COLLECTED MATHEMATICAL PAPERS
OF
HENRY J. S. SMITH
Bonbon
HENRY FROWDE
OXFORD UNIVERSITY PRESS WAREHOUSE
AMEN CORNER, E.G.
Qtttr
JIACMILLAN * CO., 66 FIFTH AVENUE
7
MATH; AI RS
ENRY JOHN STt
LATE SAVILIAN PROFE 1 -
J. W. L.
WITH A MAT ,f^ BIOC
AT DON
THE COLLECTED
MATHEMATICAL PAPERS
OF
HENRY JOHN STEPHEN SMITH
M.A., F.R.S.
LATE SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD
EDITED BY
J. W. L. GLAISHER, Sc.D., F.R.S.
FELLOW OF TRINITY COLLEGE, CAMBRIDGE
WITH A MATHEMATICAL INTRODUCTION BY THE EDITOR, BIOGRAPHICAL SKETCHES
AND A PORTRAIT
IN TWO VOLUMES
VOLUME I
ffc
OXFORD
AT THE CLARENDON PRESS
1894
jforfc
PRINTED AT THE CLARENDON PRESS
bY HORACE HIST, PRINTER TO THE UNIVERSITY
3
CONTENTS OF VOLUME I
BIOGRAPHICAL SKETCH
By DE. CHAELES H. PEABSON.
RECOLLECTIONS OF HENRY J. S. SMITH
By PBOFESSOB JOWETT
By LOED BOWEJJ ....
By MB. J. L. STBACHAN-DAVIDSON .
Note by ME. ALFBED KOBINSON
iNTRODrCTION TO THE MATHEMATICAL PAPERS .
By DB. J. W. L. OLAISHEB.
PAGE
ix
XXXV11
xlvi
li
liv
Ixi
LIST OF PAPERS.
I. On some of the Methods at present in use in Pure Geometry .
Transactions of the Ashmolean Society, Vol. II. No. xxv. Read Dec. 1, 1851.
II. On some Geometrical Constructions ... . .
Cambridge and Dublin Mathematical Journal, Vol. VII. pp. 118-126. May, 1852.
III. De Compositione Numerorum Primorum formae 4A + 1 ex Duobus Quadratis .
Crelle's Journal, Vol. L. pp. 91, 92. 1855.
IV. On the History of the Researches of Mathematicians on the subject of the
series of Prime Numbers ...... .
Proceedings of the Ashmolean Society, Vol. III. No. xxxv. pp. 128-131. Read
March 2, 1857.
V . Report on the Theory of Numbers. Part I .....
Report of the British Association for 1859, pp. 228-267.
VI. Report on the Theory of Numbers. Part II
Report of the British Association for 1860, pp. 120-169.
1
25
33
35
38
93
v J ( < ) \TEXTS.
PAGE
VII. Report on the Theory of Numbers. Part III
Report of the British Association for 1861, pp. 292-840.
\ 1 1 1. Report on the Theory of Numbers. Part IV
Report of the British Association for 1862, pp. 503-526.
IX. Report on the Theory of Numbers. Part V . 263
Report of the British Association for 1868, pp. 768-786.
X. Report on the Theory of Numbers. Part VI
Report of the Britisli Association for 1865, pp. 822-376.
XI. On Systems of Indeterminate Linear Equations . 365
Report of the British Association for 1860. Sectional Proceedings, p. 6.
XII. On Systems of Linear Indeterminate Equations and Congruences 367
Philosophical Transactions, Vol. CLI. pp. 298-326. Received Jan. 17 ; Read Jan. 31,
1861.
XIII. On the Criterion of Resolubility in Integral Numbers of the Indeterminate
Equation / = aa? + 'z' 2 + a"x"* + 2 bx'x" + 2 I'xx" + 2 b"x'x = . 410
Proceedings of the Royal Society, Vol. XIII. pp. 110, 111. Received Jan. 20 ; Read
Jan. 28, 1864.
XIV. On the Orders and Genera of Quadratic Forms containing more than Three
Indeterminates ......... 412
Proceedings of the Royal Society, Vol. XIII. pp. 199-203. Received March 22 ;
Read April 21, 1864.
XV. On Complex Binary Quadratic Forms 418
Proceedings of the Royal Society, Vol. XIII. pp. 278-298. Received May 18; Read
June 16, 1864.
XVI. On a Formula for the Multiplication of Four Theta Functions . . . 443
Proceedings of the London Mathematical Society, Vol. I. No. 8. pp. 8-14. Read
Mny 21, 1866.
XVII. On the Orders and Genera of Ternary Quadratic Forms .... 455
Philosophical Transactions, Vol. CLVII. pp. 255-298. Received Feb. 21 ; Read
Feb. 27, 1867.
XVIII. On the Orders and Genera of Quadratic Forms containing more than Three
Indeterminates .......... 510
Proceedings of the Royal Society, Vol. XVI. pp. 197-208. Received Oct. 80 ; Read
Doc. 5, 1867.
XIX. On some Geometrical Constructions 524
Proceedings of the London Mathematical Society, Vol. II. pp. 85-100. Read May 28,
1868.
XX. Observatio Geometrica 541
Annali di Matomatica, Ser. II. Vol. II. pp. 818-321. 1869.
XXI. On the Focal Properties of Ilomographio Figures ..... 545
Proceedings of the London Mathematical Society, Vol. II. pp. 196-248. Read April 8,
1869.
NX II. On the Focal Properties of Correlative Figures . . . . . . 603
Proceedings of the London Mathematical Society, Vol. III. p. 12. Read Dec. 9, 1869.
CONTENTS Vll
VOLUME II
PAOS
XXIII. Memoire sur Quelques Problemes Cubiques et Biquadratiques . . 1
Annali di Matematica, Ser. II. Vol. III. pp. 112-165, 218-242. Memoire Couronne'
par 1'Academie Royaje des Sciences de Berlin, avec une moiti<5 du prix Steiner en
.Tuillet, 1868.
XXIV. Arithmetical Notes 67
Proceedings of the London Mathematical Society, Vol. IV. pp. 236-258. The three
papers which form these Notes were read on Jan. 9 and Feb. 13, 1873.
XXV. On the Integration of Discontinuous Functions ..... 86
Proceedings of the London Mathematical Society, Vol. VI. pp. 140-153. Read
June 10, 1875.
XXVI. On the Higher Singularities of Plane Curves 101
Proceedings of the London Mathematical Society, Vol. VI. pp. 153-182. Read
June 10, 1875.
XXVII. Mathematical Notes 132
Proceedings of the London Mathematical Society, Vol. VII. pp. 287, 238. Read
Dec. 9, 1875. First printed in the Messenger of Mathematics, Vol. V. pp. 143, 144
(Jan. 1876).
XXVIII. Note on Continued Fractions 135
Messenger of Mathematics, Ser. II. Vol. VI. pp. 1-14 (May, 1876).
XXIX. Note on the Theory of the Pellian Equation, and of Binary Quadratic
Forms of a Positive Determinant 148
Proceedings of the London Mathematical Society, Vol. VII. pp. 199-208. Read
May 11, 1876.
XXX. On the Value of a Certain Arithmetical Determinant . . . . 161
Proceedings of the London Mathematical Society, Vol. VII. pp. 208-212. Read
May 11, 1876.
XXXI. On the Present State and Prospects of Some Branches of Pure Mathe-
matics 166
Proceedings of the London Mathematical Society, Vol. VIII. pp. 6-20. Read
Nov. 9, 1876.
XXXII. On the Conditions of Perpendicularity in a Parallelepipedal System . 191
Proceedings of the London Mathematical Society, Vol. VIII. pp. 88-103. Read
Dec. 14, 1876.
XXXIII. On the Conditions of Perpendicularity in a Parallelepipedal System . 213
Philosophical Magazine, Ser. V. Vol. IV. pp. 18-25. Read before the Crystallological
Society, June 14, 1876.
XXXIV. Sur les Integrates Elliptiques Completes 221
Atti della R. Accademia dei Lincei. Transunti, Ser. III. Vol. I. pp. 42-44. Read
Jan. 7, 1877.
viii CONTENTS.
XXXV. Me*moire sur Ics Equations Modulaires . .
Atti delta K. Accademia doi Lincei. Memorie della classe di Scienze fisiche, mate-
matiche c natural!. Ser. III. VoL I. pp. 186-149. Bead Feb. 4, 1877.
XXXVI. On the Singularities of the Modular Equations and Curves . . .
Proceedings of the London Mathematical Society, Vol. IX. pp. 242-272. Read Feb. 14
and April 11, 1878.
XXXVII. Note on a Modular Equation of the Transformation of the Third Order
Proceedings of the London Mathematical Society, Vol. X. pp. 87-91. Bead Feb. 13,
1879.
XXXVIII. Note on the Formula for the Multiplication of Four Theta Functions .
Proceedings of the London Mathematical Society, Vol. X. pp. 91-100. Bead
Feb. 18, 1879.
XXXIX. De Fractionibus Quibusdam Continuis .......
Collectanea Mathematica (in memoriam Dominici Chelini), Milan, 1881, pp. 117-143.
The paper is dated 1879.
XL. On some Discontinuous Series considered by Riemann ....
Messenger of Mathematics, Ser. II. Vol. XI. pp. 1-11 (May, 1881).
XLI. Notes on the Theory of Elliptic Transformation .....
Messenger of Mathematics, Ser. II. Vol. XII. pp. 49-99 (August-November, 1882).
XLII. Notes on the Theory of Elliptic Transformation . . . .
Messenger of Mathematics, Ser. II. Vol. XIII. pp. 1-54 (May- August, 1888).
XLIII. Memoir on the Theta and Omega Functions ......
XLIV. Me'moire sur la Representation des Nombres par des Sommes de Cinq
Carre's ............
M6moires pre'sente's par divers savants a I'Acad^mie des Sciences de 1'Institut
National de France, fol. XXIX.
PAOB
242
274
279
287
312
321
368
415
623
APPENDIX.
I. Address to the Mathematical and Physical Section of the British
Association at Bradford in 1873 .681
II. Arithmetical Instruments . 691
III. Geometrical Instruments and Models 698
IV. Introduction to the Mathematical Papers of William Kingdon Clifford . 711
BIOGRAPHICAL SKETCH
THE short record of Henry Smith's life, which I have compiled at the request
of his sister, is chiefly based upon a Memoir by herself, which I was anxious
to give in its entirety, but for which she has desired me to substitute my own
words. I have to thank Professor Irving for a letter containing his recol-
lections of Henry Smith at Balliol in his first years as Fellow and Lecturer,
The Memoir and Letters from which I have worked are unfortunately
most defective during the years of my own absence from England, 1871-
1883 ; and I must ask my readers to bear in mind under what disad-
vantages I have attempted to perform a sacred duty. Happily, Henry
Smith's character, about which there is really no diiference of opinion,
exhibits an unbroken continuity of growth. As a boy he seemed to have
something of the mature wisdom of a man ; and to the day of his death he
retained the simplicity and high spirits of a boy. My own estimate of him,
based on the close intimacy of more than twenty years, represents, I hope and
believe, what his friends thought and would wish said. To those who did not
know him, it will perhaps appear that my judgment has been influenced by
friendship. Those who knew him will notice points I have missed or excel-
lences I have slurred, and will condemn my inadequacy.
The original plan of this Memoir assumed that it would be supplemented
by the publication of a large number of Henry Smith's letters. This was
over-ruled in Oxford while I was in Australia, and cannot now be reverted to.
Of these letters a few only have been published. Neither, unfortunately,
%> i
*/ b
BIOGRAPHICAL SKETCH.
have I leisure or strength to recast the Memoir altogether. I have done
what I can in that direction, and can only hope that the difficulties under
which I have worked may be borne in mind.
CHARLES H. PEARSON*.
MELBOURNE, 1888.
LONDON, 1893.
HENRY JOHN STEPHEN SMITH was the second son of John Smith, a
distinguished though short-lived Irish barrister, who, after graduating at
Trinity College, Dublin, was law pupil, at the Temple, of Serjeant Henry John
Stephen, the learned editor of Blackstone's Commentaries. Mr. John Smith went
back to practise in Dublin, and in 1818 married Mary Murphy. By this marriage
he had four children a daughter, who died of consumption in 1834 at the age
of fifteen; a son, Charles, who died also of consumption in 1843, being then
a cadet at Addiscombe ; a daughter who still lives ; and the subject of this
memoir, who was named after his father's old tutor, and who was born on the
2nd of November, 1826.
Mr. Smith died in 1828, of abscess of the liver, and his widow was left for
a time in very straitened circumstances. Fortunately, after delays which seemed
interminable, the Courts affirmed the validity of a bequest of 10,000, which
had been made to Mr. Smith by his cousin the Marchioness of Ormond, and
which her husband disputed. With this money, and with that produced by
the sale of a house which Mr. Smith had just built, the widow had wherewithal
to provide adequately for her family ; and partly to escape from sorrowful asso-
ciations, partly to secure for her children that good education which it had been
their father's earnest wish they should receive, Mrs. Smith resolved within
six months of her husband's death to pass over into England. The family
wandered successively to the Isle of Man, 1829 ; to Harborne near Birmingham,
1829-30; to Leamington, 1830-31 ; and then to Ryde in the Isle of Wight,
where nine or ten years were spent.
Probably no widow left in charge of a young family could have been better
fitted to train them for eminence in after life than was Mrs. Smith. A tall
* Dr. Pearson's death has deprived this ' Sketch ' of the benefit of the author's final revision.
BIOGRAPHICAL SKETCH. XI
distinguished-looking lady, who retained the traces of great beauty to the
day of her death, and who united a certain stateliness of manner and reserve
of temperament to Irish ease and kindly charm of manner, she was also
considerably more than an accomplished and clever woman, for she possessed
powers and learning such as are rare even among able and learned men.
Henry Smith inherited genius on both sides. He was a sickly child, and he
was also short-sighted from a very early age, perhaps partly from being allowed
to read too much when he was quite young. In 1831, when he was only
four years old, he was able to follow English and French lessons. He
also picked up an old Greek grammar of his mother's, rendered additionally
formidable by contractions, and learned the alphabet, the nouns, the adjec-
tives and the pronouns for his own pleasure. ' His practice,' says his sister,
' was to lay himself at full length on his stomach on the floor with the
book he wished to study under his chin to suit his sight. When he was
between seven and eight I remember Prideaux' Connection being for long an
absorbing study.' As soon as his mother found out how marked his taste for
language was, she took him into her own hands for the classics, and for the
next six or seven years he owed all his chief training to her. In 1838 the
pupil had got so far that his mother thought it desirable to call in other aid.
She was fortunate enough to meet with a highly trained tutor in the person
of Mr. R. Wheler Bush, who has put his recollections of Henry Smith on
record in the following terms :
' In the years 1838-39 Henry Smith, then a boy of eleven years of age,
read with me for about nine months at Hyde, in the Isle of Wight. He had
been previously taught by his widowed mother a remarkably clever and highly
educated woman. After reading with Henry Smith I had a large experience
of boys during a head-mastership of more than thirty-three years, but I have
often remarked that the brilliant talents of Henry Smith prevented me from
ever being really astonished at the abilities of any subsequent pupil. His
power of memory, quickness of perception, indefatigable diligence, and intuitive
grasp of whatever he studied were very remarkable at that early age. What
he got through during those few months, and the way in which he got through
it, have never ceased to surprise me. From a record which I have before me
I see that during that short time he read all Thucydides, Sophocles, and Sallust,
twelve books of Tacitus, the greater part of Horace, Juvenal, Persius, and
b2
x ii BIOGRAPHICAL SKETCH.
several plays of jfcschylus and Euripides. I see also that he got up six books
of Euclid, and algebra to simple equations ; that he read a considerable quantity
of Hebrew ; and that, among other things, he learnt all the Odes of Horace
by heart. I could scarcely understand at the time how he contrived at his
early age to translate so well and so accurately the most difficult speeches of
Thucydides, without note or comment to guide him. He was a deeply interesting
Ix.y, singularly modest, lovable, and affectionate.' (Times, Feb. 12, 1883.)
Scarcely less valuable for the boy's development were the abundant leisure
that he enjoyed, and the comparative isolation. His lessons never occupied more
than five hours a day, and the obligatory ' constitutional ' was only of an hour.
During the rest of the time the brothers and sisters were turned out to play by
themselves. Their story books were limited to Robinson Crusoe, Evenings at
Home, Sandford and Merton, and Miss Edgeworth's Frank ; their toys consisted
of hoops and tops, and one or two dissected games. They grew up like the young
Brontes, in a world of their own, improvising plays from Robinson Crusoe or
combats from Homer. In one of these fights Henry had his finger badly hurt
by an arrow from the bow of Achilles, his elder brother, and the surgeon's aid
had to be called in. These amusements could not occupy their whole leisure.
In idle hours the children became diligent students of animal and insect life,
learning much about the habits of bees and ants and spiders and wood-lice and
garden moths. They were directed in these pursuits by two books, Insect
Architecture and Insect Transformations, from the Library of Entertaining and
Useful Knowledge, and assisted in them by two neighbours, a lady who was
something of a botanist and a conchologist, and a Mr. Jacques, who had some
knowledge of chemistry. That the interest they took in these matters was
more than cursory seems proved by the fact that they supplied Dr. Blomfiekl,
who was engaged on a Flora of the Isle of Wight, with several new homes
of rare plants.
In 1839 Mr. Bush was called away to a head-mastership. It proved difficult
to supply his place, though an excellent mathematical master was found at
Newport, who came over twice a week and carried his pupils through the advanced
parts of Arithmetic, elementary Algebra, and Euclid. Henry continued to be
a very docile pupil, sometimes asking, when he received an order which displeased
him, whether he was ' forced' to obey it, but never demurring if he understood
that obedience was required. In 1840, however, he lost his chief fellow-student,
BIOGRAPHICAL SKETCH. Xlll
through his elder brother going to Addiscombe, and Mrs. Smith decided on
moving to Oxford, where it was certain that better teaching could be found than
in the Isle of Wight. The Oxford of those days was comparatively a small place.
Resident professors, married tutors, and married fellows were almost or quite
unknown, while the Heads of Houses, then the governing body of the Univer-
sity, formed a little society of their own. Consequently, the widow lived in
comparative solitude, though even so she could not avoid hearing something of
the war of opinion that was beginning : of the angry opposition provoked by
Tract 90, of Newman's sermons, of the coalition of Evangelicals and Liberals
against Puseyites, and now and again of the few Liberals who stood outside the
strife of the Churches. Meanwhile she had been exceptionally fortunate in the
tutor she secured for her son. The Rev. Henry Highton, Fellow of Queen's and
then Curate of St. Ebbe's, was a sound though not a brilliant scholar, and a
really good mathematician, far above the average of Oxford in those days. No
one could be better fitted to develope Henry Smith's varied capacities, and in
Mr. Highton's class-room Henry, for the first time, was able to measure himself with
boys of his own age. In the summer of 1841 Mr. Highton received the offer of a
Mastership at Rugby, which at that time was chiefly valuable when a boarding-
house was attached to it. Mr. Highton accepted the offer, which allowed of his
marrying, and proposed that he should take Henry Smith with him as his first
boarder. Mrs. Smith agreed, and Henry was thus launched into school life under
the most famous teacher of the day, Dr. Arnold.
