fundamental problem
of the theory of algebraic functions ; see the same author's
paper in the Journal fUr MathematiJc, vol. 108, p. 142.
12. Mr. Felix MtJLLEB, of Berlin : On literary enter-
prises adapted to facilitate the study of mathematics. The
speaker pointed out the desirability oi an introduction to the
bibliography of mathematics ; complained of the want of
subject-indexes in the most prominent mathematical journals ;
gave an account of the progress of recent bibliographical
works ; and expressed a regret that the continuation to Pog-
gendorf's Dictionary of Authors has not yet appeared. He
also laid before the Section a plan for a new Mathematical
Dictionary for which he has been collecting the material for
the last 20 years ; it contains about 4000 mathematical terms
and over 1200 names.
100 ANNUAL MEETING OF GERMAN MATHEMATICIANS.
13. Mr. Dyck, of Munich : On the forms of the systems
of curves defined by a differential equation of the first order^
%n particular on trie arrangement of the curves of principal
tangents to an algebraic surface; will be published elsewhere.
14. Mr. David Uilbebt, of Konigsberg : On full systems
of invariants.
" Let c/,, e/., . . . , t7,-i be integral rational invariants of a
binary ground-form of the «th order, of the degrees Vj, v„
. . . v,_„ respectively,' in the coefficients of the ground-form ;
and let these invariants be so selected thab all other integral
rational invariants of the ground-form are integral algebraic
functions of those n— 2 invariants. Then the intenul rational
invariants of the ground-form form the integral functions of
a body (K6rper)of algebraic functions ; let ^ be the degree
of this body. Then the following formula can be shown to
hold:
\ V. . . . v..r 2. nllW (l/\2 /
+ . . .
for even n,
,-:^i(i)--a)(i-r-
G « - 1) \
for odd w.
15. Mr. ScHOENFLiES, of Gottingen : On Configurations
that can he derived from given space-elements by the operations
of cutting and joining alone. Kef erring to Dr. Wiener's paper
(7) the speaker showed that prop. (2) (PascaPs hexagram
applied to two lines) cannot be derived by the operations of
joining and cutting alone.
16. Mr. Minkowski, of Bonn : On the geometry of num-
bers. The author ^ves the name numher-frame (Zahlen-
gitter) to the totality of all those points of space whose
rectangular coordinates are three integral numbers and con-
siders certain solids and their relation to the frame. The
two most important cases are as follows. (1) Solids having
the origin of coordinates as centre and bounded by a surface
that appears at no point concave from without. For such
solids it can be shown that if the volume be ^2' the solid
must contain other points of the frame besides the origin.
(2) Solids containing the origin and bounded by a surface
which as seen from tne origin shows no double point. If the
NOTES. 101
volume of such a solid be ^ 1 +oi+«7 + 77+ . . . , it is
always possible to indicate deformations of the solid for which
the volume remains constant, the origin remains fixed and all
straight lines of the solid remain straight while all points of
the frame excepting the origin are found outside the solid
after deformation.
17. Mr. Peitz Kotter, of Berlin : On the problem of rota-
tion treated by Mrs. Kovalevsky. The paper develops some-
what farther the formulae given by Mrs. Kovalevsky in the
12th volume of the Acta Mathematica for a certain integrable
case of the problem of rotation of a heavy body about a fixed
point.
18. Mr. PiLiTZ, of Jena : A question in the theory of num-
bers. After an introductory discussion of the necessity for a
new calculus, or at least of a new way of conceiving of the
combination of elements in the problems of the theory of
numbers and the theory of functions, the speaker gave a
proof of the proposition announced by Riemann as probably
true : that the complex 0-points of the function C (s) all have
i as their real part.
19. Mr. F. Stackel, of Halle : On the bending of curved
surfaces under certain conditions.
20. Mr. A. Wangkrin, of Halle : On the development of
surfaces of rotation with constant negative curvature on each
other,
21. Mr. WiLTHEiss, of Halle : On some differential equa-
tions of the theta functions of two variables.
22. Mr. 6. Cantor, of Halle : On an elementary questio7i
in the theory of manifoldnesses.
23. Mr. GoRDAN, of Erlangen : Remarks on a proposition
of Mr. Hubert.
Alexander Ziwet.
NOTES.
A regular meeting of the New York Mathematical
Society was held Saturday afternoon, December 5, at half-
Sast three o'clock, the president in the chair. Mr. Wiley,
[r. Snook, and Dr. Pupin were appointed a committee to
report at the annual meeting, on December 30, nominations
for the oflBcers and other members of tlie council for the cal-
endar year 1892.
Dr. "Pupin read an original paper entitled " On a peculiar
family of complex harmonics, in which he deduced several
102 NOTKS.
useful properties of certain complex harmonic curves, explain-
ing briefly their application to polyphasal and continuous
current generators. Mr. Steinmetz and Dr. Webster made
remarks upon the physical side of the paper. Mr. Steinmetz
said that he had actually obtained in experiments with
dynamos, curves which very closely resembled those given in
Dr. Pupin's paper. T. s. f.
The following courses of lectures, extending through the
first half year, are being delivered at Clark University, Wor-
cester, Mass. :
By Professor Story : (1) Enumerative geometry and the
theory of coloring maps ; (2) Historic development of arith-
metic and algebra.
By Dr. Bolza : (1) Klein's icosahedron theory; (2) Definite
integrals and calculus of variations.
By Dr. Taber : Modern higher algebra.
By Dr. White : (1) Theta-f unctions of three and four vari-
ables ; (2) Modem synthetic geometry and higher plane
curves.
By Mr. de Perrott : Application of analysis and group
theory to the theory of numbers.
By Professor Michelson : Optical theories.
By Dr. Webster : Dynamics.
At the weekly mathematical conferences conducted by Pro-
fessor Story, non-euclidean geometry has been the subject of
systematic discussion, tojrether with less extended considera-
tion of topics of interest from other departments of mathe-
matics. At a recent meeting the new Amsler's planimeter
which has been added to the mathematical apparatus of the
University was exhibited and its theory explained by Professor
Story. Careful measurements on known areas give results
accurate to within one- twentieth of one per cent.
J. W. A. Y.
NEW PUBLICATIONS. 103
NEW PUBLICATIONS.
COMPILED BY B. WESTERMANN & CO., NEW YORK.
Annuaire de I'Observatoire municipal de Montsouris pour Tan 1891.
In-16. Gauthier-Villars. 2 fr.
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Lieferung 4. Stuttsart 1891. Lex. 8. pg. 42 u. 1-81(5 m. 11
Tafeln u. 204 Abbildungen. M. 12
Bryans (Q. H.) and Edmundson (T. VV.). Intermediate Science Mixed
Mathematics Papers, 1»77-91. (Univ. Cor. Coll. Tutorial Series.)
Cr. 8vo. Clive. 2s. 6dr
Ohave y Castilla (J.V Ensayo de una nueva Teoria de la Proporcion-
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planches. M. 2.60
Christiansen (C). Laerebog i Pysik. Band I (2 Hefte). Heft 1.
KjQbenhavn 1891. 8. M. 4.50
Cooke (T.). On the Adjustment and Testing of Telescopic Objectives.
York 1891. 8. M. 5.50
CzuDER (E.). Theorie der Beolmchtungsfehler. Leipzig 1891. gr. 8.
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D6RH0LT (K.). Die Enveloppe der Axen der einem Dreieck einge^hrie-
benen I^arabeln. Rbeine 1891. 4. 38 pg. mit 1 Tafel. M. 1.50
Eder (J. M.». Die photographischen Objective, ihre Eigenschaften und
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Flammarion (C). I j' Astronomic et ses Fondateurs. Copemic et la
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figures. 1 fr.
Flammarion (C). Qu*est-ee que le ciel ? In-16 avec 64 grav. Flam-
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FouRTiER (H.). L'Astrophotographie. Paris 1891. 8. 20 pg. M. 1.50
Galilei (G.). Opere, ristampate fedelmente sopra la Edizione Nazionale
con approvazione del Ministero della puolica Istruzione. (In 2i>
volumi.) Volume II. Pirenzo 1890. 8. 613 pg. c. tavole. M. 5
Gore (J. E.). Star Groups; a Student's Guide to the Constellations.
Ijondon, 1891. small 4. w. 30 maps, cloth. M. 5.;i0
Gore (J. H.). Geodesy. London 1891. 12. 220 pg. w. figures, cloth.
M. 5.30
Grassmann (R.). Die Zahlenlehro odor Arithmetik, der niedere Zweig
der Analyse. Erster Zweig der Formenlchre oder Matheraatik.
Stettin 1891. gr. 8. 12 u. 242 pg M. 4
GuiLHAUMON (J. B.). Elements de Navigation et de Calcul Nautique.
Precedes de Notions d' Astronomic. 2 volumes (I. Astronomic et
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M. 10
104 liEW PUBLICATIONS.
GuiUiEMiN (A.). Electricity and Magnetism. Trans, from the Frenoh.
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Koy. 8vo. pp. 998. Macmillan. 81s. 6d.
(Iexchie (E. J.). Elementary Treatise on Mensuration, containing
numerous solutions and examples. 2. edition. London 1891. b.
12 a. 224 pg. cloth. M.'8.80
Janren (W.). Die Krciselbewegung. Untorsuchung der Rotation von
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Berlin 1891. gr. ». 54 pg. m. Abbildungen. M. 1.80
KiRcnHOFF (G.). Vorlesungcn Qber matbematische Physik. Band III.
Elektricitat und Magnetism us, herausgegeben von M. Planck. Leip-
zig 1891. gr. 8. 10 u. 228 pg. m. IG Holzschnitten. M. 8
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286 pg. M. 5.80
Lanolet (S. p.). The new Astronomy. New edition. Boston 1891.
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dungen. M. 1.50
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KLEIN'S MODULAR FUNCTIOlirS. 105
KLEIN'S MODULAR FUNCTIONS.
Felix Klein, Vorhsungen uber die Tfieorie der elliptischen
MbdtUfunctionen, auscearbeitet und verrollst&ndigt von Dr. Robert
Fricke. Erbter Band. Grundlegung der Tbeorie. Leipzig, Teub-
ner, 1890. 8vo, pp. xix + 764.
The mathematical public is under great obligation to Pro-
fessor Klein's former pupil. Dr. Robert Fricke, for his able pres-
entation of the theory of the modular functions. His clearness
of treatment and skillful grouping of the many intricate feat-
ures of the subject have rendered this theory now thoroughly
accessible. Beside the work of arrangement, in itself a labor
of no small magnitude. Dr. Fricke has contributed many of the
intermediate steps necessary to the symmetry and complete-
ness of the subiect. His task has been performed through-
out with a highly creditable degree of conscientiousness and
ability.
The theory of the modular functions and the allied branches
has been one of the chief series of investigations to which Pro-
fessor Klein has devoted himself in the period of some twentj
years over which his scientific activity now extends. It is
characteristic of these investigations that they are not in-
cluded as a subordinate part in any of the great mathematical
theories heretofore commonly so recognized. Their distinc-
tive tendency is in the direction of the combination and unifi-
cation of the latter into a broader method of research. This
idea has been developed by Klein to an extent and with an
elaboration which have long since entitled it to recognition as
an independent, and in the highest degree productive mathe-
matical point of view. In the present paper some attempt
is made to sketch the general outlines of the new method, so
far as it concerns the modular functions, and to illustrate it
more definitely by the consideration of some of the more
important details.
Historically, Klein^s work has developed accurately along
the lines of a thoroughly predigested plan, the bolder features
of which arc already sharply defined in his earliest publica-
tions.* On this ground, then, a brief semi-biographical, semi-
scientific sketch of his career may properly find place here.
It is to be observed that this sketch makes no pretension to
completeness. It confines itself mainly on the scientific side
to the development of the theory of the regular bodies and
of the modular functions.
Klein's first productive activity dates from his relation to
* Cf. the preface to the ** Ikoaaeder,^* and the Eintrittaprogramm
mentioned on the following page.
106 KLEIN'S MODULAR PUKOTIONS.
Jalins Plucker^ as the latter's assistant in physics at Bonn.
On Plucker's death the preparation of the second Tolame of
his posthumous work on line geometry* was entrusted to
Klein, who was then at the age of nineteiBn. The first volnme
was edited by Clebsch. Haying completed this task, and
having taken the doctor's degree at Bonn, Klein studied in
Berlin and in Paris until the outbreak of the Franco-German
war, which compelled his return to Germany. Soon after-
ward he was appointed Privat-Docent at Gottingen, where
Clebsch was approaching the close of his brilliant career. In
1872 he was called to the ordinarius professorship of mathe-
matics at Erlangen. His Bintrittsprogramm \ prepared on
the occasion of assuming this chair is certainly a most re*
markable production for a young man of twenty-three, con-
taining, as it does, not merely a foreshadowing, out actnally
a systematic program, conceiyed with perfect maturity and
denniteness, of the scientific work to which he has since
deyoted himself. It is with the theory of operations that he
is here concerned ; not the formal theory of operations in
themselyes, but entirely with reference to the content to which
the operations are conceived to be applied, in particular
when this content is a geometrical configuration. Two such
theories were already in existence : the theory of invariants
and covariants, which deals with the effect of the entire system
of linear transformations of two or more homogeneous varia-
bles, and the theory of substitutions, in which the operations
are the permutations of a finite system of elements. These
two theories can be regarded as extreme types, between which
an infinite series of others can be inserted. A definite com-
plex of these intermediate types has furnished the field to
which Klein^s labors have thus far mainly been devoted. The
Fintrittsprogramm appears as a preliminary survey of the
general aoctrine of operations, with reference to geometrical
configurations. It involves not only the discontinuous opera-
tions, within which Klein's specific work has been included,
but also the continuous systems, which belong with differential
equations, and the theory of which has been mainly devel-
oped by his friend and fellow-student. Professor Sophus Lie.
In iJrlangen Klein formed the acquaintance of Uordan, to
whose personal friendship and scientific cooperation a high
tribute is paid in the ppeface to the IJcosaeder. It was here
and at Munich, to which city Klein was called in 1875, that
* Nexie Oeometrie des Iiauine8,gefffilndet at^f die BetracMung der getru
den Linie ale RaumeUment, von Dr. Julius PlUokeb. Leipzig, Teubner,
1869.
f Vergleichende Betraehtungen fiber neuere geometrisehe Foreehungen.
von Dr. Felix Klein, o. 6 Professor der Matnematik an der Universitftt
Erlangen.
KLEIN'S MODULAR PUNCTIOKS. 107
the theory of the icosahedron and the other regular bodies
was i^radaally developed in a series of papers in the Mather
matiBche Annalen, of which Klein had become an editor in
1873. It may be noted that nearly all of his writings are
pablished in this journal^ which is, indeed, distihctiyely the
organ of the school of which its editor is the leader.
In 1881 Klein was called from Mnnich to Leipzig, where
he remained nntil 1886, when he was appointed to the chair
in Oottingen, vacated by the death of Enneper, which he
still holds. At Munich he numbered among his students
Hurwitz, Bohn, and Dyck, all of whom have made a name
among mathematicians. In Leipzig the first American stn-
dents were admitted to his Seminar, Messrs. Irving String-
ham and Henry B. Fine, now professors of mathematics at
the University of California and at Princeton, the precursors
of a numerous throng, amon^ whom the writer had the good
fortune to be included. During the stay in Leipzig negotia-
tions were at one time pending toward inviting Klem to
Sylvester's vacated chair at Johns Hopkins. Various consid-
erations, relating mainly to his health, never very robust, led
him to decide in favor of remaining in Germany. A circum-
stance which must have contributed greatly to his decision
was the fact that he had already gathered about him a band
of talented and mature young mathematicians the direction
of whose vigorous development was a most gratifying task*.
Among the members of his Seminar in 1884-5 were Pick
now at Prague, Holder now at Gottingen, Study at Marburg,
and Pricke the editor of the Modulfunctionen.
The mana^ment of the Seminar has always been excep-
tionally efficient, even among the German models. It is
Klein's custom to distribute among his students certain por-
tions of the broader field in which he himself is engaged, to be
investigated thoroughly under his personal guidance and to
be presented in final shape at one of the weekljr meeting.
An appointment to this work means the closest scientific m-
timacjr with Klein, a daily or even more freouent conference,
in which the student receives generously tne benefit of the
scholar^s broad experience and fertility of resource, and is
spurred and urged on with unrelenting energy to the full
measure of his powers. When the several papers have been
presented, the result is a symmetric theory to which each
investigation has contributed its part. Each member of
the Seminar profits by the others' points of view. It is a
united attack from many sides on the same field. In this
way a strong community of interest is maintained in the
Seminar, in addition to the pleasure afforded by genuine
creative work.
The theory of the icosahedron appeared in book form in
108 KLEIK'S MOBULAB FUlfOTIOIfS.
1884 * The inyesti^tionB inclnded under this title trayerse a
definite^ self-liraited field, identified on the one hand with the
^oaj)8 of rotations of the regular bodies and the correspond-
ing nnite groups of linear transformations of a complex yari-
able, and on the other with the theory of the algebraic equa-
tions of the first fiye degrees and certain other special types.
The close relation of the subject to the theory of the modular
functions is also so far touched upon as to indicate the direc-
tion of the more extended theory which has culminated in
Dr. Pricke's book. Prom the point of yiew of this relation
the Ikosaeder appears as a first step in the systematic treat-
ment of the modular functions, for which it is also to serye as
a model. In the meantime Klein's lectures were forecasting
the coming theory, already dcyeloped in many of its features
in articles in the An7ialen. Beginning with the general
theory of functions, he treated in successiye semesters the
elliptic functions, the elliptic modular functions, the new
geometry, the hyperelliptic functions of deficiency 2, and the
Pliicker line geometry. Of these lectures the first three, to-
gether with the later lectures on algebra at Odttingen, relate
fergely to the present theory, while the others were to a con-
siderable extent preliminary to the theoiy of the general equa-
tions of the sixth and seyenth degrees.
In all these inyestigations it is again the theory of opera-
tions that furnishes the guiding principle and the general
outline of the subject, fiat the field of research being onoe
mapped out, it is characteristic of Klein to bring to bear on
it every instrument that modern mathematics can provide.
The theory of functions, invariants and covariants, differ-
ential equations, modem geometry, in short every method is
put under requisition and made to render its contribution to
the symmetry of the result. No advantageous point of view
is neglected, and not until the subject is traversed in every
direction, and until its external and internal relations are
clearly pictured to the mind, is the investigation to be re-
garded as complete. Por example the Ikosaeder discusses in
successive chapters the rotations of the regular bodies, the
corresponding groups of linear transformations and their in-
variants, the actual solution of the problem by the aid of a
class of differential equations, the algebraic phases of the
subject, based on the theory of substitutions, the general posi-
tion of the theory in reference to other correlated fields, its
historical development, the coordinated geometrical problems,
etc. The same plan obtains in the lectures and the Seminar.
* F. Klein, Vorlesungen aber das Ikosaeder und die Aufldsung der
Oleichunoen vom fUnften Grade, Leipzig, Teubner, 1884.
English translation by G. C. MoaRios. London, TrQbner, 1888.
kleiht'b hobulab funotioks. 109
The value to the stndent of this breadth of treatment is sim-
ply inestimable. No one can study long under Klein without
obtaining an intelligent comprehension of most of the ffreat
tendencies in mathematics. This is at present particularly
desirable in view of the extreme degree of specialization whicn
has come to prevail among mathematicians. Of all the great
services which Klein has rendered to mathematics, there is
none more valuable than his successful unification of its here-
tofore rapidly diver^ng branches. Lately he has turned his
attention to mechanics, in which new fiela, under the applica-
tion of the same principle, we have every reason to expect
from him another series of brilliant results.
Turning to the Modulfunciionen, one cannot but admire
the simplicity and perfect proportion with which Dr. Pricke
has developed the subject. So far as possible, everything is
traced from first principles. The network of interwoven
theories is constructed with a painstaking elaboration of
details and a rare ^enicQity of method. A clearer and more
scholarly presentation than that before us could hardly be
imagined.
The work divides into three principal investigations : (I.)
the theory of the modular functions in the narrower sense as
a specific class of elliptic transcendents ; (II.) the formal defi-
nition of the general problem, as based on the doctrine of
groups of operations ; (III.) the union of these two methods
and the further development of the subject in connection with
a class of Eiemann's surfaces.
We turn our attention for the present to the second division
of the subject. The operations with which we have to deal
belong to that fertile field of modern mathematics, the linear
transformations. The characteristic system on which the
theory of the modular functions turns is composed of all the
linear transformations of a complex variable
(1) ,' = ei±^,
or, in homogeneous form, of the binary linear transforma-
tions
(2)
z\=: az^ + /3z^,
for which the constants a, /3, y, 6 are real integers, subject
to the further condition that
a6 — /3y =4-1.
110 ELEIK'S HODULAB FUKCnONS.
The equations (1) and (2), and other similar types^ must be
regarded throughout^ as already indicated^ as denning opera-
tions ; namely, the operation of passing in each ease irom the
initial values z to the transformed values z'. Bestricting our-
selves for the present to the general linear transformations of a
single complex variable, z' = --^, if the values of z are
o ^ * yz + 6*
represented in the ordinary manner by points in the complex
plane, the transformation is to be conceived as carrying every
f^oint z to the corresponding position z'. The result of the
ransformation is therefore to effect a rearrangement of the
position of the points of the plane, and this geometric con-
ception, to be presently more fully developed, not only serves
to picture the corresponding analytic formula, but may often
witn great advantage entirely replace it.
cfz + /S
If now any transformation z' = ^ is followed by a
a z' + 6
second z" = -^ j^, the relation of the points z" to the
y,z 4- cJ,
original points z is directly defined by the equation
(OS ^n __ '\yz-\-dJ
^ {a^a 4- /3,y)z + a,/3 +^^6 _ a^z -f /?,
{y,a + d^y)z -f y,fi + 8^6 y,z + tf/
which is again linear. The combination, or *' product,*' of
two linear transformations of a complex variable is therefore
itself a linear transformation of a complex variable. The
total system of these transformations accordingly forms a
** group/' this name being applied to any system of opera-
tions of whatever kind such that the product of any two of
them is itself an operation of the system. If, furthermore,
we confine our consideration to those transformations for
which a, fiy v, 6 are real integers, this more limited system
is clearly still a group, which in reference to the including
general group just considered is designated as a " subgroup "
of the latter. Again wo obtain a subgroup of this subgroup
by selecting from the latter all those operations for which the
determinant aS — /3y = +1. For, on referring to (3), we
have at once for the product of any two of these operations
(4) a/, - ^,y, = {a a + ^^y) {y^fi -f 6,6)
-{a^ft^-^Mr.oc + c^,;^) = {a,6, - ft,y,)(a6 - fty) = +1.
kleik's hodulab fckgtionb. Ill
Bj way of contrast we may obserre that those transformations
with real integral coefficients for which
ad — /5y = — 1
do not form a group, since the product of any two of them
has for its determinant (— 1) (— 1) = + 1. If however we
combine the two systems ad — /Sy = ± 1, the result is again
a group.
The ^roup composed of the transformations (1) or (2) is
called simply the modular group, and is denoted by F. The
two forms (1) and (2) are distinguished as the non-homoge-
neous and the homogeneous groups F respectively. We note
that under the condition ad — /3y = + 1, a simultaneous
change of sign of all the coefficients is admissible, but that
the coefficients cannot otherwise be multiplied by a common
factor. The change of sign is of no effect on tne form (1),
but alters the form (2). It appears therefore that the opera-
tions of (2) are precisely twice as numerous as those of (1).
The modular group has itself a great variety of subgroups,
and it is precisely the theory of these subgroups which deter-
mines the formal character of the entire theory of the modu-
lar functions. The problem of establishing all these sub-
groups presents extreme difficulties and is not yet solved.
Much is to be hoped, however, from the powerful general
method of attacking the subject, devised by Klein and based
on the theory of Kiemann s surfaces.* The known sub-
groups are, with a few elementary exceptions, the " congru-
ence groups/' and their theory is exhaustively developed in
the Modulfunctionen, These groups are aefined by the
additional condition that
(5) a=6=±l, /3 = y = 0, (mod. w),
where n is any integer. f Under this condition we have,
referring again to (3),
a,fi + fi^6^y,a + 6,y = 0, (mod. n),
from which the group character is verified. In the case of
the non-homogeneous transformations it is plainly sufficient
to employ only the upper algebraic sign of ± 1. It is also a
fact of great interest that those substitutions of the homoge-
* Modulfunctionen, II., 5.
t More correctly, this is the definition of the Haupteongruemgruppen,
For other cases rf. Modulfunctumen, II., 7, § 6.
118 klbik's KODUJCJLB FUVOnOHB.
neoQB congmence group for which the tipper sign holds form
a subgroup of the latter, which therefore agrees operation
for operation with the non-homogeueous group.
Of the various characteristics of a ^up its order, ue. the
number of operations which it contains, is of prime impor-
tance. In the present case both the modular group and the
congruence subgroups are of infinite order, and the ^u^rtion
theirefore presents itself here in a modified form, vtx. it re-
quires the determination of the ratio of the oider of the
entire group to that of the respective subgroups. This ratio
is termed the '' index '^ of the subgroup and for the modular
n is denoted by /i (n), the corresponding sub^up being
designated by /^(n). The value of the function // (fi) is
deducible from purelv arithmetical considerations, if we
define as congruent (mod. n) all those transformations for
which either of the relations hold
(6) a' = - a, ?' = -1 /?,>' = 1! ^, (J'= 1 (J, (°^^- «)•
the value of /i (n) is equal to the number of incongruent
(mod. n) systems of solutions of
ad-^ /3y= -hi.
If n is a prime number/?, it is readily found that
For a compound n = g^/i • q^*"* • g/> . . . the calculation is
more complicated. The result is found to be*
") '•W = t(i-^)('4-)('-,7)-
Leaving the specific theory of the modular group at this
point for the moment, we have next to consider the position
of the present investigations relatively to the general theory
of linear transformation.! If we regard n elements z^, z^,
... Zn ss coordinates in an (n — l)--dimensional space or
maiiifoldness, the projective ^eometiy of this space is iden-
tical on the formal side with tne theory of the general group
of linear transformations
* Moduffuneiionen, II., 7, § 4.
t Of' ikoioeder, Chap. V.
kleik's mobulab functions. 113
(8) z\ = a,, «, 4 a„ z, 4- . . . + a„ «„
«', = a,j z^ + a^^z^ + , ., + Unu ««.
It is a principal problem of this theory to determine the
full system of invariant, covariant, and other concomitant
forms belonging to any configuration of the space^ defined by
any given set oi equations
/, (z„ 2„ ... zjS = 0,
Jt \^i» ^«» • • • ^«/ ^^ ^y
• • y
J* \^i> ^s» • • • ^«) ^^ "•
Prominence is also giyen to the determination of the iden-
tities which may exist among these concomitants. The
theory of the subgroups of the general group of transfor-
mations is, however, not usually considered.
On the other hand, the theory of substitutions deals with
the permutations (substitutions) of n given elements z^, z,,
. . . z^. Such a substitution is commonly and most conven-
iently written in the cycle notation
{Z, Zi, Zg , , ,) {ZaZfi Zy ...)...
the effect of the substitution being precisely to permute the
elements of each parenthesis cyclically. Written, however, in
theform ^
(where the subscripts t„ i„ ... t, are identical, apart from
their order, with 1, 2, ... n), the substitution is obviously
interpretable as a collineation of an {n — l)-dimensional
space. Prom this point of view, we may regard the theory of
substitutions as a special field within a general projective
geometry ; in other words, the groups of substitutions of n
elements may be considered as subgroups of the general group
of linear transformations of n coordinates. An important
characteristic of the substitution ^oups is the fact that they
are all of finite order, the latter being, in fact, always a divisor
of n ! On the other hand, every substitution group, like the
general linear group, possesses a system of invariants. These
are the "functions belonging to the group," i.e. such
rational integral functions of the n elements as are unchanged
in value by all and by only the substitutions of the group.
The invariants belonging to any substitution group G are all
8
114 KLEIN'S MODULAB FUKGTIOKS.
rational integral functions of any arbitrary one among them,
with coefficients which are symmetrical in the n elements z.
Every such group possesses, therefore, only a single indepen-
dent invariant.
Again suppose JJ to be anj subgroup of with an invari-
ant ^. If all the substitutions of are applied to tp, the
latter will take a series of values ^, (= tp), ^„ ^,, ... ^*, their
number k being the (always integral) ratio of the order of O
to that of ff. To every one of tnese values belongs a group
of the same order as H and similar to H. These k groups are
the '^conjugates" of JJwith respect to G. In special cases
they may all coincide ; ff is then a " self -con jugate *' group
{ausgezeichnete Untergruppe), and every f/> is a rational func-
tion of every other. A case of especial importance is that for
which H reduces to the identical operation alone, ff is then
obviously self-conjugate, since it is the only group of order 1.
In the general case, if we apply the substitution of 6 to the k
values ^', the effect is simply a permutation of these values.
In the particular case where H = 1 we have then on the one
hand a group of substitutions of the 0's of the same order as
0, and on the other, as an algebraic equivalent, an equal
number of rational processes by which the ^'s proceed from
one another. These processes also form a group. By a
proper choice of ip, it happens in certain cases tnat these
rational relations become linear. The group 0, which origi-
nally represented a system of collineations in an {n — i)-
dimensional space, appears then under a new form as identi-
fied with an equal group of linear transformations in the
complex plane. Such a reduction in the dimension (as it
may be called) of the group is obviously a step of the greatest
importance. In the Ifcosaeder and the Modulfunctionen this
problem is reversed, the groups being taken at the start in
their reduced form.*
Within the general theory of groups of linear transformations
the projective geometry and the theory of substitutions appear
as special cases, which possess the advantage of historical
precedence and a correspondingly high degree of elaboration.
To these Klein has now added in the AfodulfuncHonen a
third system of certainly comparable interest and importance.
There still remain an unlimited number of other special types
which will undoubtedly in the future furnish one of the most
fertile fields of mathematical research. The general problem
has hardly yet been touched upon. In systematic form it
requires the determination of all the binary, then the ternary,
quaternary, and higher groups. In each dimension the groups
of finite order naturally attract immediate attention. The
* Cf. Ikosaeder, I., 4, §§ 8-4, and I., 5, § 5 ; also Math, Ann, XV.
Klein's modulab FUKcriONa 116
researches of Poincarg, Jordan^ and Klein have shown that
the namber of finite groups is surprisingly small, and this field
is accordingly narrowly limited. The problem of the finite
binary groups is completely solved in the Ikosaeder. Of the
finite ternary groups which are not reducible to binary forms,
one of order 432 belong with the theory of the points of
inflection of the plane cuoic, and has been repeatedly discussed
from this point of view. Another of order 168 is treated in
the Modulfunctionen,* The Borchardt quaternary group
belongs with the theory of the general equations of the sixth
and seventh degrees, as the icosahedron group does with that
of the fifth degree, f On the other hand, of the infinite binary
groups, which naturally succeed the finite cases, the simplest
instance is precisely the present modular group.
Beturning to the modular group in particular, we can now,
from analog with the theory of substitutions, state briefly the
nature of the problem involved. It requires the determination
of all the subgroups of the modular group and of the corre-
sponding invariants, together with the systematic examination
of the functional relations between their invariants, particu-
larly when these relations are algebraic. An important dis-
tinction from the theory of substitutions lies in the fact that,
as the groups under consideration are for the most part of
infinite order, their invariants are no longer rational, but
belong to the family of integral transcendental functions with
linear transformations into themselves. It must also be noted
here that the homogeneous groups here considered have in
every case not one invariant, but three, which are then con-
nected by an identity, precisely as in the Tkosaeder. In regard
to the entire theory we observe further that from Klein's
Eoint of view it is not to be regarded as an isolated subject,
ut is to be connected as closely as possible with other
mathematical fields. It is to be examined from every possible
side, and, in particular, it is to be made tangible by the aid of
geometric representation.
At the outstart the theory of the elliptic functions is put
under requisition. To obtain the invariant of the modular
^roup the direct construction is not necessary. Such an
invariant is already at hand. It is known that the periods of
the elliptic integral of the first species, which we write
throughout in the Weierstrass form
J a/42*
dz
^^'-gj^-g!
♦ Modulfanctionent III., 7.
f Klein, Math, Ann. XXVIII., Zur Theorie der aUgemeinen Oleieh-
wigen aechtUn und Hebenien Ghrades.
116 KLBIH*8 XODULAB FmSTCrtlOVB.
are all linear combinations of two among them, o^^ and 09^
Asain 09, and 09. can be expressed in the same way in terms
01 a?', and g/, n and only if ord — /3y = ± 1. Accordingly
all the systems
famish primitiye period pairs. If we consider the ratio of
snch a pair, all such ratios are expressed in terms of any one
of them by
oaf = — ^ f ad - >5;V = ± 1, GO = —», (»= — ! ),
yao + o \ "^^ a?, car J
On the other hand, we haye at once in the three inyariants
a^y g^y and the discriminant A ^g^ — 27flr," of the binary
biquadratic form 4«j*«, — gji^z^ — gj^^ the three invariants of
the homogeneous modular group, wnue the absolute inyariant
•T" = ^ is the invariant of the non-homogeneous gronp. The
Eeriods a?,, a?, and their ratio oo are also invariants of the
iquadratic form, the latter being like J an absolute invari-
ant. In i*eference to the modular CToup these quantities are
again invariants belonging to the identical subgroup. Their
calculation from cr,, g^ and from J respectively are already
furnished by the theory of the elliptic functions. Of the con-
gruence subgroups (5) an invariant is also directly known in
the case of modulus 2. This is the anharmonic ratio \ of the
roots of the biquadratic form, or for the homogeneous CTOup
the three finite roots «j, «„ «, (where, in agreement with the
generfd principle above stated, ^^ 4- g, + e, = 0). Under the
operation of the modular group A assumes the six familiar
values
A, J, 1-A, ^ ^-^ 1
which furnish again a linear group. The latter is in fact a
dihedron group. The six values of A are connected with J
by the equation
JiJ-l :1 =
4(r - A + !)• : (A + iy{\ - 2)'(2A - I)': 27A'(A - l)'.
kleik's modulab fungtioks. 117
To obtain a comprehenBion of functional relations which in
the present state ox udyancement can be regarded as in any
way satisfactory, recourse must be had to Riemann's surfaces.
Perhaps no more brilliant exemplification of the Talue of this
gometric instrument exists than the theory of the modular
notions, as Klein has created it.* Beginning with the
period ratio oo, regarded as a function of J, we suppose the
values of co to be laid off in one complex plane ana those of
/in another. Since howeyer to every value of /correspond
an infinite number of primitive period pairs, and consequently
an infinite number of values of a?, we must, in order to secure
a one to one correspondence between the go and the / points,
suppose the /plane to consist of an infinite number of leaves
which are connected in cycles about certain junctions ( Ver"
zweigungspunkte). In the present case, as in the Ikosaeder,
these junctions are three in number and lie at / = 0, / = 1,
and / = 00. The leaves are joined at these points in cycles
of three, two, and an infinite number respectively. If now
we suppose the /point to pass along the upper side of the
real axis from — oo to 4- oo, the corresponding co points de-
scribe in every case three circular arcs bounding a curvilinear
triangle. To every upper half leaf of the / plane corresponds
the interior of one of these triangles, in the sense that if the
CO point takes successively every position in such a triangle,
the corresponding / point will take successively every position
in an upper half leaf. The co triangles then produced fill
just half of the upper half leaf of the co plane. Between
them lie an infinite number of empty spaces, of the same tri-
angular form, and these new triangles correspond to the lower
hafi leaves of the / plane. The triangles become infinitely
small and are crowded infinitely close together as we approach
the real axis, which is in fact a "natural boundary" beyond
which the function (jo{J) cannot be extended.
An immediate connection presents itself here with the
theory of the modular group, f The effect of the latter on
the systems of triangles is obviously merely to interchange
the two sets corresponding to upper and lower half leaves each
among themselves. Given any one of the triangles we can
obtain every other belonging to the same system by applying
to the former all the modular operations. We observe that
it is only those operations
, aco 4- /3
^ = r-5
yco 4- o
for which a6 — /3y = +1 that are here admissible ; those
* Cf, ModiUfuncHonen, II. \ IbU., II., 2.
118 KLEIN'S MODULAB FUNCTIOKS.
for which nrtf — /?^ = — 1 simplv convert the npper half leaf
of the GJ plane into the similarly divided lower half leaf.
This again a^ees with the fact that the product of two of the
latter operations belongs not to these bat to the modular
group.
Having now obtained a means of generating all the triangles
of either system from a single one among them^ the question
naturally presents itself how the one system can be ootained
from the other. We have seen that the real axis of the J
plane corresponds to the circular arcs bounding an a? triangle.
By a linear transformation oo' = ^ we can convert any
circle in the oo plane, for example, one of the sides of the a>
triande into the real oo axis. The function co'{J) has then
an infinite series of real values corresponding to real values of
J, Consequently conjugate imaginary values of J correspond
to conjugate imaginary values of co'{J). The co' triangle
corresponding to the lower J half leaf is therefore the reflec-
tion of that corresponding to the upper J half leaf on the a?
real axis. Retransforming now to the original oa triangle,
the reflection becomes the operation of inversion on the corre-
sponding circular boundary; i.e. given the one triangle, if we
construct the inverse of every point within it with respect to
one of its sides we have a triangle of the second ^stem. In
fact every triangle of the co plane can be obtained from any
one by a series of such reflections on bounding lines.
The preceding considerations furnish an entirely new point
of departure for the present and for a more general theory.
We may suppose any triangle bounded by circular arcs to be
a priori given, as corresponding to a half plane of the com-
panion Riemann's surface, the analytic connection of the two
variables being for the present purpose left out of immediate
consideration. From the given triangle we then construct all
possible others by the operation of reflection. In order that
these may just fill the complex plane without overlapping, the
angles of the given triangle must all be submultiples oi 27t :
— , — , — . Three cases are distinguished according as
%n In %K < ^ . 1 1 1 < ^
- -f - 4- -=-27r, i,e. — + - -f — — 1.
y, v^ V, > K, V, r, >
In the case 1 ^4- > 1 only a small number of systems
1^ 1^ V
of integral values v ^ v„ v, are possible. These lead to Wi^ finite
linear groups. Tne number of triangles is in this case also
finite, and they cover the entire plane. On the other hand, if
KLEIK*S MODULAB FUNCTIOITS. 119
— H +• — <lan infinite number of solations are possible.
The three circular boundaries of the given triangle have in
this case a common real orthogonal circle. This circle is
moreover the orthogonal circle of every triangle of the system.
The latter all lie within this circle and are crowded more and
more closely together as they approach the circumference.
In the case of the modular group the circumference is exactly
the real axis. This group is distinguished among other
types by the criterion that v^ = 2, v, = 3, v, = oo. The
remaining case, where the sum of the three angles of the tri-
angle is %ny leads to the theory of the periods of the elliptic
function.*
Turning now to the subgroups of the modular group, we
observe that these too have in each case a '^ fundamental
domain" (Fundamenial-Bereich), This is composed of a
system of the co triangles equal in number to the index of the
froup. This fundamental region, like the double oo triangle,
as tne property that from its points every other point in the
complex plane can be obtained by the operations of the cor-
responding subgroup, and that it is the smallest region which
has this property. Every co triangle can be converted by the
subgroup into one and only one tnangle of the fundamental
region. If now we suppose every co triangle to be represented
by its '^eauivalenf triangle in the fundamental region, the
effect of all the operations of the modular group is simply to
permute these representative triangles among themselves.
These permutations again form a group, the group C?^(n).t
In accordance with the entire tendency of the subject, the
question at once presents itself whether quantities refeiTed to
tne several triangles of the fundamental region can be found
such that the group (?^(n) transforms them linearly. This
is actually the case, and in fact for n = 2, 3, 4, 5 the corre-
sponding linear groups are identical with the dihedron group
of order 6, the tetrahedron, the octahedron, and the icosahe-
dron groups respectively.
For these cases, which exhaust the possibility of binary
groups, the "deficiency" of the fundamental region is 0.
For the next important case w = 7, the deficiency is 3 and
the corresponding linear group is ternary. It is in fact the
g'oup repeatedly treated by Klein in connection with the
ordan plane curve of the fourth order J
* Cf, throughout Ikoaaeder, L, 5 and Moduifunctioneiiy I., 8.
{Modtilfunctionen, II., 4, §6,
/Wd., UI., 7.
120 PBBTURBATIONS OP THB POUB INKER PLANETS.
With this brief and imperfect aeconDt we must now regret-
fully leave the subject, consoling ourselves with the reflection
that Dr. Fricke's book contains in itself that which will most
certainly attract deserved attention to this most beautiful of
E^lein's creations.
P. N. OOLB.
Ann Abbor, Decemiber 30, 1891.
PERTURBATIONS OP THE POUR INNER PLANETS.
Periodic Perturbations of the Longitudes and Radii Vectores
of the Four Inner Planets of the First Order as to the
jofasses. Computed under the direction of Simon New-
GOMB. Washington, Navy Department, 1891 ; 4to, pp.
180.
This work forms the concluding part of volume III. of a
series of astronomical researches, published under the general
title, '^Astronomical Paper Sy prepared for the use of the
American Ephemeris and Nautical Almanac."
During the past twelve years, one of the principal works
which has been in progress at the office of the Nautical Al-
manac is that of collecting and discussing data for new tables
of the planets. The most recent existing tables, which are
now used in all European Ephemerides, are those of Leverrier,
the construction of which was the greatest work ever under-
taken by that celebrated astronomer. The first tables pub-
lished, those of the Sun, were issued in 1858 ; those of Uranus
and Neptune appeared about 18 years later. The whole work
probably took about 25 years in preparation and publication.
Yet the number of observations on which the tables Were
actually based was only a few hundred in the case of each planet,
about 500 being used for Venus, 800 for Mars, and probably
yet fewer in the cases of the other planets. The results were
not completely discussed, and, in consequence, different data
were employed in different tables, making it extremely difficult
for future astronomers to derive the results of comparing them
with future observations. None except those of the Sun and
Mercury, which were the first issued, have shown a satisfactory
agreement with subsequent observations. The error in the
geocentric place of Venus at the time of the recent transit was
surprisingly great, amounting to no less than nine seconds in
longitude.
The actual number of observations now available for each
of the principal planets is several thousand. The recent ones
PBBTUBBATIONS OF THB FOUfi IKNE& PLAKETSi 121
«ie of <)onrse better than those ayailable thirty years ago. It
therefore seemed desirable to undertake the construction of
tables founded on ^1 these observations which could be of
value, and on uniform values of the masses of the planets and
pther 'elements.
As it was necessary to determine the masses from the peri-
odic perturbations, the first requisite was a determination of
the coefficients of these perturbations which should be beyond
doubt Although Leverrier's computations of these coeffi-
cients were carried out more fully than those of any of his
predecessors, some doubts of their entire accuracy hod been
expressed. In such intricate computations, which necessarily
proceed by successive approximations, and can never pretend
to mathematical rigor, the possibility of sensible quantities
being omitted can be avoided only by independent computa-
tions by different investigators using different methods. The
present paper is entirely devoted to the computation of these
coefficients. The adopted developments are so radically dif-
ferent from those of Leverrier that there can be no source of
error common to the two. The agreement throughout may
be called perfect, when compared with the probable error of
the best observations. Bare^ does a discrepancy amount to
the hundredth of a second of arc.
The principal point in which the development differed from
that of Leverrier is, that the eccentric anomaly is used, in
the beginning, as the independent variable. In this way the
series are made, in the first place, more rapidly convergent,
and it is thus more easy to be sure of including all sensible
terms. The use of this method requires, however, that the
eccentric anomalies be changed to mean anomalies by the
Besselian transformation. It was supposed that this trans-
formation was one which could be effected with ease and
rapidity. But in practice it proved so laborious that it is now
doubtful whether the terms saved in the development will
compensate for the labor of applying it,
A more radical change from Leverrier's method is, that
the perturbations are computed by direct integration of the
differential equations of motion, instead of employing the
method of variation of elements. Notwithstanding the theo-
retical elegance of the latter method as developed by Lagrange,
it becomes excessively prolix when we attempt to compute
the periodic perturbations bv it But when the equations are
directly integrated, the coeMcients admit of being found with
great facility, when once the development of two derivatives
of the perturbative function in terms of the mean anomalies
is effected. Altogether the method is a combination of the
purely numerical process of development employed by Hansen,
and tne purely analytic one employed by Leverrier.
122 PBBTUBBATIOKS OF THE FOUR IKKEB PLAJfTEBlS.
It is still a question whether the adopted method was acta-
ally the shortest, and whether mnch lahor wonld not haye
been saved by employing the purely numerical development
from the beginning.
The volume of wnich the above paper forms a part is wholly
devoted to the developments of celestial mecnanics. The
opening paper is the development of the perfcurbative funo-
tioD in smes and co-sines of multiples of the eccentric anom-
aly which was employed in computinj; the perturbations.
This is followed by a determination of the inequalities of
the Moon's motion dne to the 6rare of the Earth,^ prepared
by O. W. Hill. This is the most elaborate determination of
these difficult inequalities that has ever been made, no less
than 165 terms in the Moon's lonritude, and yet more in the
latitude, being computed. Nearly half the computed terms
are, however, entirely insensible, even in the fourth place of
decimals of seconds.
The third paper is on the motion of Hyperion, the seventh
satellite of Datum. In it is developed the theory of the
curious relation between the mean motions of Hyperion and
Titan, which H. Struve has since extended to one or more of
the inner satellites.
This is followed by another paper by Mr. Hill, bein^ a
conipntation of certain lunar inequalities due to the action
of Jupiter. The inequality in question was first discovered
empirically from observations, and was traced by Mr. Neison
to a sort of evection due to the action of Jupiter. Mr. Hill's
coefficient is, however, onlj[ 0".90, while observations gave
1".50. Probably the theory is more nearly correct, as the un-
certainty of observations of the Moon is much greater than in
the case of other heavenly bodies, and it is difficult to sepa-
rate the effects of an inequality of this kind from those of
numerous other causes affecting the observations.
It is now expected that the tables of the four inner planets
which are founded on the theories developed in the Astro-
nomical Papers, and on the great mass of observations made
since 1750, will be ready for the press in a little more than
two years, S. N.
HATHEHATIOAL PB0BLBH8. 1^
MATHEMATICAL PROBLEMS.
Solution of Questions in the Theory of Probability and Aver-
agt$. Appendix IL to Maihematicai Questions and oolutions from
the Educational Times, Vol. LV. By Profeesor G. B. Zebr, M.A.
This pamphlet of fifty-six pages contains solutions of more
than forty problems in geometrical probability and mean
valaes and of some other mteresting mathematical problems.
The soWer was also the proposer of most of the problems.
His solations show skill and perseyerance in eyaluating many
complicated definite multiple integrals.
Seyeral problems relate to mean yalues of magnitudes deter-
mined by choosing random points with certain restrictions in
a circle. For example. No. 11,153 is to find the ayera^e area
of the dodecagon formed by joining twelye points taKen ut
random in a circle, three in each quadrant. The expression
found for this area is the quotient of two multiple integrals
of twelve variables each. This is finally reduced to
2'V fyrl _ 704 \ / _409\ 2V' /5^ _ 29\
Ux' \16 1575/ \ 105/ "*" ;r' V 64 45/
■^ Stt* \3'Z 315/ \2 105/
Problem 11,037 is as follows : — '* Two points are taken at
random in the surface of a given circle. An ellipse is
described on the distance between the two points as major
axis. If a point be taken at random in the left-hand half of
this major axis, and with this point as a centre a circle is
described at random, but so as to lie wholly within the ellipse,
find the average area of the ellipse described on that portion
of the major axis between the ri^ht-hand extremity and the
circumference of the random circle. '^ The result obtained is
Trr* / 2205;r + 2012 \
1280 \ 15;r + 17 /
This solution involves the assumption that the major axis a
of an ellipse being given all possible ellipses should be in-
cluded by taking the unknown minor axis as an independent
variable with the limits and a. All possible ellipses might
with equal propriety be included in other ways ; as, by taking
the eccentricity as the independent variable with the limits
and 1. The result would be altered by such a change.
Problem 11,130 implies an assumption of the same nature.
f
124 NOTES.
'^ A chord is drawn at random across a circle, and two points
are taken at ran^dom within the circle ; find the chance that
both points lie on the same side of the random chord/' The
128
result 1 ~ j^:— J, is obtained by treating the distance of tKe
chord from the centre as the independent variable. Why
would it not be e<][ually proper to take the arc subtended by
the chord as the independent variable ?
These problems, as stated, are indeterminate. The modes
of including all possible ellipses and of choosing random
chords must be fixed before the problems become definite.
It seems strange that an incorrect construction should be
S'yen for so elementary and easy a geometrical problem as
o. 10,512. ^* Oiven four lines (in magnitude), construct *
two similar triangles each of which shall have two of the given
lines as sides.''
Edwabd L. Stabler.
New Tobk, December 10, 1891.
NOTES.
The annual meeting of the New Yobk Mathematical
Society was held Wednesday afternoon, December 30, at four
o'clock. Professor Van Amringe presiding. The following
persons having been duly nominated, and being recommended
by the council, were elected to membership : Mr. Edwin
Mortimer Blake, Columbia College ; Professor Mary E. Byrd,
Smith College ; Professor Susan J. Cunningham, Swarthmore
College ; Mr. A. E. Kennely, Edison Laboratory ; Mr. Alex-
ander Kinseley, Lafayette, Ind.; Professor Anthony T.
McKissick, Alabama Polytechnic Institute ; Professor George
D. Olds, Amherst College ; Professor M. L. Pence, State Col-
lege of Kentucky ; Miss Amy Bayson, New York, N. Y. ;
Professor Benjamin Sloan, South Carolina College. The
secretary reported that the membership of the Society was
210, of whom 37 lived in New York city and the immediate
vicinity and were able to attend the meetings regularly. The
treasurer's report having been read, an auditing committee
was appointed to examine his accounts.
The nominating committee reported the following ticket
for the oflBcers and council of the Society for the ensuing
year : — Pl^sident, Dr. Emory McClintock ; Vice-President,
Professor Henry B. Fine ; Treasurer, Mr. Harold Jacob^ ;
Secretary, Dr. Thomas S. Piske ; other Members of Council,
Professor J. E. Bees, Processor W. Woolsey Johnson, Professor
NOTES.
m
J. H Oliyer, Professor J. H. Van Amringe, Professor Thomas
Craig. A ballot being taken this ticket was unanimonsly
elected. At the invitation of the chair, Mr. B. S. Woodward
briefly addressed the Society, referring to .its work and aims,
and expressing his hopes for its success and prosperity.
Condensed Report or the Teeasueeb fob the Tear 1891.
Beeeipts.
Balance from 1890 $20.80
Net receipts from members
and snucribers 974.24
$996.04
Est^enditures,
Printing Constitntion $41.52
Bulletin 868.19
Glrcnlars, stationery, etc — 195.55
Postage and miscellaneous. . 118.78
Balance 271.00
$995.04
Harold Jacobt, Treasurer.
We have examined the treasurer's acconnts, and found the same
correct
G. L. Wiley,) j„j.v..«^
T. E. Snooe, y ^^^^^^ .^
Jas. Maclay,) Committee.
A BEGULAB meeting of the New Yobk Mathematical
Society was held Saturday afternoon, January 2, at half-past
three o'clock, the president in the chair, rrofessor D. S.
Jacobus, of the Stevens Institute of Technology, haying been
duly nominated, and being recommended by the council, was
elected a member. The following original papers were read :
'^ Application of least squares to the development of func-
tions,'' by Mr. Frank Oilman; "On the computation of co-
yariants oy transvection," by Dr. Emory McChntock. In Mr.
Oilman's paper a method was given for finding a rational
entire algebraic function of the n-i\i degree witn numerical
coefficients, which should approximately represent the value
of a given function between certain limits of the variable,
and which should furnish, in general between these limits,
more accurate results than the first n + 1 terms of its ordinary
expression as a power-series. The numerical coefficients of
the approximation were determined from the true values of
the function calculated for values of the variable uniformly
distributed between the limits, by the principle that the sum
of the squares of its residuals should be a minimum. Dr.
McOlintock's paper contained an account of the general
method for the computation of covariants of which a simple
example, illustrating a special case, was riven at the end of
his article " On lists of covariants," publisned in No. 4 of the
Bulletin, pp. 85-91. t. s. f.
126 N0TB8.
Ik coDnection with Professor A. S. Hathaway's article
" Early history of the potential " in No. 3, pp. 66-74, it may
be remarked that the mistake of ascribing the discoyerjr of
the fundamental property of the force-function, or potential,
to Laplace instead of Lagrange is a common one. In addition
to the places mentioned by Professor Hathaway it is retained
in the second edition of Maxwell's Electricity and Magnetism,
vol. I. (1881), p. 14, and in the new edition of Thomson and
Tait's Natural Philosophy, vol. I., part IL (1883), p. 28. Atten-
tion was called to this mistake by B. JBaltzer in his note
" Zur Oeschichte des Potentials/' in Crelle-Borchardfs Jour-
nal, vol. 86 (1879), p. 216. The matter is also discussed
in E. Heine's Kugelfunctionen, second edition, vol. 11. (1881),
p. 342, and in M. Bacharach's Geschichie der Potentxalthearie
(1883), pp. 4-6.
None of these authors, however, mention the memoirs of
Lagrange preceding that of October 2, 1777 ; ao that it is of
no little interest to see the first idea of the property of tke
force-function traced back in his writings to as early a date
as 1763. Professor Hath away 's reference to Oayley's British
Association Report for 1862 must be due to some oversight.
The matter is not discussed there, nor is there any reference
to Lagrange's memoir Sur Viquation sSculaire de la lune, of
1773. A. z.
In regard to the preceding note I have to state that Cayley's
report on dynamics to which I intended to refer is in tne
British Association Report for 1857, p. 3. Besides the refer-
ence to Maxwell given by A. Z. there is another to page 74,
where the error is repeated. A note just received from Profes-
sor P. G. Tait with reference to nahla, which is the quaternion
vector-operation p, and not — p' = ^^ + ^-j + ^, encloses
a copy of his address " On the importance of quaternions in
physics," Philosophical Magazine, January, 1890, p. 92. We
quote: ''Hamilton did not, so far as I know, suggest any
name. Clerk Maxwell was deterred by their vernacular sig-
nification, usually ludicrous, from employing such otherwise
appropriate terms as sloper or grader ; but adopted the word
nahla, suggested by Eooertson Smith from the resemblance
of p to an ancient Assyrian harp of that name." A. 8. H.
John Wiley & Soks have in preparation " An elementary
course in the theorj^ of equations '' by Dr. 0. H. Chapman of
Johns Hopkins Univeraitv.
Leach, Shewell & Sanborn have just published ''A trea-
tise on plane and spherical trigonometry ^' by Professor E.
Miller of the University of Elansas. T. 8. F.
KBW PUBLIOATIOKS. 187
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THE MECHANICAL AXIOMS OR LAWS OF MOTION. 129
THE MECHANICAL AXIOMS OB LAWS OF MOTION.
BT FKOF. W. WOOLSBT JOHNSON.
The three laws of motion were called by Newton Axio-
mata, sive Leges Motus. Professors Thomson and Tait in
their Natural Philosophy y section 243, say : " An axiom is a
proposition, the truth of which mnst bo admitted as soon as
the terms in which it is expressed are clearly understood.
But, as wo shall show in our chapter on * Experience/ physi-
cal axioms are axiomatic to those only who have sufficient
knowledge of the action of physical causes to enable them to
see their truth/' They then proceed to give Newton's three
laws, remarking that '^ these laws must be considered as rest-
ing on couTictions drawn from observation and experiment,
not on intuitive f>crception."
Whether this bo accepted as a proper definition of a physi-
cal axiom or not, it is at least desirable to include among the
axioms of mechanics the smallest basis of postulated princi-
ples upon which it is possible to construct the science by
ri^d mathematical reasoning.
The laws of motion, in the classic form given them in the
Princi'piay admirably express such a basis of [)ostulated princi-
ples, although the charge of redundancy has been brought
against the first and second laws ; but, in the case of the third
law, the tendency has been, on the other hand, to make it
include too much, and to assume, either under its authority or
directly, as axiomatic, principles like the ^' impossibility of
perpetual motion " which ought rather to be shown to follow
directly from the three laws of motion.
It is here proposed to discuss the question of the mechani-
cal axioms, beginning with an examination of the laws as
presented by Newton.
The first law is that ** Every body keeps in its state of rest
or of moving uniformly in a straight line, except to the
extent in which it is compelled by forces acting on it to
change its state."*
Newton had already defined vis impressa as an action
from witliout changing a body's state of rest or of moving
uniformly in a straight line, f and in the scholium to the
Definitioties had pointed out that our measure of time
depends upon the assumption of the first law of motion ; so
♦Corpus oinne perseveraro in statu suo guiescendi vel movendi unifor-
miter in directum, nisi quatenus illud a viribus impressls cogitur statum
BQum mutare.
f Definitio IV. Vis impressa est actio in corpus exercita, ad mutan-
dum ejus statum vc] quiesccndi vel movendi uniformitor in directum.
130 THE MECHANICAL AXIOMS OB LAWS OF HOTIOH'.
that with respect to a single hody there is a logical '^ yicions
circle." We define those luteryals of time as equal in which
eqnal spaces are described when there is no action: we say
there is no action when equal spaces are described in equal
intervals of time. But the physical truth expressed is that
aU bodies undisturbed by action from without describe equal
spaces in the successive intervals in which any one such body
describes equal spaces. We then define these intervals as
equals and the action^ which in any other body causes a
departure from this normal state, as a force.
The second law is that ^* Change of motion is proportional
to the moving force acting, and takes place in the direction
of the straight line in which the force acts."*
The simplest form of the physical truth involved is this : —
A given force acting upon a given hody produces the same
acceleration in its own direction, in equal intervals of time, no
matter how the body may be moving, nor what other forces
may be acting at the same time. The law as expressed by
Newton involves the obvious deductions that the force is pro-
portional to the acceleration produced in a given interval, and
that to produce a given acceleration the force must be propor-
tional to the mass ; for motion had been defined as propor-
tional to mass and velocity conjointly. The vis motrix is the
whole force acting on the body, and is defined as "propor-
tional to the motion which it produces in a given time," f in
distinction from vis acceleratrix which is defined as pro-
portional to the velocity produced in a given time. These
definitions imply the second law, just as those of vis insita
and vis impressa imply the first law. But the essential
point is the constancy and independence of the effects of
force, which effects are therefore suitable to be taken as meas-
ures of the force.
It has been objected that the first law is unnecessary
because it is included in the second which implies that no
force produces no change of motion. It appears inevitable
that the expression of the second law should thus include the
first ; but it is nevertheless fitting that the normal state of
the body suffering no action from without, a departure from
which constitutes the '* change of motion " when action takes
place, should be stated in a separate axiom. The first law is,
m fact, far more axiomatic than the second, or, in the lan-
fuage of the definition quoted above, requires a much smaller
nowledge of the action of physical causes to enable one
♦ Mutationem motus proportionalera esse vi motrici irapressae, & fieri
secundum lineam rectam qua vis ilia imprimitur.
f Definitio VIII. Vis centripetse quantitas motrix est ipsius mensora
proportionalis motui, quern dato tempore general.
THK HECHAinCAL AXIOMS OB LAWS OF MOTION. 131
to see its trath. It is only necessary to get a clear notion
of the absence of force to be in the mental state to admit the
truth of the first law : but it requires a considerable famil-
iarity with geometrical notions eyen to apprehend the man*
ner in which the efFect of a force upon a moving body is to be
compared with its efFect when the body is at rest, and the
manner in which the effects of forces acting at an angle with
one another are to be separated. Yet clear conceptions in
these matters must be obtained before an intelligent assent
can be given to the second law of motion.
The most axiomatic proposition involved in the second law
is that two opposite equal forces acting upon a body at rest do
not produce motion, which in some old treatises is taken as
the first proposition in statics, and in others as the definition
of the equality of forces.
The tnird law is that *' There is always a reaction opposite
and equal to an action : or the actions of two bodies npon one
another are always equal and oppositely directed."*
The physical truth expressed is that every force acting upon
a body is the action upon it of another body which in turn
is acted upon equally bjf the first body, the action taking place
in the straight line which joins tlie two bodies. In other words,
the law asserts that no forces exist which consist simply of
tendencies in certain directions, as the ancients supposed in
the case of the " gravity" of certain bodies and the ** levity "
of others. This granted, the equality of the two phases of
the action follows readily, by the aid of the notion of tne trans-
missibility of force which is intimately connected with this
law. Compare Newton's remarks on attraction, quoted below.
But the language of some writers concerning the " numerous
applications of this law" indicates the view that something
more than the equality of pressures is implied in it.
Of the illustrations whicn in the Principia immediately fol-
low the third law, the first is that of a stone pressed by the finger,
when in turn the finger is pressed by the stone. \ Here there
is no intervening body between the two bodies in question.
The resistance of the stone to the finger is, by the second law,
equal to the force communicated to the finger tip by the mus-
cles because it prevents its motion. By the third law, this is
the same as the force communicated by the finger to the
stone. If we go a step further and consider the etjuilibrium
of the stone, the resistance of the support upon which it rests
is equal to the last named force, and is the reaction of the
•■ .- MIIIIBIII BII^M ■ M ■ II ^
* Actioni oontrariam semper & lequalem esse reactionem : sive oor-
pomm daoram actiones in se mutuo semper esse cequales & in partes con-
trarias dirifii.
t Si qois lapidem digito premit, premitur & hujus digitus a lapide.
132 THE MECHANICAL AXIOMS OR LAWS OF MOTION.
same force regarded as the action of the stone upon the sup-
port. Thus toe force is transmitted through the substance
of the stone.
In the next illustration, that of a horse drawing a stone
attached to a rope, there is a body through which the force is
transmitted. ** The rope stretched both ways will by the same
endeavour to relax itself urge the horse toward the stone and
the stone toward the horse ; and will impede the progress of
the one as much as it promotes the progress of the other.*' *
The next illustration is from impact. "If any body im-
pinging upon another body, by its own force in any manner
changes the motion of that body, it will also in turn suffer in
its own motion (on account of the equality of mutual pressure)
the same change in the contrary direction. *' f Here the third
law is cited in the clause in parenthesis, and the equality of
the actions is indicated by the effects which are produced in
accordance with the second law. Newton proceeds in fact to
^Jf "By these actions equal changes are made not of veloci-
ties but of motions." These comments on the third law close
with the words : " This law holds also in attractions, as will
be proved in the next scholium."
The passage in the scholium here alluded to is as follows : —
''In attractions I thus briefly show the matter. Any two
bodies A and B mutually attractinff each other, conceive
some obstacle to be interposed, by which their approach is pre-
vented. If either one of the bodies A is drawn more toward
the other body B than the other B toward the first A, the
obstacle will be urged more by the pressure of the body A
than by the pressure of the body B, and hence will not re-
main in equilibrium. The stronger pressure will prevail,
and will cause the system of the two bodies and the obstacle
to move in a straight line in the direction toward By and by
an ever accelerated motion in free space to pass to infinity.
Which is absurd and contrary to the first law. J
♦ Funis utrinque distentus eodem relaxandi se conatu urgebit equum
versus lapidem, ac lapidem versus equum ; tantumque impediet pro-
gressum unius quantum promo vet progressum alterius.
f Si corpus aliquod in corpus aliud irapingens, motum ejus vi sua
quomodocuuque rautaverit, idem quoque vicissim in motu proprio eandem
mutationem in partem contrariam vi alterius (ob fequalitatem pressionis
mutuaj) subibit.
X In attractionibus rem sic bre\iter ostendo. Corporibus duobus qui-
busvis A, B se mutuo trahentibus, concipe obstaculum quodvis inter-
poni, quo congressus eonim impediatur. Si corpus alterutrum A magis
trahitur versus corpus alteram B, quam illud alterum B in prius A, ob-
staculum magis urgebitur pressione corporis A auam pressione corporis B;
proindeque non manebit in aequibrio. PraevaleDit pressio fortior, faciei-
que ut systema corporum duorum & obstaculi moveatur in directum in
Sartes versus B, motuque in spatiis liberis semper accelerate abeat in in-
nitum. Quod est absurdum & legi primas contrarium.
THS KSCHAKICAL AXIOMS OB LAWS OF KOTIOK. 133
In this passage Newton, so far from making the third law
imply anything more than equality of pressures, shows that
this equality of the two phases of an action follows from the
simple assumption that in a system of bodies preserving their
relative positions the mutual actions of any two cannot result
in a tenaency to motion. The axiomatic portion of the law
consists in this assumption. A tendency to motion is in New-
ton's system always the action of an external body.
We find, however, in Thomson and Tait the following pas-
sage : '' Of late there has been a tendency to split the second
law into two, called respectively the second and third, and to
ignore the third entirely, though using it directly in every
dynamical problem ; but all who have done so have been
forced indirectly to acknowledge the completeness of New-
ton's system, by introducing as an axiom what is called
D'Alemoert's principle, which is really Newton's rejected
third law in another form. Newton's own interpretation of
his third law directly points out not only D'Alembert's prin-
ciple, but also the modern principles of work and energy."*
In support of this the authors remark farther on,f after
commenting upon the third law, ''In the scholium appended,
he makes the following remarkable statement, introdacing
another description of actions and reactions subject to his third
law, the full meaning of which seems to have escaped the
notice of commentators :" — [Here follows the passage from
the scholium, of which the authors give the following trans-
lation, in which "activity" and ''counter-activity" are put
for actio and reactio,]
" If the activity of an agent be measured by its amount .
and its velocity conjointly; and if, similarly y the counter-
activity of ike resistance he measured by the velocities of its
several j^arts arid their several amounts conjointly, whether
these arise from friction, cohesion, weight, or acceleration; —
activity and counter-activity, in all combinations of ma-
chines, will be equal and opposite/^ J
Again Professor Tait in the article Mechanics in the
Encyclopaedia Britannica, 9th edition, quotes the passage and
remarks : "This may be looked upon as a fourth law. But,
in strict logic, the first law is superfluous. . . . Hence
there are virtually only three laws, so far as Newton's system
is concerned."
The "scholium to law III." is afterward referred to as
♦ NcUurai PhUosopJiy, Section 242. \ Section 268.
X Nam si festimetur agentis actio ex ejos vi & velocitate conjunct! m ;
& similiter resistentis reactio sestimetur conjunctim ex ejus partium
singularum velocitatibus k viribus resistendi ab earum attntione, coh»-
sione, pondere, k acceleratione oriundis ; erunt actio & reactio, in omni
instrumentorum usu, sibi invicem semper aequales.
134 THE MECHANICAL AXIOMS OB LAWS OF MOTIOK.
giving us the principle of the " transference of energy from
one body or system to another."
We shall be better prepared to estimate the import of the
passage last qnoted if we briefly consider the connection in
which it stands in the Princtpia. The comments on the
third law, quoted nearly in fall above, are followed by six
corollaries and a scholium, which complete the chapter en-
titled Axiomafa sive Leges JUotus, and immediately pre-
cede the treatise De Motu Corparum. Cor. I. ffives the
parallelogram of forces as dedaced from laws II. and I.
Cor. II. states the composition and resolation of forces.
** Which composition and resolution is abundantly confirmed
by the theory of machines."* The resolation of forces is
then applied to prove that the efficiency of a force to turn a
wheel is the product of the force and its arm — *'the well
known property of the balance, the lever, and the wheel '* —
and so on for tne other simple machines, which are thus cited
to confirm the truth of the laws of motion. Cor. III. proves
by laws III. and II. that the quantity of motion ^'directed
toward the same parts" is not altered by internal actions
between the bodies of a system. Cor. IV. derives the con-
servation of the motion of the center of gravity. Cor. V.
shows that the relative motions of a system oi bodies en-
closed in a given space are the same, whether the space be
at rest or moving uniformly in a straight line, and cor. VI.
extends this to the case in which the bodies are also acted
upon by equal accelerating forces in the direction of parallel
lines.
The scholium which follows is not a scholium to the third
law exclusively, but is occupied with the experimental verifi-
cations of thelaws ; beginning with the discovery by Galileo,
by means of the first two laws, that ** the descent of heavy
bodies is in the duplicate ratio of the time, and that the
motion of projectiles takes place in a parabola, experiment
confirming, except so far as these motions are somewhat
retarded by the resistance of the air." f Then follows an
account of experiments on the impact of bodies, showing that
experience agrees with deductions drawn from the three laws,
which ends with the words, ''And this being established, the
third law so far as impacts and reflexions are concerned is
confirmed by a theoiy which plainly agrees with experience." X
* Qu© quidein compositio & resolutio abuiide coDfirmatur ex mechan-
ica.
f DesccDsum gravium esse in duplicsta ratione temporis, & motuin
project ilium fieri in parabola; conspirante experientia, nisi quatenus
motus illi per aeris resistentiam aliquantulum retardantur.
I Atque boo pacto lex tertia quoad ictus & reflexiones per theoiiam com-
probata est, que cum experientia plane congruit.
)
THE MSCHANICAL AXIOMS OR LAWS OF MOTION. 135
The paragraph concerning attractions, quoted above, comes
nelty and is lollowed by a statement of Newton's own experi-
ments with magnetic attraction, directly confirming the tnird
law.
The paragraphs concerning attraction are followed in the
scholium by one opening with these words : ^^ As, in impacts
and reflexions, those bodies have the same efficiency of which
the velocities are reciprocally as the innate forces: so in
mechanical instruments for producing motion, those agents
have the same efficiency, and by opposite endeavours sustain
one another, of which tne velocities estimated in the direction
of the forces are reciprocally as the forces.'^* The vires
insitm or vires inertia are' proportional to the masses, as
explained in Definitio III., so that the meaning is this : — Just
as bodies having equal momenta are of equal efficiencv in the
case of impact, so, in machines, agents are of equal power
(and if opposed produce equilibrium) when the products of
the force and the velocity of the point of application in the
direction of the force are the same for each.
This is nothing more nor less than the " principle of virtual
velocities," — a succinct statement of "the whole theory of
machines diversely demonstrated by various authors," already
cited in the second corollary as a confirmation of the laws of
motion, because on the one hand deduciblo from them, and,
on the other hand, in agreement with experience.