I have always regarded it as singularly fortunate for Henry Smith that he
was at Rugby in its best days, and that he was not there long enough to
acquire that part of its tone which was not generally popular. Whether
that sweet buoyant nature, with its supreme sense of proportion, and lively
humour, could ever have been really spoiled, made pedantic or harsh, is
perhaps more than doubtful ; but I cannot doubt that the years of travel on
the Continent, which two chances, that seemed unkindly, substituted for school
and Oxford life, were really of the greatest use to the sufferer. He carried on
his studies abroad less methodically, but quite as profitably, as he could have
done at home ; he learned French, German, and Italian, and he gained some
acquaintance with foreign ideas and methods. Meanwhile his first years at
Rugby were certainly profitable to him. It was a rule of the school that no one
should be in the Sixth Form until he was sixteen, and in deference to this rule
BIOGRAPHICAL SKETCH.
Henry Smith was kept for a year doing work below his capacity in the Upper
Fifth and the Twenty Form, though by a curious anomaly he was allowed to act
as praepostor in Mr. Highton's house, where he was the senior boy. His Report
for the first half-year, which was spent in the Fifth Form, has been preserved ; and
might have been written of him at almost any time : ' Classics : In extent and
variety of knowledge he is certainly the best in the Form, and he is particularly
fortunate in combining accurate and literal construing with an excellent choice of
English words. His composition also (though it has sometimes been careless) is
spirited and clever. Mathematics: Very good. Modern Languages: He has
made great progress in German, and is getting on very well. He has been late
for morning prayers oftener than I like ; and I should wish him to get rid of a
few trifling irregularities, such as occasional inattention at lesson and inaccuracy
in saying his lines. G. E. L. COTTON, Master of the Fifth Form.' When
in the Twenty he came under Mr. (afterwards Professor) Bonamy Price, con-
fessedly the ablest teacher on the very able staff of which Rugby then boasted,
and probably never surpassed as a teacher of classics. In the Midsummer exam-
ination of that year, Henry Smith passed into the Sixth, and was accordingly
entitled to bid the Doctor good-bye. A few days later he received a letter from
Mr. Highton (June 12) : ' You hardly supposed that when you bid Dr. Arnold
" good-bye " on Friday it was for the last time. He was taken to his rest at six
this morning. . . . You may imagine how the loss is felt here. It is almost as if
a common parent were taken away. I felt it so quite myself.'
The true education of a boy at a public school is even more in the play-
ground than in the class-room. What Henry Smith was in this regard has not
come down to me. Going to Rugby just before he left it, I remember to have
heard ' Highton Smith,' as he was popularly called, spoken of with vague
reverence for his great ability, but in no other way. Nevertheless there are in-
dications from reports and letters that he was abundantly capable of healthy
enjoyment, and not merely what the Rugbeans used to call a ' swat ' or book-
worm. Mr. Highton twice reports of him in his first half-year that he was not
working as hard as he ought ; and his sister says that ' he came home for his first
holidays " astonishing " us by the buoyancy of his spirits and even more by
a propensity for " grub," unknown to the ascetic days of his childhood. By one
who learned so easily as he did, a little idleness was easily made up for.' In the
examination of June 1843 he obtained a Junior Scholarship, being ineligible for
BIOGRAPHICAL SKETCH. XV
the University Scholarship because he had not been three years in the school.
In July 1843 the new head master, Dr. Tait, wrote to Mrs. Smith to say : ' There
is no young man in the Sixth Form from whose abilities I am led to expect
more than from him ; and I have formed a very high opinion of his
character and conduct generally.'
Rugby however was not to keep him. In September 1843 his elder brother
Charles died of rapid consumption, and the uncle, who was also guardian and
adviser, declined under these circumstances to consent to Henry's remaining any
longer at a school in a bleak part of England. The boy bore the blow to his
ambition with his unvarying sweetness, and wintered with his family at Nice,
while he spent the following summer by the Lake of Lucerne. These were
months of steady reading, though his books were few, and he was even unpro-
vided with a Greek lexicon. In the autumn of 1844 he went back to his old
friend, Mr. Highton, for a month, that he might be ' coached ' for the Balliol
scholarship. He won it easily, and, as he was not to go into residence till Easter,
went back to join his family at Home. His journey was a series of disasters.
He missed the mail-boat at Dover, had his pocket picked at Paris, and, even
after pledging his books, could only muster funds enough to carry him in the
roughest way to Rome. This misadventure involved a journey of seventy hours,
on the outside of a diligence during a severe frost, to Marseilles ; and, after a third-
class passage from that port to Civita Vecchia, he arrived in Rome with both his
feet frost-bitten, and was laid up for a long time. Presently came an attack of
small-pox. ' All the same,' writes his sister, ' the winter was a time of intense
enjoyment, and a gathering and growing time.' By Easter he was well enough
to go to Oxford, and spend his first term there.
When the Long Vacation came Henry Smith rejoined his mother and sister
in Italy. Unfortunately they arranged to spend the summer at Frascati in the
Alban Hills, and Henry soon became languid and ailing, and at last ill enough to
need a doctor. The doctor who came, an Italian physician of eminence, declared
after three weeks that his patient was undoubtedly consumptive, and ordered
him to the sea at Naples by way of the Pontine Marshes. Even in their alarm
the family were discreet enough to substitute the hill route for the deadly road
along the plains ; but this involved a four days' journey, during which the sufferer
became delirious, and, when Naples was at length reached, the English doctors
had all left the city. One however was to be found at Castellamare, and he,
X vi BIOGRAPHICAL SKETCH.
when he was called in, declared that the disease was nothing but long-neglected
malaria, which an ordinary Italian doctor should have recognised. It was now
thought right to revert to the use of strong tonics. Severe inflammatory attacks
subsequently came on. These have been, since then, attributed to the presence
of gall-stones, which may have possibly laid the foundation of his latest illness.
Moreover, with spring (1845), the malaria itself returned, and it became necessary
to leave the South. He himself at a later time described his illness to me as a sort
of euthanasia, in which he seemed to be gliding painlessly out of life. The sister
who helped to nurse him remembers that he was too weak even to put up his
glass that he might look at an eruption of Mount Vesuvius. Still he was able
to enjoy being read aloud to, and his mother used to read to him incessantly from
English newspapers and standard authors, but especially from the Latin and
Greek classics, while on Sundays he would honour the day by forbearing to cor-
rect or even to shudder at a false quantity. The move from Naples was to
Wiesbaden, and there the waters restored him to comparative health. It was
thought better however that he should not return to England, and accordingly
the next winter (1845-6) was spent at Paris, where Henry Smith attended
several of the courses at the College of France or the Sorbonne, and derived
especial advantages from the lectures of Arago and of Milne Edwards. By this
time his strength was thoroughly re-established, and, though he went a second
time to Wiesbaden in the summer of 1847, he had already resumed work at
Oxford (Easter, 1847), and never afterwards needed to suspend it. His health,
as I remember it for more than twenty years of unbroken intimacy, during which
I was constantly seeing him, was always good, though never what could be called
robust. He suffered especially as a young man from weak eyes ; and he had to
l>e a little careful of himself in diet and exercise ; but he was rarely depressed,
and he habitually worked beyond what most men could have endured without
breaking down. There was one attack of low fever in 1856, the result of course
of the earlier Roman fever, in consequence of which he was ordered to ride, and
the obligation to take horse exercise was undoubtedly very good for him, and
contributed a great deal to his enjoyment of life.
The Oxford into which Henry Smith was now thrown had almost recovered
from the strong ferment which ended in Newman's going over to the Church of
Rome. The leaders of the High Church party had either followed their captain,
like Christie and Bowles, or had satisfied themselves, like Mark Pattison, that
BIOGRAPHICAL SKETCH. XV11
Protestantism does not admit of a divided allegiance. There were still High
Church cliques among the undergraduates, such as Newman has sketched with
caustic subtlety in Loss and Gain, which discussed Church matters from the
Anglo-Catholic point of view, but they rarely got beyond a mild dilettanteism.
Even this was not always treated with a proper tolerance. I remember a de-
bating society of young Churchmen, which so irritated the Protestantism or the
common sense of a rather sporting College by carrying a resolution that ' St.
Augustine's interference with the British Church was uncatholic and uncalled
for,' that at its next meeting the orators were dispersed by the agency of hot
pepper thrown into the room, and saluted with a baptism little short of total im-
mersion as they left the quad. Of the fast men of that day, it need only be said
that they have been inimitably limned for good and bad in Tom Brown at Oxford.
Outside these two sets, which have perhaps attracted more attention than they
deserve, and also outside the common and obscure men, were the abler young
men of the University, Conservative or Liberal in their politics, as tempera-
ment or training determined, but mostly with a wholesome share of English
secularism, and neither High Church, except in rare instances, nor aggressively
Protestant, nor to any appreciable extent Freethinkers. Lord Salisbury, Lord
Kimberley, Lord Brabourne, Sir M. E. Grant Duff among politicians ; Goldwin
Smith, Sellar, Grant, Sandars, Poste, and Conington among men of letters or
scholars ; Spottiswoode and Rolleston among men of science ; Chitty among
judges ; Sandford and Ducane among officials, were some of the Oxford men of
Henry Smith's day, and with most of these he was more or less intimate at
some time, while Grant Duff and Conington were among his dearest friends.
Whatever time or thought men of this type could spare from work for the
schools, was divided between politics and literature ; and Henry Smith's Univer-
sity letters are a singularly faithful reflex of the spirit of the period. They are
more mature and temperate than perhaps any one but himself could have written,
but they show the enthusiasm for intellectual eminence which is the salt of
Oxford life ; and the admiration evinced for Mill, and the praise, however
qualified, of Robert Chambers, are evidence that the writer was already to be
numbered among the few on whom Carlyle had no hold.
Of Henry Smith's Oxford career it may briefly be noted that in 1848 he won
the Ireland University Scholarship, the blue ribbon of classical scholars ; was a
double first-class in the Lent Term of 1849 ; was elected Fellow of Balliol in
c
xviii BIOGRAPHICAL SKETCH.
November 1849 ; and gained the Senior Mathematical Scholarship in 1851. He
was unable through absence to stand for the Hertford Scholarship, which falls to
the best Latin scholar of the year ; or for the Junior Mathematical ; and he was
beaten for the Senior Mathematical Scholarship in 1850, the first time that
he stood for it, by Mr. Ashpitel of Brasenose ; the single defeat of the kind,
I believe, which Henry Smith sustained.
Balliol College, to which Henry Smith belonged, was far away the best
in the University during the time of his residence, and for some years afterwards.
A variety of circumstances had contributed to build up its pre-eminence. The
first cause was the far-sighted integrity of the old Master, Dr. Jenkyns, who was
almost singular among the Heads of his day in regarding it as the first duty
of a College to promote intellectual distinction, and who waged an incessant
war with privilege, abolishing gentlemen-commoners and throwing open close
endowments as far as he legally could. Dr. Jenkyns could not have done much
single-handed, but he gradually found or created men, often no doubt abler than
himself, who were glad to carry on his work in the same spirit ; and the late
Master of Balliol, Mr. Jowett, then one of the tutors, was undoubtedly the soul
of the College during the whole time of Henry Smith's connection with it.
At the time of Henry Smith's election, the College wanted a mathematical
lecturer. There is no doubt, I think, that he was chosen in the well-warranted
expectation that he would consent to reside and lecture. In this way began
his own lifelong union with Oxford, for until then he had been a mere bird of
passage. Having once decided to accept the office thrust upon him, he gave
himself up heart and soul to doing his work well.
It was a common story in Oxford at that time that Henry Smith, being
uncertain after he had taken his degree whether he should devote himself to
classics or mathematics, had solved the doubt by tossing up a halfpenny. His
sister remembers how he actually expressed a wish that some one would do this
for him. He was, in fact, the last man on earth to have committed any im-
portant decisions to chance ; and he has himself told me that his choice was
partly determined by the fact that having at that time weak sight he found
he could do more work in thinking out problems than in any other way without
using his eyes. The decisive reason was of course a pre-eminent genius for
mathematics the born aptitude that is itself fate and the cause why the
determination was made at that particular time may have been this offer of
BIOGRAPHICAL SKETCH. XIX
a lectureship. Nevertheless the Oxford tradition is so far valuable as it testifies
to the general belief that Henry Smith could have made his mark in any study
he embraced. ' I do not know,' Professor Conington once said to me, ' what
Henry Smith may be at the subjects of which he professes to know something ;
but I never go to him about a matter of scholarship, in a line where he professes
to know nothing, without learning more from him than I can get from any one
else.' Once it seemed as if he would be attracted into chemistry. The College
demanded of him that he should give chemical lectures (1853), and Henry
Smith accordingly became a pupil under Professor Story-Maskelyne, who then
occupied a laboratory under the Ashmolean Museum and gave instruction in
chemical analysis. Here H. Smith showed that in delicate manipulation and in
accuracy of work he possessed a sort of instinctive faculty. A lifelong friendship
grew out of the hours spent in this way ; although the demands of a new
and engrossing science on his time were too great to permit his sacrificing to
chemistry the many other important subjects and duties that filled up his life.
Even then, however, I remember, his idea was to seek numerical relations
connecting the atomic weights of the elements and some mathematical basis
for their various properties*, so that we might anticipate experiments by the
operations of the mind an ambition which was very interesting to Alexander
von Humboldt, when Sir M. E. (then Mr.) Grant Duff told him of it. Ultimately
Henry Smith of course found that science is too jealous a mistress to admit of
a divided allegiance ; and, though his reading was always wide and various and
singularly well digested, he practically devoted himself to mathematics, and as
I understand to two or three great subjects with which his name will always
be associated, the Theory of Numbers, the Theory of Elliptic Functions, and
certain new processes of Geometry.
One point for which the generations younger than Henry Smith and John
Conington will always remember them gratefully was the way in which they
mingled in undergraduate society. The distinctions of academical rank were
at that time rather jealously marked in Oxford. If the tutors and fellows were
* His conviction that such a numerical and mathematical basis underlay the phenomena of chemistry
was even stronger in the case of crystals. At my euggestion he undertook the discussion of the
principles involved in the parallelism of zone-axes and face-normals in a crystal system with rational
indices ; the results of which were given by him to the London Mathematical Society in vol. viii of its
Proceedings. N. S. M.
C 2
XX BIOGRAPHICAL SKETCH.
not, like the Heads of Colleges, in inaccessible isolation, they were scarcely better
known to us through the formal breakfast parties which were submitted to
on both sides as a very irksome duty. Here and there a man, like the
admirable Charles Marriott of Oriel, made it a duty to invite young men, in
order that they might feel at liberty to go to him if they needed religious
advice, which was never obtruded ; but men of this stamp were never, as far as
I remember, more than genial hosts. Henry Smith and Conington, men of the
most opposite temperament though devoted to one another, threw themselves
with such unaffected simplicity into our interests and occupations, that we all
came to regard them as personal friends, and to talk as freely before them
as before one another. Looking back I can see that their position was that
of very helpful and sympathetic seniors at a children's party, and I can conceive
how Smith's playful sense of fun and Conington's grave humour must often have
been tried by the obligation to treat our criticisms of men and things or our
forecasts of the future seriously. That the intercourse was begun and carried
on on their part from a conscientious desire due partly to Arnold's teaching
to convey a serious interest into everyday life, I at least cannot doubt. As one
who profited by the association let me record that meetings of this kind were
not only the most pleasurable part of a chequered Oxford life to many of us,
but unquestionably did more to stimulate thought and form character than the
more formal influences of the chapel and the lecture-room.
A letter from one of Henry Smith's old pupils, Professor Irving, of
Melbourne, will complete the description of this part of his career, and speaks
with authority on some matters which I only knew from report.
Melbourne,
1st September, 1888.
MY DEAR PEARSOM,
It is by no means an easy task to carry back the memory nearly forty years, and recall
scattered reminiscences of one from whom you have been altogether parted, with whom
you have not even kept up communication by letter for more than thirty years.
Yet such in the old Oxford days was my affection for, and so highly have I ever
honoured him whom I was then proud to call my friend, that I must accede to your request
and do what little I can towards your presentment to the world of Henry Smith, Scholar
and Mathematician.
My first introduction to him was, I think, in my Freshman's Term at Oxford, Michael-
mas 1849, the term in which he gained his Fellowship at Balliol ; but our intimacy really
BIOGRAPHICAL SKETCH. XXI
dated from the beginning of 1850, when he became our Mathematical Tutor. From the
first there had been this link between us, that I had inherited the little third-floor rooms in
the inner quad of Balliol, which had belonged to Scholar Smith, as he was called in those
days to distinguish him from sundry others of the same patronymic.
I remained his pupil in Mathematics throughout my undergraduate time, and was able
to do him the credit of winning the Junior Mathematical University Scholarship in 1850.
Of his power as a Class Teacher I cannot speak, as I was almost alone with him : but my
individual experience was that as a Teacher he was all a learner could desire, most patient
with all one's difficulties, most clear and full in his explanations. But he was too kind to
me. He sympathized strongly with my disappointment when the authorities refused me
permission to compete for the Ireland in the spring of 1850, and knowing how keen my
desire was to win that distinction in the following years, he did not force me to work
at Mathematics as I ought to have done, and so I failed to do justice to his teaching, and
attain his own class in the Final Honour Schools.
Through Smith I made another valued friendship, that of John Conington, Professor of
Latin, and might have gained access to the intellectual circle in which they moved, and of
which they were such brilliant ornaments. But I must honestly confess that my own work
for the Schools was sufficient mental exercise for me, and that I sought my relaxation, not
in other spheres of thought, but on the river.
Yet in our frequent intercourse there was quite opportunity enough for me to learn
and to appreciate the manysidedness of Henry Smith's mind. All of my generation were
prepared to look up to and to admire one who had done so brilliantly as he had in Oxford :
and when you came to know him personally you could not look upon that splendid fore-
head of his without assurance of the powerful intellect it betokened : you could not con-
verse with him without realising that he was one of whom it might be said that ' omne
scibile novit.' And what he knew, he knew, not as so much stored up learning to be brought
forth as required, but he had made it all his own, he had thought as well as read.
Still with all his vast erudition, and his great intellectual power, he was the humblest,
the gentlest of men. Ignorance, even if pretentious, was not to him something to be crushed
with sledgehammer Johnsonian blows, but a thing to be pitied and kindly enlightened.
In fact were I asked to select his peculiar moral characteristic, I should say he was the
most gracious man I ever knew.
Reading over these lines, I recognize with regret how very imperfectly are therein ex-
pressed the love and admiration I felt for him, how feebly they serve to set him forth.
Yet inadequate though they are, they may help somewhat. If I cannot do better
' His saltern accumulem donis, et fungar inani
Munere.'
I am, my dear Pearson,
Yours ever sincerely,
M. H. IRVING.
XX ii BIOGRAPHICAL SKETCH.
In 1855 some of the abler Oxford and Cambridge men determined to publish
a yearly volume of Oxford and Cambridge Essays, and Oxford led the way, T. C.
Sandars, I believe, being the guiding spirit and editor. Henry Smith contributed
an essay on the Plurality of Worlds to this publication. He took for his theme
WhewelTs then unacknowledged book on that subject, and Sir David Brewster's
fiery answer, ' More worlds than one, the Creed of the Philosopher and the Hope
of the Christian.' The subject was a fascinating one, and Henry Smith was in
many respects admirably calculated to do it justice. He wrote as simply and
clearly as Herschell or Tyndall, and he was skilful in dissecting arguments of
every kind with faultless impartiality, so as to reduce them to their absolute
value or no-value. A single passage will serve as an illustration of his method.
Whewell had argued in the spirit of Lucretius that there was, so to speak, a law
of waste traceable in the Divine economy, and supported this position by facts
from the origin of species. ' If we found,' Henry Smith remarks, ' that Jupiter's
four seasons differed so slightly from one another that they hardly deserved the
name, and that they could not be conceived to be of any use to his hypothetical
inhabitants, we should be reminded by those "rudimentary seasons" of the osteo-
logical facts on which the essayist dwells so much, of the rudimentary fingers in
the hoof of a horse, or the rudimentary paws with which a snake is said to be en-
dowed. But the one thing we should not be prepared to find would be a wasted,
imperfect, uninhabitable planet. We should know of no facts in Zoology with
which to compare such an occurrence. The crust of our earth is filled with the
remains of departed life, but we find not a vestige of imperfect attempts, of forms
moulded after the vertebrate type, and yet incapable of animation.' It will be
remembered that both essayist and critic wrote in the days before Darwin, and
that Henry Smith had never made any special study of comparative anatomy.
With this allowance it must, I think, be recognised that Whewell's conception
of a general law producing a single successful result and failing in every other
case was substantially and hopelessly wrong, and that his critic's conception of
' a law uniformly asserted in a multitude of individual cases, and uniformly pro-
ductive of variously perfect results,' was a singularly correct expression of all
that science was then able to teach.
There are passages of characteristic irony scattered through the essay.
We are told that in one of Plutarch's Dialogues, ' the lunar world is connected
with the future destiny of the human soul, after a manner which, we conceive,
BIOGRAPHICAL SKETCH. XX 111
Sir David Brewster would allow to be highly creditable for a heathen, and on the
whole corroborative of his own opinion.' Of the literary form of Whewell's essay
it is said : ' In the dialogue at the beginning of the essay the earlier letters of
the alphabet, who appear as the objectors, conduct themselves so much like
simpletons that one wonders at their being thought worthy of so long an inter-
view with the enlightened Z.' Of theological arguments intruded into the domain
of science it is observed : ' We cannot imagine a more painful spectacle of
human presumption than that which would be afforded by a man who should
sit down to arrange " in a satisfactory way " a scheme for the extension of
Divine mercy to some distant planet, and who, when he found "great diffi-
culty in conceiving" such an extension of the Divine attribute, instead of
desisting from his vain attempt, should go a step further still, and infer
that no such scheme can exist because he fails to discover a modus operandi
for it.'
Still, though the article on the ' Plurality of Worlds ' was read with
pleasure and spoken of with esteem, its success was not so unquestionable as to
tempt its author into larger literary work ; and his only other contributions of
any length to English prose are, I believe, a review of Mr. Freeman's Federal
Government, which he wrote for the National Review in 1864, and the Memoir
of Professor Conington, which was prefixed to the volumes of his works in 1871,
and which is a very perfect example of skill in recording a quiet life so as
to invest it with interest. It would have been a great misfortune for science if
a man capable of enlarging its boundaries had wasted his powers upon mere
criticism or exposition ; yet, considering Henry Smith's unambitious tempera-
ment, which made him careless of personal fame, and his invariable readiness
to oblige friends, I cannot doubt that he might have been seduced into Quarterly
Reviewing or some other form of ephemeral literature if he had possessed some
of the minor qualifications of a journalist. The fault of his argumentative writing
is a disposition to hold the balance and to avoid summing up ; and it is in
keeping with this quality that his style, though it has the subflavour of irony
and the point inseparable from lucid concentration, is not epigrammatic or what
would be called strong. The writer's tenderness of disposition had something to
do with this characteristic. He who as a boy of fifteen had stopped himself in a
caustic criticism in a private letter because the subject of it was ' somebody's
bairn ' carried the same thought for others into his words and dealings through
xx iv BIOGRAPHICAL SKETCH.
life. Keserve on matters that lay near the heart was another modifying influence.
To many men, the scholar who identified himself with every movement for religious
or intellectual or political freedom in the University was still more or less a
sphinx because he never propounded his convictions as a topic for conversation.
The world that reads is rather like the world that listens to a platform orator.
It likes its instructors to be positive, even where they cannot be certain, and to
have its conclusions presented to it in the form of short and simple aphorisms,
which it may swallow and retain without trouble. Henry Smith could not have
attained to this ideal, and it is matter of some satisfaction that he did not
aspire to it.
In 1857 Mrs. Smith died. The attachment of mother and son to one
another had been deeper than is common, and the course of their lives had
drawn them nearer together than can often be the case. It was now arranged
that Miss Smith should keep house with her brother in Oxford, the two
spending the term together, and each being allowed complete liberty of move-
ment during the vacations. I cannot doubt that this arrangement contributed
very much to Henry Smith's happiness. He was eminently domestic and
hospitable, and having the cares of household life taken off his hands, and being
supplemented by one who was almost another self, was able to fill his house
with friends, who were certain of an Irish welcome, however unseasonably they
might arrive to ask for a dinner or a bed. He was also able under his own
roof to gratify his passion for pets Persian cats of distinction and two aristo-
cratic dogs to which there are frequent allusions in his letters. During the
vacations he often visited the Continent, going once to Sweden and Norway ;
more than once to North Italy; to Spain with Grant Duff in 1864; and to
Greece hi 1872 with Mr. and Mrs. Grant Duff. From time to time he paid
visits to an old friend of the family, Miss Theodora Price, who had lived with
his mother during the whole time of her widowhood, and who, on Mrs. Smith's
death, established herself at Tunbridge Wells. For some years, too, Henry
Smith was a prominent figure at the various meetings of the British Association
in England, Scotland and Ireland. It will be seen that his life was in no sense
that of a recluse ; and it may be added that he entered with zest into every
form of social enjoyment hi Oxford, from croquet parties and picnics to dinners.
That the irregular, desultory life, with its frequent breaks, suited his health is
probable ; and, as he possessed a rare power of utilising stray hours so as never
BIOGRAPHICAL SKETCH. XXV
to intermit work altogether, even when his distractions were most numerous,
it seemed possible that he might be among the singular few who have combined
residence in an English University with unswerving devotion to the claims of
abstract research.
Fortune appeared to favour this anticipation. In 1860 Mr. Baden Powell,
the Savilian Professor of Geometry, died, and Henry Smith became a candidate
for the vacant post. Some years before, he had told a friend that to occupy
this chair was the great ambition of his life. He said that the two Savilian
Professorships were the most honourable offices in the University : they were
open to the whole world of Mathematicians, and had usually been held by
distinguished men. His wish was now to be gratified in the pleasantest way.
Those who would naturally have been his rivals, the other Oxford mathema-
ticians, were the first to draw back in his favour and sign a common
testimonial to his pre-eminent claims*. I well remember the generous warmth
with which two of the senior professors, Mr. Walker and Mr. Bartholomew
Price, but especially the latter, expressed themselves to me at the time
about Smith's undoubted genius and the chances that he would one day
leave a name to be remembered beside those of Newton and Laplace. The
electors for the chair chose him, as I have understood, without hesitation,
taking the view that as no other Oxford man was a candidate, and as Henry
Smith was pre-eminently qualified, it was needless to scrutinise the testi-
monials of outsiders. He himself was a little troubled by a doubt whether
the claims of an older man, Dr. Boole, of Queen's College, Cork, ought not to
have received further consideration. That Henry Smith justified his electors
by the magnificent work he did later on, is beyond question. He was
also a very successful teacher, having what must be considered large classes
in a University where mathematics have, at least in recent times, attracted
comparatively few students. Passages in his letters prove how keenly he
* I can express no opinion worth having on this subject ; but I see from a notice in the Academy
(Feb. 17, 1883), written evidently by a personal friend, that much of the work given to the world in
later years had been produced before he was thirty-five. ' He (Professor Smith) communicated
at different times a good many notes and papers to the Mathematical Society, especially during
his Presidency in 1874-76 ; and we believe that all the results he gave he had had in his possession for
fifteen years.' His work on the Theory of Elliptic Functions and the Introduction to Professor
Clifford's Remains belong however to the last seven years of his life.
d
BIOGRAPHICAL SKETCH.
promoted the well-doing of his pupils, and what his views were about the
reforms desirable in mathematical teaching.
The present volumes show what splendid contributions Henry Smith made
to science during the short twenty-nine years of his speculative activity. Never-
theless, it must, I think, be admitted that his unrivalled powers were often
employed upon work that scores of able men might have been found to do
efficiently, and which his friends should not have asked him nor he have consented
to undertake. From 1850 to 1870 he was Lecturer at Balliol, not being able to
afford to give up his Fellowship, and having scruples about retaining it if he did
not teach, as the number of Fellowships was limited and the stipend of a Lecturer
was too small by itself to remunerate any one for the work. It must be borne
in mind that during part of this time he was also Savilian Professor, and during
the whole of it he was constantly doing other College and University work*,
assisting backward men, or taking part in examinations, or serving on Univer-
sity Boards and Committees. In 1873 he freed himself from the worst of this
drudgery, the College Lectureship, by accepting a flattering and generous offer
from Corpus Christi College of a Fellowship upon that foundation f. Not long
afterwards he obtained the Keepership of the University Museum, left vacant
by the death of Professor Phillips. The office gave him a pleasant house, a
small stipend, and not very uncongenial duties, half as master, half as servant,
' The Master and Fellows of Balliol College, for instance, once asked him to give a course of lectures
on the Schoolmen ; and he complied.
t A friend at Ba'liol writes : ' We knew perhaps better than others how necessary this relief was
to Henry Smith, and we rejoiced that it had come to him ; we knew likewise the perfect loyalty
toward) his old College which prompted his resignation. Nevertheless, it was a grievous thing to us
that he should be obliged to leave our body. Never had we felt so bitterly the difference between a
poor foundation and a rich one. Henry Smith, as Steward of Common Koom, was our chairman
on social occasions, especially at our annual " gaudy " on St. Catharine's Day. The last speech he made
in this capacity was immediately before his migration to Corpus. He assumed a playful tone, and tried
to amuse us by various quaint comparisons into forgetting our loss, but he was quite unable to subdue
his own emotion, and he was weeping himself before he had made us laugh. This was the only time
that I ever knew him break down. Though ceasing to be a Fellow, he continued to give us the benefit
of his presence and counsel at our College meetings. By the next St. Catharine's Day the keen and
constant interest which he took in our affairs had somewhat reconciled us to the change, and this feeling
was warmly expressed by the Master in proposing Henry Smith's health. I remember the Master's con-
cluding words, which struck me at the time as a note of warning, and which have now a sadder
significance : " I will only venture to express the hope that he will not suffer himself to be numbered
among those men of varied powers and charming manners who have given up to society and business
what was meant for science and posterity." '
BIOGEAPHICAL SKETCH. XXV11
which sate lightly upon one who was genial and full of instinctive tact. Never-
theless, it cannot be said that his work was sensibly lightened for any long
time. Partly, he was himself to blame. He had a speculative element in his
nature, and had invested so much money in mines almost always, I am afraid,
unremunerative that it became important now and again to eke out his
regular income. I remonstrated with him very strongly, when he added the
duties of Mathematical Examiner at the University of London to his other
heavy work (1870), for he seemed to be breaking down at the time he undertook
it, and I felt sure that whatever he did for the day's need was so much taken
from more enduring labours. It seemed however as if the world was in a con-
spiracy to force duties of every kind upon one whose talent was so flexible and
whom men of all opinions agreed to welcome as a coadjutor. He was for years
a member of the Royal Commission on Scientific Education, having been
appointed in 1870, and he drafted a large portion of its report. In 1877 he
became a member of the Oxford University Commission under Lord Salisbury's
Act ; and in the same year he agreed to be chairman of the new Meteorological
Office, the governing body of which was practically nominated by the Royal
Society. This latter work was specially congenial, and the associates were so
considerate and able as to give a charm to toil ; and Henry Smith enjoyed the
fortnightly visit to London, and the temporary rest from the turmoil of Oxford
business. Still, when all is said, it can hardly be doubted that the labours of all
these various offices meant a partial interruption of nobler toil and may have
hastened a premature death.
It may perhaps be said, and not without some truth, that those who knew
of the condition of his health should have refrained from heaping work upon
him and should even have compelled him to take a long term of real rest. But
in fact these demands came on him from several and distinct quarters, and
what might seem to each person or group of persons making the demand
a light and congenial undertaking for the always gracious counsellor of ' golden
speech,' became, when added to the aggregate of such undertakings, a serious,
perhaps even a fatal burden. The truth is that his presence was always
welcome on Boards and Committees ; for he possessed the rare gift of suggesting
some middle course which would often effect an agreement between persons who
had been advocating opposite points of view, and of so bringing about a welcome
end to a weary discussion.
d 2
xxviii BIOGRAPHICAL SKETCH.
It will be a melancholy satisfaction to some of the friends with whom Henry
Smith went on holiday expeditions, to reflect that these intervals of rest were not
only periods of unmixed enjoyment to him, but probably helped to prolong his
life. The letters he wrote from Greece in 1871 will not bear reproduction, but
are full of the pleasure he experienced. 'We certainly had a most perfect
voyage.' ' Except Pylos, I don't think we missed seeing anything we could have
seen. Grant Duff, in a place he has not seen and wants to see, is quite perfect ;
and we shall now work morning and night, till we have done Athens well.'
' All that is in Athens we have done to the greatest perfection,' is the
comment in another letter. If only excursions of this kind could have been
more frequent! The last I find commemorated is a visit to Rome in 1879,
when Henry Smith represented the Meteorological Council at the International
Meteorological Congress.
In 1878 Mr. Gathorne Hardy, who then represented the University of
t ).xf<>rd, was raised to the Peerage ; and the Oxford Liberals determined to bring
forward Henry Smith for the vacant seat, in the hope that his great personal
popularity and unrivalled academical reputation might win over many votes
from moderate Conservatives. Moreover Henry Smith was not emphatically
opposed to the Jingo or war policy of the Beaconsfield Ministry ; the test
by which Conservatives especially weighed politicians in that particular year.
He did not expect success, and he hardly desired it, but he would not
shrink from a fight if he was asked to stand forward as the representative of
a principle. I am told his friends were sanguine of success for a time. Friends
are bound to be ; but no sane looker-on could have anticipated any other
result than that which actually took place, that the Conservative candidate
would be elected by an overwhelming majority. I have never felt that, in this
particular instance, the rejection of an eminently good and wise man was uncon-
ditionally to be regretted. Personally, I have no sympathy with the doctrine
that scholars are out of place in Parliament a doctrine which would have
excluded Macaulay, Gladstone, Cornewall Lewis, Goschen, Fawcett, Grant Duff,
Morley, and Bryce among office-bearers of recognised ability, as well as Grote
and Mill and a host of others who have added distinction to tho House of Com-
mons although they never attained to office. I am convinced that Henry Smith
would have been as popular in the House of Commons as he was everywhere
else, would always have been listened to when he spoke, and would have spoken
BIOGRAPHICAL SKETCH. XXIX
with effect. Still I cannot persuade myself that his magnificent powers could
have been adequately employed in debating or administering : and I am
certain that dozens of inferior men would have played their part as usefully in
St. Stephen's ; while there was no one but himself in England so peculiarly
fitted to increase knowledge in one very difficult and abstruse department
of enquiry.
If however Henry Smith had ever gone into Parliament, he would have
been something more than a representative of learning or even of academic
Liberalism. He had a genuine sympathy with the poor of his own land ; and
his last public appearance anywhere was in the Oxford Town Hall to support a
resolution by Mr. Arch in favour of giving the franchise to the agricultural
labourer. The speech then delivered will bear reproduction for its own merits,
and is a good specimen of the speaker's style ; that of a man thinking aloud in
simple words, yet with an instinctive perception of rhetorical effect.
Professor Henry Smith said it was as a Liberal that he would say a few words with
reference to the resolution before them. In the course of his life he had found himself
sometimes on the extreme left of the Liberal party, sometimes verging towards its right
for every party had a right and left and pretty often about in the middle of it. He was
bound to say that his belief was, in the first place, that the whole of the Liberal party, right,
left, and middle, was unanimous in thinking that the National Agricultural Labourers'
Union had rendered a great service to the United Kingdom. He further believed that the
whole of the Liberal party rejoiced to think that the great benefit which that Union had
conferred upon the agricultural labourers of this country would remain for ever associated
with the name of Mr. Joseph Arch. He would endeavour to support the resolution by an
argument different a little from those they had heard. The extension of the household
franchise to the counties was inevitable ; whether they liked it or not, it was a thing which
must be done. He believed there were but few men in this country sensible men, men
who looked at what was around them, and who listened to what was said but felt that it
was inevitable. He was one of those who, when it was clear that a thing must be done,
believed that the sooner it was done the better ; and if it were for that reason only, he
would heartily support the resolution. But, in addition to that, he did believe that the
extension of the franchise to the great classes who now were excluded from it would, as
had been well put before them already, exercise a beneficial influence upon the future
course of their legislation. It might be true that some persons might ask what would the
agricultural labourer and the rural artisan do with the franchise when they got it ? They
would do like other people. He feared they would do some mischief, for he knew no class
of his countrymen among whom there were not some who did mischief with any right that
was entrusted to them, but he firmly believed on the whole they would exercise the franchise
XXX BIOGRAPHICAL SKETCH,
for good. If ho were told they would exercise the franchise for selfish objects, for objects
peculiar to their own class, he would say let it be so ; but if so, what would they say to the
..ther classes who already possessed political powers? Could any one of them send a
representative who could say that his hands were free from selfish legislation? If they
must count for, alas, they must upon some strain of selfishness in their common nature,
at least let them take care that their representation was not one-sided, but that at any rate
each class had a fair means given it of defending itself from others. It was for these
reasons that he for one most heartily supported the resolution. He hoped to see the
household franchise extended as rapidly as possible to the counties, and he was not one
of those who shrank from a still greater consequence which would come when that great
measure of enfranchisement should be followed by an equally sweeping measure of redis-
tribution of seats.
From that platform Henry Smith went home to die. Overwork and sedentary
work had gradually undermined his constitution. When I last saw him in 1879
he still looked well, and in some respects, from having filled out a little, less
delicate than as a young man, but I noticed that he was less capable of sustained
effort. In 1881 premonitory signs of a break-up of the constitution showed
themselves, and were unhappily not heeded as they should have been. First he
suffered from his digestion, and had to put himself under Sir H. Thompson's care ;
and then a stoppage in one of the veins of his leg confined him for many months
to the sofa, and made all but occasional carriage exercise impossible. He seemed
to be tiding over this illness, when a rush of University work threw him back
again into the condition of an invalid.
When he spoke at the Town Hall meeting he was suffering from a cold.
The exposure and excitement were followed by congestion of the liver, which
was the more dangerous after the severe attacks which had followed on the
Roman fever, from which he suffered in 1845. On the morning of Thursday
(February 8) there seemed to be a change for the better, but at noon the worst
-\ mptoms returned ; and Sir William Gull, who had been telegraphed for and
who arrived about eight o'clock, held out little hope. About four o'clock next
morning (Friday, February 9, 1883), the patient's state was declared desperate,
and three hours later he passed painlessly away.
He was buried (writes a friend who was present) at St. Sepulchre's Cemetery
in Oxford on Tuesday, February 13. So great a concourse of undergraduates
us well of senior members of the University and friends and strangers from
a distance has rarely been seen on an occasion of the kind in an English
BIOGRAPHICAL SKETCH. XXXI
University. An academic funeral is always an impressive spectacle, and the
long line of the procession and the scene in St. Paul's Church and around
the grave in the cemetery will never be forgotten by those who were present.