After applying this principle of virtual velocities in detail
to the several simple machines, Newton continues : ** But to
treat of mechanism does not belong to the pi-esent design. I
wished only to show by these things how widely extends and
how certain is the third law of motion." f Then follows
the passage quoted by Thomson and Tait (see above) in which
the only new idea involved is the inclusion of the resistance
to acceleration among the " reactions."
The scholium shows indeed that Newton had a clear con-
ception of what we now know as *' D'Alembert's principle" J
as well as the "principle of virtual velocities," but does not,
as it seems to me, indicate any intention to postulate a new
axiom.
*Ut corpora in concursu & reflexione idem pollent, quorum velocitates
sant reciproce ut vires insitse : sic in movendis instrumentis mechanicis
agentia iaem pollent & conatibus contrariis se mutuo sustinent, quorum
velocitates secundum determinationem virium sBstimatse, sunt reciproce
ut vires.
f Gseterum mechanicam tractare non est hujus instituti. Hisce volui
tantum ost^ndere, quam late pateat quamque certa sit lex tertia motus.
}It must be remembered, however, that we call such propositions
"principles,*' not when they are presented simply as demonstrated the-
orems, but when thev are made the basis of a systematic method of
applying analysis to the solution of problems.
186 THB ICXOHASIOAL AXI01C8 OB LAWS OF MOnOlT.
Professor Tait * lesards the first words of the scholiam-*
'^ Up to this^ I have bid down principles receiyed by mathe-
maticians and confirmed by experiments in great nnmoer ** f —
as claiming for Newton the discovery of what, as stated aboye,
he regards as a f onrth law : as if Newton were about to proceed
to some new axiom not yet known to the men of science of
the day. Yet we have seen that the scholium treats <rf a
variety of topics at great length, before comiug to what is
alleged to be the new axiom. The context rather shows that
the matter new to mathematicians, to which Newton impli-
citly refers in the words quoted, is the body of the treatise
Db Motu Corporum, which immediately follows the introduo-
torj chapters — ^the Deflnitiones, and the Axiomata and Oorol-
lanes.
At the close of the article Meehanies Professor Tait sum-
marizes the third law proper thus — '' Every action between
two bodies is a stress. '' He subsequently pointe out in the
simple instance of a falling stone bow force may be regarded
either as '^ the apace^ate ai which energy is transformed/* or
<< the time-rate at which momentum is generaiedy** and says
(§ 294) that these are '' merely particular cases of Newton's
two interpretations of action in the third law.'' He then
proceeds to connect them analytically as follows: — ''if s be the
space described, v the speed of a particle,
••— *-.^_^^__ ^^
"" " dt" ds' dt" ds*
Hence the equation of motion (formed by the second law)
•• •
ms = mv=f,
which gives /as the time-rate of increase of momentum, may
be written in the new form
dv d .. ,. ^
giving/ as the space-rate of increase of kinetic energy."
Is It not equally true in the (general case that the so-called
two interpretations of action, so far from being the subjects of
separate axioms, are demonstrably equivalent by virtue of the
equality of the two phases of a stress and the second law of
motion ?
* Article Mechanics, Enc^clopaddia Britannica, §12.
f Hactenus principia tradidi a mathematicis recepta & experientia mol-
tiplici coDfirmata.
THB KEOHAKIOAL AXIOMS OB LAWS OF MOTION. 137
Another instance of the unnecessair assamption of ph^i-
cal axioms ocoars in the Natural Phimovhy, The principle
that '^ the perpetual motion is impossible is introduced as an
axiom to prove that ^' If the mutual forces between the parts of
a material system are independent of their yelocities, whether
relative to one another, or relative to any external matter, the
system must be dynamically conservative. For if more work
is done by the mutual forces on the different parts of the
system in passing from one particular configuration to
another^ by one set of paths than by another set of paths, let
the system be directed, by frictionless constraint, to pass from
the first configuration to' the second by one set of paths and
return by the other, over and over agam forever. It will be
a continual source of enerCT without any consumption of
materials, which is impossible/'*
Again in Williamson and Tarleton's Dynamics (p. 397J,
the same demonstration is given, closing with the words
"This process may be repeated forever, and thus an inex-
haustible supply of work can be obtained from permanent
natural causes without any consumption of materials. The
whole of experience teaches us that this is impossible/'
Thus we find these authors appealing to the general prin-
ciple of the conservation of energy in proof of what is really
but its simplest form, namely the equivalent transference of
energy from body to body of a material system, and from
the kinetic to the potential form, a proposition which is
easily shown to be a consequence of Newton s laws of motion.
The appeal to experience is in fact only necessary to estab-
lish the hypothesis laid down in the above quotation from
Thomson and Tait, namely, that the forces do not in any
way depend upon velocities, or, let us say, that the stress
between two bodies depends only upon the distance between
them.
We conclude this examination of the physical axioms with
a brief sketch of the steps by which the conservation of
energy, in its mechanical forms of kinetic and potential
energy of masses, may be established directly from the axioms
of 'the independent accelerative action of force,' *the duality
of stress,' and * the dependence of its intensity solely upon
distance.' The steps 4 and 5, which establish the nght to
deal with the kinetic energy o* relative motion, are developed
more in detail, because the point does not seem to be suf-
ficiently developed in the usual text-books.
1. Ijet the conservation of the motion of the centre of
gravity be deduced, as in the Principia, from the second and
third laws of motion.
• Natural PhUo$ophy, Art 272.
138 THE MECHAKICAL AXIOMS OB LAWS OF MOTION.
2. The kinetic energy of a body may be decomposed into
parts corresponding respectively to its component velocities in
two rectangular directions.
3. A moving body acted upon by a force, directed to or
from a fixed point and in magnitude a function of the dis-
tance of the body from that pointy experiences a gain or loss
of kinetic energy equal to the loss or gain of potential energy
relative to the fixed point.
4. Although fixed centres of force do not exist, yet when a
stress exists between two bodies (in magnitude a innction of
the distance), the centre of gravity being fixed, and dividing
the distance between them in a fixed ratio, the actual chanse
of potential energy is equal to the sum of the changes in the
potential energy of the two bodies, each with reference to the
centre of gravity as if it were a centre of force. Hence
the sum of the potential energy and the kinetic energies of
the two bodies is constant.
5. The total kinetic energy of a system of bodies may be
decomposed as follows : — First, decompose the energy of each
body into parts corresponding to the velocities perpendicular
to and along the lino m which the centre of gravity is mov-
ing. Put u for the first of these components, v for the
second taken relatively to the centre of gravity, and V for
the velocity of the centre of gravity. Then the total kinetic
energy is
From the property of the centre of gravity 2 mv = : there-
fore the total kinetic energy is
The first term is the sum of the kinetic energies correspond-
ing to the motions relative to the centre of gravity, and the
second is the kinetic energy corresponding to the total mass
as if situated at and moving with the centre of gravity. Thus,
the total kinetic energy is equal to the internal kinetic energy
of the system relative to the centre of gravity as a fixed point,
and the external energy of the system due to the motion of
the centre of gravity.
6. When a stress exists between two bodies whose centre of
gravity is in motion, the stress causes at every instant the
same gain of kinetic energy in one as loss in the other, if the
distance is unchanged. But, when the distance is changing,
we find by considering the external and internal energy of the
system, tliat the former is unchanged by 1, and that the change
in the latter is, by 4, compensated for by that in the potential
energy connected with the stress, so that in either case the sum
BIOHT-FIGUBB LOOABITHM TABLES. 139
of the two kinetic energies and the mntnal potential energy is
nnchanged.
7. Hence in any sjrstem of bodies, between pairs of which
stresses exist whose intensities depend solely upon the dis-
tances, the sum of the kinetic energies and the potential
energy due to their relatiye positions is constant.
EIGHT-FIGUBE LOGARITHM TABLES.
Tables des Loaarithmes a huit decimales des nombres entiers
deld 120000, ei des ainua et tangentea de dix aecondes en dix aeeondes
d'arc dans k aysthme de la divUion cenUaimale du piadrant. Pub-
ii^s par ordre da MinJstro de la guerre. Paris, Imprixnerie Nationals,
18»1. 4to., pp. iv. + 628.
Advocates of the decimal subdivision of the quadrant will
be much pleased by the appearance of the above work, which
contains the most extensive set of tables of the kind as yet
issued. It is not intended in the present notice to enter upon
the respective merits of the several systems of dividing the
circle, but to consider the volume as a table of logarithms
simply. As such itpresents marked points of difference from
the usual types. These differences are found almost exclu-
sively in the trigonometric portion of the tables, that contain-
ing the logarithms of numbers being similar to the customary
form. The logarithms of the four trigonometric functions
appear on each double page in four separate tables, instead of
the usual arrangement in parallel vertical columns. The
interval of the argument is the same throughout the entire
quadrant, no dimmution being found near the beginning of
tne table. The auxiliary quantities for obtaining sines and
tangents of small angles by means of the table of number
logarithms are given ; but they are placed upon the pages
devoted to the trigonometric functions. It would probably bo
more convenient to find them as usual at the bottom of the
pages containing the number logarithms.
The decimal progression of the argument allows the trigono-
metric tables to bo arranged in the form usually adopted for
the logarithms of numbers. But instead of ten columns
headed with the digits to 9, we find eleven columns, of which
the first ten are headed to 9. The eleventh, which has no
heading, contains a repetition of the column headed 0. This
makes it unnecessary to look back along the horizontal line,
when we wish the difference between column 9 and the next
one. Yet the size of the volume is somewhat increased by
this system, and the tables containing the number logarithms
140
8PHBRI0AL AHB PSAOTICAL ASTBOKOMT.
are rather widely spaced in oonseqnenoe. In fact the Tolnme
18 more ponderous than might be expected in comparison with
other logarithm books, notwithstanding that the adopted
interral of ten decimal seconds is mnch smaller than that of
ten ordinary seconds of arc. This is made plain if we com-
pare the dimensions and weights of the following well-known
logarithmic tables {bound):
XAXS.
LEHOTH.
BBBAOTH.
Tmcunm.
wmnmr*
GauBs 5-fig.
fimhiis 7-fig.
Franch 8-fig.
Vega 10-flg.
mm.
241
257
870
841
mm.
150
175
298
288
mm.
14
80
48
86
grams.
895
1189
8818
9080
Negative characteristics are given thronghont for the log-
arithms of the trigonometric functions^ when the oorrespona-
ing numbers are less than unity. This departure from the
usual custom of increasing the logarithms by 10 can hardly be
regarded as an improvement. The greatest possible care has
been taken to secure the accuracy of the tables ; and in this
respect thev may be greatly commended. The actual num-
bers have been copied from the great manuscript tables of
Ftotlj, which are preserved in the archives of the Paris ob-
servatory. The typographical work, which is excellent, was
executed at the Imprimerie NationcUe,
Habold Jaooby.
SPHERICAL AND PRACTICAL ASTRONOMY.
An Introduction to Spherical and Practical Astronomy.
By Dascom Greene, Professor in the Rensselaer Polyteohnio Insa-
tuto, Troy. Boston, Ginn & Co., 1891. 8vo, pp. viii. 4- 158.
Pbofessoe Gbeene has written this work to supply the
needs of those students who wish to begin the stady of
spherical and practical astronomy, and have but very little
time to give to such study. The work is a stepping stone to
Doolittle's and Chanvenet's books. The anthor deals only
with ** those practical methods which can be carried out by
the use of poi-table instruments/' and in describing those
methods he is very brief, frequently altogether too brief.
The order of subjects is as follows : definitions ; spherical
problems; conversion of time ; hour angles; the transit instni-
BPHEBIGAL AND PRACTICAL ASTRONOMY. 141
meat ; the sextant ; finding time by obserrationy which
includes time by transit obseryations^ by equal altitudes and
by single altitudes ; finding differences of longitude, which
includes the methods by the electric telegraph, by transpor-
tation of chronometers and by moon culminations ; finding the
latitude of a place by a circumpolar star, by a meridian alti-
tude, by a zenith instrument, by a prime vertical instrument,
by a single altitude and the corresponding time, and by cir-
cummeridian altitudes ; finding the azimuth of a ^iven line,
by the elongation of a circumpolar star, by observing a bodj
{sic) at a given instant, by observing a body at a given alti-
tude, and by observing a body at equal altitudes.
These suojects are discussed in the first 95 pages. Then
follows a very short (20 pages) treatment of the figure and
dimensions of the earth, in which the author gives some of
the fundamental formulsB of the spheroid, the elements of the
spheroid as determined by measurement, the polyconic pro-
jection, spherical excess of triangles on the earth's surface,
and geodetic determinations of latitudes, longitudes, and
azimuths. The book has an appendix (pp. 115-150) on the
method of least squares — and three tables on (I.) the correc-
tion for refraction, (II.) equation of equal altitudes of the
sun, and (III.) for computing the reduction to the meridian.
The simple mention of the contents will show how inad-
equate the treatment must be. It seems to the writer that it
is far better to use always a book that encourages the student
to study thoroughly a given subject rather than one that
tempts him to be satisfied with very brief statements. In a
worfc of this kind the discussion of the method of least squares
is hardly appropriate.
The equations are not numbered consecutively throughout
the book but the numbering begins anew with each chapter.
lu the discussion of equatorial interval no account is taken of
the formula used when the declinations are 80° and over.
On page 48 in the third paragraph there is a confusion of
index correction with index error. In (1) on page 147 the
rule should show that both the measured sum and each of the
measured magnitudes should be adjusted.
For a work on practical astronomy it seems to me that the
examples given are too few and insufficient, as in the chapter
on the transit instrument. However, the author's idea seems
to be that the instructor should supply such details. To
colleges and technical schools where spherical and practical
astronomy are given but little time, this work may prove
quite acceptable as a basis of study.
J. E. Bees.
142 NOTES.
NOTES.
A BEGULAB meeting of the New Yobk Mathematical
Society was held Saturday afteraoon, February 6, at half-
past three o'clock^ the president in the chair. The following
pei'sons having been duly nominated, and being recommended
by the council, were elected to membership : Mr. Ernest
William Brown, Haverford College ; Dr. William S. Dennett,
New York ; Mr. Armin 0. Leuschner, University of Califor-
nia ; Professor Oscar Schmiedel, Bethany College ; Professor
Laenas Gifford Weld, State University of Iowa. Notice was
given by the council that it was proposed to amend Article III.
of the Constitution so as to read : The officers of the Society
shall be a President^ a Vice-President, a Secretary, a Treas-
urer, a Librarian, and a Committee of Publication, which
shall consist of two members, either or both of whom may at the
same time hold any other office or offices. An original paper
on the ** Transformation of a system of independent variaoles,'*
by Professor J. C. Fields, was read. This paper has been
transmitted to the Amsrican Journal of Mathematics for
publication.
We have to record the deaths of Leopold Kroneckerat Ber-
lin, December 29, in his sixty-eighth year ; of George Biddle
Airy at Greenwich, January 2, in his ninety-first year, and of
John Couch Adams at Cambridge, January 21, in his seventy-
second year.
The paper '* On a peculiar family of complex harmonics,**
read by Dr. Pupin before the New Yohk Mathematical
Society, December 5, 1891, has been published in full in the
Transactions of the American Institute of Electrical Engi-
neers for December, 1891, in connection with another paper,
*' On polyphasal generators," by the same author.
T. 8. F.
The suggestion made in the article " On lists of covari-
ants" (Btdletin, No. 3, last paragraph of p. 89), has been
superseded in the best possible way. Professor Cayley writes
that he has all but five or six of the forms of the sextic
complete, and adds: "I think of giving these tables in my
volume V." e. m.
Dr. Artemas Martin desires to call attention to two errors
in Degen^s Canon PeUianus.
On page 88 of Degen's Tables, in the line of denominators
NEW PUBLICATIONS. 143
of partial fractions for square root of 853, for '* 15 ^' read 14 ;
80 that the line will be
29, 4, 1, 5, 1, 2, 4, 1, 1, 14, 19, (2, 2)
instead of
29, 4, 1, 5, 1, 2, 4, 1, 1, 15, 19, (2, 2).
Page 98, sqaare root of 929, for
**30, 2,11, 1, 2, 3, 1, 5, 2, 1, 6, 1,14, 2, 1, (2, 2)
1, 29, 5, 40, 19, 16, 40, 10, 20, 38, 8, 50, 4, 22, 32,(20, 20)^'
read
30, 2, 11, 1, 2, 3, 2, 7, 5, (2, 2)
1, 29, 5, 40, 19, 16, 25, 8, 11, (23, 23)
Degen's values of x and y in both cases are correct. H. j.
Maokillak & Co. have in press a treatise on the '* Appli-
cations of elliptic functions" by A. G. Greenhill. They have
in preparation a work on '* Hydrostatics " by the same author,
ana one on the " Theory of heat '' by Thomas Preston.
T. 8. F.
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1891. 8vo, 499 pp. M. 14
Caylby (A.). The Collected Mathematical Papers. Vol. 4. 4to. Cam-
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Chappuis & Barget. Le<?on8 de physique ^n^rale. Tome III. Acous-
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DnxMANK (C). Astronomische Briefe. Die Planeten. 1892. 8vo, 228
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DuNis (N. C). Recherohes sur la rotation du soleil. 1891. 4to, 78 pp.
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HuTGENs (Ch.). CEuvres completes. Tome IV. 1891. 4to, 587 pp.
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vom Geschlecht Nail, df reD coordinaten eindeutige doppelperiodische
Functionen dcs Bogcns der Curve siDd. 1891. 8vo, 74 pp. M. 1.80
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Flouchon (J.). Theorio des mcsures. Introduction £ Totude des sys-
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Point ab£ (H.). Ijcs methodes nouTcllcs do la mdcanique c61dste. Tome
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PoiNCAR^ (IT.). Thcrmodynamique. 1891. Svo, 4:]3 pp. 16 fr.
Prado (F. de) y Sersix (A.). Tdiculo de los numcros aproximados y
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SOlfE RECENT ELEMENTARY WORKS ON MECHANICS. 145
SOME RECENT ELEMENTARY WORKS OX
MECHANICS. I.
TTie Laws of Motion, an elementary treatise on dynamics.
By W. If; Laverty, late fellow of Queen's College, Oxford. Lon-
don, Rivingtons. 18b9. 8vo, pp. 212.
In a recent number of the Bulletin (No. 2, pp. 48-50)
Professor T. W. Wright complains of the confusion existing
in the nomenclature of elementary mechanics. It would be
easy to answer his questions from a purely theoretical point
of view ; indeed, in theoretical mechanics no diflBculty is
encountered in this respect. But it must be admitted that in
elementary works, particularly in those of a more ** applied "
character, the confusion is great, both as to the use of terms
and the way of presenting tlie fundamental laws.
By reviewing somewhat at length a few of the better recent
works on elementary moclmnics it may perhaps be possible to
** fix the ideas " and arrive at some conclusions, at least as to
what is the best modern usage in treating the subject.
Mr. Laverty's little work is rather different from the ordi-
nary English text-book. There is no reference in the preface
to tne " examinations of the Science and Art Dep.irtment for
the elementary stage," nor any gentle hint to the reader that
*^ most of the examples are taken from actual recent examina-
tion papers."
" The object of this treatise," says the author (p. v.), " is to
put the subject of dynamics on* a thoroughly sound basis,
avoiding unsatisfactory illustrations and definitions which do
nothing towards defining, and to endeavour to give the stu-
dent such an accurate idea of the subject that he may be able
e.g. to give explanations and illustrations of the laws without
just merely copjring these from the book."
The author's objections to definitions that do not dc6ne, to
inadequate illustrations of the fundamental laws, and to the
loose and confused ways of stating these laws found so often
in elementary works are certainly well taken. The book is
evidently the result of careful independent thinking and treats
a well-worn subject in a fresh and original way. Newton's
laws are given in good English and in modern scientific lan-
guage ; the discussion of their meaning and interdependence
18 noteworthy in many respects.
The outward appearance of the book is pleasing ; the little
volume is neatly printed and furnished with an alphabetical
index in addition to an ample table of contents. The matter
is well arranged and distributed into sections of convenient
size ; every subject is illustrated by a few *' worked " examples
146 SOME BEOENT ELEXB17TABT W0BK8 OV KBOHAHICe.
followed by a large number of exercises for which the answers
are given at the end of the volume.
Before discussing the points of principal importance a few
minor matters might be mentioned which could readily be cor-
rected in a second edition.
In art. 9, the terms '' standard " and ''unit'' are used as if
they meant the same thing. It is preferable to make a dis-
tinction. Thusy the standard of mass in the 0. O. S. system
is the kilogramme, that is a certain bar of platinum preserved
in Paris, while the unit of mass is a gramme, that is any
mass equal to a one thousandth part of that kilogramme* —
The statement of art 267 that ''the laws of friction between
bodies, as found by experiment, are surprisingly simple/' giyes
a surprisingly optimistic view of the case. — ^The factor 2 in
the first expression for n—n' on p. 170 is a misprint; it
shonld be dropped. — ^In example E, pp. 110-111, the factor ff
should be inserted in the expression for the work, or rather in
the problem itself "42400000 ergs" should read "42400000^
ergs.'' —
The numerical data in the exercises are usually so selected
as to lead to answers expressible in round numbers. This
method has obvious advantages for class work and examina-
tions ; it saves time and allows a certain display of ingenuity
in arranging the numerical work conveniently for cancelling.
But it accustoms the student to methods that are far from
being the best in examples as they occur in actual practice.
If the working of numerical examples is to be of any value it
should lead the student to understand the bearing tnat every
quantity involved in the formula has on the final result. The
beginner should in particular loam to select for any constant
the proper number of decimal places necessary in order to
obtain the required accuracy of the result ; he shonld also de-
termine from the data the accuracy obtainable with the data
of the problem. Thus, on p. 7 we find the problem : "How
many metres are there in a mile, if there is .305 of a metre in
a foot ? " The answer is correctly given as 1610.4. But actu-
ally there arc 1609.3 metres in a mile ; the ^ven constant .305
is not suflficiently exact to give the result correct within a
decimetre. Would it not be better to refer the student to
the more exact value of the constant given on a previous page
(p. 5) and require him to select the proper number of decimal
places ?
It must be said, in general, that the author has an excessive
fondness for such merely speculative problems as the follow-
ing : ''If the unit of area and time be 10 acres and 10
seconds ; what is the unit of velocity expressed in miles per
hour?" (p. 27.) Such meanindess problems occur in great
number throughout the book. With this exception the cxer-
SOME BECENT ELEHBNTABY WORKS ON MECHANICS. 147
cises are yery well selected and constitute a valuable feature
of the book.
lu the matter of symbols and names for the units Mr.
Laverty is unusually radical. He manufactures them with-
out the slightest compunction. The British unit of velocity
(foot a second) is called /as, the C. G. S. unit (centimetre a
second) cas ; similarly the unit of acceleration are sfas, seas ;
those of momentum (a/as in a pound) if asp and casgram ; of
kinetic energy : f aspen and casgrammen ; of impulse : bim and
cim ; of force : sfasp and scasgram. This new notation is as
ingenious as it is simple ; bim for impulse strikes one as partic-
ularly happy. But will it be possible to bring this brilliant
new coinage into circulation ? And before this is accom-
Slished, what is the poor student to do tis soon as he leaves
[r. Laverty's class-room ? Nobody will understand him when
he begins to talk of casgrammen and sfasp, and he will have
difiScmty in understanding the old-fasnioned rest of the
world.
A new notation of this kind is entirely out of place in an
elementary text-book. Originality is no doubt a good thing ;
but in a work for beginners it is to be used with moderation ;
an over-dose may become fatal. It is another question whether
the uotation is in itself good and its acceptation desirable.
It may be seriously questioned whether there is any actual
need for special names and symbols for all these units. The
British Association Committee on Units suggests the name
Jcine for **a speed of 1 cm. per sec.;" J. B. Lock uses vel
and eel for the units of velocity and acceleration ; the term
** quickening'* has been proposed for unit acceleration. Mr.
Laverty's scheme has the advantage over these separate efforts
of being methodical and comprehensive; it also lends itself
readily to farther extension. A "mile an hour" might be
called a mahy a ''yard a minute" a yamy etc. But the fact
of the matter is tnat these numerous symbols and names can
be of use almost exclusively in the elementary text-book. Later
on we can get along without them. In most cases the unit
can be understood from the context, as when the phjsicist
says that the acceleration of gravity at a certain nlace is 981,
meaning '* centimetres per second," or when the engineer
gives the angular velocity of his fly-wheel as 25, meaning
"revolutions per minute." It is mere pedantry to require the
unit to be stated explicitly under such circumstances. In
other cases it is best to state the unit completely.
Mr. Laverty says, in regard to his notation (preface, p. x.) :
"These words should be looked upon simply as abbreviations
(perhaps in some cases as aids to the memory) ; I have no
diesire to add new words to the language." But if they are
not to become new words of the language, what is their use ?
148 SOKE BECEirr ELEXBHTABY WORKS OV ICBOHAVIOa.
Are they to be learned only to be forgotten as soon as possible ?
And does not Mr. Layerty himself use them thronghont as if
they were new words oi the language ? Let ns are the
student in the elementary text-book nothing but the most
approved notation of the soience and the less perfect equiva-
lents used in the applications ; he will have enough to do in
mastering these.
All these slight strictures, however, do not detract materi-
ally from the value of Mr. Laverty's work, which gives an
admirable presentation of Newton's three laws of motion.
After explaming the ideas of velocity and acceleration in the
simplest cases, the idea of mass is introduced ; the funda-
mental equations v^af, fv* = cnr are multiplied by m
and the quantities mVy ^mv', ma are given the names
momenium, kin$iic eneray, and massHtoceknUian, respeo-
tiyely. There is no good reason why the term f(Mrc$ should
not be used here instead of mass-acceleration, if force were
thus defined, the fundamental relations
mv = ma.ty ^mv = ma^x
would at once show that force is the rate of change of mo-
mentum with the time, or the rate of change of kinetic
energy with the distance.
The author prefers to call force that which produces change
ofmomentum. At the same time he objects to calling this
a definition offeree. If this be not what the logicians call a
definitio realis, it certainly is a definitio nominalis: we
observe in nature a change of momentum, and to the cause
of this change we giyQ the name force.
Newton's first law is stated very clearly in the following
terms (p. 46) : *^Tho momentum in a mass (or system of
masses) cannot be increased or diminished except by the
action of external force.'* This becomes a self-evident truth
with the above definition of force as the cause of change of
momentum ; for when there is no cause there can be no effect.
But unfortunately Mr. Laverty neglects to give a definition
of force. And yet what concept needs definition more than
force ? In ordinary language the term is used in a variety
of meanings ; and on the other hand, force itself cannot bio
directly observed in nature (excepting the case of muscular
force with which we are not concerned here), it is only its
effects, i.e. changes of momentum, that can be directly meas-
ured. In all other respects Mr. Lavertjr's explanations and
illustrations of the first law can only be commended.
The second law is given in this form (p. 68) : ''When mo-
mentum is produced, it is by the action of force ; and the
SOKE BECENT ELEMEKTABT W0BK8 ON MECHANICS. 149
amount of momentum produced in a given time is propor-
tional to and in the direction of the force."
It will be noticed that the first clause is but a re-statement
of the first law ; and the author very justly remarks (preface,
p. VII., foot-note) that the first law might be dropped and
the science of mechanics be based on only two laws, ** the law
of force (or momentum), and the law of work (or energy).*'
While the first law merely states that we shall give the
name force to the cause of any observed change of momen-
tum, the second law defines force more accurately by saying
that this cause is proportional to the effect produced and that
the direction of the force shall mean the direction of the mo-
mentum produced. It also implies the independence of the
action of two or more forces applied at the same point. Thus
it follows that the parallelogram law applies to forces just as
it applies to velocities and accelerations.
The third law is expressed as follows (p. 86) : *^The work
done by a force (or any agent) on any mass {ox system of
masses) has its equivalent in the kinetic energy exhibited, and
in the work done against molecular forces, gravity, and Ifric-
tion." The usual short form *^ action and reaction are equal
and opposite " is rejected as meaningless as long as action and
reaction are not carefnlly defined. " The fact is," says the
author, p. viii., *'that, if by * action' and M'eaction' are
meant force and resistance, the third law is but an easy deduc-
tion from the second ; while if d'Alembert's principle is really
to be ultimately deduced from the law, it is better to enunci-
ate it at once in proper form, and not in the usual indefinite
and undefined terms/' Thus, the third law in the simplest
ease is expressed by the equation
max = ^mv^,
while in the most general case it leads to d'Alembert's prin-
ciple (p. 92) : " The internal pressures of any system of rigid
bodies are in equilibrium amongst themselves."
After discussing each law for itself the author devotes
several sections to illustrations and applications of the laws ;
these embrace the theory of the pendulum, Atwood's machine,
the inclined plane, collision, projectiles, and circular motion.
Only the most elementary mathematics are used throughout
the Dook.
As a point not usually touched upon in elementary text-
books it may be mentioned that Mr. Laverty calls special atten-
tion to the lact that the parallelogram law would not hold for
forces if they were not defined as they are by the second law,
viz. as the time-rate of momentum, but e.g. as the time-
rate of kinetic energy. It is well known that on this point
150 GENERAL COMPLEX KUMBEB8.
turned the long controversy on the nature of force and energy
between Descartes, Leibnitz, and their followers.*
The closing section contains some interesting general re-
marks on the nature of the three laws and the ways of testing
their truth.
Alexakdeb Ziwet.
Asjx Arbor, Michigan, January 1, 1892.
WEIERSTRASS AND DEDEKIND ON GENERAL
COMPLEX NUMBERS.
Wbierstrass f — Zur Theorie der aus n HaupteinheUen gebUdeien
plexen Ordssen. Gdttingen NacJvriclden^ 1884.
Dedekixd — Zfjtr Theorie der aua n Haupteinheiten geb&deten complextn
Grds8en. . Gdttingen Naehrirhien, 1885.
Dedekind — Eriduterungen zwr Theorie der aogenannten aUgemeinen eonim
plexen Ch-dssen, Odttingen Nachrichten, 1887.
Ik closing his second memoir on biquadratic residues | Oanag
makes this remark : ^^ Our general arithmetic, which goes so
far beyond the limits of the geometry of the ancients, is entirely
the creation of recent times. Starting with the notion of whole
numbers its field has widened little by little. To whole num*
bers came fractions, to rational numbers the irrational ones;
to the positive camo the negative and to the real came the
imaginary."
Once convinced thiit y^^^ was properly an algebraical
quantity and that it had a meaning, mathematicians began to
look for other quantities of a similar nature. *'Why,' they
asked themselves, ** should algebra yield an imaginary unit
which makes it possible to represent two dimensions of space
analytically ; and fail to yield a second imaginary unit which
can be used to represent the third dimension?^' The thing
needed only to be sought for apparently, and at first they
looked amongst the functions of y'^H^. Unfortunately it
turned out that even the most promisingly irrational of these
could all be broken up into a real part and ^^\ times a
second real quantity ; algebra had done her best ; if mathe-
maticians wanted more imaginaries they must invent them.
From the time of Gauss, then, until the present day the
architects and the masterbuilders have turned occasionally
* See for instance E. Mach, Die Mcchanik in ihrer Enttviekelung, Leip-
zig, Brockhaus. 1889, pp. 254-259.
I Extract from a letter to Schwarz.
i Wcrke, II., p. 175.
GBKEBAL COMPLEX KUHBEBS. 151
from their labors upon the theory of functions, that monu-
ment which of all tnat human hands have built will rise the
highest and stand the longest, to try their skill in construct-
ing systems of imaginary, or complex, numbers.
Gauss himself was of the opinion that no complex numbers
except those of type x 4- \/^^ y would be found admissible
into arithmetic,* but does not state his reason for the opinion.
The occasion of the articles cited above was an inquiry into his
most probable reason, an inquiry which involved a funda-
mental investigation into the properties of the hyper-complex
[Uber-complex] numbers, as Dedekind calls them. After full
and interesting researches, of which this paper aims to give a
sketch, these great mathematicians came to opposite conclu-
sions. The fact that in the field of complex numbers the
product xy may vanish when neither x nor y is zero, a fact
made public by Peirce long before,! seemed to Weierstrass so
unlike anything in ordinary mathematics that he concluded
this must have been Gauss's reason for excluding hyper-com-
plex quantities from arithmetic. On the other hand Dedekind
asserts that it is quite a common thing in ordinary arithmetic
for such a product to vanish, and concludes that Gauss's rea-
son for excluding quantities of a nature different from x -\- itfl
was the fact that such quantities, conditioned as they must be,
do not exist.
To construct a complex number Weierstrass writes down a
system of n units ^„ e„ . . . e^ and multiplies each by an
ordinary real number Sr ; then the expression x = S^e, + . . .
4- Sn^n 18 a number of the kind considered. His first under-
taking is so to define the fundamental operations of arithmetic
for quantities of this kind that x -^ y, x — y, xy, x/y may all
be linear expressions of the same form as x ; and that the com-
mutative, associative and distributive laws of addition and
multiplication may hold good for them. It appears that the
multiplication table for the units maybe constructed in an
infinite number of ways so as to satisfy all these requirements.
Of course the fundamental condition is the first one, which
comes to the same thing as this, that every rational function
of the units shall be expressible in the form
Division is defined by the equation
I = r.^i 4- . . . + Xne, = y.
* Werke, II., p. 178.
\Am. Journ. Math'., vol. IV. (1881), p. 97.
X X and y real ; % = \^- 1.
163 GBITEBAL COXPLBX XTUMBBBS.
Mnltiplying both members by b and equating the ooeflSoients
of 0jy • . . Bn on both sides, a set of n equations is obtained*
linear in y^, . . . y^. If their determinant yanishes identi*
cally, it is impjossible to determine y^, . . . y^, and therefore
all multiplication tables are exclud^ which would bring this
to pass. But even then there will be certain values of b for
which this determinant will vanish. Suppose such a value
chosen ; we can then find a value of y such that by shidl
vanish, both b and y being different from zero ; for by =
leads to a system of n equations linear and homogeneous in
Vif ' • ' X* ^^^^^ determinant vanishes. The quantities b
having this unique and wonderful property are called by
Weierstrass *' divisors of zero'* [Theikr aer ifuUL
It turns out that when } is a divisor of zero there are an
infinite number of quantities y such that by = 0, and thence
it is an easy inference that the equation
ka + kbx + Jkcjf + . . . + khr =
has an infinite number of roots if ib is a divisor of zero. We
have, in fact, only to make
a -h bx -h . . . +far = ^
where ff is any one of the infinite number of quantities satis-
fying the relation kg = 0.
'^The existence of these divisors of zero which are not
themselves zero, seems/^ says Weierstrass, ''to make a real
distinction between ordinary arithmetic and the arithmetic of
hyper-complex * numbers " ; but ordinary algebraic eauations
exist which have an infinite number of roots, namely those
whose coefficients are all zero. As to this point then there is
a good enough correspondence between the numbers of our
common arithmetic and hyper-complex numbers.
The author now obtains a multiplication table of beautiful
simplicity by the following process. He expresses the first,
second, . . . , (w + l)-th powers of x, where
linearly in terms of e , . . . e^; then, excluding the case when
the determinant of the right members of the first n equations
vanishes, we can express e„ . . . e^ in terms of the first n
powers of x ; and substituting these values in the last equation,
obtain a relation among the powers of x of the form
where ^^ is the determinant just mentioned.
* Weierstrass does not use this term.