For once there was no discordant criticism over a grave. Not only did all
agree to speak with tenderness and admiration of the dead man, but there was a
singular consent of opinion as to his character and pre-eminent intellect. The
funeral procession that carried him to his resting-place was nearly a quarter of a
mile in length, and included every man of position or note in Oxford and many
others distinguished in their various ways who had come from every part of
England to pay the last honours to a dead friend. The Times wrote of him
as 'one of the most remarkable men of his day,' and the Spectator declared
that ' it would be difficult among the world's celebrities to find one who in gifts
and nature was his superior.' ' Some of us,' said Professor Stubbs (now Bishop
of Oxford), in a University sermon, ' can remember the youth of brilliant promise,
of almost unparalleled achievement in all our studies ; all of us have before our
eyes the manhood of indefatigable energy, of most generous devotion, of most
kindly and effective sympathy with all good work ; the entire expenditure of
consummate accomplishments and of every bright gift on the work of Oxford.'
Perhaps however no words were more frequently before men's eyes or in their
thoughts in connection with Henry Smith's death, than a tribute which Sir M. E.
Grant Duff had once paid him in the House of Commons, in commenting on his
nomination as one of the Oxford University Commissioners. He said :
'Professor Henry Smith is not merely in the first rank of European mathematicians,
but he would be a man of very extraordinary attainments even if }-ou could abstract from
him the whole of his mathematical attainments. He was the most distinguished scholar
of his day at Oxford. . . . But Professor Smith's extraordinary attainments are the least
of his recommendations for the office of Commissioner. His chief recommendations for that
office are the solidity of his judgment, his great experience of Oxford business, his services
on the Science Commission, and his conciliatory character, which has made him perhaps
the only man in Oxford who is without an enemy, sharp as are the contentions of that very
divided seat of learning.'
To myself, who am no mathematician, and who therefore cannot estimate
Henry Smith's intellectual power in the departments where it was highest,
it has seemed also, as it seems to Sir M. E. Grant Duff, that I have never known
his equal or perhaps one who could be classed with him. What always im-
X.\X ti BIOGRAPHICAL SKETCH.
me however was not so much his marvellous versatility or his thorough
mastery of everything he touched, or his conversational brilliancy though none
of all these can be separated from my recollections of him as his singularly clear
judgment, combining insight into the essential truth of whatever he examined
and balance in the summing up of it. Never did genius more completely take
the form of sublimated common sense ; and this effect was undoubtedly enhanced
by his unassuming manner. What he had to say was never thrown into a
doctrinaire form, half dogma, half epigram, but was stated in the simplest
possible words. Sometimes no doubt an opinion given in this way would attract
less attention than it deserved and it would certainly be less effective than
a brilliant paradox. Gradually but surely those who met Henry Smith, or who
came to him for counsel, perceived that his insight was unerring, and learnt
to defer to his judgment, the less reluctantly as ' he had the great art of never
pressing a victory home, and of bearing defeat with pleasant equanimity *.'
Perhaps it was this faculty of judgment which kept him from being over-
weighted with his learning. He had read many books which even scholars
rarely open, and he never forgot what he had once read. I remember for in-
stance how he gave me on one occasion a most amusing account of the Letters
of Synesius, which Kingsley's Hypatia had, I think, induced him to look up.
His knowledge of Protestant Hymnology was curiously intimate and wide :
and, when he assisted a friend to compile the University Hymn-book, his
recitals from memory of whole hymns by Wesley and others impressed those
who were present as very remarkable. Even his private friends, however, only
learnt by rare glimpses what his acquirements were ; and in general society,
though he never affected to be other than a scholar, he impressed those who met
him as a man of the world with perhaps unusual cultivation.
His friends sometimes compared him to Pascal, with whom he had many
points of resemblance, the combination of mathematical and general ability,
a keen wit, an extreme reserve, and an unfortunate Tack of personal ambition.
There was, however, one remarkable difference. Pascal, who has recorded the
opinion, ' Diseur de bons mots mauvais caractere,' meaning, I suppose, that an
epigram is a truth pared to a point and twisted into a barb, was yet seduced by
his genius into endowing the world with a book that scathed and blasted the
* Spectator, Feb. 17, 1883.
BIOGRAPHICAL SKETCH. XXXlll
cause and the men he assailed. Henry Smith's tenderness of feeling interfered
with his command of literary form. He had a feminine instinct for avoiding what-
ever would give pain, and never allowed his buoyant spirits to betray him into
a word that might seem harsh, or his inimitable persiflage to pass the boundary
line into sarcasm. Those who heard him talk were conscious of wit that played
round every subject with a perpetual sparkle, and that left a delicate aroma
behind ; but no one ever knew it employed as a weapon of offence. Reading
over his private letters I find the same kindliness, the nearest approach to
personal satire being perhaps the description of a heavy dinner, 'with four
pieces de resistance, not including X and Z,' two rather overwhelming talkers.
Therefore if Henry Smith had ever written on any of the subjects on which
he felt strongly and he was an ardent Liberal on every University question
and on almost every political topic of interest I cannot doubt that he would
have adopted a style of earnest simplicity, and would have trusted for effect
to argument, enhanced at most by a restrained eloquence. Bearing in mind
that he was confessedly one of the most brilliant talkers of his day, so that
every obituary notice dwelt lingeringly upon this trait, and considering how
easily the playful but keen humour might have been transformed into caustic
satire, I can only wonder at the mixture of kindliness with strong self-
discipline that prevented even an occasional lapse. Both in this matter and
in his judgments of men and things, a singularly fine character gave law to the
intellect. He was clear and just in expression because he was accurate and
truthful in thought ; he was irreproachable in speech, because he never allowed
himself to cherish an ill-natured thought.
Any one who has been often in the society of brilliant talkers can fiardly
have failed to notice how little of the best conversation is of a kind to bear
record or is practically remembered. Dr. Johnson was singular in attracting
an unrivalled biographer, who took notes unblushingly, and was skilful enough
in literary form to polish up what he took ; and Sydney Smith's fertility was
so great that some of his mirth has survived him : but of George Selwyn and
Luttrell, of Fox and Canning, of Macaulay and Bagehot, we know disappointingly
little. Two or three trifling instances may serve to show what Henry Smith's
manner was. He was once winding up a mathematical lecture by explaining
a new solution of an old problem. ' It is the peculiar beauty of this method,
gentlemen,' he concluded, 'and one which endears it to the really scientific
XXXIV .BIOGRAPHICAL SKETCH.
mind, that under no circumstances can it be of the smallest possible utility.'
He was obliged to pass through France in 1870, when fortune had just turned
against the French armies, and the cry of treachery was raised everywhere.
A guard noticed the tall Englishman with a blonde beard and spectacles, and
instantly denounced him as a German spy. A suspicious crowd collected in
a moment around the carriage. 'Gentlemen,' said Henry Smith with an
amused smile, ' I speak French very badly, but not I hope with a German
accent.' The proof and the speaker's impressive serenity carried conviction,
and the crowd melted away. ' You take tea in the morning,' was the remark
with which he once greeted a friend, ' If I did that I should be awake all day.'
A friend mentioned to him the enigmatical motto of Marischal College. ' They
say; what say they; let them say.' 'Ah,' said Henry Smith, 'it expresses
the three stages of an undergraduate's career. In his first year he is reverent,
and accepts everything he is told as inspired : " they say ; " in his second year
he is sceptical and asks " what say they ? " and " let them say" expresses the
contemptuous attitude of his third year.' At a time when English society
was perhaps extravagantly fluttered by Lord Beaconsfield's apparent success
at the Berlin Conference, Henry Smith reduced the event to something like
its proper proportions. ' Dizzy,' he said, ' has taken John Bull to Cremorne,
and the old gentleman is rather pleased to have been there.' On the news that
a distinguished friend, who was also markedly pessimist by temperament, had
been appointed to a high post in India : ' How fortunate ! ' was the remark ;
' it will give him another world to despair of.' He summed up X, a brilliant
writer but inconsecutive thinker, in the criticism, ' X is never right and never
wrong ; he is never to the point.'
It is sometimes said of loveable men, that they diffuse their affections
BO evenly as to be incapable of strong personal attachments. With Henry
Smith to be a friend once was to be a friend for life. The masterly biographical
sketch which he wrote as an introduction to Conington's Miscellaneous Writings
will give a measure of one friendship that lasted from school days till it was
interrupted by death. Professor Rolleston, who could hardly ever speak of him
without some epithet such as ' the golden-mouthed,' confided his family when
he died to Henry Smith's care, and the trust was accepted and discharged
with exemplary fidelity. Probably no other great student was ever so ready
as he always was to put aside books and papers when a friend entered the house.
BIOGRAPHICAL SKETCH. XXXV
Yet his nature, genial and hospitable in the extreme, was not what is called an
effusive one. He has noticed in the life of Professor Conington, that that great
scholar, after he underwent a spiritual conversion, used to speak of his experi-
ences unreservedly though in the simplest language. On this as on every
subject of delicacy, Henry Smith was absolutely reticent. He would discuss
religious topics if they were started as matter of interest, but I never knew him
talk of his own faith, and I should be slow to believe that he ever did. My
impression is that he accepted Christianity not only as a habit and a conviction,
but as a rule of life, and in fact his character can scarcely be explained, except
by ranking him with those who feel that they are ' ever in the great Task-
master's eye.' I think he regarded much popular theology as irrational, and
much fashionable doubt as a mere winnowing of chaff. Some of the weightiest
words I ever heard from him were on religion. Beyond this I can say and
surmise nothing, and his letters are not more unguarded than his speech was*.
The one question as to Henry Smith's character that appears to be still un-
decided is, whether his inaptitude for self-assertion, his scorn of personal ambition,
his severe acceptance of duty in whatever shape it came to him, are to be
regarded as blemishes or excellences. The distinguished friend who wrote about
him in the Spectator^ has shown in thoughtful and wise words how much there is
admirable in the 'philosophic life' 'life of exemplary moderation, far removed
from even a suspicion of worldliness and vanity.' ' Great moral gifts,' as the
writer goes on to say, ' can be found when occasion demands them ; talents grow
on every tree. But the serenity of heart which enables its possessor to wear the
gifts of genius with sobriety, and to use them nobly and well, without seeking to
expend them in the purchase of fame, or wealth, or of advancement, is a quality
which modern society little cultivates and seldom sees.' It may seem to
those who ponder this temperate and lofty apology, that it is a sufficient answer
to the regrets I have freely expressed in this sketch over genius that was
often lavish "of itself on work for the moment's need or work of ordinary compass.
Let me say for myself and for those who think with me, that we never desired
wealth for Henry Smith except in such measure as might free him from sordid
* He on more than one occasion spoke to me on these subjects. His position was perhaps most
simply expressed in a conversation in the course of which I remember his saying that the essential
features of the Christianity held to-day were held in the time of Justin Martyr. N. S. M.
t The article is printed at p. xlvi.
e 2
BIOGRAPHICAL SKETCH.
necessities, or fame except as a recognition of what he might achieve ; and that
the world's opinion or the state's honours could not have raised him in the
estimation of those to whom he was already above all men. Our feeling about
him was essentially what Newton expressed when he said that ' if Cotes had
lived, we should have known something.' It is very possible that those who
saw how much of his time Henry Smith gave to Examinations, and Boards, and
Commissions, and who unconsciously estimated the range of human effort by
their own measure, did injustice to the special capacity I have noted in him
for carrying on consecutive work in stray moments. It has been said by one
who can speak with authority on such a subject that Henry Smith was ' the author
of mental achievements in the most abstract and complicated of the sciences,
which will rank as scarcely second to any in the century.' On this matter the
collected works will be conclusive evidence. What, however, surviving friends will
and must feel is that genius is as rigidly bound to husband its powers as mere
capacity, and that nothing can be spared from the supreme work of life without
loss *. Prove to us that Henry Smith's work was indeed scarcely second to any
in the century, and we are constrained to assume that, had his energies been more
severely economised, it must have been second to none. Certainly the ordering of
these things is not in our hands. He who gave the perfect intellect gave also
the fine temperament, the tenderness that shrank from disobliging, the modesty
that esteemed no duty undignified, the absolute disregard of self. To us who
knew him, let me repeat, the man was always greater than any possible work he
might do, though we set no limits to its possibilities ; and to us the ever-green
wound of his loss is partially compensated by the remembrance of an ideal
character. What we grieve for is that generations that did not know him
as we have known him will try him by the only standard possible, that of
his completed work, and will give him less than the measure of his real
capacity, though they can never refuse to number him among the great names
of the century.
' I may add, however, to this that, frequently as I urged on H. Smith to turn a deaf ear to some of
the too many supplicants for his time, and to give up some of his less important occupations, his answer
tome always was that lie did ' get all that he could out o himself as it was ; that in truth his greatest
work could only be done now and then, and could not be reeled off the mind indefinitely. Much
interval was necessary to him. N. S. M.
RECOLLECTIONS OF PROFESSOR SMITH.
I. BY PROFESSOR JOWETT *.
MY recollections of Professor H. J. S. Smith extend over about 40 years.
I first heard his name mentioned in the year 1843, by the late Archbishop
of Canterbury, then recently elected Head Master of Rugby, who told me that
there was a boy in the School quite deserving of a place by the side of Conington
and Walrond, who were the great names of Rugby in those days. He was going
to try for the Balliol Scholarship. At the end of the year, on that occasion, he
beat all the other candidates, of whom one was the late Sir Alexander Grant,
elected Scholar at the same tune with him. I remember him in the viva
voce part of the examination, a youth of eighteen, rather overgrown and stiff, as
youths of eighteen are apt to be, construing Latin and Greek authors in a pious
and evangelical tone of voice, which provoked a smile in the Examiners, but with
never-failing accuracy. The old Master, as we used to call him, took up his
English Essay and showed it to me, saying, in his emphatic way, ' There 's mind
in that.' The subject given for Latin Hexameters at that examination was
the Pelasgi, of whom he did not forget to mention in his verses that 'they
worshipped nameless gods.' Meeting Arthur Stanley on the Woodstock Road
the day after the election, he congratulated me on our having chosen a youth
whose fame had preceded him at Oxford.
He more than justified the promise which he had given. Though not a poet
or creative genius, he was, I think, possessed of greater natural abilities than
any one else whom I have known at Oxford. He had the clearest and most
lucid mind, and a natural experience of the world and of human character
hardly ever to be found in one so young. He took up all subjects at the right
end ; he knew whereabouts the truth lay even when he was imperfectly ac-
quainted with the facts. And he was the most amiable and good-natured of
* Professor Jowett's death has deprived these ' Recollections ' of the author's final revision.
XXXVlii RECOLLECTIONS OF PROFESSOR SMITH
young men. I might apply to him the words in which Plato describes the
youthful Athenian Mathematician, Thesetetus, where he says : ' In all my
acquaintance, which is very large, I never knew any one who was his equal in
natural gifts. He had a quickness of apprehension which was almost unrivalled,
and he was exceedingly gentle. There was a union of qualities in him which
I have never seen in any other, and should scarcely have thought possible, for
quick wits have generally quick tempers . . . but he moved surely and smoothly
and successfully in the paths of knowledge and enquiry. He flowed on silently
like a river of oil. At his age it was wonderful. He was also surprisingly
liberal about money, though his fortune was only moderate' (Thesetetus, 144).
The facility with which the youthful Scholar of Balliol picked up all sorts
of knowledge was equally wonderful. During the first two years after his
election to the Scholarship he was struck down by a serious and almost fatal
illness, and did not come up to Oxford until what is usually the third year of
residence had commenced. In the interval, residing in Italy he acquired a con-
siderable knowledge of Roman inscriptions and antiquities, and also of modern
languages. Within the year and a half which remained of his Undergraduate
course he obtained the Ireland Classical Scholarship, and a double first. Two
years later, the Senior Mathematical Scholarship was awarded to him. Similar
honours have only been gained by one other person the late Very Rev.
G. H. S. Johnson, Dean of Wells, an eminent man, but little known, who, from
ill-health, was unable to do justice to his great natural talents.
After he had taken his degree, he at one time thought of going to the Bar,
for which he was very well suited he would have risen rapidly to the high
places of the profession. But the feebleness of his constitution when a young
man led him to abandon this intention, and he soon settled down in the post
of Mathematical Lecturer at Balliol College. The other tutors were the late
Rev. E. C. Woollcombe, Dr. Lake, the present Dean of Durham, and myself.
The Bishop of London (Temple) who preceded him in the office had just left us.
In those days he was almost equally a lover of Classics and Mathematics. There
was a time when he was quite divided in his allegiance between them, and used
to say, in his free and easy way, that he ' must toss up a shilling to decide.'
Even in the last years of his life he was in the habit of taking with him Greek
books to read during the Vacation. In conversation he left tlie impression of
being a well-read scholar, and a real critic, who was never led away by ingenious
conjectures or uncertain fancies. For some time he was intending to edit the
Timseus of Plato for the Clarendon Press, but he never had leisure to carry out
BY PEOFESSOR JOWETT. XXXIX
this project. He finally determined, and probably he was right, to make
Mathematics the chief work of his life.
The Mathematician is more cut off by his pursuits from his fellow-men than
the student of any other branch of knowledge. He has interests which are
locked up in his own breast, pleasures and also pains which he cannot communi-
cate to others; the better part of him is moving about in a world of numbers
and figures which have no connection with ordinary life (cp. Plato, Thesetetus,
1 84 D). His study is apt to become a passion with him and affects his character.
I am sure that this was true of Professor Henry Smith. It was the smaller part
of him which we knew or could appreciate. His mathematical speculations could
have been shared by a very few, not more than two or three, of his contempora-
ries at Oxford. Yet he did not withdraw himself from business or society. He
was not the silent philosopher who is lost in reverie, or who, while acknowledged
to be a mathematical genius, is pointed at by mankind as a poor and eccentric
mortal. He was a thorough man of the world and greatly liked by everybody.
He was very manly in his bearing, and quite free from shyness and nervousness
in any company. He had a kind greeting for servants, and felt a real kindness
for them they were devoted to him. His manner of behaviour towards all
sorts and conditions of men might be described as exhibiting a singular 'urbanity.'
He was decidedly good-looking, and there was a certain intellectual distinction
in his features and expression. It is necessary to combine these various aspects
of him if we would duly estimate him. He was everywhere, and known to every-
one, the life and soul of a social gathering. But he was also a thorough student,
and an omnivorous reader, passing several hours of the day in abstruse Mathe-
matics, but nevertheless acquainted with all new books, and on a level with
every recent scientific enquiry.
He went on teaching at Balliol College as Mathematical Tutor for about
thirteen years ; at the end of that time he was appointed Professor of Geometry ;
he then combined the duties of Tutor and Professor. While only a Tutor of
Balliol he had hardly any pupils worthy of him. The College, having at that
time no Mathematical Scholarships, had seldom any good Mathematical
students (those who were being usually men who read for double honours). His
duties were, for the most part, confined to the preparing and examining men for
Pvesponsions. But he never thought it beneath him to take pains with any one,
and he was an admirable teacher. He used to have his pupils on a Sunday
afternoon to be examined by him, and would tell them that 'it was lawful on
the Sabbath Day to pull an ass out of the ditch.' The better men were of
xl RECOLLECTIONS OF PROFESSOR SMITH
opinion that they learned more from him in a few minutes than from another in
a whole hour. He was constitutionally apt to be late and irregular in lecture,
and on occasions of business as well as at a dinner party was often the last
to arrive, but every one was very willing to wait for him. The circumstances
of the University hardly admitted of his raising up a School of Mathematical
pupils, but he was the life and centre of the study while he was with us.