GENEBAL COMPLEX KdHBEBS. 153
^ Dividing by A^x this becomes, if we replace x by the par-
ticular value gy
Here e^ is a quantity which satisfies the conditions
for any number of the system. Its value is in factg/g ; which
is determinate so long as ^r is not a divisor of zero. We are
now in a position to put every number a in the form
a = a,e^ + a^g -\- a^ -h . . . + a,g*-^ = « (fl^) *
and the product of any two numbers takes the same form.
Consider now the algebraic equation f{S) = formed by
replacing g in f{g) by S ; form also the function a{S) by
replacing ^ by ^ in a (g). There is no diflSculty in seeing
that the product a (S) . d (S) will vanish if it contains the
factory (5^). If /(f) = has a root of multiplicity A, it cau
be indicated by writing
and the arbitrary function g>(S) =f^ {S) . F{S) . (p^ {S) will
be of such a nature that (p^ {S) is divisible hjf(S) and there-
fore vanishes ; but if S be replaced bv ^ in <p {^) we obtain a
hyper-complex q^uantity x whose A-th power is obtained by
replacing S byg m <p^ ^). The A-th power of x will therefore
vanish. Hence, if /(^) has a multiple root, the equation
0^ =
can be satisfied in as many ways as there are different choices
of the function tp (S) ; but this number is infinite. It is our
intention, however, to allow an algebraic equation an infinite
number of roots only when each of its coefficients is a multiple
of the same divisor of zero f ; matters must consequently be so
arranged that f{S) = shall have no multiple roots. To
effect this, the original multiplication table must be so con-
stituted that the discriminant oif(S) shall not vanish. This
imposes another restriction upon the freedom of choice of the
coefficients e^* in the equations
ei€j = ^i' e^^Cu . (i, y = 1, 2, . . . n).
1
The simplified multiplication table is now in sight. Take
any function tp {S) of degree n — 1 and with real coefficients
* This is a departure from the notation of Weierstrass.
t Weierstrass, he. eit.y p. 899.
154 GEX£BAL COMPLEX XUMBEBS.
and break up into partial fractions the quotient of ^ (£) by
f(S) . This yields the ec] nation
f{s) i-b, ^-b,'^ • • • "^ e^ufi^kj • • •'
the quadratic denominators corresponding to pairs of conjn-
gate imaginary roots of/ (5) = 0.* The quantity y__i ^®
a polynomial in ^ of degree n — 1 and may be changed into a
hyper-complex quantity, c^, by replacing ^ by ^ as aboTC. In
the same way >r ^,' leads to another quantity c,. The prod-
' f(S)AS)
uct c^c^ is obtained by replacing ^ by ^ in A^A^ .^_2, (/^ \ \ ;
but this product vanishes and, in consequence, c^c^ = 0. If
then/(^) = has the m real roots 6^, . . . ft,, we may con-
struct m hyper-comp!ex quantities c„ c,, . . . c^ such that the
product of any two of them Tanishes. MoreoYer we can
obtain
where B, is a constant. This reduces to B, ' L^ \ and we infer
that <r,' is equal to r, times a real quantity. Moreover this
real multiplier cannot be 0.
Again the product -^~- — ^fl^^il ^'i^l> when S is replaced
by g, yield two hvper-complex quantities, c«+i, c^^x since
6', and i>, are both arbitary. These quantities form the
doubly extended manifoldness ^,r,„.i 4- Z'/'m+i ; and each pair
of conjugate iniaginary roots oif[B) = enables us to form a
similar manifoldness. Repeating the reasoning already given
we find that the product of any two quantities belonging to
different manifoldncsscs vanishes ; thus
(r:c„.,,-f DrC',,,,:) {C\c,„+, + A^'.+O =
whether A and D, be different from zero or not ; and that the
product of two quantities belonging to the same manifoldness
also belongs to that manifoldness. Suppose the whole num-
ber of ))artial fractions to be r; each fraction yields a simple
or com])lex quantity a^, . . . a^ and any hyper-complex quan-
tity whatever can be expressed in the form
^/'l 4- . . . + CtrCLf.
* Tho notation of Wciorstrass is here altered for simplicity.
GENERAL COMPLEX KUMBEBS. 155
K y be any other quantity
then the rule for multiplication is
If now in x the coefficients ^j, 5,, . . . ^* all vanish, and in
y the coefficients z;*, 7;*+i, . . . y/r all vanish, then ary will
vanish while neither x nor y is zero.
An equation of the form
a -\- px -k- yx^ 4- . . . + o^^ =
breaks up into ;• equations of the form
{B) a^ 4- /i^a:^ + . . . + ci^x^,^ =
where o'^, /?^, . . . , ir^ are ordinary quantities. Equation (-5)
can have an infinite number of roots — only in case a/*, /S^,
. . . G7^ all vanish. Suppose they do vanish : then
Taking any quantity
k = k^a^ + . . . 4- ^V-i ^^-1 + ^V+i «^+i + • • . 4- kr^r,
we can put a in the form
a = ka where a ,e. = -r—^ = -P e. or a, = -,- ;
' • ^•,aJ ){;, • ' k/
similarly" for a\, and so on. But a'^ = 0/0 ; that is it may
be anything we ])lea8e.' Proceeding in this way, the equation
can be put in the form
ka 4- k^'x 4- . . . + koj'x^ —
where k, having one coefficient zero, is a divisor of zero.
Equation (B) having an infinite number of roots, of course x,
of which each root of (B) forms a part, has an infinite num-
ber of values. We thus see why it is that in this system an
equation must have an infinite number of roots when each co-
efficient is a multiple of the same divisor of zero.
Closing this section of his letter the distinguished author
remarks that very likely Gauss's only reason for excluding
from arithmetic these hyper-complex quantities was that he
regarded the vanishing of xy when nefther x nor y is zero as
an insurmountable difficulty ; otherwise " it could hardly have
escaped him that an arithmetic of these quantities can be con-
structed in which all the theorems are identical with those
156 EMILE MATHIEU^ HIS LIFE AND WOBES.
concerning ordinary complex quantities, or at least analogous
to them. **In lact/^ he continues, ^' the arithmetic of
hyper-complcx quantities can lead to no result which could
not be reached by processes known in the theory of ordinary
complex quantities."
The views of Dedekind upon this last point quite coincide
with those of Weierstrass ; but for an account of his beau-
tiful method of generating systems of complex quantities, the
reader is for the present referred to the memoirs cited above.
C. H. Chapman.
Johns Hopkins Univkhsity, February 8, 1892.
EMILE MATHIEU, HIS LIFE AND WORKS.*
If it were asked what tyranny in this world has least foun-
dation in reason and is at the same time most overbearing and
capricious, none could be found to answer better to this de-
scription than fashion ; that fashion which makes us admire
to-day what but yesterday would have excited astonishment,
and which may provoke ridicule to-morrow. We all know
that this sovereign whose iron rule is so much more keenly
felt on account of its injustice governs the thousand and one
details of cvery-dav life ; that it is supremo in literature and
in the arts, mit tiiose who have not watched closely the life
of the scientific world may perhaps be surprised to hear that
even there if you would please you must bend the knee to
fashion. What ? might exclaim the stranger to the world of
science, can it bo true that the mathematician knows other
laws than the inflexible rules of logic ? Does he care to obey
other orders than the invariable commands of reason ? — Well,
yes. Of course, to have a mathematical production accepted
as correct^ it is sufficient that it conform to the precepts of
logic ; but to have it admired as beautiful, as interesting, as
of importance, to gain honor and success by it, more is re-
fj^uired : it must then satisfy the manifold and varying exac-
tions imposed by the prevailing taste of the day, by the prefer-
ences of prominent men, by the preoccupations of the public.
Thus it comes to pass that, in mathematics as elsewhere,
fashion will sometimes award the laurels to those who have
not deserved the triumph and make victims of men whose
lack of success is an injustice. In every country there are
such victors and such victims ; but nowhere perhaps are they
* Translated from tho MS. of the author by Professor Alexakdeb
ZlWET.
EHILE MATHIEU, HIS LIFE AND WORKS. 157
more numerous than in France. In this country where cen-
tralization is carried to an extreme, nothing is accepted unless
it receive the sanction of Paris, or rather of certain constituted
bodies, of certain official peraons residing in Paris. Those
who have been so fortunate as to have their work noticed by
these persons and approved b^ these bodies, who have been
granted admission to the chairs of the capital, form in the
opinion of the French public the only men of science worthy
01 honor. The others, relegated to the provinces, are left to
oblivion, almost like those seigneurs in the age of Louis XIV..
whom a caprice of the monarch relegated to their country
estates. Such are the reflections suggested to my mind by the
contemplation of the life and works of Emile Mathieu. An
indefatigable and productive worker he leaves behind him
the results of a lifework, partly as newly acquired possessions
of science, partly as suggestions that will open new paths to
the seeker after truth. After a life full of disappointments,
he died at a time when the official men of science hardly bad
begun to suspect that somewhere in the provinces, far away
from the capital, there lived a mathematician whose works
were an honor to his country. These works had one defect :
the subjects they treated, the methods they employed, were
not in fashion I
Emile Mathieu was bom at Metz, on the 15th of May, 1835.*
From early youth he showed a taste for study. While attend-
ing the hjc6e at Metz he was year after year awarded, by the
consent of his fellow-pupils, tne prize for scholarship and con-
duct. His uncle Aubertin, colonel of artillery and director
of the gun foundries at Metz, was there to point him the
way to the Ecole Polytechnique, But at this period it was
not in mathematics he excelled, but in the study of the clas-
sics ; the prizes he took again and again in those early years at
the lycie were for Latin and Greek compositions. However,
his special aptitude for the abstract sciences 6oon developed
itself. From the time he reached the higher grades at the
lyc6e, he continued to rank first in mathematics. He entered
the Polytechnic School at an early age. There he devoted him-
self exclusively to mathematical studies ; and a few months
after leaving this institution he resigned his commission in
the army to give himself entirely to scientific work. While
yet at the Polytechnic School he had published an interesting
paper f in which he extended to finite differences the algebra-
* The biographical data contained in this article are for the most part
taken from the Notice stir E, Mathieu, sa vie et ses travaux, prepared by
his coUeague G. Floquet for the Bulletin de la SocUii dee sciences ae
Nancy*
t ** Nowoeaux thioremes sur les Squationa algibriqtiee" in Now>, Aim*
de math., vol. 15 (1856), pp. 409-430.
158 XKILB MATHIEU, HIS IJFB AKD W0BK8.
ioal theorems of Descartes and Badan regarding deriirafciTeB
and differentials. This pai>er preyed of some senioe to
Mathieu when he presented himself for the decree of Bachelor
of Science, which lie had neglected to do before. As Dnhfr-
mel began examining him in algebra the candidate presented
to him a copy of his pamphlet, and the professor after glano-
inff through its pages declared the examination finished.
scarcely eighteen months had elapsed since this filst examinar
tion when Mathieu, who had not yet reached the a^ of twenty-
four, took the d^rce of Doctor of the Hathematic»l Sciences.
On the 28th of March, 1859, he defended before the Sorbonno
his thesis On the number of values a function eanasmrne^
and on the formation of certain muUip^ transitive fum^ians.
This thesis was very favorably received oy the Facolty. The
theory of snbstitations which formed the snbjeot of this thesis
furnished the yonn^ mathematician material for two otherim-
Krtant papers,'* which were published in LiouviUe's Journal
tween the years 1859 and 1862. In these papers Matbiea
investigates more fully the idea of multiply transitive f anc-
tions. He studies in particular the various classes, of mul-
tiply transitive functions whose degree is a power of a primo
number or such a power increased l)y one. In the course of
this study he discovered the curious fivefold transitive func-
tion of 12 elements. This function and the fourfold transitive
function of 11 elements which he also investigated form two
entirely isolated cases in the domain of transitive functions
as was shown by C. Jordan.!
In another memoir^ published in 18G2 in the Annali di
Matematica, Mathieu undertakes to apply to the solution of
equations whose deffree is a power of a prime a resolvent
function which stands in the same relation to these equations
as does Lagrange's resolvent to the equations whose degree is
a prime- number. These important researches concerning
the most difficult parts of algebra and appearing within so
brief a period coula not fail to attract the attention of the
scientific world to the young mathematician; and this atten-
tion soon manifested itself in the most flattering manner. In
April, 1862, the Paris Academy of Sciences had to elect a
• ' * M^moire siir le nombre de valeurs que peut aequirir une fincOon
quand on y permute ses variables de touteslea mani^rea po98ijble8*^ in Liou*
villous Joum. de math., 2 series, vol. 5 (1860). pp. 0-43 ; and ** JUmaire
sur Vitudc dee foncHon* de plusienre qttantitis, eur la manure de Us
former et sur kesubstUutiofte qui lee laisserU invariablee,** ib., voL 6 (1801),
pp. 241-323.
t ** liee/ierches sur les subetituiions,'* in Liouville's Joum,, 2 ser., toL
17, p. 851.
X ** Mimoire sur la resolution dea iqucUions dont le degri est une pttis-
sanee d'un nombre premier,** in Tortolini's Ann. di mat,, voL 4, pp.
118-182.
JSMILE MATHIEU, HIS LIFE AND W0KK8. 169
member in the section of geometry. LamS who at the time
was dean of the section asked that the name of E. Mathieu
be placed on the list of candidates. Nor was Lame alone
with his opinion in the Academy ; Liouville fully approved it.
This lienor conferred upon a young man of not yet twenty-seven
years of age who had not taken any steps to solicit such dis-
tinction was indeed a brilliant promise for the future. Who
would then have predicted that he who so early received this
promise was to die at the ago of fifty-five, after a life wholly
consecrated to the advance of science without being admitted
bv the Academy even among the number of its correspondents ?
Mathieu at that time held no oflScial appointment. While en-
gaged in the profound researches which gained him the favor
of Lame he was compelled to make a living by devoting him-
self to the exhausting and thankless work of a private tutor.
Prouhet who was examiner (rSpetiteur) at the Polytechnic
School procured him employment as assistant, or "quiz-
master,^ in the li/cee St. Louis, the lycie Charlemagne, and
various private schools. These unremitting labors brought on
a serious illness from which he at length recovered thanks to
the care of his devoted mother.
In 1863 Mathieu first entered upon the study of mathemati-
cal physics. In a note On the flow of liquids through tubes of
very small diameter, published in the Comptes rendus of the
Academy of Sciences,* he shows that the adhesion of a very
thin layer of liquid to the walls of the tube is suflScient to ac-
count for the results of Poiseuille's experiments. A few years
later, in 1866, he published an important paper On the dis-
persion of light. \ In the same year Lame, whom ill-health
prevented from continuing his course at the Sorbonne on
mathematical physics and the theory of probability, presented
him as his substitute to Duruy, then minister of public in-
struction, lie was however not appointed as the minister
had already made his selection. This chair of mathematical
. physics at the Sorbonne was to remain for Mathieu the never-
attained goal of his ambition. In 1867, at the Congress of
the Scientific Societies, a gold medal was awarded him for his
fruitful researches. At the same time J. Bertrand published
his well-known Report on the progress of mathematical anal-
ysis. The following passage is found in this report :
• " M. E. Mathieu "has studied far more fully than had been
done before, the idea of transitive functions, first introduced
by Cauchy, and his memoir deserves quite special mention,
* '^Bwr le fnouvement dea liquides dans Us tubes de trh-peiit diamHre,*^
in Comptes rendus, vol 57 (1868). pp. 820-324.
t ** Mimoire sur la dispersion de la lumilre" in Liouville's Joum.y 8
fler., vol. 1 1 (1866), pp. 49-103.
160 EMILE MATHIEU, HIS LIFE AND W0BK8.
on account both of the importance of the new results it con-
tains and of the ingenious form of the proofs. Other memoirs
by M. Mathieu, relating to mathematical physics, give evi-
dence, like his algebraical researches, of acute penetration and
broad learning. An account of these memoirs will be given
in another report, whose author, I trust, will heartily join me
in calling attention to a young man who tinily possesses the
gifts of a mathematician, but has so far, in spite of the esti-
mation in which he is held by all, remained outside the sphere
to which his remarkable investigations ought to gain him ready
access. "
Toward the end of the year 1867, M. Duruy, influenced
no doubt by the high reputation attained by the name of E.
Mathieu, offered him the complementary course in mathe-
matical physics just then created at the Sorbonne. This was
an entrance to public instruction : the young mathematician
accepted with eagerness. He published later, in 1872, the
substance of this complementary course in a work to which
we shall have to return. This work shows him thoroughly
imbued with the teachings of the great masters, Fourier,
Laplace, Poisson, Lame. Ho proves himself fully conversant
with their methods of integration and knows how to use them
for the treatment of questions as yet unapproached, such as the
difficult problem of the cooling of a planetary ellipsoid. Ma-
thieu had already turned his attention to mathematical phys-
ics, having published a memoir on the theory of light, when
this appointment determined him to devote his main efforts
to the applications of analysis to mechanics and physics. He
did not, however, completely abandon the pursuit of pure
matliematics. Tims, in 1807, he published an important
paper On the theory of biquadratic remainders.* Gauss had
found by induction that the biquadratic character of a prime
number depends on its decomposition into the sum of two
squares ; but he did not succeed in discovering the law of
this dependence of which he says : '^At lex hujus distrihu-
tionis absfrusior videtur, ctiamsi qucedam generalia prompte
animadvertantxir. *^ Mathieu in his memoir actually dis-
covers and proves this law. Later, in 1873, we see him return
to the theory of substitutions and investigate the relations of
his fourfold transitive function of 11 elements to Kroneckers
function of 11 elements. f
But those investigations in pure analysis must henceforth
be regarded as constituting only an incidental part of Ma-
* ^' Memoir e sur la Morie dcs rSsidns inqiiadratiques,^^ in LiouvUlo's
Joum.. 2 scr., vol. 12 (1807), pp. 377-438.
f •* Sur la fonction cinq fins transitive de 24 qiuintites" in LiouviUe's
Joum,, 2 ser., vol. 18 (1873), pp. 25-40.
EMILE MATHIEU, HIS LIFE AND WORKS. 161
thien's work. Sesearchcs in analytical mechanics, in celestial
mechanics, in mathematical physics become the constant ob-
ject of his meditations. In spite of the importance of the
results obtained by him in the domain of the theory of sub-
stitutions and the theory of numbers, it can therefore be said
that what characterizes his scientific individuality is his work
in applied mathematics. " Not having found the encourage-
ment I had expected for mv researches in pure mathematics,
I gradually inclined toward applied mathematics, not for the
sake of any gain that I might derive from them, but in the
hope that the I'esults of my investigations would more engage
the interest of scientific men.'' In this hope he was deceived.
The death of Lam6 resulted in finally bringing mathematical
physics into discredit in Prance. D Alembert, Clairaut, La-
grange, Laplace, Legend re, Fourier, Poisson, Cauchy, Navier.
Presnel, Ampere, Sadi Camot, Clapeyron, Lam6, accumulated
in the course of a century the discoveries that had grown out
of the fruitful union of mathematical speculation and the ob-
servation of nature. Reactions are abrupt and extreme in
the country that had brought forth this succession of men
of genius. Suddenly, the path they had laid open was for-
gotten ; the results of their researches were no more known
to their successors ; the problems that had occupied their
minds were regarded as futile and childish; and while the
higher minds took refuge in the realm of mathematical com-
binations devoid of all reality, the great mass of students
turned to the ascertainment of facts, to experimentation with-
out theory, without idea.
It is this forgotten, despised, and scorned tradition of the
great mathematical physicists that E. Mathieu had the am-
bition and the honor to follow, in the face of the indifference
of his time. All these great authors he studies with passion,
he expounds and compares them, he corrects their errors, he
elucidates and complements the most rigorous propositions
they had obtained. He is imbued with their spirit ; he fully
appreciates whatever in their ideas is imperishable, and once
in a while, as in the preface to his Course of mathemntical
physics y he has a smile of pity for those who pretend to de-
spise that with which they are unacquainted. There is one
among these masters to whom he has a particular affinity ; it
is Poisson, — Poisson who is too fertile in resource, too power-
ful in genius, to be appreciated at his full value by those, so
numerous to-day, who dread long memoirs and difficult ana-
lytical processes. Mathieu had studied him thoroughly ; he
might be said to be his successor. Such tendencies were not
calculated to secure Mathieu in the good graces of his con-
temporaries. His researches might require great intellectual
qualities ; they might be fraught with beautiful resalts \
162 EMILB XATHIEU, HIS UFE AND WORKS.
what of it ? He was the champion of a science that was out
of fashion.
The complementary course to which he had been appointed
did not promise him a yery stable position. The future
seemed so little assured that when the chair of pure mathe-
matics at Besangon became yacanty Matiiieu did not hesitate
to apply for it. The scientific men who constituted the
Oonncil for the Improyement of the Pol;^technic School unani-
mously recommended him for the position, and he receiyed it
without difSculty. Four years later, in 1873, he was trans-
ferred in the same capacity to Nancy. From this time on he
finds himself relented to profound and undeseryed obliyion.
Seyeral times chairs become yacant at the Sorbonne, at the
ColUge de France; by his memoirs and his books, he is just
the man to fill the place ; and yet nobody thinks of serioasly
considering his candidacy. The Academy of Sciences forgets
that, somewhere in France, there liyes a man who through
the whole of his scientific work, throngh the adyance pro-
duced by it in physics, is fully entitled to its rewards and
honors. And only a few months before his death is he
at last allowed to adorn his button-hole with that decoration
which is so stingily bestowed upon those who honor their
country, and so profusely on those who reap honor and profit
from their fatherland.
It had been Poisson's desire to giye, in a series of works, a
connected yiew of all that is rigorously known of mathematics
as applied to the study of nature ; but he had not time to
publish more than four yolumes of this gigantic undertaking :
nis Treatise on mechanics, his Theory of neat, and his Theory
of capillary action. Mathieu coneeiyed this same idea whose
execution, owing to the broad adyances made in all branches
of mathematical physics since Poisson's time had assumed far
wider dimensions though, on the other hand, the task had
perhaps become more sim])le and easier to accomplish on
account of tho comprehensiyeness and generality of modem
analytical methods. More fortunate than Poisson, he was
able to carry the work much farther than his predecessors ;
but he, too, died before accomplishing his purpose. Eight of
the eleven volumes that this work was to comprise have been
Eublished. Tho first of these volumes appeared in 1873. It
ears the title Course of mathematical physics ; * but as the
author himself remarked later on, it should have been called :
On the methods of integration in mathematical physics. It
represents the final form of the analytical introduction to the
Physical sciences that Mathieu had given his students at the
orbonne in 1867-1868. In 1878 appeared the Analytical
* Gouts de physique mathimatique, Paris, Oautbler-Villars, 1874. 4to.
EMILE MATHIEU, HIS LIFE AND WOBES. 163
dynamics,* a kind of introdnction to celestial mechanics. In
1883, Messrs. Gauthier-Villars published the Theory of capil-
larity, \ in sumptnoQS typographical execution. This was
followed, in 1885-1886, by the two Tolumes on the Theory of
the potential and its applications to electrostatics and mag-
netism I ; in 1888, by a volume on the Theory of electro-
dynamics § ; finallj, in 1890, by two volumes on the Theory
of elasticity of solid bodies. ||
At the time of his death Mathieu was actively engaged on
the theory of the elasticity of the ether, that is on optics.
We have examined with painstaking care the papers left by
the indefatigable worker when the chain of his meditations
was broken forever. But the hope of obtaining at least some
fragments of the work he had planned was not realized ; the
notes we had in onr hands did not bear the stamp of the
author's ffenius. We could only trace out from them the
general plan of this treatise which was intended to give an
exposition of the traditional science of optics as elaborated,
after Fresnel, by Green, MacCullagh, Newman, Lamg, and G.
Kirchhoff. Within the narrow limits of this article it would
be impossible to give a full account of the numerous new
results dispersed throughout the memoirs and books published
by Mathieu. We shaU only try to sketch the general ten-
dencies which mark his character and individuality as distinct
from that of the army of mathematical physicists.
Like Poisson and Lam6, Mathieu is skilful in treating
particular problems of mathematical physics, in integrating
the partial differential equations in certain special cases. Let
us here mention for illustration a few of the more difficult
among the questions of this nature that he succeeded in solv-
ing. As early as in 1868 we find him engaged on a problem
presenting great difficulties ; it is the theory of the oscillatory
motions of a homogeneous elliptic membrane subjected to an
equal tension in all directions.! He succeeded in determining
completely the sounds it produces and the shape of its nodal
lines. The only cases whose solution was known before this
* Dynamiqus arudytique, Gauthier-Villars, 1878. 4to.
4 TMorie de la eapiUaritS, ib., 1888. 4to.
X TMorie du poUrUiel et ses applications d VileetrosUUique et au ma-
gnititme. I^partie: Thioris du poterUiel,ib.,l&S5. II* partie: Elec-
trostatique et nutanSHsms, ib., 1886. 4to.
finiorie de VUectrodynamique, lb.. 1888. 4to.
TMorie de VUctstieiii de$ corps solidea, J^ partie : Considiraiions
girUrcUes 8ur rSktaticite ; emploi des eoordonnSes cumilignes; probUmes
retaltfs d ViquUibre de VHiastidU; plaquee mbrantee; ib., 1890. IL
partis: Mouvemente vibrcUoiree des corps eolidee ; ifuilibre de ViUuUciU
duprisme rectangle ; ib., 1890. 4to.
if " MSmoire sur le mouvement vibratoire d^une membrane de forme eUip-
tique," in Liouville's Jaurn., 2ser., vol. 18 (1868), pp. 187-208.
1C4 EHILE MATHIEUy HIS LIFE AND WORKS.
are the Tibrations of rectan^lar'and triaogular membranes
whose theory was shown by Lam6 to be intimately connected
with certain delicate qnestions in the theory of numbers, and
the yibrations of circiuar membranes whose properties depend
on BesseFs fanctions. In his Course of mathematical physics
Mathieu treats another difficnlt problem which had 'been
pointed out to him by Lam6 and which is somewhat allied to
the precedinj2[ question, viz. the cooling of a planetary ellip-
soid. There is a problem to which Lam6 attached great im-
portance ; it is the theory of the deformations of a rectangular
parallelopipedon whose six faces are subjected to forces dis-
tributed m any waj whatever. After a long and unsuccessful
study of the question he made it repeatedly the subject of one
of the prizes of the Academy of Sciences, but without result.
Mathieu succeeded in solying, if not the general problem, at
least a rather comprehensive special case : the extremities of
the prism pressing against two fixed walls and the external
force being the same along a generating line of the prism.
We may also mention amon^ the special problems solved
by Mathieu the diCBcult question of the distribution of the
electric currents in a rectangular prism or a rectangular lam-
ina. But in spite of the analytical power displayed in the
solution of such particular cases, they do not constitute the
most important and characteristic "paxt of Mathieu's work, the
part that distinguishes his work from that of Fourier, of
Poisson, of Cauchy. In the first place it must be said that,
while full of respect for the tradition of these men of genius,
Mathieu does not allow this reverence to become a supersti-
tion ; he knows where to depart from their views. In the
theory of elasticity he does not hesitate to abandon Poisson^s
favonte theory of molecular attraction to follow the more
rigorous ideas of Lam6. In optics he shows that the calcula-
tions by which Cauchy thought to have unfolded the nature
of the ether and its relations to ponderable matter lead to in-
admissible conclusions; and he Doldly modifies the differen-
tial equations by which the great master had represented the
motion of light in an absorbing medium. In the second place,
Mathieu is far more mindful of the generality of the methods
he uses than was the custom with the great mathematicians
of the beginning of the century. In the very preface to his
Course of mathematical physics we see him proclaim his ideas
on this point with perfect distinctness : "As the domain of
science broadens and expands it becomes more and more
necessary to expound its principles with clearness and concise-
ness and to substitute for artificial processes, however skilful,
the transformations that can be accounted for by the nature
of the subject. This is clearly illustrated in comparing the
Micanique analytique of Lagrange with the VorUsungen of
XMILE MATHIEU^ HIS LIFE AND WOBKS. 165
Jacobi on the same subject. Exaroinin^ the treatment of
certain problems in each of these works the results obtained
will often be found to be the same ; the difference consists in
the fact that in the latter work the calculations are performed
according to rules laid down in advance/'
The ideas expressed in this passage are everywhere kept in
view in Mathieu's treatises, in his Analytical dynamics he
introduces at the very beginning the general methods due to
Hamilton and Jacobi. In his Theory ofcapillarity he lays aside
the direct consideration of the capillary forces employed by
Poisson^ and follows Gauss in establishing the equations for
the various problems by seeking to determine the minimum
of the potential of the active forces. In the Theory of the
elasticity of solid bodies he invariably uses the principle of
virtual velocities to throw the problems of equilibrium into
equations. This care for generality is also Mathieu's guide in
the solution of problems requiring the use of hypotheses that
are uncertain or only approximately true. Following a method
which in our opinion could not be too much recommended, he
always begins by establishing the equations of the problem and
treating them as long as possible without making use of those
hypotheses so as to introduce them only at the end. In this
way ho has treated the motion of a projectile in the air and
the equilibrium of rods. General methods have the advantage
of bringing into clear perspective the principles that serve to
solve the problems, and in this way thev will frequently lead
one to recognize the possibility of attacking a problem which
might seem to escape the treatment by more special processes.
Mathieu has thus succeeded in throwing a clear light on cer-
tain theories not hitherto approached. We may mention two
examples.
The oscillatory motion of a plane lamina had been treated
before ; but not that of a curved lamina. Without solving the
Sroblem of the vibrations of an absolutelv general curved plate,
lathieu prepared at least the way for the solution of the gen-
eral problem, reconnoitring, so to speak, the ground in two im-
portant directions, by studying the vibrations of a cylindrical
plate of any cross-section, or curved lamina^ and those of a
Elate of revolution, or hell. Let us briefly examine the results of
is memoir On the oscillatory motion of bells,* The thickness
of ordinary bells is not generally the same throughout. Hence,
to obtain a theory applicable to ordinary bells, the thickness
must be assumed to vary in passing along any meridian from
the top of the bell to the base. There is an essential distinc-
tion between the vibratory motion of a bell and that of a plane
* "Mimoire sur le mouvement mbrataire des cloches,^* Id Joum, de
l^£cole PolyUchn., Oah. 51, pp. 177-247.
166 EXILE XATHIEU, HIS UPE AKD W0SK8.
lamina. In the latter, as is well known, the longitudinal or
tangential motion and the transyersal or normal motion are
giyen by different equations. In a bell, the normal and tan-
^ntial yibrations are given by three equations which are not
independent of each other. Another distinction from the
ease of a plane lamina lies in the fact that the pitch of a bell
does not change if its thickness be varied throughout in the
same ratio, since the terms depending on the square of the
thickness in the differential equations are ffenerallyyer^ small
and may be neglected. This, at least, will be the case if only
the deepest sounds produced by the bell are taken into consid-
eration. When a bell yibrates under the strokes of the clapper
the tangential yibrations are generally of the same order of
magnitude as the normal yibrations. The author has exam-
ined whether it be possible to so select the meridian of a bell
as to give it a purely tangential yibratory motion ; and he has
shown that this is only possible in a spherical bell of constant
thickness. Although the differential equations of the most
general yibratory motion of a spherical bell present them-
selves under a rather complicated form, Mathieu has sue-,
ceeded in integrating them by means of formula of remarka-
ble simplicity.
In connection with this theory of the yibrations of a bell
which Mathieu owes to the g^enerality of his methods we may
mention another result which, though of a more special
nature, deserves to remain classic on account of its intrinsic
importance as well as of the elegance of the demonstration.
This is the complete determination of the action produced by
the capillary forces on a solid body partly immersed in a
liquid. Poisson had only succeeded, by very complicated
though skilful processes^ m determining the vertical upward
pressure produced on a solid of revolution whoso axis is ver-
tical.
But we must now speak of what is perhaps the most note-
worthy part of Mathieu's work. Most problems of mathemati-
cal physics depend not only on one or more partial differential
equations but also on so-called boundary conditions adapted
to determine the arbitrary functions introduced by the inte-
gration of the equations. This science requires therefore the
investigation of partial differential equations^ not taken by
themselves, but in connection with such boundary conditions
and taking into account the form of any such conditions
that may be given. This method has proved a remarkably
fruitful source of beautiful results in analysis. It will bie
sufScient to mention the theory of the potential and the
numerous theories connected with the principle of Dirichlet.
If investigations of this kind have for some time been neg-
lected by pure mathematicians, they have always called forth
SMILE KATHIEU, HIS LIFE AND WOBKS. 167
the efforts of the physicists. Poisson, Helmholtz^ Kirchhoff
have accnmulated the results in the study of the equations
that occur in the theories of sound and light. They haTe
thus prepared the way for the resumption of those researches
which, owing to the labors of H. A. Schwarz and E. Picard,
are at the present day again coming into j^eneral favor. And
in the work of maintaining this now triumphant tradition
only gross injustice could refuse to recognize the important
part taken by E. Mathieu. Ho has devoted a large number
of memoirs to researches of this nature concerning the equa-
tions of sound, of elasticity, and of heat.
As early as in 1868, in his memoir on the vibrations of an
elliptic membrane, he indicates or foresees a part of the
results established later on by H. A. Schwarz and E. Picard
in their beautiful researches on the equation
At a later period he rediscovers and systematizes the results
obtained by Helmholtz in studying the equation
^tt + * V = 0.
Similar considerations he extends to the equation
Au = h
6t
But his most noteworthy researches in this field are those on
the partial differential equation of the fourth order
AAu = 0,
which governs the components of the pressures and the com-
Eoneuts of the displacements at the interior of an isotropic
ody in the state of elastic equilibrium.* Designating aa first
potential the function usually denoted simply as potential, the
author considers under the name of second potential another
analytical expression differing from the former in having the
distance of two points substituted for the inverse of the dis-
tance ; and he develops the entirely new theory of this second
potential. Concerning the partial differential equation of the
fourth order which expresses the equilibrium of elasticity he
proves the following theorem : Every function that satisfies
this equation at the interior of a surface and is there continu-
ous itself as well as its derivatives of the first three orders is
♦ **Mimoire sur Viquation aux difirences partiellea du gtiatrthne ordre
AAv, = 0, ei 8ur Viquilibre d'Uasttciti d^un carps solide," in Liouville's
Journ,, 28er., vol. 14 (1869), pp. 378-421.
168 EXILE XATHIEU, BIS LIFE AND WORKS.
the sum of the first potential of a layer coyering the bounding'
sorface and of the second potential of another layer spread
over the same surface.
I here conclnde this exposition of Mathien's work, not for
want of material, for I have said nothing of his researches in
the theory of perturbations and regaraing the problem of
three bodies, but in order not to exceed the limits of this
article. Besides, whoever desii'es to obtain accarate informa-
tion on the state in which his predecessors had left the science
of mathematical physics aYid on the adyances made in it by
himself can do better than read me by reading him. I haye
known E. Mathieu only as a man of science ; and I have
spoken of him only as sach. To describe the man I mnst
borrow the testimony of one of those who haye best known
him, one of his colleagues at the Faculty of Sciences of
Nancy * : ^^ Of an essentially straightforward, sincere, and
generous nature, he was kindness itself. He possessed the
devotion that seeks to be ignored. In July, 1890, when the
&ta] disease had already attacked him, he succeeded in con-
cealing his ill-health from his colleagues, being unwilling to
leave to them the burden of his examinations. In Septem-
ber, on his deathbed the same anxiety agitated his mind with
respect to the October examinations. His loyal and trust-
worthy character made him esteemed and beloved by all ; in
the Faculty at Nancy he had none but friends. Sensitive to
any kindness, touched by the slightest mark of sympathy, he
belonged to those who ai*e most easily satisfied. " He lived in
simple style dividing bis time between his lectures and his
mathematical researches.^'
P. DUHEM,
Lecturer in Maihemaiicai Physics
and CrystaUography at the Faculty
of Scimces of L%Ue,
♦ G. Floquet, loc. dt.
NOTES. 169
NOTES.
A REGULAR meeting of the New York Mathematical
Society was held Saturday afternoon, March 5, at half-past
three o'clock, the president in the chair. The following per-
sons having been duly nominated, and being recommended
bv the council, were elected to membership: Professor Arthur
Cayley, Cambridge University, England; Professor J. de
Mendiz4bal Tamborrel, Military College, Mexico ; Professor
Truman Henry Safford, Williams College; Professor Ed-
mund A. Engler, Washington University. The secretary
read letters from Professor Cayley and Professor Sylvester
expressing interest in the work of the Society. — The follow-
ing original papers were read : '* On exact analysis as the basis
of language, '' by Professor Alexander Macfarlane ; ** A geo-
metrical construction for finding the foci of the sections of a
cone of revolution, '' by Professor Edmund A. Engler. Mr.
Maclay made some remarks upon the locus of the centers of
curvature of parallel sections of a ruled surface at i)oint8 upon
the same generatrix.
Professor J. J. Sylvester has been compelled to apply
for leave of absence from Oxford on account of ill health.
Mr. J. Griffiths of Jesus College, Oxford, will lecture on the
" Recent geometry of the circle and triangle " for the pro-
fessor.
We learn from Nature that a memorial is to be presented
to the University of Oxford by the council of the Association
for the Improvement of Geometrical Teaching in regard to
the Pass Examination papers in geometry. These generally
consist entirely of propositions onunciatea without any varia-
tion from the ordinary text of Euclid, and scarcely any
attempt is made to discover whether a student's answers are
other than the outcome of a mere effort of memory. The
Association is of the opinion that such papers have the effect
of a direct incentive to unintelligent teaching, and respect-
fully asks for the introduction of simple exercises and of simple
questions suited to promote the rational study of geometry.
The Register Publishing Co., Ann Arbor, announces that
it has completed the publication of Professor Cole's transla-
tion of the " Theory of substitutions and its applications to
algebra,*' by Professor E. Netto.
NaturcB Novitates announces the death of Dr. H. B.
170- NOTES.
Sohroeter, Professor of Mathematics at the University of
Breslan, on January 3, in his sixty-third year. x. s. f.
Sib Bobebt Stawell Ball, Astronomer Boyal for Ireland,
and Professor of Astronomy at Trinity College, Dublin, has
been elected Lowndean Professor of Astronomy at Cambridge
XTnirersity, to succeed the late Professor John Couch Adams.
H. J.
KEW PUBLICATIONS. 171
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THE ALGBBEAIO EQUATION. 173
KEONECKER AND HIS ARITHMETICAL THEORY
OF THE ALGEBRAIC EQUATION.
Leopold Kronecker, one of the most illustrious of con-
temporary mathematicians, died at Berlin on the 29th of last
December in his 68th year.
For many years he had been one of the famous mathe-
maticians of uermany and at the time of his death was senior
active professor of the mathematical faculty of the University
of Berlin and editor in chief of the Journal fur reine una
angewandte Mathematik (Crelle).
He was the last of the great triumvirate — Kummer, Weier-
strass, Kronecker — to be lost to the university. Kummer
retired nearly ten years ago because of sickness and old age^
and recently "Weieratrass followed him ; but vounger than
the other two, Kronecker was overtaken by deatn in the midst
of the work to which his life had been devoted. Despite his
years he was much too early lost to science. The genius
which had enriched mathematical literature with so many
profound and beautiful researches showed no signs of weak-
ness or weariness.
Kronecker was born at Liegnitz near Breslau in 1823.
While yet a boy at the Gymnasium of his native town his fine
mathematical talents attracted the notice of his master,
Kummer, whose distinguished career was then iust beginning.
Kummer's persuasions rescued him from the business career
for which he was preparing and brought him to the univer-
sity.*
He studied at Breslau, whither in the meantime Kummer
had been called, Bonn, and Berlin, making his degree at
Berlin in 1845 with a dissertation of great value : De unitati-
iu8 complexly.
Of his instructors besides Kummer he was most influenced
by Dirichlet, owing in part to Dirichlet's commanding
abilities, in part to the strong arithmetical bent of Kronecker
himself. As long as Diricnlet lived Kronecker^s relations
with him, as with Kummer, were those of the closest personal
intimacy. *
From the university Kronecker returned for a number of
years to business and the management of his estates. But
* Kronecker makes this graceful acknowledgment of his debt to Kam-
mer in the dedication of tne Festsehrift with which he honored Eum-
mer's Doctor Jubeldum (OrundzUge einer arithmetigehen Theorie der
alffebraisehen Grdssen) : *' In Wahrheit verdanke ich Dir mein mathe-
matisches Dasein ; icn verdanke Dir in der Wissenschaft die Du mich
frQh zugewendet wie in der Freundschaft die Da mir frUh entgegenge-
bracht must, einen wesentlichen Theii des Gldcks meines Lebens."
174 EBOKECKBB AND HIS ARITHirETICAL THEOHY
his mathematical actiyity was continuous and his fame grew
apace. In 1853 he communicated to the Berlin Academy the
solution of the problem : to determine all abelian equations
belonging to any assigned " domain of rationality," and in
1857 the first of nis famous memoirs on the complex multipli-
cation of the elliptic functions. His letter to Hermite : sur
la rSsolution de Vequation du bme degri in which his solution
of the equation is indicated appeared in 1858.
In 1861 ho was made a member of the Berlin Academy of
Sciences and in 1867 corresponding member of the Paris
Academy.
His election to the Berlin Academy was an eyent of the first
importance for his subsequent career, inasmuch as it was the
occasion of his resuming the academic life. As member of
the Academy he had the right to lecture at the University, and
of this right — following the example of such men as the
brothers Grimm, Kiepert, Jacob! and Borchardt — he forth-
with availed himself, beginninff in the winter of 1861-62
those lectures on Algebra which naye for many years been one
of the chief glories of Berlin. In 1883 his relations with the
University were made closer still through the appointment
^'Professor ordinarius" and director — with Kummer and
Weierstrass — of the mathematical Seminar.
The range of Kron^cker's productiye activity was very
^eat. Besides distinguished work in the theory of definite
mtegrals, he did work of the first importance in no less than
three great departments of mathematics : the theory of num-
bers, algebra, and elliptic functions. As an aritnmetician
his name is associated with the great names of Gauss, Dirich-
let, and Eisenstein ; as an algebraist with those of Abel and
Galois.
Some idea of the scope of Kronecker's contributions to
mathematical literature may be convejed by the following in-
complete list of his more important memoirs : Dissertatio de
unitatibus complexis (1845) ; Zwei Sdtze Uber Oleichungen
mil ganzzahligen Coefficienten (1857) ; Ueber die algebraisch
aufldsbaren Gleichungen (1853, 1856) ; Sur les facteurs irrS-
ductibles de V expression a;* — 1 (1854: ) ; Ueher elUptische
Functioncn fur welche complexe MultiplicaUon stattfindet
(1857, 1802); Ueher complexe Einheilen (1857); Sur hi
resolution de V equal ion du bme degri (1858) ; TJeber lineare
Transformalionen ( 1858 ) ; Ueber die Theorie der algebra-
ischen Functionen (1861) ; Ueber die verschiedenen Factor en
der Discriminant en von Eliminationsgleichiingen (1865) ;
Ueber den Affect der Modular gleichungen (18G5) ; Ueher
bilinearc Formen (1868) ; Ueber Systeme von Functionen
mehrer Variabeln (1869, 1878) ; Ueber die verschiedenen
Sturmschen Reihen und ihre gegenseitigen Beziehungen
OF THE AXGEBBAIO EQUATION. 175
(1873) ; Zur Theorie der Elimination einer Variabeln aus
ztaei algebraischen Gleichungen (1881) ; Zur Theorie der
Abelschen Oleichungen (1882) ; Zur aritkmetischen Theorie
der (Hgehraischen Eormen (^1882) ; Ueber die BernouilWschen
Zahlen (1883) ; Ueber bihneare Forrnen mit vier Variabeln
(1883) ; GrundzOge einer arithmetischen Tlieorie der alge-
oraischen Orossen (Kummer Jubcldum 1882) ; Zur Theorie
der elliptischen Functionen (1883-1891) ; Ueber den Zahlbe*
ariff (Zeller Jubeldum 1887). Most of his writings were pnb-
lisned in the Berichte der jBerliner Akademie or in the Jour"
nalfUr reine und angewandte Mathematik.
Among the finest of Kronecker's achievements were the
connections which he established among the various disciplines
in which he worked : notably that between the theory of q^uad-
ratic forms of ne&;ative determinant and elliptic functions,
through the singular moduli which give rise to the complex
multiplication of the elliptic functions, and that between the
theorv of numbers and algebra, by his arithmetical theory of
the algebraic equation.
He discovered * that to each class of quadratic forms corre-
sponds a singular modulus which allows of complex multipli-
cation ; to the aggregate of classes of the same determinant,
an algebraic equation with rational coefficients which he
showed to be irreducible; and, in fine, that the theory of
quadratic forms was an anticipation of the theory of elliptic
functions, the two theories being so closely related that one
could have derived the notions of class and order and other
fundamental properties of the quadmtic forms by investiga-
tion of the properties of the elliptic function.
He was above all things the great arithmetician and no-
where does this appear more clearly than in his algebraic
writings. It is not merely that the purely arithmetical prob-
lems growing out of algebra were attractive to him — ^he " arith-
metized'* algebra itself. In the Zeller Festschrift , after
declaring his allegiance in the words of Gauss : " Die Math-
ematik sei die Konigin der Wissenschaften und die Arith-
metik die Konigin aer Mathematik," he writes "Und ich
flaube auch, dass es dareinst gelingen wird den gesammten
nhalt aller dieser mathematischen Disciplinen (Algebra and
Analysis) zu * arithmetisen ', d. h. einzig und allein auf den
im engsten Sinne genommeuen Zahlbegriff zu grunden, also
die Modificationen und Erweiterungen dieses Begriffs wieder
abzustreifen, welche zumeist durch die Anwendungen auf die
Oeometrie und Mechanik veranlasst worden sind.*'
Kronecker arrived at the conception of an arithmetical
theory of the algebraic numbers and functions very early.
* Cf. Hermite: Note sur M. Kronecker, Comptee JBendus, Jan. 4, 1802.
176 KBOKECKEB AKD HIS ABITHMEnCAL THEOBY
There are indications of it even in a letter to Dirichlet written
in 1856. As its yarious salient concepts and theorems were
discovered they were announced in the BericMe of the Acad-
emy or deyeloped in his lectures. But he did not arrange the
whole into a consecatiye and complete body of doctrine until
1882 in his Orund»iige einw ariihmeiischen Theorie der
' aigebraischen OrdMMn. Within the limits of this brief sketch
it would be impossible to convey any adequate notion of this
monumental work. I can attempt only to indicate the salient
points of the first and more elementary of the two parts into
which it is divided.
The ''domain of rationaltv'' (R R\ •) of any system of
quantities R B!' embraces aU rational functions of the R%
with integral coefficients.
These ^s may be quantities of any sort whatsoever, algebraic
or transcendental constants or variables. In particular all
the R% may equal 1 when the domain is that of rational num-
bers in the ordinary sense, or all the R^ may be independent
variables. In either of these cases the domain is said to be
bounded naturally.
An integral function of one or several variables is irreduci^
lie in the domain (R R\.) when it contains no factor having
coefficients which belong to this domain.
Every root of an irrraucible algebraic equation of the n^
degree with coefficients which belong to the domain {RR'. . )
is called an algebraic function of the n^ order of the R*%y the
n roots of the same equation bein^ called conjugate functions.
If a single such root be '^adjoined'' to the J^'s the domain
{G,RyR' . .) is the domain of the " genus '^ (Gattung) (?, the
genus itself embracing those functions of the domain which
are, like 0, functions of the w** order.
If O and 0' be algebraic functions of different genera, but
such that all functions of the genus belong to the domain
of 0', the genus is said to be contained in the genus O';
and the order of 6^ is a divisor of tliat of 0'.
More than one may of course be adjoined to the R% but
it is shown that any number of G's may be replaced by a single
such function which indeed is but a linear function of the ^iven
G% with integral coefficients. In terms of this Q and the
B's all functions of the domain {G', G" .. : R^ R" ..) can be
expressed rationally: or the domain {G', G" ,.: R\ R'..) is
equivalent to a domain (G, R', R'y . .). This is a theorem of
fundamental importance. For from it follows that in the
discussion of all algebraic questions there may be selected as
** elements " Ry B," ,. of any domain of rationality whatsoever,
a number of variables or inaeterminates and a single algebraic
function of them.
A quantity x is called an integral algebraic function of the
OF THE ALGEBRAIC EQUATION. 177
iPs when it satisfies an equation in which the coeflBcient of
the highest power of 2: is 1 and the remaining coefficients are
integral functions of the R'8 with integral coefficients. It is
a fundamental theorem of the theory that for every genus
there exists a finite number of such integral functions
x', x", . .X (•+*^ in terms of which all other integral functions of
the genus can be expressed linearly ; i.e. in the form
where the ^'s are integral functions of the R's with integral
coefficients. Such a system of functions x', x'\ . . . a?^"**"*^
is called 2k fundamental system of the genus. In special and
important cases m can equal 0.
The square of the determinant of any set of n of the func-
tions x'y x", . . . a;^'"'""^ and their conjugate functions is
called the discriminant of these n functions. The aggregate
of the discriminants of every set of n of the functions a: , x' , . .
2.(»+«) constitutes a system of rational functions of the R'^,
snch that whatever properties are common to them all belong
also to the discriminant of every set of n functions of the
genus and are thus characteristic of the genus itself, forming
a complex of 'invariants '^ of the genus in a higher sense 01
that word.
If there exist no algebraic relations among the i?'s, i.e* if
the domain of rationality be the natural domain, there exists
always an integral function of the R's with integral co-
efficients which is a common divisor of all the discriminants
of the fundamental system of the genus and may therefore be
appropriately called the discriminant of the genus itself. If
m equal the discriminant of the n elements x\ Qif\ . . . x^*^
is itself the discriminant of the genus.
The discriminant of the eenus is a divisor of the discrimi-
nant of every equation of the genus, i.e. of every equation a
root of whicli is a function belonging to the genus, and the
greatest common divisor of the discriminants of all these
eq^iiations is a divisor of the ^n [n — l)th power of the dis-
criminant of the genus.
Again if the genus G' be contained in the genus its dis-
criminant will be a factor of the discriminant of G.
And finally the discriminant of the genus to which a set of
functions belong which are defined by a system of equations
-Fj = 0, i^^ = 0, . . . ^, = is identical with the discrimi-
nant of this system of equations.
The demonstration of this last theorem as well as the fur-
ther development of the theory necessitates a general investi-
gation of elimination, the principal outcome of which is that
the complete *' resolvent'' of a system of m eauations in n
quantities x\ x\ x"y , . . a;^"^ is an equation of tne form
178 KSOKBOKBB AVD HIS AiOXHMBTIOAJi THEOBY
FSx, of,. .. aK— »)) JP, (a?, ar', . . . «<—•>) . . . i^. (a?) =
where x^u^x^ + n,a;, + . • . + ti» a;., the n's being indeter-
minates.
The system of equations or '^ partial resolvents'' F^ = 0,
F, = 0, . • • F, = is the complete equivalent of the ^ven
system. Each partial resolvent ^4 = represents a mamfold-
ness of n — ik dimensions, so that speakmg geometrically a
given system of equations in n quantities may define simul-
taneously systems of points, lines, surfaces, eta
Furthermore every divisor of the product F^. F^ . . . F^
set equal to constitutes the entire resolvent of a certain sys-
tem of n + 1 equations Whence the important theorem : the
total content of every divisor of the resolvent of a system of
equations in n quantities can be represented by a system of
only n + 1 equations, and therefore also a system of any
number of eNquations can be replaced by one of only n + 1.
Any algebraic curve of double curvature, for instance, can be
represented by a system of four algebraic e^quations.
Another most important result of this investigation of
elimination is the demonstration that the concept of the
algebraic function does not require any extension when
systems of equations instead of single equations are brought
under discussion.
This doctrine of elimination brinjipi out the true significance
of Galois' theory of algebraic equations.
Let c,, c^ . . . c^uQ quantities belonging to the domain
{B\ R'\ . . ) and
{A) af — (?,ar-* + c,ar-»- . . . ±c. =0
an irreducible equation with the roots 5„ f „ . . . 5».
Further let / (a? , a;, . . . a:.), /, (a;,, a?„ . . . a:,), . . .
f% (^19 ^,y • • - ^n) be the elementary symmetric functions
defined oy the identical equation
(x — a:,) (a; - a;,) ... (a: - a;,)
= a:- -/»ar-» +/,a?— ~ . . . ±/,.
Then the n quantities S^y £„ . . . Sn may as well be re-
garded as determined by the system of n equations
{B) /m (a:,, a;, ... a:,) = Cj, (* = 1, 2, . . . n)
as by the single equation {A).
If F (x) = be the resolvent of this system {E) or an ir-
reducible part thereof, and, as above,
a; = ttj a:, + w, a?, + . . . + w.a:,,
the coefficients of F {x) are integral functions of the indeter-
OF THB ALQEBBAIO EQUATION. 17d
minates u and rational functions of the R'b. And since the
equations (B) can be satisfied only by systems of values such as
2J, — ^r|> fl'j — ^rty • • • ^n — ^i
r»
where r, r,. .r, is some permutation of the numbers 12 . . .«,
F {x) is simply the product
n {X — U^Sr^—U^Sr^ — ' - — UnSm)
extended over certain of these permutations.
If now
be the ** Galois equation '' whose n 1 roots are the n ! functions
u^Xi^ + u^Xi^ -\- . . .-\-UnXu gotten by forming the n 1 permu-
tations t\ t, . . . t« of 1 2 . . . n, the coefficients of x, fy . .^, in
^ (^> /i'Jf«' •••/•) ^^® integral functions of the w's with inte-
gral coemcients, and one of the irreducible factors of (a?, c„
c„ . . c^) must be the same with F{x). Such a factor is there-
fore an integral function of x and the indeterminates u„ u^y
. . . ?^„ with coefficients belonging to the domain of rationality
{R\ R" .,) and may be represented by g (a;, «„ w,. . w,).
^ (a;, u,y «„ . . w,) = n {x—u^Sr^—u^Sr^—. . .-u.Srn)
or, if the terms of each factor be arranged with reference
to the ^'s instead of the u'b
g {x, w„ tt„ . . w,) = n(x-Ur,S,—Ur^S^—. . .-ttr,5,) ;
that is to say g {x^, w„ w„ . . m,), regarded as a function of the
indeterminates u, is a function which remains unchanged for
certain permutations of these u% those represented by r^,
' j> • • ' «•
In this manner^ starting with any special equation (A) one
is led to general functions of indeterminates which are char-
acteristic of the equation and have the property of maintain-
ing their yalues unchanged for certain permutations of these
indeterminates.
The true significance of Galois* principle thus lies in the
fact that it takes as basis for the iuTestigation of an equation
the system of equations which define its conjugate roots
simultaneously.
The functions g to which it leads may themselyes be made
the starting point of the discussion. The problem then is
when one replaces the indeterminates w„ u^, . . i*, by x^,
x^ . . x^\ the investigation of integral functions of n inde-
terminates a;,, a:,, . . a;, with respect to the changes which
they experience when the ar's are permuted in all possible
180 KROKEOKEB AND HIS ABITHMBTIOAL THEOBY
ways, the investigation taking its place in the general arith-
metical theory when one regards the a^'s as algebraic functions
of the n elementary symmetric functions /r.
If the fs take the place of the R's so that the domain of
rationality is (/j, /., . . /,) every rational function of the
x's is an algebraic function of tnis domain and as such be-
longs to a definite genus, called simply genus of functions
oj a? , x^y , . Xf^,
The order of the genus of any single one of the x's is n,
that of w, a?j + w, a?, + . . . + i*, a;,, the " Galois gen us, ^^ n I
This genus contains all others and therefore their orders are
all divisors of w ! If p be the order of a genus g and — be r,
r is the "number of permutations of the genus g,'^ i.e. the
number of permutations of the x^a for which any function of
the genus g remains unchanged.
A genus g is said to be a genus ** proper " if after it is ad-
joined to the domain of rationality tne equation
remains irreducible. When such a genus g is adjoined, so
that the domain of rationalitj[ becomes (/i, /i, . ./., g), the
algebraic character of a function defined by a:* — /i a;"~^ +
. . . ± /« = is changed, it falls into a special "class'' of
algebraic functions. All algebraic equations belong to the
same class which go over into each other by rational transfor-
mation and for which the functions of the roots belonging to
a definite genus g belong also to the given domain of ration-
ality {R\ ir . .)
This characteristic property of the class of an algebraic
eqiiation and the function which it defines may be called its
affect. An irreducible equation
a:" — Cix"-^ 4- . . . ± c„ =
whose coefficients belong to the domain (R', R", . . ) is said
therefore to have a special affect where there exists a special
function of its roots, which may be called the affect-genuB,
which likewise belongs to the given domain. The group of
permutations of this genus is called the Galois group of the
equation.
The affect-genus being g (xi, x,, . . xj), it is the system of
w 4- 1 equations
9 = ^0, fk = Cky (^ = 1,2,.. n)
which by Galois' principle takes the place of the single given
equation. This system is satisfied only by the r systems of
values
T
OF THE ALGEBBAIC EQUATION. 181
which correspond to the r permutations of the genus g. Its
order is therefore r and it constitutes the irreducible part of
the system A = Ct, (* = 1, 2, . . n) whose order is n I
The n I functions
afiix^ . . . a:J-^i (A^ = 0, 1, . . w — i ; * = 1, 2, . . n—1)
are the elements of a "fundamental system" of the Galois
genus ; but the number of elements can be reduced to p, the
order of the genus, if fractional numerical coeflBciente be
allowed.
If the discriminant of the genus x^, t,e.
n (rr< — X]^) {i, k = 1, 2, . . n ; i >< k),
»
be D, the discriminant of the Galois genus is />*•'. There-
fore, since the discriminant of everjr other j?enus is a divisor
of that of the Galois genus and D is irreducible, the discrim-
inant of every ^enus is a power of D. Prom this fact it follows
that for any given set of values of fi,ft, . . /n for which D
does not vanish, an infinite number of special functions of
each genus can be determined all of whose conjugates differ
from one another, and in terms of which every other function
of the same genus can be expressed rationally. Moreover
this theorem leads to a remarkably simple demonstration of
the "arithmetical existence ^^ of the roots of algebraic equa-
tions.
Upon the profound researches of the second part of the
OrundzUge we cannot now enter, though this contains the
heart of the arithmetical theory. Here, by aid of the ^^ Mo-
dul'Systeme'' and the principle of "association " the distinc-
tively "arithmeticar^ properties of the integral algebraic func-
tions are developed, their properties, namely, when consid-
ered with respect to their divisibility by other integral
functions of the same genus ; and the nnal step is taken in
the " reduction '^ of the domain of rationality, whereby the en-
tire theory of tho algebraic functions is reduced to a theory of
the integral functions of variables and indeterminates with
integral coefiBcients.
Thus Kronecker's theory completes that of Galois. For it
carries the general theory of equations back to a theory of
indeterminates, which, before Gfalois, it was always assumed
to be in the superficial and false sense, that the coefficients,
and therefore the roots, of any equation may be treated as
indeterminates.
The fine quality of Kronecker's work is even more notable
than its range or the importance of its results. It possesses
the rigor and elegance of the theory of numbers.
Early in the w'undziige,when defining an irreducible func-
182 KBOKEOKEB AKD HIS ABITHinnOAJi THBOBY
tion, Kroneoker remarks : '' Die Definition der Irredaotdbili-
tftt entbehrt so lange einer sicheren Gmndla^ als nicht eine
Methode angegeben ist, mittels deren bei einer bestimmten
Torgelegtcn Function entschieden werden kann, ob dieselbe der
anteoBtellten Definition gem&ss irreductibel ist oder nicht,''*
and proceeds therewith to supply the missing test
This criterion, according to which no definition may be
considered justified, no theorem established, until a method
is supplied for determining in every given concrete case
whether the definition or theorem actually applies or not, he
everywhere insisted upon, scrupulously meeting its require-
ments in his own work and sharply criticisin|B; lul failures to
meet them in the works of others. A definition which did
not stand this test he denominated the invention of a mere fic-
tion, an artificial abstraction for which there should be no
place in mathematics.
This is the rigor of the ancient Greek geometry — ^in re-
jecting hypothetical constructions Euclid reoogniied a sim-
ilar criterion — and though far enough from oeiiijg alvroys
realized in the modem analysis, must cbaractenxe every
mathematical theory in its finite form. For until it has been
attained, either the ultimate elements of the theory haye not
been reached or the artificial concepts with which it has aided
itself in its growth have not been set aside and the theory de-
duced directly from these elements.
Closely related to this fine conception of mathematical rigor
are the other salient traits of Eronecker's work.
It possesses that high artistic merit which consists in the
perfect adaptation of means to ends. His methods are al-
ways pure, fit, direct, and the simplest which the reauire-
ments of absolute rigor will allow. Writing to Dirichlet in
1856 he says of a method which he has discovered for deduc-
ing the properties of solvable equations of prime degree that
it meets all the proper requirements of simplicity and rigor,
^^ denu die Methoae verlangt keinen irgend hoheren Stund-
punkt mathematischen FossnngsvcrmoKens als das Problem
selbst, welches dadurch erledigt wird.^f And again for his
principle of *' association" he claims: **Sie gewahrt den
^ einf achsten ' erf orderlichen und hinreichenden Apparat, um
die arithmetischen Eigenschaften der allgemeinsten alge-
braischen grossen ^ volls&ndig ' und ^ auf die einfachste Weise '
darzulegen,"! adopting the phrases which are Quoted from the
first proposition of Eirchhoff^s Mechanics. Tnis ^' Einfach-
heity to be sure, is of a kind which it oftentimes requires
* OrundzUge, etc., p. 11.
+ Qdttinger ^aehrichUn, 1885, p. 864.
; OrundzUge, etc., p. 98.
OF THE ALGEBBAIO EQUATION. 183
mnch reflection to appreciate. He was a foe not only of arti-
ficial concepts but of all artificial methods and of all artificial
or purely formal tendencies in mathematics. He would have
rid matnematics of the artificial numbers and of its '* sym-
bolic " methods^ and the devising of new functions seemed
to him a foolish waste of energy. " God created numbers and
geometry/' I once heard him say, **but man the functions.''
It was bis boast that he was tne most practical of mathema-
ticians. He said whimsically to me one day last summer :
*' It is a pity that you, Americans, do not know me better. You
would surely appreciate me, I am so practical." And in a
somewhat transcendental sense of the word, to be sure, he was
profoundly practical. Ho sought to ayoid all mere abstrac-
tions and to ffiye his theories concrete form. Thus in the
Galois theory he replaced the abstraction, a group of substi-
tutions, by concrete functions which remain unchanged
for the substitutions of the group. Neither definition,
theorem, nor method had value in his eyes which could not
be applied to concrete cases, which could not be made to yield
concrete results. On this account he did not set great store
by the services of the theory of substitutions to algebra.
With all its beauty, he would urge, it is only formal, it does
not show how to construct the group of a given equation.
Kronecker influenced the mathematics thinking of Ger-
many as much through his lectures as through his published
writings. He was a very stimulating and interesting lecturer.
To an unusual degree he took his hearers into his confidence
and allowed them the privilege of watching the actual evolu-
tion of his thoughts. His lectures were not overprepared, but
the details of even important demonstrations were left to take
their chances in the lecture room. Occasionally there would
be a disastrous slip in the reckoning or argument, or the
outcome would be tne discovery that the theorem sought to be
established was false. But that only afforded opportunity to
see the marvellous quickness with which he would run an
error down and recover himself.
His lectures were always fresh. The principal courses were
on determinants, theory of numbers, algebra, and definite
integrals, and one of these in its turn he delivered each se-
mester. But he never merely repeated himself. If a lecture
did not differ from all its predecessors in content, it surely
did in point of view or method. It was always the most re-
cent product of his mathematical thinking.
In his lecturing, moreover, he avoided the excessive concise-
ness, which is the chief cause of the difficulty of his published
writings.
Personally, Kronecker was most charming and amiable, a
polished gentleman and man of the world. He was very gen-
184 MULTIPLICATION OF SERIES.
erons with his time and thoughts, loving to talk to an appre-
ciative listener of some favorite doctrine, or of the famous
mathematicians with whom he had been associated.
He was a man of rare genius, a mathematician of the first
rank in this century of great mathematicians.
Henry B. Pine.
PsiNCETON College, April 20, 1892.
MULTIPLICATION OF SERIES.
BY FBOF. FLORIAN CAJORI.
The salient feature of the new era which analysis entered
upon during the first quarter of this century is vividly illus-
trated in the history of infinite series. Extending from that
time back to Newton we have a formal period which gave
rise to general theorems, the validity of which was not
thoroughly tested. Thus, in series, there were put forth
during that epoch the binomial theorem, the theorems of
Taylor, Maclaurin, John Bernoulli, and Lagrange. Infinite
series were used by Newton, Leibnitz, and Euler in the study
of transcendental functions. As a rule, the convergency of
expressions was not ascertained, and the confusion which
prevailed in the theory of series gave rise to curious para-
doxes. But with the advent of Gauss, Cauchy, and Abel,
began the new era which combined dexterity in form with
rigor of demonstration.
In the multiplication of series, mathematicians of the ear-
lier period considered simply the form of the products and
hardly ever thoiight of inquiring further into the validity of
the operation. Reliable tests for convergency were unknown.
The product of any two infinite series was accepted with
nearly the same degree of confidence as was the product of
finite expressions. Thus, De Moivre * extended the binomial
formula to infinite series and deduced the following formula :
{az + bz^ + . . .)"»
1 1 4i
This was accepted as true without any limitations what-
ever.
* A method of mising an infinite multinomial to any given power, or
extracting any given root of the same. PhUowphiccU Transactiondt
No. 2JJ0, 1697.
HULTIPLIOATION OF SERIES. 185
The first to cry " halt ** to these reckless proceedings was
Baron Cauchj. fie instituted for the first time a painstak-
ing examination of the principles of series and strove to intro-
duce absolute rigor. He is the founder of the theory of
convergency and divergency. He pointed out that if two
series are convergent, their product is not necessarily so.
Thus,
V2 a/3 V4
is convergent, but its square
1 ^ /*^ 1\/^ ^ \
= +
is divergent. Not only did he discriminate between conver-
gent and divergent series, but also between what we now call
** absolutely convergent series " which are convergent even
if all the terms are made positive, and *' semi-convergent
series '' which cease to be convergent when the terms are all
made to have like signs. In his Cours d^analyse algSbraique
(1821) Cauchy proved rigorously the following celebrated
theorem : If 2u^ and 2^v^ converge absolutely to values
U and V respectively , then the series 2{uoV^ + ^x^n-, +
. . . + UnVo) converges to the value TJY, So far as the
researches of Cauchy went, two absolutely convergent series
appeared to be the only ones which could be multiplied by
one another with absolute safety. This same theorem was
proved also by Abel in course of his demonstration of the
binomial formula,* but in the same article he took a giant
step in advance by establishing the following theorem : If the
series 2u^ and ^v^ converge to the limits U and V respec'
tively, then if the series 2(uoV^ + w,^«-i + . . . + w„Vo) be
convergent, it will converge to the product UV. The beauty
of this theorem lies in the fact that all three series in ques-
tion may be semi-convergent. Strange to say, this result, so
remarkable for its simplicity and generality and put forth by
so prominent a mathematician as Abel, was for nearly half a
century almost universally overlooked. Schlomilch's Cbm-
pendium der hoheren A7iah/sis knows it not, nor does Ber-
trand's Traite de calcul diffireyitiel,
AbeFs theorem would dispose of the whole problem of
multiplication of series, if we had a universal practical criterion
of convergency for semi-convergent series. Since we do not
* Crelle's JoumcUy Bd. 1, 1827 ; also CEuvret competes de N, H, Abel,
Tome 1, p. 66 et eeq.
186 uxrurrmcAxiov of SHEns.
possess snoh a criterion, theorems have been established
which remove in certain cases the necessity of applying tests
of convemncy to the prodnct-series. Snch a l^heorem is
that of Mortens * who in 1875 demonstrated that Oanchy's
theorem still holds tme if, of the two conyergent series to
be multiplied together, only one is absolately convergent.
Thus, if the absomtely convergent series
l + J + i+i+. . .
is multiplied by the semi-convergent series
the product will surely converge to the value 2 log 3. A still
more comprehensive but more complex theorem was given by
Mr. A. Pringsheim, in 1882 if If U =: 2ru^,r - Spv^ be
convergent series of which one^ say U^ hoe the ^operiy that
its terms, arranaed in certain groups containing always a
finite number of terms, constitute an absolutely convergent
series, i.e. thai-
be absolutely convergent, then we have
UV= JSy Wp ^ W, where Wp = 2x UxVp^xi
provided that the series 2^ UyVy be absolutely convergent and
remain so when any number of factors Uy, tv is replaced by
other factors of higher indices. That is, in any number of
terms, UyVy, the factors Vp (or Up) may be erased and any
other factors v,, + „ (or Up + ») put in tneir places, with the
single restriction that none of the indices be repeated in
the series. Whenever applicable, the above theorem excels
Gauchy's in this, that the often difficult determination of the
convergency of the product-series is replaced by the easier
determination of the absolute convergency of 2p v^v,. In
illustration of this theorem I give the following example.
Let
- (_j^ 1_ 1 ^ 1 )
(4K + 1 4v + 2"^V4v + 3 V4»'+4)
* Crelle's Journal, Bd. 79. Proofs of this theorem and of AbeVs theo-
rem will be found in Chrystal's Algebra, Part II., p. 127 and p. ISIS.
f Matlhematische Annalen, Bd. 21, p. 827.
MULTIPLICATION OF SERIES, 187
This semi-convergent series becomes absolntely convergent
when its terms are grouped thus,
jj^^ j_i _j_) , ^^\ i J I
The series
. (__1 1 1 1 )
^=f''j(4v + l)l (4.' + 2)1 + 4FT3 ~ i7T4j
is semi-convergent. Each fraction in the first series stands for
a term Uy and each fraction in the second for a term Vp. Hence
7 7 ( (4r + l)f ^ (4v + 2)« ' (4v + 3)»
"^(4v + 4)1 j'
which is absolutely convergent and remains so if any number
of terms, Up or v^, be replaced by others occurring later in
the series. Hence the product, W, of the two series con-
verges toward UV.