He was very desirous to promote the interests of Natural Science in Oxford,
and was in favour of some measure which would have made the knowledge of a
portion of some one of the Natural Sciences the condition of obtaining a degree.
The teachers of these sciences had long been fighting a battle against the older
traditions of the University ; they had now become the study of a few, but he
clearly saw that they could never truly flourish until an interest in them was more
generally diffused and they had a congenial atmosphere. But he was also the best
friend that the older studies then had in the University, for he could speak with
authority, and he was firmly convinced that in education Science should not
supersede Literature. He deplored equally the want of literary culture which
he observed in many scientific men, and the gross ignorance of the most general
facts of Science which prevails in the world at large, especially at English Uni-
versities and Public Schools. In a similar spirit he was anxious to encourage at
Oxford the study of Medicine and also of Engineering, thinking that they would
supply a missing link between the Physical Sciences and the older studies of the
University.
A considerable portion of his time was devoted to College and University
business. Though he transferred his name to C. C. C. about ten years before
his death and nominally ceased to be a member of Balliol College, he con-
tinued to show the same earnest interest in its concerns which he had always
done. He took the same part in its Examinations and College Meetings the
only difference being that he no longer received the stipend of a Fellowship from
it. There was never any one more affectionately regarded by the Fellows, or
whose opinion had greater weight with them. He had the art not only of doing
business well, but of making it pleasant, often with a slight jest or play of words
smoothing away difficulties. I do not remember his ever having had a quarrel or
difference with any one in the University. It will be easily understood that such
a man was well adapted to keep men together and to carry things forward. At
the Hebdomadal Council, where he usually appeared rather late in the day, he
gave life and animation to every discussion. He seemed to say things in a better
way than anybody else, and in an argument there was no one who was a match
BY PROFESSOR JOWETT. xli
for him. When a new measure had been put into form by the Council he was very
often selected to carry it through Convocation, his popularity and his manner
of speaking having great weight in that assembly ; and it was whispered that
' the Council relied for the success of their measures too much upon Henry
Smith's oratory.' Though well aware that the order and discipline of the Uni-
versity must be maintained, he was always a very earnest supporter of freedom,
and a great enemy to the imposition of useless restrictions upon Undergraduates.
He was indulgent to the failings of young men, and felt a humane pity for
persons who had lost their character. He was one of whom it might be said
that ' he would have stood by a friend, not only in adversity, but in disgrace.'
Two occasions on which he distinguished himself were long remembered by those
who heard him, once in the Common Room, more than thirty years ago, when
some of the elder members of the College sought to impose a new-fangled test
upon the undergraduates instead of the time-honoured Thirty-Nine Articles.
He pleaded earnestly for the retention of the latter, alleging that ' old chains
were smoother and easier to the wearer of them.' The other occasion was in
Congregation, about twelve years ago [1880], when he introduced a measure
granting privileges to Colonial Universities, and drew a sketch of the growth
of the London University, and of the mistaken policy of Oxford and Cambridge
in their opposition to it.
He was not an orator, but a very good speaker, who had the faculty of
thinking when on his legs, never faltering for a word, able to strike out, right and
left, good-humoured and telling blows. His speeches were clear and luminous,
and they also had the merit of keeping up the attention. Above all, he had
tact. He said what he ought to have said, and abstained from giving needless
offence. As a writer, he never attained to considerable eminence. He was the
author, when quite a young man, of a very clever review in the Oxford Essays
of Sir David Brewster's ' More Worlds than One.' In this paper the fallibility,
both of men of science and of theologians, was impartially exposed, and I re-
member that Bishop Temple remarked, after reading it, that ' the author could
do many things well, but that he would write better than he did anything else.'
The prophecy was not destined to be fulfilled. His mind was drawn in
another direction, and he had not the poetical gifts which seem to be indis-
pensable in a great writer, whether of poetry or prose.
He was wanting in initiative. Though a very able supporter of the plans of
others, he rarely, if ever, took the first step in introducing a measure himself.
He was easy-going, not burnt up with a fiery zeal for change, but satisfied in
f
xlii RECOLLECTIONS OF PROFESSOR SMITH
general with the world as it is, and really, I think, somewhat deficient in prac-
tical originality. He was contented to be a follower rather than a leader in most
of our University contests. When he was brought forward a few years ago,
rather against his will, as a Candidate for the representation of the University
in Parliament, he was, I believe, absolutely indifferent to the result. I never
knew a man, possessing so much ability, with so little ambition. Hence he inter-
fered with no one, and as no suspicions were entertained of him, everybody was
willing to do justice to his great merits. Nor had he any sympathy with new
opinions in politics or theology. In politics he would have professed himself a
Liberal, but he was not an advanced one. He was willing to talk about them,
and his views were always worth hearing, because they were not strained out
of newspapers, but the result of his own reflections. He was an acute political
economist, a disciple of Mill and Ricardo, not much interested in the wider field
which is sometimes claimed for the science. Nor was he at all disposed to under-
value the influence of theology. He was well acquainted with the results of
German criticism on the Scriptures, but they seemed to make no difference
to him. When he first came up to the University he was an Evangelical,
and, for a while, retained his old belief. Indeed, some years after, on the
occasion of a high-church sermon at St. Mary's, he would say, with indignation,
' that was not the sort of religion in which he had been brought up.' But, in
time, the old clothes of his youth naturally fell off he had out-grown them,
and there remained a blameless character, a singular kindness and generosity,
a love of justice and fairness, and a sense of religion which was wrapped in im-
penetrable silence it was one of the subjects of which he least desired to talk.
He was very reserved. Like many other persons who pour themselves out
freely in conversation, there was the appearance of abandon, but there were
many subjects about which he rarely, if ever, spoke. One of these was himself.
He was probably the confidant of many, for no man could give better advice in a
difficulty, or was more willing to assist others. There was a feeling that he could
be absolutely trusted, and even if a foolish thing were said to him, that he would
not repeat it. His insight into human character was said, by one of his friends,
to be ' terrible,' but it was never used by him except for some kind purpose.
He could see through the vanity and folly of a friend, and yet retain a never-
changing affection for him. Of his own life, he seldom or never spoke ; he
was not an egotist, and his own sayings or doings did not seem to interest him
afterwards.
It is difficult to give an idea of his conversation. It was gay rather than
BY PROFESSOR JOWETT. xliii
serious, full of life and chaff, arising naturally out of the circumstances of the
hour. If a stranger had come across him in a railway train, or had been his
companion on a voyage, he would probably have found that this unknown person
was one of the most agreeable men he had ever met. It was a great pleasure to
have a tete-a-tete with him, for he was not one of those who required a company
in which to show off. I have often decoyed him into my room for the sake of
having a chat with him, and when once there, he was very willing to stay, for
he was one of those who like to have a talk out and did not hurry away when
the clock struck. In society he was ready to talk to every one, and every one
was ready to talk to him. He had the art of setting people at their ease. He
would at times break out into fits of laughter and joviality, which showed that
the original Irish nature was not extinguished, but only kept under by him.
Stories were repeated of his performances at Meetings of the British Association,
which must greatly have enlivened that sedate assembly. He was certainly a
wit, but his good sayings were of too delicate a fibre to be transplanted. To use
Boswell's expression, his bans mots ' would not carry.' But they were the delight
and admiration of those who heard them at the time. They possessed also one
of the highest qualities of wit and humour, spontaneity. They were made on the
spur of the moment with reference to something which was said or done at the
time. And this very quality tended to impair then: effect on those who were
not present when they were first uttered, and did not know the occasion which
had given rise to them. A light irony seemed to be always playing about his
mind. It was the form under which he inclined to regard all human things, for
he was very unimpassioned. An old school-friend would sometimes be the target
at which he aimed. The great scholar, Professor Conington, a man so unlike
himself, that their mutual friends wondered what could be the tie which united
them, was often the butt of his humour. But the slight humiliation to which he
was subjected was more than made up to him by the constancy and faithful
attachment of his friend, who afterwards collected his literary remains and wrote
his life. He was a little provoking to some others, especially to those who were
too earnest or of too pushing a temper. He knew how, in Aristotle's language,
'to overcome seriousness by laughter,' or in other words, to make such persons
appear slightly ridiculous. An enthusiastic friend might have thought him
deficient in sympathy, but he was really always kind and considerate.
I hardly venture to repeat some of his good sayings, lest, detached from their
surroundings, they should seem not to justify the high opinion which has been
expressed of his conversational gifts. They are not of course of the quality of
fa
xliv RECOLLECTIONS OF PROFESSOR SMITH
the best sayings of Charles Lamb or Sydney Smith, yet they are such as might
have been said by them. The reader is requested to bear in mind their im-
promptu or occasional character. He who made them could have made many
such every day of his life, and never aspired to be a wit, but only to amuse
himself and his companions. At any rate they may serve to remind his friends
of pleasant hours which they passed with him, never to return.
A friend told him of a rather ponderous jest made by Sir George Lewis,
who, when Minister of War, once proclaimed in the House of Commons in a loud
voice that he had ordered experiments to be tried respecting the comparative
effect of ' short and long bores.' To this heavy piece of artillery Henry Smith
instantly replied by asking whether he was not aware that ' smooth bores ' were
the most deadly of all. Another friend said to him : ' What a wonderful man
Ruskin is, but he has a bee in his bonnet.' ' Yes,' replied Henry Smith, ' a whole
hive of them ; but how pleasant it is to hear the humming!' The Lectures of
a certain College Tutor were reported to be ' cut and dried.' ' Yes,' said Henry
Smith, ' dried by the Tutor and cut by the men.' A dispute arose at an Oxford
dinner-table as to the comparative prestige of Bishops and Judges. The argu-
ment, as might be expected at a party of Laymen, went in favour of the latter.
' No,' said Henry Smith ; ' for a Judge can only say, " Hang you," but a Bishop
can say, " D n you." ' The next is of a higher class of wit. Speaking of an
eminent scientific man to whom he gave considerable praise, he said : ' Yet he
sometimes forgets that he is only the editor and not the author of Nature.'
The two remaining ones are autobiographical. He once said to a friend :
' C., I was kept in bed by illness when quite young for six weeks ; I then began
to study mathematics, and I wish I had remained there ever since.' Speaking
to a newly elected Fellow of a College, he advised him, in the low whisper which
we all remember, to write a little and to save a little, adding : ' I have done
neither.'
These slight jests may perhaps be thought disappointing : it is probable
that they are marred in the telling. They were the bubbles which were always
rising to the surface of his mind, and though but poorly reported, may help to
give to those who did not know him personally a faint idea of the charm of his
character and conversation.
Though not rich, he was extremely liberal. He never seemed to think
either about gaining or spending. He used to say that not enough money was
to be had in Oxford to make it worth while to take trouble about it. Yet a
certain love of speculation which was latent in his nature once led him into an
BY PROFESSOR JOWETT. xlv
unfortunate venture, from which he extricated himself by taking the affairs of
a Company into his own hands, and at a considerable loss. For his services as a
College Tutor he received a very moderate remuneration, but, having enough for
his wants, he never seemed to desire that it should be increased. He did not
wish to impose on the College a burden which it could ill afford to bear.
I have endeavoured, in a few pages, to give a sketch of one with whom
I was in daily intercourse during thirty-five years of his life, and who I think
may be regarded, without exaggeration, as one of the most remarkable persons
of his time. Yet he lived and died almost unknown to the world at large.
I have sometimes asked myself what was the reason of this contrast between his
reputation and his real merits. It has been said that ' the world knows nothing
of its greatest men,' but this familiar line, whether true or not, is not the whole
account of the matter in his case. The explanation is partly to be sought in his
own character. He had no ambition, he had not a strong will, and he had never
made himself known to the public. He was once reproached by a friend for
'giving up to society what was meant for mankind,' and the reproof, as far as it
applied to his life at Oxford, was not without foundation. He was not the author
of any considerable work. His Mathematical writings, on which his fame chiefly
rests, await the judgment of time. Though he managed, in great part, the
affairs, not only of the University, but of several other great institutions
such as Winchester and Rugby Schools, University College, Bristol, the Univer-
sity Commission, the Meteorological Office, the Oxford Museum, of which he was
Keeper, and the Ashmolean Society, of which he was the Secretary, he could
hardly be said to have left his mark upon any of them, however valuable
his services have been to those institutions. To understand his superiority over
his contemporaries, it was necessary to have lived with him and known him, to
have heard him lecture, to have been with him at a College Meeting, to have
enjoyed his society at a dinner-party, or on an excursion of pleasure. He never
offended you, never disappointed you, he was never tired or out of humour. His
greatness was shown in the peaceful continuity of a private life, not in great
actions, or on striking occasions.
B. JOWETT.
xlvi RECOLLECTIONS OF PROFESSOR SMITH
II. RECOLLECTIONS BY LORD BO WEN*.
GREAT statesmen, successful generals, famous authors, distinguished men of
science, eminent theologians all those who have been raised by industry,
talent, or the caprice of fortune, to prominence in a profession become by
degrees actors on whose movements our attention rests, and whose familiar
figures are part of the spectacle of life. The public they have interested
during their time bids them, when they die, a kindly and sympathetic farewell,
retraces their career, counts up their successes, and assesses their general
apparent value. Professor Henry Smith, whose loss this week casts a shadow
both over Oxford and through many circles of educated men and women,
belonged to none of these categories. To by far the greater number of
Englishmen, his name is probably unknown. Some will vaguely recollect it
as that of a candidate put forward unsuccessfully a few years ago by Oxford
Liberals for the representation of the University. Many even of those who
are aware that a man in the fulness of his powers is just dead, whose
brilliant intellectual attainments have probably not been surpassed by any
other of their English contemporaries, may, nevertheless, be surprised at regret
so widely felt and so loudly expressed over the loss of one who wrote no
great books, patented no great invention, amassed no fortune, made no famous
speeches, and led no conspicuous movement, political or social. Measured
by the popular measure of publicity and fame, Professor Henry Smith would
hardly seem, to most of us, to have been one of the great men of the time.
Yet it would be difficult among the world's celebrities to find one who in
gifts and nature was his superior. Generally speaking, there is a rough
justice in the sentence passed upon intellectual men who achieve no definite
worldly success. We surmise, and often with truth, that some weak spot
somewhere in their powers has been the cause of their failure to acquire
those sublunary distinctions and rewards which coarser and more practical
people manage to secure. To the case of Professor Smith, this kind of
criticism would be inapplicable, for he possessed both the qualities and the
character which might have made him famous in many active walks of life.
His mental attainments were of the highest order. A finished classical
* [Reprinted (by permission) from The Sjxxtator, Feb. 17, 1883.J
BY LORD BOWEN. xlvii
scholar, a mathematician, in some respects of European distinction, a con-
siderable metaphysician, a trained master of most branches of knowledge,
literary, economic, and scientific, an adequate linguist, and a man of sound
judgment, perfect temper, and wise aptitude for affairs, he combined with
his other special excellences a delicate gaiety of spirit, a brilliant conversational
power, which made him one of the most accomplished and attractive ornaments
of any educated company in which he moved. To what eminence in public
or professional life accomplishments so varied might not have led him, it is
difficult to feel sure, if only he had ever plunged into the stream of competition
or adventure. But some delicate touch of indifference to worldly success
mingled itself with his genius, and he remained to the last content with
playing, and with playing well, whatever part fortune brought to him to
play. Incessantly occupied in the discharge of duties both of a public and
a private kind, that thickened round him as years went by, he was satisfied
with what had fallen to his share in the lottery of life, and neither solicited
nor ostentatiously avoided anything beyond. The ' note ' of personal ambition
seemed absent from his composition. And so it happens that the great public
which takes its knowledge of men from newspapers and books, from debates
in Parliament and the records of our Law Courts, hardly knew if, indeed, it
knew at all Professor Henry Smith.
As the personal ' note ' was wanting in Smith, so, on the other hand, the
intellectual or academic ' note ' was one which he possessed in, perhaps, its
most attractive form. Vanity and self-seeking, every form of mental intem-
perance and extravagance, seemed to have no place in anything that he ever
said or did. The last, the rarest triumph of education, is when it destroys
the desire of self-assertion in a man of genius, and substitutes in its place
the crowning flower of perfect moderation and equanimity. The greatest of
Greek philosophers, in the greatest of moral treatises, has elaborated a theory
that virtue consists in a golden mean, and in the avoidance of dangerous extremes ;
but when driven into a corner for a standard by which the mean is to be measured,
the illustrious moralist has no better compass to furnish for our guidance than
this, that the golden mean in each case must be that which is defined by the
reason of some thoroughly temperate man. The result of Henry Smith's genius
and culture combined seemed to make him the very man required by a philosopher
for his human measuring-rod. A University life sometimes spoils and sometimes
perfects natural capacities, but it usually leaves its mark upon them, whether
it be for good or evil. Nobody could doubt but that Henry Smith, as he
xlviii RECOLLECTIONS OF PROFESSOR SMITH
issued from the Academic mould, was a natural genius, with an impress of
his University stamped distinctly upon him ; and Oxford has, perhaps, never
had a more happy specimen to produce of her best influence than the late
Savilian Professor of Geometry.
Smith came from Rugby to the University as a remarkable boy, and won
the blue ribbon in all the great intellectual competitions of his undergraduate
days. He became in due course a Fellow of Balliol, and joined a Common Room
which consisted of a small group of very distinguished men. The present* Master
of Balliol was already conspicuous in the society of Balliol Fellows, as the most
successful and most energetic tutor of the first of the Oxford Colleges of the
period. Among the rest were names of academic fame Mr. Lake, the present
Dean of Durham ; Riddell, an accomplished hero even among Shrewsbury
scholars, whose beautiful character and refinement of mind were prematurely
lost to the University by an early death ; Archdeacon Palmer, not the least
distinguished of a trio of brothers with all of whom Oxford had reason to be
content ; Lonsdale, Wall, Woollcombe, Walrond, and a few years later, Newman
and Green. These were the days when Oxford, always passing through some
phase or other, was entering on a new situation. The Tractarian movement had
subsided, but the University was not at rest. A reforming Parliamentary Com-
mission was troubling the waters. The old system of close Scholarships and
Fellowships was slowly giving way, and like the rotten boroughs of a past
political period, the close preserves of the Colleges were being either extin-
guished, or thrown open to public competition. But Oxford was still Conser-
vative at heart. Leaders of the old school and their followers held the University
pulpits, dominated Congregation, monopolized the best preferments, resisted to
the best of their powers all local change, and were ready on provocation to
ostracize unorthodox reformers for being, like Socrates, the corrupters of youth.
Married Fellows were as yet unknown ; it had not yet become necessary to build
whole suburbs of semi-detached villas to receive the feminine colonists of the
future. But there was a stir and an agitation throughout the Academic world
which the sense of changes, present and to come, had produced. University
politics and polemics were, as always, of absorbing interest. Mansel and Goldwin
Smith tilted against each other in debate before an admiring and competent
academic audience. Oxford was, in fact, at war, a war, it is true, polite,
polished, and courteous.
* The late Professor Jowett.