The importance of inquiring whether ^^WrVi* remains ab-
solutely convergent after the substitution of higher terms in
f)lace of the lower, is brought out by Pringsheim in the f cl-
ewing example. Take the series U, given above, and
r-~_vJ ^ - ^ .^> l_l
\ V4v 4-1 v'4r + 24v + 3 4r-h4)'
In this case
2^ UyVp = 2v
- -■"" - („ + 1) I
which is absolutely convergent, while
" * ( 1 1
-f '"'"'*' = . i(4v + l)(4v + 3) + (4k + 2) (4k + 4)
^ , „!
V(4v + 3) (4v + 6) ^ \/74^
+ 4) (4k + 6))
is divergent I It is indeed found that in this case the prod-
uct UV^ cannot be represented by the series W.
In proving his theorem, Pringsheim shows in the first place
that an obviously necessary condition for the convergency of
If, namely ^'ifi, Wv = 0, is satisfied. His theorem, like that
of Oauchy and of Mertens, offers sufficient conditions for the
188 MITLTIPLICATION OP SEEIB8.
applicability of the rule of multiplication, but they are not at
the same time necessary conditions. He shows that Cauchy's
and Mertens' theorems are included in his own.
Pringsheim then considers the multiplication and con-
yergency of sp^ecial classeB of semi-conver^nt series, of which
we shall mention one. He shows that if U and V are conyer-
gent series and one of them, say U^ is so constituted that
is absolutely convergent (as is the case when the terms Uw
never increase and have alternating signs), then
/f"„ «;. =
is a necessary and sufficient condition for the convergency of
W. Mr. A. Voss* has treated similarly the more general
case when the series U, expressed in the form
Cr= (tto + Wi) + (t*, + «,) + •• •
is absolutely conver^nt and has shown that in this case the
necessary and sufficient condition for the convergency of W
lies in the two following relations :
n =*"« (WoVi- + UtVt^i + . . . + W„Vo) = ;
n^^^(UiVt^^ + UtV^^ + . . . 4- W,»_,V,) = 0.
Mr. Pringsheim reaches the following interesting conclu-
sions : The product of two semi-convergent series can never
converge absolutely, but a semi-convergent series, or even a
divergent series, multiplied by an absolutely convergent
series may yield an absolutely convergent product. Thus, the
product of the ubsolutely convergent series
and the serai -convergent series
log 2 = ^ — J + I — . . .is
2 (log 2)' = 1 + 4)(-^)^- (. + 2)V + 3) frT^[>
which series converges absolutely. Again, the absolutely con-
vergent series
-1 4- -1- -i —
■^ 1.2 "^ 2.3 "^ 3.4 +
• • •
* Math. Annalen, Bd. 24, p. 42.
OK EXACT AKALYSIS AS THE BASIS OF LAKOUAGE. 189
mnltiplied by the divergent series
1 + J + i -h . . .
gives an absolntelj convergent product. The strangeness of
this last conclusion is removea when we consider that the
series
-1-11-
1*2 2^ •
= - 1 + (1 - i) + a - i) + . . . = 0.
Since one of the factor-series is zero, we may well have a
product-series with a definite limiting value. This value in
this case is itself zero^ as is seen from the following expression
for the productHseries
- "1
W= — Cx -¥ 2y (cy — Ck+i), where c •= 2x —. — — ; r.
I 1 a? (v 4- 1 — a:)
Colorado Colleos, March 28, 1892.
ON EXACT ANALYSIS AS THE BASIS OF LAN-
GUAGE.*
BT A. MACFAHLANG, 80. D., LL.D.
Abstract.
A SCHEME for an artificial langnage was published in the
Philosophical Transactions of the Royal Society for 1668 by
Bishop Wilkins. Since, however, it presupposes a complete
enumeration of all that is or can be known, it would bo over-
thrown by every considerable advance in knowledge. The
mathematician and philosopher Leibnitz devoted much
thought to what he called a specieuse gSnSrale, which he
hoped would be an aid in reasoning and invention ; but he
died without publishing even an outline of his system. The
new universal language Volapuk, invented by J. M. Schleyer
of Constance, is built upon a purely linguistic basis, bemg
derived from a comparative study of the chief natural lan-
guages. In this paper it is proposed to show that the proper
and necessary basis for an artificial language is scientific
analysis and classification, and two specimens of language
* Abstract of a paper preseDted to the Society at the meetiDe of
March 5, 1892.
190 car xxAci akaltbib as thb babd ov uurairAeB.
80 oonstnicted will exhibit the gmt eomjifeiBtj «f the
pfobleni.
In the notation for nnmben in Volapilk we obaerre aeriooB
defects. As regards the d^ts there u no word to expreoB 0.
Ab rupurds the expreasiona for the denominationii, an tfbitraiy
nae of the nfRx tor the plnnd denotes the denomination <0f» :
thus we have id, two ; Ms, twenty ; and the other names for
the denominations are no more systematio than the En^^ish
words. There is the nsnal jamp from thousand to million ;
we are not told whether telion means thousand million or
million million ; and no words are provided to express frso-
tional denominations. In physical works we meet with the
highest development of the notation for nnmher ; it consists
of a series of significant figores, and of a positive or nq;ative
power of ten. To vocalize this notation we reqoire an ele-
mentary word for each of the elementaiy numbers, 0, 1, 2, 8,
^ 6, a, 7, 8, 9 ; and a series of words for t^ integer powers
of ten, and for the fractional powers of ten. As there are
five elementary vowels, ten woras for the digits may be ob-
tained by prefixing the consonants b and Z.
Thus 0, 1, 2, 8, 4, 5, 6, 7, 8, 9,
ba, be, bi, bo, bu, la, h, U, lo, lu.
The word for a higher number is formed by taking the
appropriate monosyllables in succession ; for example : 11,
beoe; 23, bibo; 105, bebiUa, The integer denominations
may be expressed by affixing p to the number for the place or
power of ten, while the fractional denominations may be
expressed by adding n instead of p, thus : —
10, 10', lOS 10*, 10*, 10', 10% 10', etc.
bep, hip, bop, bup, lap, lep, lip, lop, etc. and
1 1 1 1 1 J_ i.
10' 10" 10'^ 10*' 10*' 10«' 10^'
betiy bin, bon, bun, Ian, len, lin, etc
For example, one hundred and twenty- three thousand
would be Yocalized by bebipo bop, and forty-five hundredths
by bula bin.
Some years ago in a series of papers on '^ An analysis of
the relationships of consanguinity and affinity,'' * the author
devised a system of notation both literal and graphical, and
indicated u corresponding nomenclature. On this analysis
may be oonstructea another specimen of a scientific language,
ana by the system of words it provides for such relationships
the efficiency of Volapiik may be tested.
♦ Proc, Eov. Soe. Edinb.,Yol X., p. 224; Vol. XI., pp. 5 and 162; PhiL
Mag. June 1^1 ; and Journal of the Anthrop. lost, of London for 1883.
ON EXACT ANALYSIS AS THE BASIS OF LANGUAGE. 191
Let a denote the relationship of parent and e the reciprocal
relationship of child ; by forming the different permutations
of ^hese letters we get expressions for the several compound
relationships. Those of the second order are : —
NOTATION.
GENERAL MEANINQ.
IBREDUCIBLB MEANING.
aa
ae
ea
ee
parent of parent,
parent of child,
child of parent,
child of child.
grandparent,
consort.
brother or sister,
grandchild.
The meaning given in the third column may not coincide
exactly with that riven in second ; where a reduction of the
expression is possible, that is, where a is followed by ^ or e by
a ; the special or reduced meanm^ is excluded. Thus ae and
ea each m its most general meaning includes self; when the
special meaning of self is excluded, the parent of child be-
comes consort, and the child of parent oecomes brother or
sister.
Similarly the relationships of the third order are :
NOTATION.
GENERAL MEANING.
nUlEDUCIBLB MEANING.
aaa
•
great grandparent.
great grandparent.
aae
grandparent of child.
parent-in-law.
aea
aee
parent of child of parent,
parent of grandchild.
step-parent,
chud-in-law.
eaa
child of grandparent,
child of Darent of child,
grandchild of parent,
great grandchild.
uncle or aunt.
eae
step-child.
eea
nephew or niece.
eee
great grandchild.
In the case of all these relationships, excepting the first and
the last, the general meaning includes a simpler relationship
to which it may reduce ; for example, grandparent of child
includes the simpler relationship of parent. In the same
manner the relationships expressed by four, five or any num-
ber of elements may be exhibited.
To change this notation into a nomenclature, all that is
necessary is to insert some consonant as d between the vow-
els ; for then each combination can be easily pronounced. In
the systematic language so derived ada means grandparent,
ade consort, eda brother or sister, ede grandchild, adadd great
grandparent, adade parent-in-law, adma step-parent, and so
on.
192 OK KXAOT AKALY8I8 AS THB BA8IB OV LAKOUAQB.
Each genuB of relationship is divided into species by intro-
ducing the distinction of sex. Let the consonants m and /
denote mde and female respectiYely, then the species of the
first order are ma father, /a mother, me son, /a aaaghter. If
we introduce the distinction of sex after the yowel we obtain
such relationships as mam father of man, maf father of
woman, mef son. of woman. The species of the second order,
obtained by introducing the distinction of sex before the first
Yowel only, are, e.a. ; mada grandfather, feda sister, fede
granddaughter. If the distinction of sex is introduced
before the second Towel also, we may obtain : mama paternal
SEindfather, mame father of son, fema sister-german, fe/e
ughter of daughter, etc. Thirty-two species may be formed
by introducing the distinction of sex after the last yowel, but
four of these species reduce necessarily to the relationshin of
self; for example mamem. The double relationship inYoiyed
in full brother may be denoted by mem/a, that of full sister
hj fern fa, and that of full brother or sister by em/a. If, on
tne otner hand, we wish to express that the brotherBhip is
only half, we may replace d hj t; thus meia, half-brother ;
feta, half -sister ; and eta^ half brother or sister. These prin-
ciples suffice to supply a word for every possible relationship
of consanguinity or affinity. The nomenclature is based on
a notation which serves as the basis for a calculus,* and it
seems to me that this is a developed specimen of the kind of
language which Leibnitz had in nis thoughts.
If wo test Volapuk by the vocabulary which it provides for
these relationships wo find that the words supplied are not
founded on a scieutilic analysis, and indeed are far inferior to
the terms supplied by the English language. Almost all the
stem words, as son, son. Mod brother, involve the masculine
gender, the corresponding feminines being formed by prefix-
ing ji. Thus daughter is expressed by ji-son and sister by
ji'btod. There are no words to express the relationships
which are independent of sex. The confusion on the subject
of the more involved relationships is very great, no distinction
being made for example, between step-brother and half-
brother, both of which are denoted by lafa-hlod. The derived
relationships are not expressed by general rules for combining
the elementary relationships, but on the contrary a few words
are obtained in an arbitrary manner by attaching to the stems
comparatively meaningless prefixes and affixes. It has been
pointed out by several scholars,! that the inventor of Vola-
♦ Problems in Relationship. Proe, Roy. Sac. Edinh., 1888.
fDr. D. Q. Brinton— " Aims and Traits of a World-language ; " Dr.
Horatio Hale— '* An International Language," Proc. A. X A. 8., VoL
XXXVII.
KOTES. 193
puk makes a fundamental error in proceeding synthetically
instead of analytically^ and in this matter of terms for rela-
tionship we have an example of that fundamental mistake.
NOTES.
A REOULAB meeting of the New Yobk Mathehatioal
Society was held Saturday afternoon^ '^Ef^' ^^ ^^ half-past
three o'clock, the president in the chair. The following per-
sons having been only nominated, and being recommended by
the council, were elected to membership : Mr. B. S. Annis,
Johns Hopkins University ; Professor Samuel Marx Barton,
Emory and Henry College ; Dr. Maxime B6cber, Harvard
Universitv ; Mr. William H.Butts, Pontiac, Michigan ; Dr.T.
Proctor flail, Clark University; Professor S, W. Hunton, Mount
Allison University; Mr. W. F. King, Ottawa, Canada; Mr. B. M.
Boszel, Johns Hopkins University ; Dr. Arthur Schultze, New
York. The proposed amendment to the Constitution (Bulletin,
No. 6, p. 142.) was unanimously adopted, and the By-Laws were
amended by striking out section 2 of bv-law ix., and altering
the number of the fcllowing section, i^he following origin^
papers were read : ''The cubic-projection and rotation of a
tessaract," by Dr. T. Proctor Hall ; " On final formulas for
the algebraic solution of quartic equations,^' by Professor
Mansfield Merriman.
A tessaract is a geometrical figure generated by the mo-
tion of a cube in the direction of the common perpendicular
to its edges and faces, bearing exactly the same relation to a
cube that a cube bears to a square. It is bounded by ei^ht
cubes, and has twenty-four faces, thirty-two edges, and six-
teen vertices. Dr. Mall presented the Society with a wire
model representing the projection of a tessaract into space of
three dimensions.
The Cambridge University Press has in preparation "A
treatise on the mathematical theory of elasticity,^ by A. E. H.
Love, fellow of St. John's College, Cambridge. The first
volume of the work, which is to be in two volumes, is in
press.
Maohillak & Co. have nearly ready a work on the *' The-
ory of f auctions,'' by Professor Morley of Haverford College,
Pa., and Professor Harkness of Bryn Mawr College, Pa.
At the meeting of the Acadimie dee Sciences at Paris on
18
194 KOTES. .
March 7, committees were appointed to award the mathe-
. matical prizes of the cnrrent year. For the Orandprix de»
sciences mathimeUiques, The aetermination of the nnmber
of primes inferior to a given limits the committee is composed
of MM. Jordan, Poincar6, Hermite, Darboux, Picard. For
the Prix Bordin, The application of the general theory of
abelian functions to geometry, the committee consists of
MM. Hermite, Poincarl, Darboux, Jordan, Picard. For the
Prix Bordin of 1890, To study the surfaces whose linear ele-
ment can be reduced to the form
de' = [/ {u) - cp (v)] {du* + dv%
the time of competition for which was extended until 1892,
the committee is MM. Poincarl, Darboux, Picard, Hermite,
Jordan.
The memoir of M. PainleT6, which won for its author the
Grand prix of 1890, To perfect in an important respect
the theory of the differential equation of the first order and
first degree, has just been published in full in the AnnaUs de
V£cole Normale, while that of his competitor, M. Autonne,
which was awarded an honorable mention, is in course of pub-
lication in the Journal de VJScole Polytechnique,
We learn from Naturae Novitates that Professor H. A.
Schwartz, of Gottingen, has been called to Berlin as the suc-
cessor of the late Professor Kronecker, and that Professor
Rudolph Sturm has been invited to the professorship of
mathematics at the University of Breslau.
The second number of the current volume of the American
Journal of Mathematics was delayed through the occurrence
of a fire in the printing oflSce. In future the new volumes
will begin in January instead of in October. T. s. F.
At Johns Hopkins University during the academic year
1892-93, the following graduate courses in mathematics will
be given : by Professor Craig, (1) Theory of functions of one
and two variables, (2) Mathematical seminary, (3) Partial dif-
ferential equations, (4) Linear differential equations, (5) El-
liptic and abelian functions ; by Dr. Franklin, (6) A general
course for graduate students on the elements of modern math-
ematics, (7l Theory of invariants, (8) Metrical theory of sur-
faces ; by Dr. Chapman, (9) Mechanics and hydrodynamics,
(10) Projective geometry, (11) History of mathematics.
T. C.
KEW PUBLIOATIOKS. 195
During the coming year at Clark UniTersity, Professor
Story will lecture on the following subjects : (1) History of
algebra during the Benaissance> (2) Aavanced course on the
geometry of surfaces and twisted curves^ (3) Applications of
?uaternions, (4) Hyperspace and non-eucliaean geometry, (5)
ntroductory courses on calculus of finite differences, proba-
bility, and theory of errors. Dr. Webster will lecture on
Theory of functions according to Gauchy, Biemann, and
Weierstrass^ with applications lo functions defined by certain
differential equations. Besides, introductory courses will be
given in : Theory of numbers, Modem higher algebra, Higher
plane curves. General theory of surfaces and twisted curves,
Quarternions, and Modem synthetic geometry. 0. B.
NEW PUBLICATIONS.
COMPILED BY B. WESTERMANK ft CO., NEW YORK.
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daires des s^nces de I'Acad^mie ded Sciences, suite. Paris 1893.
gr. in-4. 446 pg. Prix de souscription, 2ef
Deter (C. G. J.). Repertorium der Differential- und Integralrechnung.
2. Auflage. Berlin 1892. 8. 118 pg. m. Figuren. M. 1.50
Faraday. — Jerrold (W.). Michael Faraday, the man of science. New
York 1892. 8. 100 pg. yloth. $0.75
Flamharion (C). La Plurality des Mondes habits. £tude oh I'on
expose les conditions d*habitabilit^ des terres celestes, discut^es an
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Lange (G.). Ueber die linearen homoeenen Differentialgleichungen,
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Fall, dass die IrrationaHt&t 8. Grades ist Halle 1892. 8. 45 pg.
M. 1.20
196 HXW FUBUOAnOVB.
IiATn (K.). BdMge tor Bwitimiining mid VanrBrthiiitt dor Ikm^
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i
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TOPOLOGY OF ALGEBRAIC CURVES. 197
TOPOLOGY OP ALGEBEAIC CUEVES.
In the Mathematische Annalen, Vol. 38 (1891), Mr. David
Hilbert of Konigsbcrg has a very interesting and suggestive
article on the real branches of algebraic curves. The simplic-
ity of the method which Mr. Hilbert employs, and the possi-
bility of its being made to yield further important results
seem sufficient reasons for presenting here, lu some detail,
that portion of the article which treats of plane curves. It
has seemed to the present writer advisable to amplify por-
tions of Mr. Hilbert^s article, with the view of making his
method more intelli^ble, and also to make some changes in
the proof of the principal theorem, in order to avoid some
slight inaccuracies that have crept into his demonstration.
The first part of the article m question is devoted to the
determination of the maximum number of nested branches
Sossible to a plane algebraic curve of order n, and of maximum
eficiency. By 7iested branches is meant a group) of even
branches so arranged that the first lies entirely within the
second, the second within the third, and so on, like a series
of concentric circles.* It should be observed that some or all
of the non-nested branches may, in perfect accord with this
definition, lie within the ring-shaped regions formed by the
nested branches. A single even branch, which neither en-
closes another branch nor is enclosed by one, may be looked
upon as a nested branch or not, according to the nature of the
question under discussion. For reasons that will presently
appear, Hilbert does not consider the even branches of the
conic and cubic as nested. Hilbert bases some of his inves-
tigations upon results previously obtained by A. narnack,f
and his method is entirely analogous to that of the latter.
Hamack had proved, in the article referred to, that a plane
algebraic curve, without singularities, of order n and of defi-
ciency p^ can not have more than p + 1, that is, J (n—\)
(»— 2) + 1 real branches ; and, further, that, for every posi-
tive integral value of n, a non-singular curve with J (n — 1)
(n — 2) + 1 real branches actually exists. Setting out from
this result of Hamack's, Hilbert shows first that a non-
singular curve can have no more than ^ (w.— 2) or ^ (n— 3)
nested branches, according as n is even or odd ; for, if it had
more, a right line could be drawn meeting the curve in more
♦ This definition is not scientific but it serves the present purpose. To
make it rigorous Mr. Hilbert needs only to define accurately what is
meant by inside and outside of a closed branch. Such a definition has
virtually been given by von Staudt, Chowietrie der Lage, § 1, 16.
f Maihematische AnnaUn, Bd. 10, Ueber die VieltheUigkeit der ebenen
cdgebraischen Curvetk
108 TOPOLOGY OF ALOEBSAIO OUBTBB.
than n points.* He proyes fnrther the following theorein :
Ibr every positive integral palue of n, a mm^ngwar ewrve of
order n exvete, having the maximum number of real tranches,
i (n— 1) (n— 18) + 1, and | («-2) or | (n— 8) nested brandkes,
aceordiftg as n is even or odd.
We shall, for sake of brevity, desijniate an even branch by
the term **oval" It is evident that all nested branchea
are oyals. Moreover^ we consider that case only where all the
nested ovals are ffronped in a single nest. We first assame
the theorem true for a curve of n^ order, O^, whose eqnation
may be written/ = 0, and we assume further that an ellipse,
E^, whose equation we write A = 0, can be constructed en-
closing one or more of the nested ovals, and cutting a non-
nested oval, b, in 2n points, whose order of succession shall be
the same upon b as upon jSL. It is evident that B^ and 0»
have no other common point. The ellipse E^ and the branch
b form, by their intersections, in renons, each completely
bounded by a single segment of JB^ and a single segment of b.
Within one of these regions there exists one or more nested
ovals. Whether this region, which we call R, contains the
nested ovals interior to Ja^ or exterior to it,t depends upon the
nature of b, and its position with respect to ^,. (When JB^
encloses all the nested ovals, it may occur that none of theae
3n regions contains a nested oval ; in that case one of these
regions will be all the plane exterior to E^ and b, and this we
desi^ate by R.) Let s be any segment of J?, determined by
the intersections of E^ and b, except that segment which forms
a portion of the boundary of R, Upon s we choose 2(n + 2)
points, none of them coincident witn the extremities oi s, ana
join by right lines the first and second, the third and fourth,
, and the (2n + 3)th and (2w + 4)th. Let the product
of the equations of these n + 2 right lines be 7 = 0. Then
for very small values of (J,
F=fh ±61 =
is the eqnation of a curve, C, + j, of order n + 2, lying very
near the degenerate curve fh = 0. This (7,+, passes through
the points common to C^ and the right lines, and through
the points common to E^ and these lines, but not through the
intersections of E^ with U^.
♦ This theorem is not true for curves of order lower than the fourth.
Moreover, it must be borne in mind that every non-singular curve with
the maximum number of real branches has at least one non-nested oval,
because i (n— 2) and i (n— 8) are each less than i (n— 1) (n— 2) + 1.
f A nested oval exterior to JS,, since it encloses those interior to j^s,
must also enclose E^ itself. Therefore, when, among a number of iso-
lated ovals, we have to consider a single one as nested, we choose as such,
one that lies in the interior of Et.
TOPOLOGY OP ALGEBRAIC CURVES. 199
We proceed now to prove :
(1) that Cn+thasp' + 1 real branches, p' being the defi-
ciency of (?,+, ;
(2) that C,+a has the maximum nnmber of nested
branches ; and
(3) that the ellipse, E^, encloses one or more of the nested
ovals of Gn+tf awd cuts one of its non -nested ovals in 2(w + 2)
points, whose order of succession upon C.+t is the same as
upon J?,.
1. Ignoring the branch b for the moment, it appears, from
the form of the equation -P = 0, that in the immediate vicin-
itv of every other branch of C„, there exists a similur branch
of Cn+u The (7, has by hypothesis i{n — l)(n — 2) real
branches, exclusive of b. These give rise, tnerefore, to
i{n — l)(n — 2) real branches of 0^ + %. Furthermore, under
proper choice of the sign of d, there exists, in the vicinity of
the complete boundary of each of the 2n regions formed by
j^,and b, an oval of C,^.,. The latter curve has no real branch
save those already enumerated. Therefore C^^., has ^{n — 1)
(n — 2) -f 2n = \{n + l)n -h 1 = p' -h 1 real branches,
2. Each of the nested ovals of (7, gives rise to a nested oval of
Gn + fl. Moreover, the oval of 67, + , engendered by the boundary
of R is itself a nested oval of O^ + ,. The latter has, therefore,
one more nested oval than does C,. Since increasing n by
2, increases the functions i(/i— 2) and i(n— 3) by 1, it follows
that (7« + 1 has the maximum number of nested branches.
3. In the vicinity of that region, a portion of whose boun-
dary is 8y there exists an oval of C, + , which cuts the ellipse in
the 2(w + 2) points already determined upon s, and the order
of succession of these 2(n + 2) points is the same upon C, + ,
as upon s.
Hence, if our assumptions concerning (7, and B^ are valid,
the curve (7„ + , has the maximum number of real branches, and
also the maximum number of nested branches. And further-
more — and this is a very important point — the ellipse B^ has
the same position with respect to C^ + , that it was assumed to
have with respect to G^. It follows, then, that we may in like
manner derive from the C^ + 1 a 0, + 4 having the same proper-
ties, and so on. If, then, we can prove our assumption valid
for one even value, and for one oda value of n, we may con-
clude that our theorem is true for all values of n.
That these assumptions are valid whenn = 4 can be demon-
strated as follows : Let / = be the equation of a given ellipse
G,, and h = that of the auxiliary ellipse B^. Let B^ intersect
C, in 4 real points ; and upon any segment, s, of B^ deter-
mined by two successive points of intersection, choose the 8
successive points, 1, 2, 3, . . . 8. Join by right lines, 1 with 2,
3 with 4, . • . , and 7 with 8. Let the product of the eqna-
200 lOFOLOOT OF ALGEBRAIC OUBYXS.
tions of these 4 right lines be Z = 0. Then, for very small Tal-
lies of Sj
fh±dl =
represents a non-siDgakrqnartic, C^^ and, by proper choice of
the sign of d, this qnartic has four ovals, one of which inter-
sects B^ in the eight points npon 8. Horeover, within JB^
there lie one or two oTfus of C^ one if 8 is exterior to C^ and
two if 8 is within (7,. Now a qnartic can haye no more than
^(4 — 2) = 1 nested oval. We choose as snch, an oval in the
interior of E^. We have then a C^ with the nuudmnm num-
ber of real branches, viz., J(4 — 1^(4 — 2) + 1 = 4 ; with the
maximum number of nested oyals, 1 ; and the ellipse jSL en-
closes this nested branch, and cuts a non-nested oval in 2(2 +
2) = 8 real points, whose order of succession upon C. is the
some as upon E^. Hence our ti88umptian m vtilia wi^n
« = 4.
That this is true also when n = 5 is similarly proved. Let
/= represent a straight line. Draw the ellipse, E^^ not cut-
tinff / = in any real point. Upon J?, choose six points and,
as before, join alternate pairs by right lines. Let the prod-
uct of the equations of these three right lines be Z = 0.
Then, when 6 is very small,
fh± 61 =
represents a non-singular cubic, C„ the oval of which inter*
sects E^ in the six points whose order of succession upon E,
and the oval is the same. Proceeding one step further, let
the equation of C^ be/ = 0. Upon any segment of E^ choose
2(3 + 2 ) = 10 points, and join alternate pairs by right lines,
the proauct of whose five equations is Z = 0. Tien, for suffi-
ciently small values of d,
fh± 6l=z0
represents a non-singular quintic, C^, and, upon proper choice
of the sign of d, this C^ has six ovals, one of which intersects E^
in ten points. Within E, lie two ovals of C^, one of which we
consider a nested oval. Moreover, C^hna an odd branch in
the vicinity of the odd branch of C,. We have then a quintic
with the maximum number of real branches, ^(5 — 1)(5 —
2) + 1 = 7 ; with the maximum number of nested branches,
i(5 — 3) = 1 ; and with a non-nested oval cut by E in 2(3 -f-
2) = 10 real points ; E^ also encloses the nestea branch.
Hence, our assumptions are valid when » = 5. The theorem
is therefore true in general,
Headers of Hilbert's article in the Annalen will notice some
TOPOLOGY OF ALOEBBAIO GUBYES. 201
minor errors in his proof. He states, for instance, that the
auxiliary ellipse may lie wholly within the innermost nested
oval ^see Annahn, vol. 38, p. 117). This is impossible, for
the ellipse could not then be made to intersect a non-nested
oval. Again, he allows the ellipse to cut any of the non-
nested branches. If the ellipse be drawn to enclose all the
nested branches and to intersect in 2n points an odd branch,
the derived C« + , will have indeed the maximum number of
real branches, but one fewer than the maximum number of
nested branches. And, lastly, Hilbert chooses the 2{n + 2)
points of B^y through which the lines / = are to pass, upon
any segment of B^. If, however, these betaken upon that seg-
ment of E^ which forms part of the boundary of A, the branch
of C, ^. , wnich has these points in common with B^ will be a
nested oval, and, though the C, ^. , will then have /? + 1 real
branches, and the maximum number of nested ovals as re-
quired, it will be impossible to carry the process further.
It will be observed that Hilbert's results apply only to
curves of maximum deficiency, and of the maximum number
of real branches, n being given. It by no means follows that
a curve of order n and of maximum deficiency, but with
fewer than the maximum number of real branches, cannot
have more than ^ (n — 2) or ^ f n — 3) nested branches. For
instance, in the case of the cnoic discussed above, if <^ be
given the opposite sign to the one there chosen, the equation
fh ± 61 =
will represent a non-singular quintic, having but three real
branches, two of which are nested.
And, in general, it is easily seen that a non-singular curve
of even order, and possessing but ^n real branches, may have
them all nested. Similarly, a curve of odd order having
only ^{n + l) real branches, may have ^(»— 1) of them nested.
Hilbert leaves untouched also the case of singular curves,
and thus excludes from his investigations a large class of
curves. It would be interesting to Know under what con-
ditions^ and in what way, the branches of a singular curve
can be nested.
Lack of space prevents any discussion of the second part of
Hilbert's article, in which the author determines some of the
properties of curves in three-fold space. I give only the re-
sult^ of these investigations. By a method entirely analogous
to that presented above, Hilbert proves the theorem : An
irreducible twisted curve of order », with the maximum num»
ber of real branches [J {n — 1)' + 1 when n is even, and J
(n — 1) (/J — 3) + 1 when n is odd] can have no more than
iv — 2, 2v — I, 2v — 1 odd branches, according as n = 4k,
202 FINAL FOBMULAS FOB THB
4r +1, 4k 4- 3. When n = 4v + 2, no odd branch can
exist. Exceptional are the cases when n = 3, ^, 5, the maxi"
mum number of odd branches being 1, 2, 3, respectively. Then,
by applying Abers theorem for elliptic functions, be proves,
for every value of n, the existence of curves with the maximum
number of real odd branches.
L. S. HULBUBT.
WoacESTER, Mass., AprU 6, 1892.
FINAL FORMULAS FOE THE ALGEBRAIC
SOLUTION OF QUARTIG EQUATIONS.*
BT MANSFIELD MERBIMAN, PH.D.
I. Final formulas for the algebraic solution of quadratic
and cubic equations are well known. Such formulas exhibit
the roots in their true typical forms, and lead to ready and
exact numerical solutions whenever the given equations do
not fall under the irreducible case. But for the quartic, or
biquadratic, equation the books on algebra do not give similar
final formulas. The solution of the quartic has been known
since 1540, and numerous methods have been deduced for its
algebraic resolution, yet in no caso does this appear to have
been completed in final practical shape. It is the object of
this paper to state the fiual solution in the form of definite
formulas.
II. The expression of the roots of the quartic is easily made
in terms of the roots of a resolvent cubic, and the cubic itself
is solved without difficulty. Yet great practical difficulty
exists in treating a numerical equation on account of the
presence of imaginaries in the roots of the resolvent. Wit-
ness the following example which is generally given to illus-
trate the method in connection with Euler's resolvent :
*' Let it be required to determine the roots of the biquad-
ratic equation,
X* - 25a;' + 60a: - 36 = 0.
By comparing this with the general form the cubic equation
to be resolved is,
y* - 50y3 ^ 729y - 3600 =
* Abstract of a paper presented to the Society at the meeting of May
7, 1892.
ALGEBBAIO SOLUTIOK OF QUABTIO EQUATIONS. !^
the roots of which are founds by the rules for cubics, to be 9,
r, = 4, an
positive^
— _ _ — — _ — _ — ^ _^
16 and 25, so that V^i = 3, ^y% = 4, and y^y, = 5. There-
fore, since the coemcient of a; is p
jB, = J(-3 + 4 + 6)=+3
ar,=i(+3-4 + 5)= +2
a:, = i(+3 + 4-5)= + l
which are the four roots of the proposed equation.''
m. Now all that can be said of this numerical work is that
it is a Yerifying instance. It is not an algebraic solution in
any sense of the word, for, as both the quartic and \i% cubic
resolvent have real roots, this is the irreducible case where the
numerical solution fails. In Euler's Algebra, 1774, where
this example was first given, the roots of the cubic are ob-
tained by the use of trigonometrical tables, but in subsequent
quotations it is usually merely stated that they are found " by
tne rules for cubics.'* This numerical example has certainly
no place in the exemplification of the algebraic solution of the
quartic equation, and yet it is so given m most mathematical
aictionaries and it may also be seen in the article Algebra in
the last edition of the Encyclopsedia Britannica.
As this Bulletin is intended for historical and critical re-
marks rather than for original investigations it will not be
well to here set foi-th the method whereby I have brought the
solution into such shape as to produce final practical formulas.
But the results may perhaps be allowed place, as their state-
ment is very brief. ' ^ ^
IV. The following are final formulas for the algebraic solu-
tion of the quartic equation,
aj* + 4aa;" + 6te' + 4ca; + c? = 0.
First, let m and n be determined by
m = a^d — %dbc + J* — &d + c'
n = {V + irf - iac)\
Secondly, let s and t be found from
; = J (m — V^* — w)*'
204 FINAL FORMULAS FOB THE
Thirdly, let w, v, and tv be derived by,
v = 2 (a* - J) - (« +
w=zv' +3 {s- i)\
Then the four roots of the given quartic are expressed by
the formulas,
ajj = — a 4- VH + y V + ^/Iv
a;, = — a+Vw — V v -h V '^
a?, = — a — Vw + yv — V?^
x^= — a — Vu — y v — ^/ w
in which the signs before the square roots are to be used as
written provided 2a' — dab + c is negative, but if this is
positive all radicals except V w blto to be reversed in sign.
V. As a numerical example, let the equation to be solved
be the complete quartic,
z* - Sx' - 10a;' + 56a: + 192 = 0.
Here, by comparing the coeflScients with the given form,
a = - 2, h = - I, c = + 14, 6? = 4- 192.
From these are first computed,
^ _ 3 2 8 3 ^ — Rg2flgS6953
27 ' 729 '
and next in order are found,
5 = 5.983, ^ = 4.350.
Accordingly 5 + ^ is 10.333, and s — t is 1.633, and then
?^ = IG, V = 1, w = 9.
Now as 2a' — Sab + c has a negative value, the formulas give
:r, =2 + 4 4- a/T+T= + 8
ALGEBRAIC SOLUTIOK OF QUABTIO EQUATIONS. 206
a:, = 2 + 4 - V 1 + 3 = + 4
a;, = 2— 4+Vl-i^=-2 + V-2
ar, = 2-4-a/1-3=-2-v'-2-
which are the simplest expressions for the four roots.
As a second example let the proposed quartic equation be
a* + 7a; 4- 6 = 0. Here a = 0, 6 = 0, c =}, and d = 6.
Then m = H and n = S. Next in order, s = 0.8091 and
^ = 0.6180, whence w = + 1.427, v= — 1.427 and «; = 2.146.
Now, c being positiye, the roots are
x^ = - 1.194 - V- 1.427 + 1.465 = - 1.388
a:, = - 1.194 + V- 1.427 + 1.465 = - 1.000
a;, = + 1.194 - V- 1.427 - 1.465
a:, = + 1.194 + V- 1.427 - 1.466
which closely satisfy the given equation.