BY LORD BOWEN. xlix
Into this atmosphere, charged as it was with considerable personal
electricity, Henry Smith was thenceforward absorbed ; for nearly thirty
years, no more attractive, brilliant, or genial figure was to be found in the
perturbed society of the University. Some happy combination of judgment
and temper made him acceptable even to those with whose opinions he had
nothing in common. He succeeded in being a politician, without wearing the
obnoxious colours of a partisan. He had the great art of never pressing a victory
home, and of bearing defeat with pleasant equanimity. His business powers,
his modesty, his wisdom, and his entire freedom from egotism and dogmatic
presumption, a delicate gaiety that never flagged, wit that sparkled without
wounding, and which rose incessantly to real brilliancy, made him not merely an
effective personage in the Oxford world, but universally acceptable in any
society, whatever the shade of its opinions. His finished persiflage, his pleasant
epigrams, will long be remembered, though the brightest conversation is often
the most evanescent, and the finesse of wit, like a musical laugh, disappears with
the occasion, and cannot be reproduced upon paper or in print. As by degrees
his attainments were recognized, both in England and abroad, his influence at
Oxford naturally deepened ; but neither within nor without the University did
he grasp at opportunities for notoriety. Such power and authority as he
possessed he held without an effort, without solicitation, apparently without any
personal satisfaction in them. In offices of friendship he was constant ; in such
public or civic duties as came in his way, assiduous ; no good or benevolent work
ever needed a helping hand, but his was at its service, without ostentation, and
without any expectation of personal advantage. He was a good speaker, without
being a rhetorician ; his death, indeed, last week was hastened by a chill caught
or increased while he was addressing a gathering of agricultural labourers.
A life like Henry Smith's, of exemplary moderation, far removed from
even a suspicion of worldliness and vanity, is seldom found in these days
in combination with intellectual powers and practical ability on so considerable
a scale. There are, no doubt, many nooks and corners in which at times may
be seen flowering 'the wise indifference of the wise.' Students, divines, men
of science or of letters, not seldom seem content to retire from the world, as if
they had measured the true value of the things we most of us eagerly compete
for, and were perfectly satisfied, of deliberate choice, to remain spectators
of the fever of mankind. Some physical inaptitude, or some constitutional
tendency, not unfrequently lies at the bottom of this apparently philosophic
temper. Patient self-possession, and a sober estimate of the world and of
g
1 RECOLLECTIONS OF PROFESSOR SMITH
what it can give, are rarely found in a man who lives in constant contact
with other men and their affairs, who shares in the interests of his generation,
occupies himself with its business, and whose genius seems to bring high honour
and success almost within his reach. Professor Henry Smith was not buried
away from his fellow-creatures in literature, or study, or contemplation ; he
was no recluse or invalid, but a man of the world, active, competent, social,
only not ambitious. Personal serenity of such a type is rather a classical
than a modern virtue ; perhaps an age different to our own may yet regard
it as one of the highest forms, not merely of intellectual, but of civic excellence.
It is the characteristic of recent civilization, that in almost all its aspects it
seems based upon a theory of personal competition. The prominent figures
on every stage are the result of a struggle, not for existence, but for success.
It is a contest which all seem satisfied to recognize as one of the conditions
of ordinary life ; which constitutes the essence of our politics, of our commerce,
of our political economy, of our laws of property themselves. In the general
race to possess more than the average share of wealth, power, fame, it is,
perhaps, a wholesome lesson to turn for a short breathing-time to the uneventful
example of the life of a man of genius, who was fitted for most distinctions,
if he had cared to seek them ; but who was unaffected by the universal fever,
possessed his soul in perfect patience, and remained to the last content to
discharge all the duties which Providence allotted to him, without affectation,
and with that composure of soul to which great gifts are not always allied.
The secret of the philosophic temperament, exhibited in this its most manly
shape, is one which is not easy to explore ; but when the phenomenon is seen,
its charm attracts us the more in proportion to its rarity. Essayists and
moralists for the last two thousand years have preached it, and inculcated it ;
some have gone so far as to boast of its acquisition, its praise, certainly, is
among all the prophets. Probably it is the product neither of Nature, nor
of education singly, but of a happy, and of an admirable combination of the
two. Among the many friends, acquaintances, admirers, whose thoughts have
in the last few days been saddened or sobered by the unexpected death of
a brilliant man of genius, there are none who will not readily accord to Professor
Henry Smith the tribute of unaffected respect for what without extravagance
may be termed his extraordinary powers of mind, his gentle and Laelian wisdom,
and the sweetness of character which never made an enemy, lost a friend, or
sought a personal advantage for itself. But besides this and beyond this,
it may not be out of place, before a personality in many ways so complete
BY MB. STEACHAN-DAVIDSON. li
fades into indistinctness, and a life ceases to be familiar to us which must
hereafter be treasured rather in the memory of his contemporaries and friends
than in the history of his time, to recognize in the Professor Oxford has lost
that special type of wholesome and of manly virtue the growth of which
is not much favoured by the rush and turmoil of these times. Great mental
gifts can be found, when occasion demands them ; talents grow on every tree.
But the serenity of heart which enables its possessor to wear the gifts of
genius with sobriety, and to use them nobly and well, without seeking to
expend them in the purchase of fame, or wealth, or of advancement, is a quality
which modern society little cultivates, and seldom sees.
III. RECOLLECTIONS BY MR. J. L. STRACH AN -DAVIDSON.
THE death of Henry Smith will be felt as the greatest loss which could
have befallen Oxford. In him the University possessed a student whose know-
ledge and genius were honoured throughout Europe. Of those amongst whom
he lived few indeed could follow him to the height of his scientific speculations.
Most of us did not know enough to understand where and how he was working
in the field of Mathematical Science. By us he is lamented as the wisest
counsellor of the University, and as the delightful companion who gave life and
charm to its society. Though his activity extended far beyond the limits of the
University, he was very constant to Oxford. Since he took his degree he did
not miss a single Term's residence. Re-elected time after time to the Hebdomadal
Council, his assistance was called for whenever any serious business required
sound judgment or delicate handling. His advice was generally followed, and
if not, the neglect of it was almost always regretted in the sequel.
In Henry Smith were united to a rare degree knowledge of business and
knowledge of men. He seemed most thoroughly in his element when swaying
and guiding his fellows. To every matter which he took in hand, he seemed to
come with a fresh mind, throwing off all the multitude of concerns which beset
him, and unburdened by care or anxiety. Then under the cover of his easy playful
manner it would soon become manifest that he had grasped all the true points
at issue, and was ready with a firm and wise decision. He always looked facts
in the face, and strove hard to distinguish the difficult from the impossible.
To the more ardent spirits among his followers it was sometimes a matter
of disappointment that he would not lead them to assaults which he saw to
lii RECOLLECTIONS OF PROFESSOR SMITH
be fruitless. Though he could fight hard when the moment for fighting came,
no one was more averse to multiply occasions of controversy. He saw things
without passion and without prejudice, and laboured quietly and steadily for all
that could advance the studies and promote the efficiency of the University.
In this spirit he accepted the thankless task of serving on the University
Commission. It is the inevitable fate of such a body that their work is attacked
at once by the criticisms of those who think that it has gone too far, and of
those who would have had it go further. Henry Smith knew well that it was
impossible to satisfy either the one party or the other. But it was a source
of keen satisfaction to him to notice that when those who joined in complaining
of the Commission came to propose alternative schemes they found that these
divided them more than that to which they had objected. In the same way he
was much gratified that it was only a minute point in the Commissioners'
arrangements which was finally contested by the University. He claimed, and
with justice, that when the Proctorial appointment of Examiners was the only
portion of the old constitution which was defended to the last, it was pretty clear
that the more important changes were acknowledged to be wise and necessary.
It may be interesting to note what Henry Smith thought of these greater
changes. He fully appreciated the charms of the old system of celibate Fellow-
ships, and never for a moment cherished the illusion that the new seven years'
tenure could ever have the value and dignity of the old. But he felt that the
old system could be practically worked only so long as the majority of Fellows
were willing to take Orders and retire to a College Living in middle life.
When it became evident that the University must either renounce the service
of its most efficient members or be content to be served by laymen, he recognized
that, at whatever sacrifice, a career must be provided into which a man could
enter as his profession for life.
Another important question often present to his mind was the effect of the
College system on the life and teaching of Oxford. He felt the difficulties
as keenly as many who urged radical changes ; but he felt likewise that it was
worth making an effort to preserve this distinctive feature of the English
Universities by transforming it to suit the new conditions. When in conver-
sation he summed up the work of the Commission, it was, 'we have given
a fresh lease of life to the College system.' He was not very sanguine that this
system could be permanent, but he was convinced that it ought to have another
chance, and that the best chance was secured to it by the reforms which he and
his colleagues had effected.
BY ME. STRACHAN-DAVIDSON. liii
It is difficult to speak of the charm of his life and conversation. The light
touch and happy play of mind with which he enlivened the most serious business,
and softened the jarring of controversy, was a source of real power, and procured
a ready acceptance for the wisdom of his practical suggestions. In social life
the same qualities shone forth at every moment. It seems hardly credible to
those who knew him best that a deep-seated disease had been sapping his life
for years. His temper was always unruffled, his spirits always gay and easy,
and his sympathy always ready. In the midst of a mass of business which
would have absorbed any ordinary man he could always find time to attend
to the interests and concerns of his friends. To cheer a sick friend with the
sunshine of his presence, to be the protector of the children of those who were
taken away, to lend a ready ear to the perplexed and a helping hand to those
who had committed themselves by any foolish action all such kindnesses seemed
so easy and natural to him, that men claimed and accepted his benefits almost as
a matter of course. He seemed to be good not in obedience to any external law
nor as the result of any internal struggle, but because goodness was the simple
outcome of his nature.
His wit and gaiety were the delight of all who listened to him. It
was not so much that he was a sayer of good things to be remembered
and repeated though of these too there was no lack but the really charac-
teristic feature of his talk was that its interest never flagged ; a certain flavour
of freshness and originality pervaded it, and revealed itself even in his commonest
remarks. To walk or ride with him was to enjoy a conversation in which not
a sentence was commonplace. There was always some new light, some refine-
ment or subtlety of thought or expression which gave a charm to the most
ordinary topics. This effect was due mainly to the keen and delicate temper of
his mind, but partly also to the wonderful breadth of his knowledge and his
interests. He knew the literature of Greece and Rome as if he had made their
study the work of his life, whereas it was really the amusement of his leisure
hours. He had the sincerest love for the classical writings and the most profound
belief in their value. His retentive memory and delicate taste made his conver-
sation on these topics a storehouse of interesting and instructive criticism.
Though the resources of his own mind filled to overflowing every moment
he could snatch for quiet study, yet he never shut himself up or held aloof from
his fellows. There was absolutely nothing stern or forbidding about him. He
seemed to take in the society of his friends the same pleasure which his presence
imparted to them. In every relation of life there was in him the perfect ease
liv RECOLLECTIONS OF PROFESSOR SMITH.
and grace which flows naturally from complete and sufficing strength. The
sweetness of his character and the perfect cordiality of his nature seemed
to offer all the rare gifts of his genius to minister to the happiness of his friends.
His death leaves dark what was a ray of sunlight in the lives of many.
He was entirely free from superstition. He held deliberately that the
questions whose solution is hidden from man, and above all the prospect of death,
should never be allowed to cast a shadow over the life and work of the present
hour. He believed that it became a man to live at his best and to labour at his
best during every day allotted to him, even as though an endless succession of
such days were in store. It was permitted to Henry Smith to give a bright
example of his theory. Till within a week of his death he was teaching from his
chair, attending to all the varied work of government and management for which
he was responsible, and living a bright and happy life which shed cheerfulness
and comfort on all around him. He always maintained, that there is no such
thing as a necessary man, and that every place left vacant can be adequately
filled. Of all that Henry Smith taught, this doctrine is the one which it seems
most difficult to realize at this moment.
[February, 1883.]
NOTE BY MR. ALFRED ROBINSON.
I HAVE been asked to give some account of the contested election in which
Professor Smith was a candidate for one of the University seats in the House of
Commons ; and I do this with much pleasure, because, although he was defeated,
the amount and kind of support which he received in the contest show how
much he was valued by men of all parties in Oxford, and how unique was his
position in the University.
In the spring of the year 1878 it became known that Mr. Gathorne Hardy,
who was then one of the University representatives, was about to be summoned
to the House of Lords. The rejection of Mr. Gladstone in 1865, and the defeat
of Sir Roundell Palmer by Sir John Mowbray in 1868, had proved that no one
but a Conservative could win in a contest conducted upon the lines of political
party. But it was thought by many persons that the Members fur the University
ought to be chosen upon academical rather than upon political grounds, and
ought to represent learning, science, and education, without special reference to
party interests.
THE ELECTORAL CONTEST OF 1878. Iv
Professor Henry Smith was brought forward as a man most eminently
qualified to represent the University in this sense. The fact that he was
a Liberal in politics of course was not disguised. He was indeed at this time
not fully in sympathy with some of the Liberal leaders. The Eastern question
then filled the political foreground, and Professor Smith, while disapproving of
the general policy of the Conservative Government, thought that Lord Salisbury
ought to be supported in maintaining against Russia in her dealings with
Turkey, the rights of the neutral Powers and the general interests of Europe.
Perhaps, also, Professor Smith was too critical, and too fond of looking at
questions from every point of view to have ever made a first-rate party man.
But still he belonged undeniably to the Liberal party, and he was not a man to
be led by any waywardness, or by any love of fads or crotchets, into a position of
political isolation. So his election by the University would no doubt have been
a transfer of a seat from the Government to the Opposition side of the House,
and this was the aspect in which the contest presented itself to the great
majority of the voters. It was not, however, on a contest of this kind that
Professor Smith's chief supporters wished to enter. In their view the special
representation of the University in Parliament was useless if the University
Members were to be party men of the ordinary type, without special qualifications
for dealing with the questions with which the University was specially concerned,
and their main object in bringing forward Professor Smith was that this view
should be put before the constituency and the country
With the arrangements for his own candidature Professor Smith had very
little to do. An old custom of the University, which had been observed by
Mr. Gladstone throughout his long tenure of his seat, precluded a candidate
from issuing any address, or from making any speech to the electors. At an
early stage in the proceedings Professor Smith was invited to stand, and he
agreed to allow himself to be nominated, but he took no part in the initiation of
his own candidature ; he was never present at the meetings of his committee ;
and his supporters defrayed the expenses of the contest, declining a request
which he made that he might at least be permitted to contribute to the
subscription list.
Professor Smith's Committee was formed in the month of April, 1878, and
consisted of two sections.
(1) The London Committee, the Chairman of which was Mr. Mountague
Bernard ; and which had for its Vice-Chairmen the then Marquis of Tavistock,
Mr. Goschen, Mr. Knatchbull Hugessen (afterwards Lord Brabourne), Mr. Dodson
Ivi THE ELECTORAL CONTEST OF 1878.
(now Lord Monk Bretton), Dean Stanley, the Dean of Canterbury, the Dean of
Durham, and the late Sir Benjamin Brodie ; and for its Secretaries, Mr. Buller
of All Souls, Mr. Ilbert, late Bursar and Fellow of Balliol, Mr. Pope, formerly
Fellow of Lincoln, Mr. Robertson, Fellow of Corpus, Mr. A. L. Smith, then
Fellow of Trinity.
(2) The Oxford Committee, having for its Chairman the Dean of Christ
Church (Dr. Liddell) ; for its Vice-Chairmen, the then President of Corpus and
Archdeacon Palmer ; and for its Secretaries, Mr. Crowder, Bursar of C. C. C.,
Professor Green, Mr. Jackson, Fellow and now Rector of Exeter, Mr. Monro,
Fellow and now Provost of Oriel, Mr. Papillon, Fellow of New College, Mr.
Salwey, Student of Christ Church.
It is not likely that any of these persons were under the illusion that their
cause was going to win. The Conservative feeling of the constituency was soon
found to be so strong that under no circumstances could any one but a Conser-
vative have been elected. And at this time the excitement of the two political
parties with reference to the Eastern Question much increased the difficulties
with which Professor Smith's Committee had to contend. On the one hand,
a large section of the constituency saw in him only an opponent of the Govern-
ment which was patriotically defending British interests against Russia. And
on the other, some well-known Liberals considered that he was too lukewarm in
his censure of the Party and of the Ministry which they associated with the
notorious atrocities in Bulgaria.
Opposition of the former kind, which insisted that the representative of the
University of Oxford must be a supporter of a Conservative government, it was
impossible to disarm. But an effort was made to conciliate the critics and
opponents who belonged to the Liberal ranks. Professor Smith was requested
by his Committee to put forward some definite statement of his views on the
Eastern Question, and in the following letter to a member of his Oxford
Committee he complied with this request.
DEAR _ , * 5 ' l878 '
I am well aware that the custom of the University imposes a great measure of
reserve upon any candidate for the honour of representing it in Parliament. But I do
not think that I shall be departing from a tradition, which I am most anxious to see
maintained, if I venture to write a few lines to you in explanation of the views which
I entertain with regard to the Eastern Question.
There has been much in the foreign policy of the Government during the last two
years of which I cannot approve. I think that they should have recognised, at a far
earlier period than they did, that the condition of the Christian Provinces of Turkey
NOTE BY MR. ALFRED ROBINSON. Ivii
had become unendurable, and that the maintenance of the status quo was no longer
possible. A grave, but long foreseen, emergency had arisen ; and this country should
have been prepared with a well-considered policy to meet it. Instead of this, the
Ministry seem to me to have drifted with the stream of events, until at last they find
themselves in a position in which it is immeasurably more difficult, than it would have
been twelve months ago, to assert the right of the neutral powers to have a decisive voice
in the settlement of a question affecting such vast European interests.
Looking at the most recent events, I have to express a general concurrence with the
main tenor of Lord Salisbury's Despatch ; and I have observed, with great satisfaction,
that it has been received with cordial approval by the Liberal Press on the Continent.
Interpreting that document, as I think I am justified in doing, by the light thrown on
it by Lord Salisbury's own conduct at the Conference of Constantinople, I do not
perceive in it any intention to restrict the liberties to be granted to the Christian
subjects of the Porte ; but I regard it as a protest in favour of the recognition of inter-
national obligations, and against any attempt on the part of Russia to dispose of the
Eastern Question in her own way.
For the reasons which I have stated, I feel that I could not enter Parliament, except
upon the condition of preserving the right to form while there an independent judgment
with regard to the future action of the Government in these important matters. If there
is a war party in England, I have no sympathy with it : but I am not for peace at any
price ; and, if any of the great interests of the country should be endangered, I should
hope to see all Englishmen, without distinction of party, united in defending them.
Believe me to remain.
Very faithfully yours,
HENRY J. S. SMITH.
P.S. You are at liberty to make any use which you may think fit of this reply to
your letter.
This letter removed some of the misapprehensions as to Professor Smith's
position, and probably produced some effect upon the canvass. Promises of
support were received from some of the Liberals who had originally stood aloof,
including one from Mr. Gladstone, which arrived a few days before the opening
of the Poll. His example, however, was not imitated by all his followers, a few
of whom, more Gladstonian than their chief, remained neutral to the last.
May 13th was fixed for the nomination. On that day the two candidates
were proposed to the House of Convocation in Latin speeches Professor Henry
Smith by the Dean of Ch. Ch., Mr. J. G. Talbot by the President of St. John's.
The Dean dwelt upon the scientific and literary qualifications of Professor Smith,
his ability in business and in debate, and his suavity and fairness of judgment,
which conciliated the regard of all. He recommended him to the electors as
a man whom the Ministry of the day had entrusted with the most weighty
h
Iviii
THE ELECTORAL CONTEST OF 1878.
affairs, and as one ' unice idoneum qui ipse academicus academicos suos in Parlia-
raento reprsesentet.' The President of St. John's, in nominating Mr. Talbot, made
some kindly remarks on the undesirableness of withdrawing Professor Smith
from the Professorial duties and from the sciences which he adorned.