VI. The above formulas for the algebraic solution of the
quartic equation are final in the sense that, like those so well
known for the quadratic and cubic, they exhibit true symbolic
representations of the roots in terms of the given coefficients,
and that they are not capable of further essential simplifica-
tion. They furnish the means of the discussion of all the
circumstances concerning the occurrence of equal roots in the
quartic, as well as of cases where the roots are connected by a
known relation. They will be found to embrace the solution
of all special and critical cases. For instance, applied to the
binomial a;* — 1 = they giv e the roots ar, = -h 1, a:, = — 1,
x^— 4- V— 1 and a:^ = — \/— 1. Again, if applied to the
form X* 4- 6Ja;' -\- d = 0, they give the same solution as that by
quadratics, for u becomes zero, v becomes — db and w reduces
to 9F — d. Lastly, they furnish ready and exact numerical
solutions whenever the proposed equation has two real and
two imaginarv roots, or when two or more roots are equal. If
there be eitter four unequal real roots or four unequal
imaginary roots, the irreducible case arises where m" — n be-
comes negative, and the formulas, although correctly repre-
senting the roots, fail to furnish numerical solutions in as
simple forms as desired.
Lehigh Univebsitt, March, 1892.
206 POIKGABi'S MifeOAKIQUE c£lE8TS.
POINCAR^'S Ml^OANIQUE ClfiLESTE.
Lbs Methodes nouvelles de la Mecanique Cileste, Par H.
PoiNCABi. Tome I. Paris, Gauthier-Villars, 1892. 8vo.
The publication of this new work on Celestial Mechanics,
embodying some of the results of the labors of mathema-
ticians in that direction during the last fifteen years, comes
as a welcome addition to our knowledge of this subject. Until
lately, nearly all treatises have been written with a special ob-
ject, that of obtaining expressions which can be used by the
i)ractical astronomer ; the mathematical aspects of the prob-
ems solved have been almost entirely neglected. These latter
have an interest of their own apart from any use which can
be made of them, and it is to the study of such questions that
M. Poincare largely devotes himself. At the same time he
points out where they can be applied usefully in the case of
the problem of three bodies. Bat this is not all. Most of
the results obtained can be applied equally to the general
problems of dynamics where there is a force function, and by
the use of a dfssipation function could doubtless be applied
to any natural problem whatever.
The applications are, however, more particularly made to a
satellite system, in the special case when the tnree bodies
move in one plane, as well as in the general case. The limi-
tation generally imposed consists in making the ratios of the
masses of two of the bodies to that of the third a small
quantity, an assumption which, nevertheless, does not limit
greatly the usefulness of the results. M. Poinear6 says,
*^ Le out final de la Mecanique celeste est de resoudre cette
grande question de savoir si la loi de Newton explique ii elle
seule tons les phenomfines astrouomiques," and for this end
to be attained it is absolutely necessary to know whether
the developments of the expressions for the position of any
heavenly body do mathematically represent that position. In
general, the series obtained must be convergent, and it is to
the questions on the convergence of such series that M.
Poincar6 has been able to give some definite answers.
In his introduction, the author points out that the starting
point of the present developments of the lunar theory, was
the publication in Vol. I. of the American Journal of Mathe-
matics of a paper by Dr. Hill entitled, '* Researches in the
lunar theory.'' It is true that in this memoir, Dr. Hill has
largely occupied himself in obtaining exact numerical and
algebraical values for certain inequalities in the motion of the
moon ; but the general considerations involved at the be-
ginning and end of it are of a far-reaching nature. In par-
POINCAIUfi'S MlklAKIQUE CELESTE. 207
ticnlar, a superior limit to the radius vector of the moon is
founds and a general study of the surfaces of equal velocity is
made. His consideration in a particular case of the moons
of different lunations with respect to the primary^ will be men-
tioned below.
M. Poincar6's book is principally based on his own memoir,
^* Sur le problime dea trots corps et les iquations de la
dynamique,'^* The arrangement is not quite the same. In
the treatise, many of the demonstrations are more completelv
explained, the a{)plication8 are more numerous^ and muca
matter that is entirely new has been added. In what follows,
I have not in anv sense attempted to give a complete account
of the book. Much that is given there is outside the scope
of an article such as this ; the results that are mentioned
are chieiSy noticed either because they can be given in a few
words, or because from their peculiar interest they merit a
somewliat longer treatment.
The first chapter deals with some general well-known the-
orems with respect to differential equations. Two types are
selected. The general form which it is necessary to consider
is shown by the system
^ = X (1 = 1,2, ...n).
The X^ are analytic and single- valued functions of the x^ and
may or may not contain the time explicitly. This type in-
cludes the system of canonical equations
dx^ __dF dy^ _ dF
dt " dyi' dt '~ dXi '
which possess a set of properties special to themselves. Some
space is devoted to the consideration of these properties, and
special attention is directed to changes of variables for
which the system still remains canonical. The proofs for
these theorems are sketched very briefly in cases where they
are well-known.
In all the particular cases of the applications of canonical
equations to the problem of three bodies, M. Poincare works
out the results with some detail. The masses are taken to be
m„ w„ wi, ; iw, is the mass of the primary while m,, w„ satisfy
* Ada MathenuUica^Yol. XIII.
808 ponroABi's MioAKianB cfLnn.
Buoh that fA is small whQe fi^fi^ remain finite qnantitiea. Itii
then possible to expand i^ in a series arranged in ascending
powers of /i :
In general F^ will be independent of one system of
elements, say; the y, •
The canonical conations giyen above (Sorreepond to n de-
grees of liberty. Ii we know an integral of the ig^tem, this
number can be lowered by one nnit in general, if we know
q integrals, Poisson's conditions must w fnlfilled between
these integrals taken two and two, in order that the number
of degrees of liberty may be lowered by ; units. The amplica-
tion of this to the general problem of three bodies is imme-
diate. The three mtegrals for the motion of the centre of
mass of the system beinff known and fulfilling the conditions,
we can reduce the number of degrees of liberty from nine to
six. The three known integrals of areas are also intqpds of
the system thus reduced, and by using two combinations of
these latter, it is possible to reduce the system toyimr decrees
of liberty ; also in the case when the bodies move m one pume,
the system can be reduced to three degrees of liberty. The
'usual transformations are then effected so as to leave the
equations still in the canonical form and to carry only
the smallest number of degrees of liberty.
The form of the disturbing function is also discussed, and
it is considered under what circumstances we can develop it
in ascending powers.
The sceona chapter deals with the general conditions for
integration in series, and in particular with the conditions that
these scries ma^ be convergent. It is here that M. Poincar6's
penetrative genius especially shows itself. The complicated
forms which appear in the lunar problem render it an almost
impossible task to attack directly the question of convergence
of the scries obtained. But by going back to the differential
equations themselves, and considering the disturbing function,
he is able to obtain definite results, with respect to the prob-
lem of three bodies, for the convergence of those series which
may be taken to represent certain particular solutions.
The notation introduced by M. Poincar6 a short time back
for dealing with questions of convergence is an especiidly
happy one. It is as follows : — If we have two functions <py ^,
expanded in ascending powers of x, y,
^ «: (arg. X, y)
denotes that the coefficient of every term in ^ is greater in
absolute value than the corresponding term in ^, the *^ argu-
POINCAR^'S MBOAlSriQUE CELESTE. 209
mentfl '^ in terms of which the expansion is made being written
as above. This can of coarse be used for any number of
arguments. An extension of this notation is given at the end
of the chapter. The coefficients, instead of being constants,
are supposed to be periodic functions of the time ; then, if
every coefficient of ^ in its expansion according to powers of
^> yy 6^^ i^ ^ga1> positive, and greater in absolute value than
the corresponding coefficient in q?,
Cauchy's general theorems on convergence are quoted and
extendea to the case in which the function is expanded in
terms of several variables. If we have a system of differential
equations
^=0{x, y, z, //), ^ = ^ (X, y, z, yu), ^ = ^ {x, y, z, //)
where 0, (p, fp are expanded in powers of ar^, y^, z^y and //, /,
there will exist three series expanded in powers of x^, y^,
z^ and ^y t which will satisfy these equations and reduce re-
spectively to .T^, y^, «o when ^ = 0. For these to be convergent
it is necessary that |a;J, lyoU lfol> My V\ should be sufficiently
small. The restriction ji| sufficiently small is evidently incon-
venient, and Poincar6 is able to get rid of it and to say that
the series are convergent if t lies between given limits pro-
vided that |ju| be sufnciently small.
In most cases, however, expansion is not made in powers of
the time, but in trigonometncal functions of it, and it there-
fore becomes necessary in the first instance to examine a system
of differential equations,
dx
-^ = fPui^ + ^'.t^^t + . . • + (Pi.nX^ (* = 1, 2, . • . «)
where the (p are all periodic functions of the time. The gen-
eral solution found is,
Xi = c, e««^ A,^, + c, «<^' A^, + . . . + <?,6V A^,
the A being periodic functions of the time only, the a< depend-
ent on the roots of a determinantal equation, and the <;< arbi-
trary constants.
These Ui are called the characteristic exponents {expomnts
caractSristiques) of the solution. On them depends the
whole nature of the various solutions. Thus if two of the ex-
ponents are equal, the time appears as a factor ; if they are all
14
210 POINCAR^'S m£0AKIQUE CELESTE.
pnre imaginaries, the general solntion contains periodic terms
only, and so on. Also, on them depend the ^' asymptotic solu-
tions/* Chapter IV. is devoted to the consideration of these
exponents.
Chapter III., which deals with [periodic solutions, is perhaps
the most interesting from the point of view of its immediate
application to some of the problems in the lunar theory. In
this connection, a periodic solution is defined as being such
that the system at the end of a finite time T comes into the
same relative position as at the beginning of that time. The
period is then T. Thus if <p{t) represent a periodic solution
of period T
also if
^(^ + T) = q){t) + 2Jc7c {Jc = whole number)
(p{t) is still said to be a periodic solution. These two types
are analogous to linear and angular co5rdinates, respectiyely.
In the canonical system of codrdinates as applied to dynamics
problems, one set of elements belongs to the first type, and the
conjugate set ia general to the second type. It is to be noted
that by defining a periodic solution in this way, the system
can, so to speak, be separated from its external relations. The
motions both of rotation and translation of the system as a
whole can be detached, and those of its various parts amongst
themselves considered.
The question which is put forward for examination is as
follows : If for /^ = we nave a periodic solution, what are
the conditions necessary in order that the solution shall still
remain periodic when ^ is not zero but a small quantity ? It
must be remembered that in this and in what follows, the
term ** periodic " has the meaning which has just been given
to it. In order to answer the question, M. Poincare considers
the system
dXi _
where the Xi are functions of the time periodic and of period
2;r, as well as of the a:,. Space will not permit me to re-
produce the argument, which finally reduces the answer
to the consideration of the properties of a certain curve
in the neighborhood of the origin. This curve is ex-
amined in certain particular cases and notably in the case
where there are an infinite number of periodic solutions for ^
zero, i.e, when the period is an arbitrary constant of the
general solution. Generally, it is found that in these cases
the equations do admit periodic solutions. In another partic-
poincare's m^canique o£leste. 211
nlar case^ the equations when /i = admit a solution of period
27ry and when /i is small but not zero^ save in an exceptional
case, the equations admit a solution of period 2k7t {k bein^ a
whole number) which is different from the solution of period
Zfty and is only not distinct from this latter when pi becomes
zero.
If the Xi are periodic with respect to the time, the solution
in general if periodic must have the same period. When
however the time does not enter into the -a, explicitly, the
period of the solution can be 'anything whatever. Suppose
that the period selected when /4 = be jT. The question
resolves itself into finding under what circumstances a solu-
tion of period 7' + r is possible when ^ is small. The argu-
ment proceeds in a somewhat similar manner as in the first
case and similar results follow.
To apply these results to the problem of three bodies, sup-
pose ^ = 6. Then two of the bodies describe ellipses about
the third. At the end of a certain period measured by the
difference of their mean motions, the system is found in the
same relative position as at the beginning of the period. The
solution tor pi = is then periodic. Wul periodic solutions
be still possible when //, instead . of being zero, has a small
positive value? From what has been proved above, we can
say that such solutions are in general possible. M. Poincar6
distinguishes three classes : — (1) when the inclinations and
eccentricities are zero, (2) when the inclinations only are zero,
(3) when the latter are not zero. He then examines these in
detail.
Under (1^ comes, as a particular case. Dr. Hill's now classic
solution, wnere the mass of one body is supposed to be infi-
nitely great and at an infinite distance, but to have a finite
mean motion, and the mass of the other is infinitely small. The
solutions are referred to axes moving with the infinitely dis-
tant body which takes a circular orbit. The period is one of
the arbitraries and can be anything whatever. When /i =
the motion is circular, and when pi is small, the curve does not
differ much from a circle, and is somewhat elliptical in shape
with its shorter axis directed constantly towards the sun. [If
the sun be not infinitelj distant, the only change in the curve
is a loss of symmetry with regard to the line joining the earth
and the sun.] Dr. Hill calculated the various shapes which
the curve takes for different values of the arbitrary period,
corresponding to gradually decreasing values of the constant
of vis viva. As this latter constant diminishes, the ratio
of the magnitude of the axes becomes greater, until for one
particular value of it a cusp appears at each end of the
greater axis. This gives what Dr. Hill calls, "the moon
of maximum lunation.^' At the cusp and therefore in quad-
212 roiNOABfi'8 MdOAXIQUB OiLESTS.
tatiiie, the moon becomes for a momeiit stationary with
reroect to the snn.
He did not pnrsne the calcalations beyond this point It
was stated however that any member of this chiss of satellites
if jnroloDfl^ beyond the moon of maximum Innation would
oscillate to and fro about a mean place in sjrzygy, never being
in quadrature. M. Poincar6 points out an inaccuracy in this
statement. The satellites which are never in quadrature,
are indeed possible but belong to a diflerent class of solution,
and are not the analytical continuation of those studied by
Dr. Hill. He shows that if we prolonged them beyond the
critical orbit, they would cross the line of quadratures six
times, cutting their own orbits twice and forming a curve with
three closed spaces. The class to which the moons without
quadrature belong has, as a limiting case, a moon which is
stationary with respect to the sun and which is always either
in conjunction or opposition.
M. roinoar6 next goes on to consider the canonical system
when ^0 is supposed independent of the y^. This is the
Soneral problem of dynamics where the forces depend on the
istances only and where we {)roceed by successive approxi-
mations. The first approximation is
Xi = const = ai^ ^ = sansi = «|.
at
If the solution is to be periodic and of period T, all the
riiT mast be multii)]es of 2;r. It is then shown that unless
the Hessian of F^ with respect to the x^ vanish, we can have
a periodic solution of period T or differing little from T
when fx is small. If this Hessian vanish we can sometimes
find a function of F^ whose Hessian does not vanish. If we
cannot do this the case must be otherwise examined. Such
an examination shows, that when the Hessian of jP vanishes,
if the mean value R of F^, with respect to t, admits of a
maximam or a minimum, periodic solutions are still possible.
In the problem of three bodies, F^ corresponds to the dis-
turbing function, and we are led to periodic solutions of the
second and third kinds. Here R does admit of a maximum
or a minimum^ and hence such periodic solutions are always
possible. The periodic solutions of the first kind only cease
to exist when n' is a multiple of n — n\ When, however,
this ratio w' : w — n' is nearly a whole number, as happens in
several cases in the solar system, a large inequality will exist
and its principal part can be calculated suitably by the help
of these periodic solutions.
In the next chapter M. Poincar6 passes on to the considera-
tion of the characteristic exponents, One solution of a
poincare's mecanique celeste. 213
system of differential equations being known, it is required
to find a solution differing little from it. The equations of
variations are formed in the usual way, and these brin^ in
the equations given above, which involve the characteristic
exponents. As an example of tho use of these equations. Dr.
Hill's work on the motion of the lunar perigee is quoted,
where he obtains the principal part of it accurately to a large
number of places of decimals.*
It is then considered under what circumstances one or more
of the exponents become zero, and their effect on the ex-
istence of a periodic solution. The argument and result
depend chiefly on two things : first, the presence in, or ab-
sence from, the Xi of the time explicitly, and, secondly, the
existence or non-existence of single-valued integrals of the
system. If canonical equations be used, the exponents arc
equal and opposite in pairs. With the limitation that F^
does not depend on the f/< , two exponents will be zero, and
unless certain conditions be fulfilled, two exponents only will
be zero. In the periodic solutions of tho problem of three
bodies, whether in one plane or not, two exponents and two
only are zero. The solutions corresponding to these ex-
ponents are called ^^ solutmis dSgSnerescentes/' and are of
the form
^i = Si'\ /;< = Ti"y
g, = ^: + / sr, lu = t: + / t;\
in which the S, 7' are periodic.
The canonical system given above has an integral which is
known, namely tlie integral of vis viva. The author de-
votes himself in Chapter V. principally to prove that, save in
certain exceptional cases, there does not exist any single-
valued algebraic or transcendental integral other than tliat
of vis viva. For this a function ^ is supposed to be analytic
and single-valued for all values of x, v, // within a certain
region, and within this region to be developable according
to powers of ju, thus :
As long as 0^ is not a function of F^, it is proved that ^ =
consL cannot be an integral of the system. If 0^ be a func-
tion of F^, it is possible to find another integral which is
distinct from F, and which does not reduce to F^ when // is
zero. In case, however, the Hessian of F^ be zero, an excep-
tional case arises, and it is in this exceptional case that the
* Acta Mathematirn, Vol. VIII. See also a note by the writer in
^fo. Not. R. A. .S., Vol. XVII. No. 6.
214 POINOABi'S M^ANIQUE oiLBSTE.
importance of the principle applied to problems in dynamics
is seen. A general set of conditions is found, necessary bat
not safiScient for the existence of another integral oi the
equations. These conditions take the form of relations be-
tween the co-efficients in the deyelopment of F.
Applying these to the problem of three bodies, the author
arrives at the conclasion, that there cannot exist any new trans^
cendental or algebraic single-valued integral of tne problem of
three bodies other than the well-kfiown ones, whether we con-
sider the particular cases of two, three, or the general case of
four degrees of liberty mentioned above. This important
result is of course applicable here to the case only when /t
is small, a restriction which nevertheless occurs in most
problems of celestial mechanics. It is pointed out, however,
that M. Bruns has demonstrated that there cannot exist any
other algebraic single-valued integral for any values of the
masses. In actual application M. Poincar^'s theorem will be
found the more useful, since he includes transcendental as
well as algebraic forms in his demonstration.
The most interesting example given to illustrate the general
theorem is that of the motion of a solid suspended from a
fixed point and acted on by gravity only. Tne distance of
the centre of mass of the body from the point of suspension
is supposed small. Two integrals are known : is it possible
that a third can exist ?* When the conditions are applied it
is found that there is nothing to prevent the existence of a
third integral, but since the conditions are necessary and not
sufficient nothing proves that it does exist ; such an integral
however cannot be algebraic.
Chapters VI. and VII. treat of the disturbing function and
M. Poincar6's asymptotic solutions, respectively. In the con-
sideration of the latter a series appears which is divergent in
a manner aualogous to Sterling's series.
Ernest W. Brown.
Haverford Colleqe, Pa., April, 1892.
* For an elementary discussion of this problem, see Routh's Rigid Dy-
namics (4th cd.) Vol. II., Chaps. IV., V.
NOTES. 216
NOTES.
A REGULAR meeting of the New York Mathematical
Society was held Saturday afternoon. May 7, at half-past
three o^clock, the president in the chair. The council an-
nounced that Professor D. A. Murray had been appointed as
librarian to hold office during the remainder of tne current
year. The following persons having been duly nominated,
and being recommended by the council, were elected to mem-
bership : Professor Louis Duncan, Johns Hopkins Uuiversity ;
Lieutenant George Owen Squier, TJ. S. A. and Johns Hop-
kins University ; Mr. Joseph Moody Willard, Johns Hopkins
University ; Miss Ella C. Williams, New York ; Professor
Robert Woodworth Prentiss, Rutgers College, The following
papers were read: *'The fundamental fonnulas of analysis
generalized for space, ^* by Professor A. Macfarlane ; "A sim-
ple and direct method of separating the roots of ordinary
equations," by Professor J. W. Nicholson.
The prize of one thousand marks offered bv the Prince
Jablonowski Society in the department of mathematics and
natural science for the year 1893, has for its subject : The de-
termination of an extensive class of invariants of ordinary
differential equations in accordance with the notation and
methods of Lie.
At the meeting of the London Mathematical Societv on
April 14, the following six mathematicians were elected hon-
orarv members : Messrs. Poincare, Hertz, Schwartz, Mittag-
Lemer, Beltrami, and Willard Gibbs.
Dr. Kurt 'Blei^qhl, jprivat-docent at the University of Ber-
lin, has been appointed to a professorship of mathematics at
that university. T. 8. f.
Professor H. Weber of Marburg has accepted a call to
Gottingen to fill the post vacated by Professor H. A. Schwartz.
Professor Frobenius of Zurich has accepted a call to Berlin.
M. Bd.
A MEETING was held on Saturday, February 20, in the com-
bination room of St John's College, for the purpose of taking
steps to place a memorial of the late Professor Adams, in West-
minster Abbey, in recognition of his brilliant discoveries in
astronomical science. H. j.
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tTHEMATICAI. SOCIETY.
VUL. 1. No. III.
EDWARDS' DIPPEEENTIAL CALCULUS. 217
EDWARDS' DIFFERENTIAL CALCULUS.
An Elementary Treatise on the Differential Calculus, with
appliccUions and numerous examples. By Joseph Edwards, M.A..
formerly Fellow of Sidney Sussex College, Cambridge. Second
edition, revised and enlarged. London and New York, Macmillan
& Co. 1892. 8vo, pp. xiii + 621.
When a mathematical text book reaches a second edition,
80 much enlarged as this, we know at once that the book has
been received with some favour, and we are prepared to find
that it has many merits. We are at once struck by Mr.
Edwards* lucid and incisive style ; his expositions are smgu-
larly clear, his words well chosen, his sentences well balanced.
In the text of the book we meet with various useful results,
notably in the chapter on ** some well known curves," and
moreover the- arrangement is such that these results are easy
to find ; and in addition to these, numbers of theorems are
given among the examples, and, this being a feature for
which we are specially grateful, in nearly every case the
authority is cited. Recognizing these merits, however, we
notice that the book has many defects, some proper to itself,
some characteristic of its species ; and just oecause it is so
attractive in appearance, it seems worth while examining it in
detail, and pointing out certain specially vicious features.
A book of this size may fairly be required to serve as a
preparation for the function theory ; at all events, the influence
of recent Continental researches should be evident to the eyes
of the discerning. Mr. Edwards' preface strengthens this
reasonable expectation, for he promises us ''as succinct an
account as possible of the most important results and methods
which are up to the present time known." But we soon find
that the " important results and methods*' are those of the
Mathematical Tripos ; and in our disappointment we utter a
fervent wish that mstead of the 'Marge number of university
and college examination papers, set in Oxford, Cambridge,
London, and elsewhere,'* Mr, Edwards had consulted an
equally large number of mathematical memoirs published,
principally, elsewhere. The Mathematical Tripos for any
given year is not intended for a Jahrbuch of the progress of
mathematics during the past year ; and as long as so many
will insist on regarding it m that light, text books of this type
will continue to be published.
Nothing in this book indicates that Mr. Edwards is familiar
with such works as Stolz's Allgemeine Arithmetik, Dini's
Fondamentiper la teorica delle funzioni di variabili reali, or
Tannery's Thiorie des fonctions d'une variable. In support
of our contention we may instance the definitions of function.
S18 BDWABDS' D lFfJBMTlA L CAUOULXJB.
m
limit, continaitv, etc. On page %, Lejenne Diriohlefs
definition of a function is adopted. Aocoidinff to this Teiy
Kneral definition, there need be no analyticd connection
tween y and z ; for y is a function of x even when the
Talues of y are arbitrarily assi^ed, as in a table. That Mr.
Edwards does not adhere to this definition is evident from his
tacit assumption that every function q}{x) can be repvesented
by a succession of continuous arcs of curves. Whatever
definition is adopted for a continuous function y of x, it is
evident that to small increments of x must oorrespond small
increments of y ; but Weierstrass has proved that there exist
functions which have this property, but which have nowhere
differential coeflScients. Tue well known example of sucli a
function is
00
f{x) =z 2 b^cos {arxir),
I»s0.
where a is an odd integer, b a positive constant less than 1,
and db greater than 1 + d7r/2. According to the accepted
definition, this function of x is continuous ; according to Mr.
Edwards' definition, it is not continuous, inasmuch as it can-
not be represented by a curve y =/(^) with a tangent at
every point.
We acknowledge that Mr. Edwards displays a considerable
degree of consistency in his view of the meaning of a contin-
uous function, but we insist that after the adoption of the
curve definition he should have been at some pains to prove
00
that the numerous series of the type 2f^{x) scattered
throughout the book Rive rise to curves with tangents, whereas
he never even takes the trouble to prove that they are contin-
uous functions of x in any sense of the term. No more dam-
aging charge can be brought against any treatise laying claim
to thoroughness than that of recklessness in the use of infinite
series ; and yet Mr. Edwards has everywhere laid himself
open to this charge. One of the most difficult things to teach
the beginner in mathematics is to give proper attention to the
convergency of the series dealt with. All the more need,
then, that a text book of this nature should set an exa mpl e
of consistent, even aggressive carefulness in this respect. We
do, it is true, find an occasional mention of convergence (pp.
9, 81, 454, etc.), but as a rule it is ignored. Mr. Edwaras
rearranges the terms of infinite and doubly infinite series,
applying the law of commutation without pointing out that
his series are unconditionally convergent ; he differentiates
f{x) = 2fJ{x) term by term, and gets f(x) = 2fJix\ im-
BDWARDS' DIFFEEBNTIAL CALOTTLUB. 219
plying that the process is nniversally valid {e.g. p. 84) ; or, at
all events, giving no hint that there are cases in which the
differential coefficient of the sum of a convergent series is
different from the sum of the differential coemcients of the
individual terms. We find no formal recognition of the im-
portance of uniform convergence in modern analysis, nothing
even to suggest that ho has ever heard of the distinction
between uniform and non-uniform convergence. We begin
to suspect that he has never looked into Chrystars Algebra.
The unreasoning mechanical facility thus acquired in per-
forming operations unhampered by any doubts as to tneir
legitimacy, naturally leads Mr. Edwards to view with favour
" the analytical house of cards, composed of complicated and
curious formulae, which the academic tyro builds with such
zest upon a slippery foundation,'^ * — and to build up many a
one. A curious and interesting specimen is
to be continued to infinity. This expression has been exam-
ined by Seidel,f who points out that Eisenstein's paper in Crette,
vol. 28, requires correction. Before such an expression can be
differentiated, a definite meaning must be assigned to it ; but
Seidel's conclusion is that, denoting xP^ by a?,, a:*» by aj„ x'^ by
a?„ and so on, then as x varies from to 1/e*, L Xftn increases
n = oo
from to 1/e, while L x^ + i decreases from 1 to 1/e;
« = 00
beyond these limits for x, the case is different. In particular
when X > eV«, the expression diverges. Our objection is not
to the non-acceptance of Seidel's conclusions, but to the un-
necessary use of a function of this doubtful character. Ex-
amples can be found to illustrate every point that ought to
be brought up in an elementary treatise on the differential
calculus without ranging over examination papers in search of
striking novelties.
Feeling now somewhat familiar with Mr. Edwards' point
of view, we examine his proofs of the ordinary expansions
with a tolerably clear idea of what we are to expect. We
find, of course, " the time-honoured short proof of the exist-
ence of the exponential limit, which prooi is half the real
proof plus a suggestio falsi''; we find in the chapter on
expansions a general aisregard of convergency consider-
ations ; we find throughout the book the assumption that
* Professor Chrtbtal, in Ifature, Jane 25, 1891.
f Abhandlungen der k. Ah. d, Wiss, Bd. zi^
S20 EDWABD6' DIFFBBBHTIAL OAUOUhXJB.
<p(a) = L q^x), and that ^0, 0) = L q^x, y) • ; we
find the nraal aflsamptions bb to expansibility in series proceed-
ing by integral powers, with disastrous resolts further on. We
find tne osnal dread of the complex Tariable, thoa^h Mr. Ed-
wards has j^yen one or two examples inyolying it, without how-
eyer explaining what is meant by f{x + iy). We can hardly
TB^ad these examples, eyen with § 190, as a sufficient roo^-
nition of the complex yariable in a treatise of this siae. We
must notice also the thoroughly faulty treatment of the in-
Terse functions. For exampk, no exptSuiation is giyen of the
signs in -^ when y = cos -' a; or sin -^ op. Mr. Edwards' attitude
towards many yalued functions is simple enough ; as a rule,
he ignores the inoonyenient superfiuity of yalues. He does,
it is true, giye in § 54 a note, clear and correct, on this point ;
but he is yery careful to confine this within the limits of the
single section, and to indicate, by choice of type, that it is
quite unimportant.
We pass on now to the second part, applications to plane
cunres ; and here we must object emphmcally to the intro-
duction of so many detached and disconnected propositions
relating to the theory of higher plane curyes. From Mr.
Bdwaras' point of yiew this is doubtless justified ; we are
quite ready to acknowledge that we know of no book that
would enaBle a candidate to answer more questions on sub-
jects of whose theory he is totally ignorant. The deficiency
of a curve, e./y., is a conception entirely independent of the
differential calculus ; bat probably this single page will obtain
many marks for candidates in the Mathematical Tripos ; these
we should not grudge if we thought an equivalent would be
lost by a reproduction of Mr. Edwards' treatment of cusps.
Our spirits rose when we remarked the italicised phrase on p.
224^ that there is " in general a cusp *^ when the tangents are
coincident. But three pages further on we find that the
exception here indicated is simply our old friend, the conju-
gate point, whose special exclusion from the class in which it
appears must be a perpetual puzzle to a thoughtful student
with no better guidance than a book of this kind. Such a
student, probably already familiar with projection, knows
that the real can be projected into the imaginary, and the
imaginary into the real. If then the acnode, appearing as a
cusp, has to be specially excluded, why not the crunode ?
But here Mr. Edwards reproduces the now well established
* See e.g, p. 122 ; and on this page note also the assamption that the
relation between A, k, while S0 4- A, y + A;, tend to the limits x, y exerts
no influence on the result.
BDWA&DS' DIFFBBBNTIAL CALCULUS. 221
error, calling tacnodes, formed by the contacfc of real branches,
doable cnsps of the first and second species, and excluding
those formed by the contact of imaginary branches ; he even
goes farther astray, introdacing Cramer's osculinflexion as a
casp that changes its species.
This matter of double cusps is a fundamentally serious one,
and not a mere question of nomenclature. This persistent
misnaming effectually dis^ises the essential characteristic of
the cusp. It is not the coincidence of the tangents that makes
a cusp. From the geometrical point of yiew it is the turning
back of the (real) tracing point, expressed by the French and
(German names, {point ae rebroussement, HUckhehrpunht] ;
from the point of view of algebraical expansions (of y in
terms of re, y = being the tangent) the essential character-
istic of a single cusp is that at some stage in the expansion
there shall be a fractional exponent with an even denominator,
so that the branch changes from real to imaginary along its
tangent ; from the point of view of the function theory, which
is really equivalent to the last, the simple cusp is character^
ised by the presence of a Verzweigungspunkt combined with
a double point. The simple cusp, that is, presents itself as
an evanescent loop. A double cusp, then, in the sense in
which Mr. Edwards uses the term, does not exist. There
cannot bo two consecutive cusps, vertex to vertex ; for the
branch if supposed continued through the cusp, changes from
real to imaginary ; and two distinct cuBpB, brought together to
give a point of this appearance, produce a quadruple point.
While on this subject, we must mention Mr. Edwards' rule
for finding the nature of a cusp. Find the two values of
T-^ ; these by their signs determine the direction of convexity
(§ 296). How does this apply e.g. to y* = x* ?
This confusion regarding cnsps is made worse bj the
assumption already noticed that when f(x, y) = is the
equation of the curve, y can be expanded in a series of integral
powers of x. This error is repeated on p. 258, where to obtain
the branches at the origin, this being a double point, we are
told to expand y by means of the assumption y =/?2; + Vr +
etc. The whole exposition of this theory of expansion is
most inadequate. In § 382 there is no hint that the terms
obtained are the be^nning of an infinite series, giving the
expansion of (say) y in powers, not necessarily integral, of x ;
there is no hint what to do when the first terms of the expan-
sion are found ; there is no suggestion of the interpretation
of the result when two expansions begin with the same terms.
A thoughtful student may by a happy comparison of scattered
223 XDWABDCr' DUVmaMSTlAL OAJUIUWB.
examples (p. 200, and ex. 3, p. 230) arriTe at the ooneot
theory ; bat he siuely desenres better gaidaDoe.
One or two more points must be noticed. The theory of
asymptotes, when two directions to infinity coincide, cannot
be satisfactorily dereloped without assuming a knowledge of
doable points ; and the only way of giving the true geometrioal
significance is to introduce the conception of the line infini^,
and to consider the nature of the intersections of the cnrre by
this line. A tangent lying entirely at infinity does not ''count
as one of the n theoretical asymptotes'' ; if counted among
the asymptotes at all, it has to be counted as the equivalent
of two out of the n. This is one of the stron^pst arguments
against iDcludin^ the line infinity in enumerating the asymp-
totes. The vanons expressions for the radius of curvature
involve an ambiguity in sign ; what is the meaning of this ?
The omission of this explanation causes obscurity, notably in
L830. The equation of a curve, referred to oblique axes,
dng (p(x, y) = 0, what is the condition for an infiexion P
As a matter of fact it is the same as in the case of rectangular
axes, given on p. 2G4 ; but as this is obtained from a formula
for the radius of curvature, the investigation is not applicable.
Throughout Mr. Edwards displays an almost exclusive pref-
erence for rectangular axes, and seems to regard the metrio
properties so obtained as of equal importance with descriptive
properties. For instance, in the case of an ordinary double
point (p. 224) instead of the three cases usually distinguished,
we have /(mr, the additional one being that of perpendicular
tangents.
In the third part we notice that in the chapter on *^ undeter-
mined forms '' there is no discussion of the case of two variables,
though it is on this that we have to rely for a rigorous proof
of the theorem , . = , -[ . We recoe^nize an old friend,
dxoy oydx °
the discussion of the limit of oo/oo, in which it is first as-
sumed^ and then proved, that the limit exists. The state-
ment of ex. 17, p. 457, is somewhat misleading ; the formula
there given for the expansion of {x -h a)" is true when m is a
positive integer ; but when m = —1, it is evidently not true
for a;= —b, —2^, etc.* The treatment of maxima and
minima of functions of two variables (§§ 497-601) is incom-
plete and incorrect. The geometrical illustration, as given on
p. 424, omits the case of a section with a cusp^ which is the
simplest case that can occur when r^ = «' ; of the more com-
S Heated cases Mr. Edwards attempts no discrimination ; he
oes not even stat^ correctly the principles that must ^ide
us in this discrimination. The inexactness of the ordinary
* Laurent, TrcM d^Analyee, m., 886.
NOTE ON BESULTANXS. '^2A
criteria (given in § 498) appears at once from the example
u=: {y* — 2px){y* — 2qz) [Peano]. The origin is a point
satisfying the preliminary conditions ; taking then for x, y,
small quantities h, k, the terms of the second degree are posi-
tive for all values except A = ; when A = 0, the terms of
the third degree vanish, and the terms of the fourth degree
are positive ; nevertheless the point does notgive a minimum^
which it should do by the test of § 498. For we can travel
away from in between the two parabolas, so coming to an
adjacent point at which u has a small negative value, while
for points inside or outside both parabolas the value of u is
positive. The truth is, the nature of the value a of the func-
tion w at a point {x^ yj at which ^ and ^ yanish, depends on
the nature of the singularity of the curve t« = a at this point.