Immediately after the nomination the Poll opened, and under the Act
governing University elections it was not to be closed until the 17th, unless
either of the candidates were withdrawn in the meantime.
The electors could vote either in person or by voting papers. At the time
when the voting began Professor Smith's Committee had received less than
a thousand promises from a constituency numbering more than four thousand
members, and the last hope of the most sanguine of his supporters had dis-
appeared. It was, however, thought best that all the votes should be recorded,
in order that the amount and kind of support with which his candidature had
been received might be accurately measured and generally known ; so the
polling was continued daily for five days in all. At the close Mr. J. G. Talbot
was declared to be elected, the numbers being Talbot, 2687 ; Smith, 989.
Defeated by a majority of more than two to one, Professor Smith's Committee
might to some extent console themselves with the thought that they had
conducted the contest with great economy. The expenses amounted to about
420, the chief item being the bills for advertising the lists of supporters in the
chief London papers. This sum was less than half of what was believed to have
been spent on the Liberal side in each of the two preceding Oxford elections.
Still more consolatory was the analysis of the Poll Book, which was pub-
lished soon after the result of the election was declared. This proved that the
majority of the electors and the working staff of the University had been ranged
under opposite banners. The following table shows how certain sections of the
constituency had voted :
SMITH.
TALBOT.
Abstained
from voting.
Heads of Colleges, including two acting Heads
10
10
2
Professors, Readers, and University Teachers
28
II
6
Tutors and Lecturers of Colleges and Halls
01
3O
is
Fellows of Colleges, resident and non-resident . .
1 5Q
82
R3
'Residents
I S2
I 1*7
Residents Members of Congregation qualified by residence i. e. all electors who were iii
residence Oct. 1876 Oct. 1877, including the parochial clergy in Oxford and others not engaged in
University work.
NOTE BY MR. ALFRED ROBINSON.
lix
Whether this table points to any practical conclusion or not may be doubted.
That the Members for the University should be chosen by those who are identified
with it as the place of their work or residence in the present, and not merely of
their education in the past, may seem reasonable, but no change could possibly
be made which would reduce the constituency to less than one-tenth of its former
number ; and perhaps the special representation of the Universities in Parliament
is more likely in the future to be abolished than to be reformed.
But whatever inferences of this kind might be drawn from the result of the
election, the Poll Book was unequivocal in its recognition of Professor Smith's
personal qualifications and eminence. And even the numbers set forth in the
foregoing table, emphatic as they are, do not fully express the estimation in
which he was held by that portion of the constituency in the midst of which he
had lived, and with which he had been officially connected. For among those
who abstained from voting there were some who remained neutral, in spite of
their high appreciation of his claims, because they thought that a seat in Parlia-
ment would be incompatible with his Oxford work ; and there were others who
on ordinary occasions would have been ranked among his warmest supporters,
but were unable at this time, when the foreign policy of the country filled the
political horizon, to vote for a man who was variously criticised as going too far,
or not far enough, in support of, or in opposition to, the Government of the day.
But, in spite of these abstentions, the preponderance of opinion in the
working staff of the University was clearly marked, and was most significant.
It may be confidently said that no other man could have enlisted at this
time among his supporters in a Parliamentary contest so many of the men who
were identified by their position or occupation with Oxford ; and it may be
doubted whether in any of the controversies, political and academic, which have
divided the University at various times in its history, so many of its resident
graduates have ever enrolled themselves upon one side.
h 2
INTRODUCTION
TO THE
COLLECTED MATHEMATICAL PAPERS
OP
HENRY J. S. SMITH.
THE present volumes contain all the mathematical papers published by the
late Professor H. J. S. Smith in his lifetime, as well as those which were in
the press or which had been written out for printing. The reader is therefore in
possession of all that he had already published, or had wholly or partially
prepared for publication at the time of his death.
The arrangement of the papers is strictly chronological, the order being that
of the date of reading or publication. The only partial exception is the Report
on the Theory of Numbers, which is printed as a whole, although several other
papers which follow it were published in the six years during which it was in
progress. It is possible therefore by merely glancing over the titles of the
papers to trace the course of Professor Smith's mathematical studies and tastes.
The first two papers, written in 1851 and 1852, show that his mind was then
occupied by Geometry. Within three years he published his first paper on
the Theory of Numbers, consisting of a characteristic proof of Fermat's theorem
that every prime number of the form 4n + 1 is the sum of two squares. From
this time until 1867 the printed papers relate almost exclusively to the Theory
of Numbers. Then follow a number of geometrical papers. In the last years of
his life he was occupied principally with the subject of Elliptic Functions.
It will be seen therefore that his work falls into three distinct groups :
(i) Geometry, (ii) Theory of Numbers, (iii) Elliptic Functions. From the fact
that the two earliest papers relate to Geometry we may infer that this was the sub-
ject which originally proved most attractive to him. The first of these papers was
written in the year in which he obtained the Senior Mathematical Scholarship,
and only a little more than a year after his election to a Fellowship at Balliol. It
would seem that about 1853 he commenced the study of the Higher Arithmetic,
INTRODUCTION.
a subject which engaged his almost undivided attention for many years, and which
was never afterwards quite absent from his thoughts. The short notes which bear
the dates of 1854 and 1857 show the tendency of his mind at this time : and in
1859 the first part of his Report on the Theory of Numbers was contributed to
the British Association. The subsequent instalments appeared in the annual
volumes of the Association for 1860, 1861, 1862, 1863, and 1865. These reports,
which contain in a very condensed form the result of an immense amount of
research, are models of clear exposition and systematic arrangement. Besides
the accounts there given of the work of others, many of the paragraphs contain
results of his own. These original contributions are not, however, noted as such,
and they can only be detected by those who are already well acquainted with
the details of the subjects to which they belong.
During the preparation of this Report he carried out elaborate researches of
his own in several important branches of the Higher Arithmetic. The principal
investigations undertaken at this time, which were completed for publication,
relate to systems of indeterminate linear equations and congruences and to the
orders and genera of ternary quadratic forms containing more than three indeter-
minates. These memoirs appeared in th^ Philosophical Transactions for 1861
and 1867. He also contributed several shorter papers to the Proceedings of
the same Society, which indicate much more extended investigation in the same
field : one especially (No. xviii, vol. i.) consists merely of a brief statement of
results which were obtained by means of a very long and delicate analysis.
A considerable part of the last instalment of the Report is concerned with
arithmetical formulae derived from Elliptic Functions, and it seems likely that it was
in this way that he was first attracted to this Theory ; for his first published paper
on the subject (No. xvi, vol. i.) bears the date 1866. The x'emaining papers in-
cluded in the first volume relate to Geometry, principally homographic figures.
In the second volume (1869-1883) there is more Elliptic Functions and
less Theory of Numbers : but the sequence of the papers no longer affords an
indication of the author's train of thought : for, in the later years of his life, he
was frequently compelled, by various circumstances, to leave the subject upon
which he was engaged, in order to prepare for publication theorems and demon-
strations that formed part of the many unfinished investigations stored up in
his note-books.
The first paper in the second volume was a prize memoir for which, con-
jointly with another memoir, the Steiner prize of the Berlin Academy was
awarded. The subject was announced in 1866, and the memoirs were to be sent
INTRODUCTION. Ixiii
in, each designated by a motto, before March 1, 1868 *. Four were received, and
the prize of six hundred thalers was divided between Professor Smith and
Dr. Hermann Kortum, of Bonn, the two memoirs being regarded as of equal
merit. The report on the memoirs received, which was laid before the Academy
by Professor Kummer on July 2, 1868, contained the following remarks relating
to that sent in by Professor Smith :
' Die vierte, in franzbsischer Sprache abgefasste Preisschrift mit dem Motto : " Haud facilem ease
viam voluit " ftthrt den Titel : " Memoire sur quelques problemes cubiques et biquadratiques," und ist in
drei Abschnitte eingetheilt. Der erste Abschnitt beschiiftigt sich mit der Theorie des Imaginaren
in der Geometric, der zweite enthalt verschiedene Methoden, die gemeinsamen Puukte zweier durch
ihre Elemente gegebener Kegelschnitte mittels des Lineals, des Cirkels und eines festen Kegelschnitts
zu censtruiren, in dem dritten Abschnitte endlich lost der Verfasser ausser einigen andern sogenannten
kubischen und biquadratischen geometrischen Aufgaben namentlich das speciell in der Preisfrage
hervorgehobene, die Curven vierten Grades betrefiende Problem. Die ganze Arbeit zeiclmet sich durch
iibersichtliche und systematische Behandlung des Stoffes aus. Der Verfasser macht bei seinen
Constructionen, wie es in der Preisfrage verlangt wird, nur von den einfachsten erforderlichen und
ausreichenden Hilfsmitteln Gebrauch, aber bei den Constructions-Methoden selbst hat er mehr auf
gedankliche als auf praktische Einfachheit, mehr auf die vollstiindige Darlegung aller Gesichtspunkte
als auf die Ausfiihrung aller einzelnen Operationen sein Bestreben gerichtet. Dadurch ist es ihm
gelungen, im zweiten Abschnitte das an sich durftige und trockene Material in gediegener und
interessanter Weise zu verarbeiten nnd im dritten Abschnitte die specielle dort behandelte Frage mit
allgemeineren zu verkniipfen. Fast uberall lasst die Arbeit zum Vortheil fur ihren wissenschaftlichen
Werth deutlich erkennen, dass der Verfasser zu seinen umfassenderen Untersuchungen durch alge-
braische Betrachtungen gelangt ist, deren genauer Zusammenhang mit dem Gegenstande der Preisfrage
schon in deren Formulirung enthalten istt.'
The origin of the long memoir on the Theta and Omega Functions the last
paper but one in the second volume was as follows. At the end of 1873, or the
* The announcement of the subject was made in the following terms : ' Fur cliejenigen geometrischen
Probleme, deren algebraische Lbsung von Gleichungen von hoherem als dem zweiten Grade abhangt,
fehlt es noch an der Feststellung der zur constructiven Lbsung derselben erforderlichen und
ausreichenden fundamentalen Hilfsmittel, so wie an den Methoden zur systematischen Benutzung
dieser Hilfsmittel.
'Indem die Akademie die Frage, die sie stellt, auf die Probleme beschriinkt, welche auf kubische
Gleichungen fiihren, wiinscht sie, dass wenigstens an einer Anzahl von speciellen Beispielen gezeigt
werde, wie diese Lucke in dem Gebiete der constructiven Geometrie ausgefiillt werden kbnne.
Namentlich verlangt sie die vollstandige Losung des folgenden Problems :
'Wenn dreizehn Punkte in der Ebene gegeben sind, so sollen durch geometrische Construction
diejenigen drei Punkte bestimmt werden, welche mit den gegebenen zusammen ein System von
sechzehn Durchschnittspunkten zweier Curven vierten Grades bilden.
' Bei der Losung sind die Falle zu berucksichtigen, . in welcheti einige der dreizehn Punkte
imaginar und demgemass nicht als individuelle Punkte, sondern als Durchschnittspunkte vorgelegter
Curven gegeben sind. Gewiinscht wird ferner, dass sammtliche geometrische Constructionen durch
die entsprechenden algebraischen Operationen erlautert werden.'
t Mmatgberichte for 1868, p. 420.
l x iv INTRODUCTION.
beginning of 1874, when I was passing through the press the Tables of the
Theta Functions which I had calculated in connexion with a Committee of
the British Association, I asked Professor Smith, who was a member of the
Committee, if he would contribute an Introduction to the volume. He replied
that he did not see his way to writing anything appropriate to the tables
themselves, but that he 'could say something with respect to the constants
at the head of the pages.' These constants were K, K', E, J, J', &c., the
numerical values of which were given for every minute of the modular angle.
The memoir grew in extent, and was subject to frequent interruptions ;
in fact a number of other papers were written and published during its
progress. These papers were generally called into existence by special cir-
cumstances unconnected with the memoir, but a few of them, and especially
the Notes on the Transformation of Elliptic Functions (Nos. xli, xlii, vol. ii.),
which immediately precede it in the volume, arose directly out of it. The
first two of these Notes were given to me in the summer and autumn of 1882
for the Messenger of Mathematics, and appeared in the numbers from August
to November of that year. The remaining Notes were printed after his death
from a draft manuscript which he had shown to me, and explained in some
detail, in October, 1882. The memoir itself, with the Notes that were con-
nected with it, formed the principal new work upon which he was engaged
from the time of its commencement until his death : most of the other papers,
published in the interval, containing results which were mainly derived from his
earlier investigations. It was left incomplete : Arts. 1-31 (pp. 415-484) had
been passed for press : Arts. 32-48 (pp. 485-535) had been revised, and Arts.
49-73 (pp. 535-585) were in type in quarto pages and had been partially
corrected. The succeeding Articles up to Art. 88 inclusive were in type in
octavo slip, and had been partially corrected in this form *. The last two
Articles (89 and 90) are printed from a manuscript found among his papers, and
which he had marked as following on after Art. 88. I believe that no more
was written, even in draft. The figures which occur in the Memoir had not
been drawn.
The object of Professor Smith's first paper on Elliptic Functions (No. xvi,
k The whole of the Memoir was originally set up all in octavo slip, and rem lined in this form for
a long time, during which it was greatly altered and extended. It was reset in quarto pages
during 1881 and 1882, and passed for press in this form. It had been intended that it should appear
as an Introduction, but it was finally decided that it should follow the tables with the title ' Memoir
on the Theta and Omega Functions.'
INTEODUCTION. Ixv
vol. i.) was, as stated in the first paragraph, to enunciate and demonstrate
a fundamental theorem, the nature of which had been indicated in a letter,
written in 1845, from Jacobi to M. Hermite, in which he mentioned that he used
it as the starting-point in his Konigsberg lectures. Jacobi died in 1851, and as
the theorem referred to had never been published, Professor Smith reproduced it,
in 1866, in this paper. Guided by Jacobi's suggestion, he multiplied together
four general Theta series and expressed the product as the sum of four terms,
each of which was the product of four Theta series with different arguments.
From this theorem he derived, as indicated by Jacobi, all the principal results of
Elliptic Functions, either as particular cases or as simple corollaries. In 1881
the first volume of the Collected Works of Jacobi was issued, and his Konigs-
berg lectures on Elliptic Functions were there printed for the first time. By com-
paring them with Professor Smith's paper it will be seen that, although the theorem
itself is of course essentially the same, still there are differences in the mode in
which it is presented which enhance the interest of the latter. Professor Smith
treated the question with great generality, and with absolute precision, and this
short paper is very characteristic of his style of work.
At the meeting of the London Mathematical Society on January 8, 1879,
Professor Cayley communicated to the Society the theorem
where a, /3, 7, S are any quantities whose sum is zero. Professor Smith, who
was present at the meeting, remarked that it was a special case of a theorem
relating to the multiplication of four Theta functions, and at the next meeting
in February he communicated to the Society the general Theta Function
formulae which dominate all results of this class. This paper {No. xxxviii,
vol. ii.), which is supplementary to that of 1866, was written out from notes
which he had had by him since that date.
The paper on the conditions of perpendicularity in a parallelepipedal system
(No. xxxii, vol. ii.) was written in response to a request from his friend
Professor Maskelyne, who was seeking for a general treatment of a problem
which, in the particular case of its application to crystallography and the dis-
tribution of molecules in a crystal, was of paramount importance.
The circumstances connected with the publication of the memoir which
concludes the second volume require a more extended notice. In February, 1882,
he was surprised to see in the Comptes Rendus that the subject proposed by the
French Academy for the Grand Prix des Sciences Mathe"matiques was the decom-
l xv i INTRODUCTION.
position of a number into five squares *. His feelings in the matter are shown
by the following extracts from letters to myself. In the first, dated Oxford,
February 17, 1882, he wrote 'The Paris Academy have set for their Grand
Prix for this year the theory of the decomposition of numbers into five squares,
referring to a note of Eisenstein, Crelle, vol. xxxv, in which he gives without
demonstration the formulae for the case in which the number to be decomposed
has no square divisor. In the Royal Society's Proceedings, vol. xvi, pp. 207, 208,
I have given the complete theorems, not only for five, but also for seven squares :
and though I have not given my demonstrations, I have (in the paper beginning
at p. 197) described the general theory from which these theorems are corollaries
with some fulness of detail. Ought I to do anything in the matter ? My first
impression is that I ought to write to Hermite, and call his attention to it.
A line or two of advice would really oblige me, as I am somewhat troubled and
a little annoyed ;' and in the second, of date February 22, he proceeded, ' You
see I take your advice entirely upon the point that he ought to be written to.
The worst of it is that it would take me a year, and a hundred pages, to work
out the demonstrations of the paper in the Royal Society's Proceedings.'
The following reply was received from M. Hermite :
MON CHEE MONSIEUE,
A urn ii des membres de la commission qui a proposd pour sujet du prix des sciences math^matiques en
1 882 la demonstration des the'oremes d'Eisenstein sur la decomposition des nombres en cinq Carre's n'avait
connaissance de vos travaux contenant depuis bien des annexes cette demonstration et dont j'ai pour la
premiere fois connaissance par votre lettre. L'embarras n'est point pour vous, mais pour le rapporteur
des memoires envoyte au concours, et si j'etais ce rapporteur je n'Wsiterais pas un moment a faire
d'abord 1'aveu complet de Fignorance oil il s'est trouv6 de vos publications, et ensuite a proclamer
hautement que vons aviez donn6 la solution de la question proposed. Une circonstance pourrait 6ter
tout embarras et rendre sa tache facile autant qu'agr^able. S'il avait en effet a rendre compte d'un
me'moire adresse' par vous-mgme dans lequel vons rappelleriez vos anciennes recherches en les com-
pletant, vons voyez que justice vous serait rendue en me'me temps que les intentions de I'Acaclcniie
" The subject of the prize for 1882 had also been announced a year previously, but the notice
had then escaped his attention. The following are the terms of the announcement :
Grand Prix des Sciences Math&natiques. (Prix du Budget.) Question proposed pour I'ann^e
1882. L' Academic propose pour sujet du prix la Tfieorie de la decomposition des nombres entiers en
une somme de cinq carres, en appelant particulierement 1'attention des concurrents sur les re^ultats
extrgmement remarquables 6nonc6s sans demonstration par Eisenstein dans une Note ecrite en langue
francaise au Tome 35 du Journal de Mathemaliques de Crelle (p. 368, annee 1847).
' Le prix consistera en une me'daille de la valeur de trois mille francs.
'Les Memoires devront etre remis au Secretariat avant le 1<* juin 1882; Us porteront nne
epigraphe ou devise rep6te dans un billet cachete qui contiendra le nom et 1'adresse de 1'auteur.
Ce pli ne sera ouvert que si la piece a laquelle il appartient est couronnee.' (Comptes Rendus, vol. xcii.
p. 622, March 14, 1881, and vol. xciv. p. 330, Feb. 6, 1882.)
INTRODUCTION. Ixvii
seraient remplies puisqu'on lui annoncerait la solution complete de la question proposed. Jusqu'ici
je n'ai pas eu connaissance qu'aucune piece ait et6 envoyee, ce qui s'explique par la direction du
courant mathematique qui ne se porte plus maintenant vers I'arithm6tique. Vous etes seul en
Angleterre a marcher dans la voie ouverte par Eisenstein. M. Kronecker est seul en Allemagne ;
et chez nous M. Poincare, qui a jete" en avant quelques id<?es heureuses sur ce qu'il appelle les
invariants arithmetiques, semble maintenant ne plus songer qu'aux fonctions Fuchsiennes et aux
equations differentielles. Vous jugerez s'il vous convient de r^pondre a 1'appel de 1'Academie a ceux
qui aiment I'Arithme'tique ; en tout cas soyez assure que la commission aura par moi connaissance de
vos travaux si elle a se prononcer et a faire un rapport a 1'Academie sur des mcmoires soumis a son
examen . . . Je vous renouvelle, mon cher Monsieur, 1'expression de ma plus haute estime et de mes
sentiments bien sincerement deVou^s.