If this curve has at (x^, y^ an isolated point of any degree of
multiplicity, we have a true maximum or minimum of u ; but if
through (a?,, yj pass any number of real non-repeated branches
of the curve, we have not a maximum or minimum ; in
Peano's example the branches coincide in the immediate
neighbourhood of the origin, but then they separate, and there-
fore we have not a minimum value for u.
We object, then, to Mr. Edwards' treatise on the Dififer-
ential Calculus because in it, notwithstanding a specious show
of rigour, he repeats old errors and faulty methods of proof,
and introduces new errors ; and because its tendency is to
encourage the practice of cramming *' short proofs'' and
detached propositions for examination purposes.
Charlotte Anoas Soott.
Bbtn Mawb, Pa , May 18, 1892.
NOTE ON BESULTANTS.
BT PROP. M. W. HASKELL.
On page 151 of Prof. Gordan's lectures on determinants*
is to be found the theorem
where /?/, ^ denotes the resultant of two functions / and ^ of
a single variable x of degree m and n respectively. This
* Vorluungen Hber Invariantentheorie, herausgegeben von Kbbschbn-
STBINEB. ErstQt Band. Leipzig, 1885.
234 KOTB OK BESULTANT8.
theorem is proved under the restriction that n is not greater
than m and that the degree of the arbitrary function ^ shall
not be greater that m — w. The author ffoes on to say : *'e8
w&re eine sch&tzenswerthe Arbeit, anch den Fall zu unter-
suchen, in welchem m < n ist, d. h. allgemein die Prage zu
behandeln : Wie hangen die Resultanten -B^ + ^.^r,^ und Bf^^
zusammen, wenn wir uber den Orad der diesbezuglichen
Functionen keinerlei Voraussetzung machen ? ''
This statement is somewhat remarkable on account of the
ease with which it may be shown that the theorem is true in
general in exactly the form riven above.
I. Let F=f'\-(l>,tp, Then, no matter what the degree of
the functions involved may be, if the degrees of i^and ^ be m
and n respectively, n is certainly not greater than m, and the
degree of ^ cannot be greater than m — n. Hence, by Qor-
dan's result as quoted,
Rf,^ = -ff^-*.^,^ = -Br.*.
That is, the theorem is true without restriction.
II. Suppose the resultant i2^, ^ to be found in the usual way
by the method of greatest common divisor. Two functions
A and B of jc, of degree n — 1 and m — 1 respectively can be
found to satisfy the relation :
1 = A.f ■¥ B.^.
The coefficients of A and B are rational, but not integral,
functions of the coefficients of /and ^, whose least common
denominator is the resultant -B/, ^.
It follows that
l^A{f^(j>,tp) + {B^Atp)(fi
and the resultant Rf ^^.y^^^ is the least common denominator
of the coefficients of A and {B — Afp). But the coefficients
of jp will evidently not occur in the denominators at all, and
the least common denominator is therefore identical with that
of the coefficients of A and B, viz. i2/,^.
Berkeley, Cal. , May 12, 1892.
COLLIlJfEATIOK A8 A MODE OF MOTIOK. 225
COLLINEATION AS A MODE OP MOTION.*
BT HAXDfE BdCHEB, PH.D.
In the following paper I have attempted togiye an account
of some Tery simple matters, which, although familiar to
many, appear to have attracted but little attention in this
country. The subject, however, has never, as far as I know,
been presented from precisely the point of view here adopted.
Perhaps the most important difference between the old and
the new geometry lies in the extended use made during the
present century of geometric transformations.! The change
which has come about in this direction is due in part to the
influence of certain branches of applied mathematics in
which one has to deal not merely with geometric configura-
tions but also with certain changes which these configurations
are forced to undergo. There are however two distinct
ways of looking at a transformation. First we may consider
the original and the transformed figure as standing side by
side, or even as occupying portions of the same space, the
latter being in a certain sense a picture of the former ; or
secondly, we may consider the original figure to be gradually
deformed accoraing to a given law into the tran^ormed fig-
ure. Each of these points of view can be tmced to a physical
origin. Perspective and allied subjects strikingly illustrate
the first, while the second will most naturally be adopted in
hydrodynamics, the theory of elasticity, etc. Now while the
first of the above mentioned ways of looking at a transforma-
tion has the advantage of introducing no unnecessary element
into the consideration, the second in turn has the advantage
of making the idea of a transformation lose much of its
abstractness, for by its aid we are enabled to see the points of
the original firare rearrange themselves by a gradual motion
into the transformed figure.
I wish to illustrate this way of looking at a transformation
as a mode of motion by considering one of the simplest of
transformations, the so-called linear transformation orcolline-
ation,t and for the sake of simplicity I will confine myself to
two dimensions.
* Lecture deliyered June 4, 1892, before the New York Mathematical
Society.
f The following remarks should be understood to apply only to point
transformations, t.d., to transformations which carry points oyer into
points.
X The word coUineation seems to be by far the best name for this trans-
formation, not only because it is as applicable in synthetic as in analytic
geometry, but also because the ambiguity which arises in speaking of a
16
226 COLLINEATIOK AS A HODB OF MOnOK.
Using anj system of tri linear coordinates {x^y x^ 2;,), a ool-
lineation will be expressed by the linear formols :
pa;/ = a, a;, + a, a:, + a, a?„ (1)
P< = ^,^. +^,2;, + c,a:„
(p being an undetermined factor of jproportionalitj).
It is however well known that m general a collineation
leaves three points of the plane fixed while all other points
are carried over into new positions. If now these three fixed
points be taken as the vertices of the triangle of reference,
the collineation will evidently be expressed by the very simple
formulsB :
px^ = ax^, pa;,' = &r„ px^'=^cx^. (2)
These formulae tell us into what position each point of the
plane is earned over by the transformation ; they give us,
nowever, no clue as to wnat path it is advisable to regard as
traversed by each point in passing from its originsd to its
final position.
To determine this, let us first consider the case in which
two of the fixed points are the circular points at infinity, the
third (finite) fixed point being denoted by the letter 0. This
collineation may be shown by a simple calculation, to consist
of a rotation of the plane as a whole about the point
combined with a uniform stretching of the plane away
from (or contraction of the plane towards) this same point.
Id the case of a rotation, however, each point will naturally
be regarded as moving from its original to its final position
along the arc of a circle whose centre is at ; in the case of
expansion or contraction on the other hand, the lines of mo-
tion will be the straight lines through 0, the motion taking
place away from in the case of expansion and towards in
the case of contraction ; the amount of the motion in any
case being proportional to the distance from 0. If, then, we
have a combination of rotation and expansion (or contraction)
the lines of motion will evidently be equal logarithmic spirals
with pole at 0. Taking as the origin of a system of polar
coordinates, the equation of this family of logarithmic spirals
will be :
r = A^'^ ,
linear transformation without specifying what system of coordinates we
use is a very real objection, as ttiere are other coordinates besides triiinear
(for example, Darboux's tetracyclic coordinates) in which linear trans-
formations are actually considered. The term '* homographic trans-
formation," introduced by Chasles, is not as expressive as the term col-
lineation used some years before by M5bius. It does not seem as though
Chasles' ignorance of the German language could justify us in adopting
his poorer names in place of the original oetter ones.
OOLUKEATIOK AS A MODB OF HOTIOK. 227
where ^ is a constant determining the shape of the spirals,
while ^ is a parameter varying from one member of the fam-
ily to another. We shall nnd it more convenient to write in
place of k the quotient A,/A,.
Let us now introduce a system of trilinear coordinates in
which the vertices of the triangle of reference are the circular
points at infinity and the point 0. We will denote these
coordinates by the letters (5, v> 0» ^^^ ^^^ ^^^ referring to
the imaginary sides of the triangle of reference through 0,
while the last refers to the line at infinity. In this system
of homogeneous "circular" coordinates the equation of the
above mentioned family of logarithmic spirals is readily found
to be :
^*,+tt, ,^-tt, ^-«*,= ^8*1,
or since k^, k^, A are any constants :
provided that a + /^ + ;^ = 0.
We can now write out at once the equations of the lines of
motion in the general case where we have as fixed points any
three points of the plane, for we have merely to project the
point and the circular points at infinity in the special case
wo have just considered into any other three points in order
that the logarithmic spirals should go over into the lines of
motion of a general collineation. The equation of the family
of lines of motion^ referred to the triangle of reference whose
vertices are the fixed points^ is then :
X- x\^ xy = G,
where G is the variable parameter of the family while the
constants a, /3, y (which are connected by the relation
« 4- /? 4- >^ = 0) depend upon the coeflBcients a, h, c of the
linear transformation (2) above.*
The fixed points of a collineation may be all real, or one
of them may be real and the other two conjugate imaginary.
The last of these two possibilities need not detain us long, as
it may be obtained by a real projection from the special case
considered above where two of the fixed points were the cir-
cular points at infinity. In it we shall have in our triangle
of reference one vertex and the side opposite real, while tne
remaining vertices and sides are imaginarjr. The lines of
motion will have a spiral form, each consisting of an infinite
* It is easily found that a, fl, y axe proportiozial respeotively to
log ?, log -, log -.
228 OOLLTRBATIOK AS A MODB 09 MOTIOK.
nnmber of coils about the real fixed point. As these coils
become larger they will become more and more elongated
in the direction farthest from the real side of the triangle of
reference, until each coil finally assumes a hyperbolic lorm,
running out on the side of the nzed point farthest from the
real side of the triangle of reference through infinity, and
completing itself on the other side of the real line in question.
These hyperbolic coils ultimately approach the real side of the
triangle of reference asymptotically.
When all of the fixed points are real, however, the lines of
motion will have completely lost their spiral character. Here
again there are two cases to consider, according as the three
coefficients a, i, c of the transformation have all the same
sign, or one of them a different sign from the other two.
The three indefinitely extended sides of the triangle of refer-
ence divide the platie into four parts, one finite and the other
three infinite. We may speak of each of these parts as *' tri-
angles,'^ in spite of the fact that each of the three infinite
triangles appears to be diyided into two distinct portions by
the Ime at infinity. Using this terminology we may say that
when all three coefficients a, b, c have the same sign, the
interior of each of these four triangles is transform^ into
itself ; but when one of the three coefficients has a different
sign from tlic other two the triangles are interchanged in
pairs. We will begin with the simpler of the two cases, in
which each triangle is transformed into itself. The lines of
motion in this case will be found to lie as follows : — *
Within the finite triangle the lines of motion all start from
the vortex corresponding to the smallest of the three coeffi-
cients,! and run without singularity to the vertex correspond-
ing to the largest of them ; at each of their extremities these
curves are tangent to the side of the triangle joining that
extremity with the vertex corresponding to the coefficient
which lies in magnitude between tne other two. The side of
the triangle joining the vertices which correspond to the
greatest and the smallest coefficient is, of course, itself a line
of motion, and the same is true of the broken line consisting
of the other two sides of the triangle.
* One way of seeing this is to consider first the special case in which
the triangle of reference consists of two lines at ri^bt angles to one
another and the line at infinity, and then to project this into the general
case. In the special case just mentioned we nave t# deal with the same
transformation which occurs in the theory of small irrotational strains
(see for instance MrNCHiN, Uniplaner Kinematics, chap. v.). It is inter-
esting to notice that this is a case in which the idea of lines of motion
is naturally suggested by a physical application.
f The coefficients a, h, c are said to correspond to the vertices opposite
the sides Xx = 0, Xs = 0, a;t = respectively.
GOLLIKEATIOK AS A MODS OF MOTIOK. 229
The lines of motion wifchin each of the other three triangles
will be precisely like those jast described, the difference in
appearance being doe to the fact that these triangles them-
selves extending through infinity, some of the lines of motion
in one of these three triangles and all of the lines of motion
in tlic other two will nin through infinity on their way from
the vertex corresponding to the smallest of the coefficients to
the one corresponding to the largest.
It should be noticeo that while these curves are in ^neral
transcendental and extend onlv between two fixed points of
the collineation where they suddenly stop, we can find special
cellineations for which the curves are algebraic and all of any
degree we please. The family of curves will not look partic-
ularly different in these cases from what it does when the
curves are transcendental, but the curves themselves will have
a different shape. They will now no longer stop at the two
fixed points just mentioned, but will continue beyond them
into another triangle, having singularities in these points
when their degree is higher than the second (in the case of
cubics, a cusp in one point, and a point of inflection in the
other). The case where the lines of motion are conies all
tangent at the extremities of one of the sides of the triangle
of reference to the other two sides deserves special mention
owing to its frequent occurrence in projective geometry.*
Coming now to the case where one of the coefficfents of
the transformation has a different sign from the other two, it
is readily seen that the lines of motion are here imaginary
although each contains an infinite number of real points.
Every point of the plane is therefore carried over from its
original to its final piosition through an imaginary path. We
are therefore unable to follow the motion of the points of the
plane. It is however possible to break up the transformation
into two parts, one verv simple, the other more complicated
but having real lines or motion. Thus for instance we can
break up the collineation :
pa;/=-2a;,, /ar,' = 3a;,, /ox,' = 6a?,,
into the two collineations :
A^, = — «, , /w,= z^, pi, = a:,;
/ox/ = 2x„ px^' = 35„ pa?/ = 65,.
* We may of course have oonioa aa lines of motion when two of the
fixed points of the collineatioD are imaginary, rotation of the plane
about a point being a special case of this. In fact, whenever the lines of
motion are conies, whether the fixed points are real or imaginary, the
collineation will be merely a non-euoUdian rotation if we take one of
these conies as the absolute.
230 OOLLIIOSATIOK A8 A MODE OF MOTIOK.
The second of these has as its lines of motion the real tran-
scendental curves discussed above, while I may perhaps be
allowed to describe the first as a " projective reflection '' with
regard to the side rr, = and the opposite vertex. The na-
ture of the transformation brought about by this projective
reflection is so simple that it need not be discussed nere, and
that we do not need the assistance of lines of motion to get a
perfectly clear idea of it.* It is of course merely the pro-
jective generalization of ordinary reflection ; reflection with
regard to the ^xis of X, for instance, in a system of rectan-
gular coordinates, being mereljr a projective reflection with
regard to this line and the infinitely distant point on the axis
of V,
It remains to mention some of the literature connected
with this subject. The transcendental curves, which we have
called the lines of motion of the colUneation, occur incident-
ally in a paper by Clebsch and Gordan in the Matheniatische
Annalen, vol. i. They were however first systematically con-
sidered bv Klein and Lie in vol. iv. of the same journal
(1871). The reader is referred to this paper for the modifica-
tion of the lines of motion which occur in the various special
cases (when the colli neation has two coincident fixed points,
etc.). The very brief indications there given can readily be
amplified as has been done in this paper for the general case.
The reader will also find in this beautiful paper an account
of some of the remarkable properties of these curves, which
thus gain an interest far above that attaching as yet to most
other transcendental curves, owing to the fact that their prop-
erties form to some extent a systematic whole, not a mass of
facts more or less ingeniously proved. More important still
however is the connection of these lines of motion with Lie's
now famous theorj of differential equations,! some of the
very earliest of Lies investigations in this direction being
contained in the paper just mentioned. By the introduction
of intinitesimal transformations it is possible to obtain the
lines of motion directly without first considering the special
case in which the circular points at infinity are two of the
fixed points. We thus find the equation of the lines of motion
as the solution of a differential equation.
In still another way must Klein's name be connected with
* It should however be noticed that a projective reflection (and there-
fore any ordinary reflection) may be regarded as having real lines of
motion, viz., conies. This will bo most readily seen if we consider that
the projective reflection with regard to the line at infinity and a finite
point is equivalent to a rotation through an angle of 1^0^ about that
point.
f See Lie's recently published book on this subject edited by Scheffers.
KOTE& 231
this subject. A few preliminary remarks are necessary to ex-
plain this. The linear transformation of a single straight line
into itself may be studied from precisely the same point of
view as wo adopted above in the case of two dimensions.
Three cases would again present themselves : one in which the
two fixed points are imaginary, and two in which they are
real. In one of these last the transformation cannot be re-
garded as a real motion, while in the other two it can. Now
the extension of our theory which suggests itself to us here
depends upon the fact that the complex points of a straight
line can be conveniently represented in a plane of which
the line is the axis of reals. The linear transformation of the
line will then give us a corresponding transformation of
the plane which of course should not be confounded with the
collineation discussed above. The coeflScients here need no
longer be real to give us a real transformation. This new
transformation of the plane may also be regarded as a mode
of motion and has been so treated by Klein in his lectures for
a number of years (see an article by Prof. Cole in the Annals
of Mathematics for June, 1890, and part ii. chap. i. of the
recently published Modulfunctionen of Klein-Fricke). The
idea cannot fail to suggest itself that the transformation of
the plane which we have called collineation should be general-
ized in a similar way by representing the complex as well as
the real points of the plane. I do not know of this subject
having been treated; it would of course lead us into four
dimensional space.
Harvard UwivBRBrrT, June, 1892,
NOTES.
A REGULAR meeting of the New York Mathehatioal
SocxETY was held Saturday afternoon, June 4, at half past
two o'clock, the president in the chair. The following per-
sons having been duly nominated, and being recommended by
the council, were elected to membership : Dr. James Whit-
bread Lee Glaisher, Trinity College, Cambridge, England ;
Mr. Ferdinand Shack, New York, N. Y. The following
papers were read : " An expression for the total surface of an
ellipsoid in terms of (T- and p- functions, inoluding an appli-
cation to the surface of a prolate spheroid, '' by Professor J.
H. Boyd ; *' On collineation as a mode of motion,*' by Dr.
Maxime B6cher ; '* On Peters' formula for probable error,'' by
Professor W. Woolsey Johnson.
282 KOTES.
The meeting of the Deutsche Mathematiker - Vereiniffung,
which will be held this Bummer as qbuuI in conjunction with
that of the OeseUsehaft deuUeher Kaiurfarecher und Aerzie,
will take place at tTarenberg, September 12 to 18. Special
interest attaches to the meeting this year on account of the
organisation by the Union of an exhibition of medals, charts,
apparatus and instruments used in pure and applied mathe-
matics. The Bayarian ffOTernment will lend its aid to the
enterprise, which has dready secured the co-operation of
several eminent mathematicians, of the leading publishers,
instrument makers etc., and of a large number of hiffh-
schools and polytechnic institutes. The object of the euii-
bition is *^ to extend the use of the various auxiliaries in the
shape of models, apparatus and instruments, which are of
advantage for instruction and invertigation in pure and ap-
plied mathematics, and to forward the interests of this kind
of scientific work.'' A recent prospectus contains a prelimi-
nary classification of articles, giving as the main heaas : (1^
geometry and theory of functions, ^) arithmetic, algebra ana
mtegral calculus, (3) mechanics and mathematicttl pnysics.
Pbofbssob Pbano, the editor of the Bwista di Maiema^
tica^ has undertaken a veir interesting work, the parts of
which will appear as supplements to his journal It is an
extended collection of the formulas and results of mathe-
matics, expressed throughout in the language or notation of
symbolic logic. The first signature of the work accompanies
the number of the Rivista for April, 1892 (vol. ii., No. 4).
The publication of the collected works of the late Professor
Weber has been undertaken by the Gottingen Academy of
Sciences. The collection will probably fill six large octavo
volumes, and it is to be completed by 1894. T. s. f.
Dr. Arthur SchOnflibs, privatdocent at the University
of Gottingen, has been appointed professor extraordinarius
at the same university.
Harvard University. Besides the more elementary
courses, the class-room work in which will amount to twenty-
three hours a week throughout the year, the following mathe-
matical courses are offered for the year 1892-93 :
By Professor J. M, Peirce ; Algebraic plane curves ; Qua-
ternions (second course) ; Theory of functions (first course) ;
Linear associative algebra, and tne algebra of logic.
By Professor C. J. White ; Planetary theory.
By Professor Byerly ; Trigonometric series, and spherical
KBW PUBLICATIONS, 233
harmonics ; Problems in the mechanics of rigid bodies (second
course).
By Professor B. 0. Peirce ; Potential function ; Wave mo-
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By Dr. Bdcher; Mathematical seminary on geometrical
topics ; Functions defined by differential equations ; Curvi-
linear co-ordinates and Lam6 s functions.
Each of the above courses extends throughout the whole
academic year, and in most of them the instructor lectures
three hours a week. A number of courses largely mathemati-
cal are also offered in the departments of Physics and Engi-
neering, as for instance a course on the mathematical theory
of electrostatics and electromagnetism by Professor B. 0.
Peirce. m. b6.
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MoucHOT (A.). Les Nouyelles bases de la g6om6trie sup^rieure (Q^o-
m^trie de position). In-8. Gauthier-Villars. 6 fr.
Mi^ELLER-EazBACH (W.). Physikalische Anfgaben fQr den mathema-
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Nassiruddin-el-Tousst, Traits da Qaadrilatdre. Texte arabe aveo
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NiBMf^LLER (F.). Anwendnnff der linealen Aasdehnangslehre von Grass-
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Panzerbieter (W.). Ueber einige LSsungen des Triseotionsproblems.
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PoiNCARi (H.). Lemons sur la theorie de I'^lastioit^, r^g^ par Emile
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Coara de la Facalt^ dei eciencen de Paris.
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EouTH (E. J.). The Advanced Part of a Treatise on the Dynamics of a
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in. Biiropa durch Gerbert. Beclin 1893. gr. 8. 5 u. 123 pg.
M. 3.00
Wbltzibn (C.)- Hcber die Bcdingnn^n, iinter denen eine gAOte ratiou-
ale Punction von mebreron Verfinderlichen die Tollstfcadige Potens
eioer anderan datstellt. Berlin 1802. 4. M. 1.00
ZOBAWSKi (K.), Uober Bicgiingsinvarianten. Eline Anwcndaag der
Page line for read
26 foot-note Rarm, Rahs.
87 25 (5.), (6,).
133 i Q. B. Zbkb, Q. B. M. Zbrb.
" 17 twelve, twenty-four.
163 1 9 Newman, NenmaDn.
May 7, April 2.
(a' _ S) _ (j + (), (a^- S) + (g + 0.
INDEX.
Acad^mie des Sciences, 198.
Adams, John Couch, 143 ; Memorial of, 215.
Airy, George Biddle, 142.
Algebra, Fme*s Number System of, G. EnestrSm, 26.
Algebraic Curves, Topology of, L. S. Hulburt, 197.
Equation, KronecKer and his Arithmetical Theory of the, H. B.
Pine, 173.
Solution of Quartic Equations, M. Merriman, 202.
Amendment to Constitution, 142, 198.
American Association for the Advancement of Science, Washington
Meeting, 80.
Institute of Electrical Engineers, Transactions, 142.
Journal of Mathematics, 55, 56, 142, 194.
Analytical Geometry, A French, C. H. Chapman, 92.
Annual Meeting of the German Mathematicians, A. Ziwet, 96.
Meeting of the Society, 124.
Association for the Improvement of Geometrical Teaching, 169.
for the Improvement of the Teaching of Mathematics and the
Natural Sciences, 31.
Astronomischo Gesellschaft, Catalo^e of the, T. H. Safford, 83.
Astronomy, Greene's Spherical and Practical, J. K. Rees, 140.
Authors of articles in tne Bulletin :
B6cher, 225. Hulburt, 197.
Brown, 206. Jacoby, 27, 28, 44, 189.
Cajori, 184. Johnson, 1, 129.
Chapman, 92, 150. Macfarlane, 189.
Cole, 105. McClintock, 85.
Davies, 75. Merriman, 89, 202.
Davis, 16. Newcomb, 120.
Duhem, 157. Rees, 140.
Enestr5m, 26. Safford, 33.
Fields, 48. Scott, 217.
Fine, 173. Stabler, 123.
Fiske. 12, 61. Wright, 46.
Haskell, 228. Ziwet, 6, 42, 96, 145.
Hathaway, 66.
Ball, R. S., 170.
Bertrand's Calcul des Probabilit^s, E. W. Davis, 16.
Bibliographv. 57, 82, 103, 127, 143, 171, 195, 216, 283.
Bibliotheca Mathematica, 26.
Bdcher (M.): Collineation as a Mode of Motion, 225, 281.
Boyd (J. H.): An Expression for the Total Surface of an Ellipsoid in
Terms of d- and p- Functions, Including an Application to the Pro-
late Spheroid, 231.
Briot and Bouquet: Lemons de G^om^trie Analytique, 92.
Brown (E. W.V, Poincar6's M6canique C61este, 206.
Caiori (F.): Multiplication of Series, 184.
Calcul des Probabilit^s, E. W. Davis, 16.
838 IKDBX.
Ounbridge University Press, 66, 108.
Canon Pellianus, Di^n, 142.
CaseT, John, 81.
Catalogue of the Astronomische Qesellschaft, T. H. Salfoid, 88.
Cajley (A.): On Lists of Coyariants, 142.
Chapman (C. H.): A French Analytical Oeometry, 98 ; WeierstraBS and
Dedekind on General Complex Numbers, 150.
Clarendon Press, 58.
Clark University, 102, 104. •
Cole (F. K.): Klein's Modular Functions, 106.
Collineation as a Mode of Motion, M. BScher, 225.
Columbia Coll^;e, 80.
Complex Numbers, Weierstrass and Dedekind on General, C. EL Chap-
man, 150.
Constitution. Amendment to, 142, 108.
Covariants, Lists of, £. McClintock, 85, 142.
Craig (TO: Treatise on Linear Differential Equations, 48.
Dav^ (J. E.): Preston's Theory of Light, 75.
Davis (£L W.): Bertrand's Caloul des Probabilit6s, 16.
Dedekind and Weierstrass on Gtoeral Complex Numbers, C. H. Chap-
man, 150.
Degen's Canon Pellianus, 142.
Deutsche Mathematiker- Yereinigung, 06, 282.
Differential Calculus, by J. Edwards, C. A. Soott, 217.
Equations, Craig*s Linear, J. C. Fields, 48.
Equations, PicaM's Demonstration of the General Theorem upon
the Ixistenoe of Integrals of Ordinary, T. S. Fiske, 12.
Doubly Infinite Products, T. S. Fiske, 61.
Duhem (P.), Emile Mathieu, his Life and Works, 157.
Early History of the Potential, A. S. Hathaway, 66; Notes on, 126.
Edward's Differential Calculus, C. A. Scott, 217.-
Eight-figure Logarithm Tables, H. Jacoby, 188.
Election of New Members of the Society, 54, 78, 124, 125, 142, 169, 108,
215, 281.
Enestrdm (6.) : Fine's Number System of Aleebra, 26.
Engler (E. A.): Geometrical Construction for Finding Foci of Sections of
a Cone of Revolution, 169.
Errata, 236.
Evans, Asher Benton, 56.
Exact Analysis as the Basis of Language, A. Macfarlane, 189.
Fargis (E. A.) and Uagen (J. G.): The Photochronograph, H. Jacoby,
44.
Ferrcl, William, 55.
Fields (J. C): Craig*s Linear Differential Equations, 48; Transformation
of a System of Independent Variables, 142.
Final Formulas for the Algebraic Solution of Quartic Equations, M.
Merrimau, 202.
Fine(H. B.): Kronecker and his Arithmetical Theory of the Algebraic
Equation, 178; Number System of Algebra, G. EnestrOm, 26,
Fiske (T. S.): On the Doubly Infinite Products, 61 ; Picard's Demonstra-
tion of the General Theorem upon the Existence of Integrals of
Ordinary Differential Equations, 12.
Fricke (R.): Felix Klein, Vorlesimgen Uber die Theorie der EllipUschen
Modulfunctionen, 105.
Frobenius, Professor, 215.
Geometry, Oxford Examinations in, 169.
German Mathematicians, Annual Meeting of, A. Ziwet. 96.
Schools, Teaching of Elementary Geometry in, A. Ziwet, 6.
Gescllschaft Deutscher Naturforscher und Aerzte, 64th Meeting, 79, 96.
INDEX. 239
Gilman (F.): Application of Least Squares to the Development of Func-
tions, 125.
Grand Prix des Sciences Math^matiques, 194.
Qreene (D.): Introduction to Spherical and Practical Astronomy, 140.
Hagen (J. U.) and Fargis (G. A.): The Photochronograph, H. Jacoby, 44.
Hagen (J. G.): Synopsis der HSheren Mathematik, 31.
HaU (T. P.): Cubic Projection and Elotation of a Tessaract, 193.
Harvard University, 232.
Haskell (M. W.): Note on Resultants, 223.
Hathaway (A. S.): Early History of the Potential, 66; Notes on, 126.
Hensel, Kurt, 215.
Hilbert (D.): Algebraic Curves, 197.
Hulburt (L. S.): Topology of Algebraic Curves, 197.
Infinite Products, on the Doubly, T. S. Fiske, 61.
Inhalt und Methode des Planimetrischen Unterrichts, H. Schotten, 6.
Inland Press, 56, 169.
Italian Mathematical Journal, A New, A. Ziwet, 42.
Jacoby (H.) : Determination of Azimuth by Elongations of Polaris, 55;
Eight-figure Logarithm Tables, 139; South American Longitudes,
by J. A. Norris and Chas. Laird, 28; The Photochronograph, by
J. G.iHagen, S. J. and E. A. Fargis, 44; West African Longitudes,
by D. GiU, 27.
Johnson (W. W.): Peters* Formula for Probable Error. 231; Octonary
Numeration, 1 ; The Mechanical Axioms or Laws of Motion, 129.
Johns Hopkins University, 81, 194.
Klein's Modular Functions, F. N. Cole, 105.
Kowalevski, Sophie, 31.
Kronecker, Leopold, 142 ;
His Aritnmetical Theory of the Algebraic Equation, H. B. Fine,
173.
Language, on Exact Analysis as the Basis of. A. Macfarlane, 189.
Laverty (W. H.): The Laws of Motion, An Elementary Treatise on Dy-
namics, A. Ziwet, 145.
Laws of Motion, An Elementary Treatise on Dynamics, by W. H.
Laverty, A. Ziwet, 145.
Leach, She well and Sanborn, 126.
Least Squares, a Problem in, M. Merriman, 39.
Light, Preston's Theory of, J. E. Davies, 75.
Lists of Covariants, E. McClintock, 85, 142.
Logarithm Tables, Eight-figure, H. Jacoby, 139.
London Mathematical Society, Honorary Members, 215.
Longitudes, South American, by J. A. Norris and Ch. Laird, H.
Jacoby, 28.
West African, by D. Gill, H. Jacoby, 27.
Lucas, Edouard, 81.
Macfarlane (A.) : Fundamental Formulas of Analysis Generalized for
Space, 215 ; On Exact Analysis as the Basis of Language, 189.
Macmillan and Company, 143, 193.
Marie, Maximilian, 31.
Martin (A.) : Errors in Degen's Canon Pellianus, 142 ; On Powers of
Numbers whose Sum is the Same Power of Some Number, 55.
Mathematical Problems, E. L. Stabler, 123.
Society of the University of Michigan, 80.
Mathieu, Emile, His Life and Works, P. Duhem, 157.
Matzka. Wilhelm, 31.
McClintock (E.) : On the Computation of Covariants by Transvection,
125 ; On Lists of Covariants, 85 ; Note, 142.
M^ani(^ue Celeste, H. Poincar^, E. W. Brown, 206.
Mechamcal Axioms or Laws of Motion, W. W. Johnson, 129.
%inuL, m ^ >. lit. 2
'. - V.--
iiTJl'C*
X-L— TTT.I.T. rA
> i'-li o IT. ill.
/■'•^ ■--■"*." J .I.* if "jiy • :»Lr Ti-irtr r..!::**"*. "^. ^fw^.^nL?. 131.
44 * ' _ -.
/'■.•:ii"t*-«ri X-iJia^ t.-j*: f.Jt.j'r*!'*, Zl "ST. 3.- w-i. i5.»L
?'."-■=--■»: • . J*- r- 1l 7. z — . *' Z. r»fc"j**. 75.
{'-'■-^ '^ '■•-: ?J" "J <t ■ri.":u:o'"rs£L >-«:>*C;. il^.
^•'''1-' '--',' V'*^^'''^ zi "-ijr* Tiff'rr "f- "-. 3. Jl ~
2>*r"n.-.i. * Tr-*.-.-* :r_ Z. '5'. Ih-a If.
:jC£tXix «£
?r>ijwrt;.ip ;« -.jjft ^KOtfj. 54, ?5. lOl. 1*4. Iii5, Ifi. :«L !«.
INDEX. 241*
Publications, New, 57, 82, 108, 127, 148, 171, 195. 216, 238.
Pupin (M. I.): On a Peculiar Family of Complex Harmonics, 101, 142.
Ouartic Equations, Final Formulas for Solution of, M. Merriman, 202.
Kecent Elementary Works on Mechanics, A. Ziwet, 145.
Rees (J. K.): Greene's Spherical and Practical Astronomy, 140.
Register Publishing Company, 58, 169.
Resultants, Note on, M. W. Haskell, 228.
Rivista di Matematica, 42, 282.
Royal Astronomical Society of London, 55.
Sanord (T. H.): Catalogue of the Astronomische Gesellschaft, 88.
Sch5nfLies, Arthur, 232.
Schotten (IL): Inhalt und Methode des Planimetrischen IJnterrichts, 6.
Scbroeter, H. E., 169.
Schwarz H. A. 194.
Scott (C' A.'): Edward's Differential Calculus, 217.
Series, Multiplication of, F. Cajori, 184.
South American Longitudes, by J. A. Norris and Ch. Laird, H. Jacoby,
28.
Spherical and Practical Astronomy, Greene's Introduction to, J. K. Rees,
140.
Stabler (E. L.): Mathematical Problems, 128.
Steinmetz(C. P.): On the Curves which are Self-Reciprocal in a Linear
Nul-System, and Their Confi^ration in Space, 78.
Stringham (1.): Classification of Logarithmic Systems, 55.
Sturm, Rudolph, 194.
Synopsis der H5heren Mathematik, J. G. Hagen, 81.
Tait (P. G.): Note on Early History of the Potential, 126.
Taylor (A. B.): Octonarv Numeration, 2.
Teaching, Association for the Improvement of Geometrical, 169.
Oi Elementary Geometry in German Schools, A. Ziwet, 6.
Of Mathematics and the I^atural Sciences, Association for ImproY-
ingthe, 31.
Topology of Algebraic Curves, L. S. Hulburt, 197.
Treasurer's Report, New York Mathematical Society, 125.
University of Michigan Mathematical Society, 80.
Weber, Wilhelm Eduard, 81, 232.
Weber, H., 215.
Weierstrass and Dedekind on General Complex Numbers, C. H. Chap-
man, 150.
West African Longitudes, by D. Gill, H. Jacoby, 27.
Wiley (John) and Sons, 56. 81, 126.
Wright (T. W.) : Nomenclature of Mechanics, 46.
Zeitschrift fur den Math, und Naturwiss. Unterricht, 81.
Zerr (G. B. M.): Solutions of Questions in the Theory of Probability and
Averages, 123.
Ziwet (A.): A new Italian Mathematical Journal, 42; Some Recent Ele-
mentary Works on Mechanics, 145; The Annual Meeting of the
German Mathematicians, 96; The Teaching of Elementary Geometry
in German Schools, 6.
16
CONTENTS.
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