CH. HERMITE.
PABIS, 26 Kvrier, 1882.
In consequence of an accident when riding, Professor Smith had been
confined to his sofa for some weeks ; but, as far as his strength permitted,
he had been working steadily at subjects connected with the memoir on
the Theta and Omega subjects, which he was very reluctant to lay aside.
Nevertheless, he thought it his duty to accede to the suggestion of M. Her-
mite, and bring his demonstrations before the Academy in the form of
a memoir sent in for the concours. For a while he divided his spare time
between Elliptic Functions and the work connected with the prize subject, but
in April he wrote : ' I fear I cannot let you have the Transformation papers
before the end of June. As I foresaw, getting the quadratic forms of n
indeterminates into my mind again, putting my proofs into a rigorous form,
and writing them out, will take up every moment till the end of May (the
paper has to be in Paris by June 1). My sole reason for taking this trouble is
that sooner or later I should have had to do it unless I was to allow my demon-
strations to perish.'
Professor Smith died on February 9, 1883, and it was not till nearly two
months after his death (at the meeting of the Academy on April 2) that the
report of the Commission was announced, two prizes being awarded, one to
Professor Smith and one to M. Minkowski, of Kb'nigsberg. The following is
the text of the report :
Grand Prix des Sciences Math6matiques (Prix du Budget).
(Commissaires : MM. Hermite, Bonnet, Bertrand, Bouquet ; Jordan, rapporteur.)
L'Acadtmie avait propose pour sujet de prix la ' Theorie de la decomposition des nombres entiers
en une somme de cinq carr6s,' en appelant particulierement 1'attention des concurrents sur les resultats
extremement remarquables ^nonces sans demonstration par Eisenstein dans une Note ecrite en langue
francaise au tome 35 du Jownal de Mathematiques de Crelle, p. 868, annte 1847.
i 2
Ixviii INTRODUCTION.
Ce probleme semble asscz restreint au premier abord ; mais on avait lieu de penser qae lea
theoremes obtenus par cet illustre geometre s'etaient offerts a lui conime consequences dernieres d'une
longue aerie de recherches, oil devaient se tronver combiners les notions d'ordre et de genre, eiablies par
Gauss pour les formes binaires, et transportees par Eisenstein dans le domaine des formes ternaires, celle
de la deruile, qu'il avait iutroduite pour la premiere fois, enfin les m6thodes infinite'simales de
Dirichlet. L'Academie etait done fondee a esp^rer que ce voyage de decouvertes impost aux concur-
rents a travers une des regions les plus interessantes et les moins explorees de I'Arithmetique
produirait des risultats fecouds pour la Science. Cette attente n'a pas ete tromp6e.
Trois Memoires ont 6te transmis au Concours ; ils portent les epigraphes suivantes :
No. 1. Quotque quibusque raodis possint in quinque resolvi
Quadratos numeri, pagina nostra docet.
No. 2. Felix qui potuit rerum cognoscere causas 1
No. 3. Rien n'est beau que le vrai ; le vrai seul est aimable.
Le Memoire No. 2 montre chez son auteur des connaissances ttendues et renferme plusieurs
resultats interessants ; mais la question pose'e par 1' Academic ne s'y trouve meme pas abordee. La
Commission a done principalement concentre son etude sur les deux autres Memoires. Tous deux sont
des ceuvres considerables, oil se trouvent exposes d'une maniere magistrale plusieurs des points fonda-
mentaux de la theorie des formes quadratiques. Les formules relatives a la decomposition en cinq
carres n'y figurent que comme consequences tres particulieres des principes gent'raux.
II est d'ailleurs aise de discerner dans ces deux Memoires, a travers les differences d'exposition, une
singuliere identite dans la filiation des idees, au point qu'il serait difficile de signaler dans 1'un d'eux
une notion ou un theoreme important qu'on ne retrouvat pas dans 1'autre, et que, pour eviter les redites
et fa ire mieux ressortir les nuances qui les separent, nous devions les analyser simultanement.
L'auteur du Memoire No. 1 montre tout d'abord qu'a une forme quadratique quelcoiique on peut
associer une s^rie de formes adjointes * ; la, valeur numerique du plus grand commun diviseur des
coefficients de ces diverses formes et leur ordre de parite servent de base a une distribution en ordres
de iiirinr determinant.
L'auteur du Memoire No. 3 ne parle pas de ces formes adjointes, si ce n'est de la premiere, que
Gauss avait deja definie ; mais il considere la serie de leurs coefficients, ce qui lui donne un resultat
identique au precedent. La marche suivie dans les deux Memoires est d'ailleurs la meme et consiste
a transformer la forme proposee en une autre cquivalente, telle que son residu par rapport a un module
donne soit ramen^ a une expression canoniqae.
Cette expression canonique contient encore des coefficients indetennines dont la valeur dependra
de la maniere de conduire les calculs ; mais de quelque facon que Ton opere, en partant d'une forme
donnee, certaines combinaisons de ces coefficients conserveront toujours un caractere quadratique
determine par rapport aux nombres premiers qui divisent le determinant et par rapport aux nombres
4 et 8. L'ensemble de ces caracteres, invariables pour toutes les formes d'une mfime classe, detininv
le genre.
Ainsi que Gauss 1'avait deja signaie pour les formes binaires, en insistant tout particulierement
sur ces circonstances, qui sont pour rArithmetique du plus haut interet, toutes les combinaisons de
caracteres ne sont pas admissibles. Les deux auteurs indiquent d'une facon precise les conditions que
doit remplir nne semblable combinaison pour correspondre a un genre reellement existant.
Ils passent ensuite a la recherche du nombre des solutions des congruences du second degr6
a plnsieurs inconnues. Cette question se lie intimement aux precedentes. La methode elegante fondee
sur 1'emploi de la resolvante de Lagrange, par laquelle elle est traitee dans le Memoire No. 3, merite
* Ces formes avaient deja ete considerees par M. Darboux dans le Journal de Liouville.
INTBODUCTION.
une mention particuliere. L'Auteur enonce ensuite cette proposition, dont il est facile de rdtablir la
demonstration : Deux classes de formes qui appartiennent au meme genre sont congrues par rapport
a un module quelconque.
Cette nouvelle definition du genre, deja formulee d'ailleurs par M. Poincar6, a 1'avantage de
s'etendre immediatement aux formes d'ordre superieur au second.
Les deux auteurs s'occupent ensuite de la representation des nombres par une forme quadratique
a n variables. Us montrent, en gen6ralisant une methode de Gauss, que cette recherche revient a celle
de la i-epresentation d'une forme quadratique a n 1 variables. Abordant ensuite ce dernier probleme,
ils font voir comment 1'ordre et le genre de la forme representee peuvent se de'duire de 1'ordre et du
genre de la forme qui la represente. Les resultats precedents leur permettent de ramener la recherche
de la densite des representations d'un nombre donne par 1'ensemble des formes d'un meme genre a celle
de la densite d'un genre donne.
L'application des methodes de Dirichlet a fourni la solution de ce probleme a 1'auteur du Memoire
No. 1 pour les formes quaternaires ; a celui du Memoire No. 3 pour les formes a un nombre quelconque
de variables dont toutes les adjointes sont des formes impaires. Mais chacun d'eux, presse par le
temps, n'a donnd la demonstration de ses resultats qu'autant qu'il etait necessaire pour resoudre le
probleme pose par 1' Academe. Tous les deux le ramenent a la semination d'une aerie infinie,
m ' TO 2
fort analogue a celle que Dirichlet avait rencontree dans son celebre Memoire sur les applications du
Calcul infinitesimal a la Theorie des nombres.
L'auteur de Memoire No. 3 s'arrgte a ce point ; celui du Memoire No. 1 donne sans demonstration
le rtsultat de la sommation, d'oii decoulent immediatement les thoremes d'Eisenstein.
De meme que nous n'avons pu separer ces deux beaux Memoires dans la courte analyse qui
precede, nous ne eaurions les presenter 1'un sans 1'autre aux suffrages de 1'Academie. Tous deux en
sont egalement dignes. Ils font faire un pas considerable a 1'Arithmetique, en fixant d'une maniere
definitive la theorie de 1'ordre et du genre dans les formes quadratiques. Le talent deploye par les
auteurs nous est d'ailleurs garant qu'ils sauront mener a terme les questions difBciles qu'ils ont du
traiter un peu hativement a la fin de leur travail.
Dans 1'impossibilite ou elle se trouve de mettre 1'un d'eux au second rang, la Commission a 1'una-
nimite emet le vceu que 1'Academie accorde a chacun d'eux la totalite du prix, si elle le juge possible.
Nous devons faire observer, en terminant, que le Memoire No. 3 est ecrit en allemand, contrairement
a 1'une des conditions du programme. L'auteur s'en excuse dans sa Preface, en disant que le temps lui
a manque pour faire la traduction de son Memoire. Nous n'avons pas pense qu'il y eut lieu de repousser
a priori, pour une irregularite de forme, un travail de cette importance. Mais, tout en 1'accueillant,
a titre exceptionnel, 1'Academie devra faire toutes reserves pour 1'application des regies ordinaires aux
concours a venir.
L' Academic adopte les propositions de la Commission et decide qu'elle decernera deux prix de
m&ne valeur aux auteurs des Memoires inscrits sous les Nos. 1 et 3.
Conformement au Reglement, M. le President precede a 1'ouverture des plis cachetes qui accom-
pagnent ces Memoires et proclame pour le No. 1 le nom de M. J. S. Smith, professeur a 1'Universite
d'Oxford, et pour le No. 3 nom de M. Hermann Minkowski, etudiant de Mathematiques a I'Universite
de Konigsberg.
It will be seen that in this report, which has been reproduced in its entirety,
no mention is made of Professor Smith's previous publications, nor is there even
a reference to his having completed Eisenstein's formulae for five squares, and
l xx INTRODUCTION.
given the corresponding formulae for seven squares, more than fifteen years before :
in fact, the report shows that the writer regarded Professor Smith's memoir as
perfectly new work called into existence by the prize competition. Under these
circumstances Miss Smith, as the representative of her brother, wrote to M. Her-
mite recalling his attention to the expression in his letter of February 26, 1882,
' En tout cas soyez assure* que la commission aura par moi connaissance de vos
travaux si elle a se prononcer et k faire un rapport k 1'Acaddmie sur des
me'moires soumis & son examen,' and expressing the hope that he would give the
explanation that had become necessary. M. Hermite replied that the omission
of which she complained was an error which was due to absolutely involuntary
forgetfulness (' ce tort ne consiste que dans un oubli, qui a 6t6 absolument invo-
lontaire') ; but he made no further statement of any kind. The award of the
prize gave rise however to a good deal of comment in the Paris newspapers. The
Academy was blamed for having been unaware of work published by the Royal
Society in 1868, and it was pointed out that the award was necessarily unsatis-
factory, in spite of Professor Smith himself having sent in a memoir, as any
other competitor might have availed himself of the indications contained in his
published writings. The striking identity between the first and third memoirs,
which is emphasized in the report, gave rise to the statement, which appeared
in the newspapers, that this had actually taken place. In consequence of
these criticisms M. Bertrand made certain explanations at the meeting of the
Academy on April 16, 1883. The proceedings commenced with the reading
of an appreciative obituary notice of Professor Smith by M. Camille Jordan, in
which special reference was made to his arithmetical researches. The account
then proceeds :
M. Bertrand demande a 1'Acad^mie la permission d'ajouter quelques mots a la lecture qu'elle vient
d'entendre.
' La Commission chargte de proposer le sujet du prix de Math^matiques avait demand^ aux
concurrents 1'einde d'un th^oreme enonce, il y a pres de quarante ans, par 1'illustre gt'ometre Eisenstein,
enleve a la science avant d'en avoir public" la demonstration.
' Un seul Memoire depuis la mort d'Eisenstein avait et6 consacr6 a cette difficile question : il 6tait
de M. Smith et, comme celui d'Eisenstein, contenait l'6nonce seulement des resultats priucipaux. Si le
conconrs propose 1 par 1' Academie n'etait pas venu reporter 1'attention de M. Smith vers ces i echerches
deja ancienues, il n'aurait, de menu- qu'Eisenstein, Iegu6 Bur ce sujet aux ge'ometres qu'un euigme
difficile a d6chiffrer.
' Sur les trois M^moires pnteente's au concours, le premier a dte ecart6 comme insuffisant.
' Le deuxieme suivait exactement la marche traces par M. Smith et donnait la demonstration de
BBS enonces ; celui des Commissaires qni a accept^ la tache d'en faire 1'examen a pu, sur ces indices,
deviner le nom de 1'auteur. Peu importait, d'ailleurs, que le Memoire fut de M. Smith ou inspire par
INTRODUCTION. Ixxi
le travail depuis longtemps livre au public par le savant professeur d'Oxford : il me'ritait incontestable-
ment le prix.
' Un troisieme M6moire risolvait la question ; il etait difficile que deux g6ometres assez habiles
pour parcourir ce terrain e'leve, mats un peu etroit, ne s'y rencontrassent pas sur plus d'un point. Les
methodes avaient de 1'analogie, mais chaque M^moire portait la marque d'un esprit original et distingue' ;
tous deux etaient excellents et il semblait impossible de donner a i'un d'eux le second rang.
'Les deux Memoires seront publics, et 1' Academic se f61icitera d'avoir donne 1 a leurs savants
auteurs, 1'un a la fin, 1'autre au debut de sa carriere, 1'occasion de montrer les ressources d'un esprit
ingenieux et la preuve, inscrite a chaque page, d'une science etendue et profonde.'
These official remarks, which are supplementary to the report of the
Commission, render justice to M. Minkowski, and offer a carefully framed defence
of the Academy, but without admitting that the subject was proposed in
ignorance of Professor Smith's work, or that the reporter was not aware of the
existence of the paper of 1867 until after the publication of the report. In
a historical statement relating to the subject and award of the prize, drawn up
a fortnight after the publication of the report, and in reply to adverse criticisms,
a full avowal of all the circumstances might have been looked for. It is right
to say that M. Camille Jordan, the reporter, was not a member of the Academy
when the subject was announced, and that it was only at the last moment that
he was charged with the duty of reporting upon the three memoirs.
It is much to be regretted that it should have been necessary to
devote so much space to the matters connected with this memoir. A very
brief notice would have sufficed if M. Hermite had communicated the existence
of the paper of 1867 to the other members of the Commission, or if after the
award he had given a brief account of the facts, or caused such an account
to be given. But the only statement made was that of M. Bertrand, and it
therefore became impossible to avoid details and quotations, as Professor Smith
would not have been willing to send in a memoir for the competition except
under the special circumstances of the case and in response to M. Hermite's
suggestion.
An Appendix at the end of the second volume contains four writings
which, though not of the same original character as the papers themselves, neces-
sarily find a place in a collected edition of Professor Smith's mathematical works.
The last of the four is a portion of the Introduction to the collected edition of
Clifford's Mathematical Papers, which was written in the summer of 1881. Only
so much of this Introduction has been included as could be of interest to a reader
who had not the book itself before him. A reference should be here added to
a review by Professor Smith of Campbell and Garnett's Life of Professor
Ixxii
INTRODUCTION.
Clerk Maxwell which appeared in the Academy for January, 1883 (vol. xxiii,
pp. 19, 35). This review, being almost wholly biographical, is not reprinted.
He contributed verbally to the meetings of the London Mathematical
Society and British Association a number of papers, which unfortunately were
never written out. The following is a list of the titles of these papers :
London Mathematical Society.
1. Construction of the last point of intersection of a cubic curve by a curve of a superior order.
March 26, 1868 (vol. ii, p. 61).
2. Geometrical note on the concomitants of a binary cubic. March 26, 1868 (vol. ii, p. 61).
3. Theory of certain systems of conies which present themselves in connexion with cubic curves.
May 28, 1868 (vol. ii, p. 67).
4. On a problem in kinematics, and focal properties of skew surfaces. April 14, 1870 (vol. iii,
p. 99).
5. On elliptic integrals. December 8, 1870 (vol. iii, p. 195).
6. On skew cubics. March 9, 1871 (vol. iii, p. 224).
7. On the partition of geometrical curves. February 10, 1876 (vol. vii, p. 90).
8. On the aspects of circles on a plane or on a sphere. April 13, 1876 (vol. vii, p. 172).
9. On some elliptic function properties. January 11, 1877 (vol. viii, p. 139).
10. On Eisenstein's Theorem. June 14, 1877 (vol. viii, p. 289).
11. Note relating to the theory of the division of the circle. April 11, 1878 (vol. ix, p. 102).
12. On a correction in Sohncke's tables. January 9, 1879 (vol. x, p. 44).
13. Upon a modular equation. January 9, and February 13, 1879 (vol. x, pp. 42 and 75).
14. Two geometrical notes relating to surfaces of the second order. March 13, 1879 (vol. x,
p. 104).
15. Two geometrical notes. June 12, 1879 (vol. x, p. 167).
16. Geometrical notes (3). February 12, 1880 (vol. xi, p. 50).
British Association.
1. On a property of surfaces of the second order . . . 1866, p. 6. Nottingham.
2. On the large prime numbers calculated by Mr. Barrett Davis 1866, p. 6. ,,
3. On a construction for the ninth cubic point . . . 1868, p. 10. Norwich.
4. On geometrical constructions involving imaginary data . 1868, p. 10.
5. On a property of the Hessian of a cubic surface . . . 1868, p. 10.
6. On the circular transformation of Mobius .... 1872, p. 24. Brighton.
7. On modular equations 1873, p. 24. Bradford.
8. On singular solutions 1875, p. 21. Bristol.
9. On the effect of quadric transformation on the singular
points of a curve 1875, p. 21.
10. On the modular curves 1878, p. 463. Dublin.
11. On quadric transformation 1878, p. 465.
12. On inverse figures in geometry 1880, p. 476. Swansea.
13. On a mathematical solution of a logical problem . . 1880, p. 476.
14. On the distribution of circles on a sphere .... 1880, p. 476.
INTRODUCTION. Ixxiii
15. Note on the skew surfaces of the third order . . . 1880, p. 482. Swansea.
16. On a kind of periodicity presented by some elliptic functions 1880, p. 482. ,,
17. On the differential equations satisfied by the modular equa-
tions 1881, p. 535. York.
18. On the equation of the multiplier in the theory of elliptic
transformation 1881, p. 538.
19. On the linear relation between two quadratic surds . . 1881, p. 538.
20. On a property of a small geodesic triangle on any surface . 1881, p. 548.
I have omitted a title from this list whenever I knew that the paper in
question was published elsewhere. Thus a paper ' On Continued Fractions' which
was communicated to the British Association in 1875 was afterwards published
in the Messenger, and forms No. xxviii. of the present reprint. It is probable
that the contents of a few others are included in the published papers.
No doubt all the results upon which these communications were founded are
contained in his note-books.
With respect to the character of Professor Smith's mathematical writings
a very noticeable feature is the arithmetical spirit that runs through the whole
of his work. The years of study which produced the Report upon the Theory
of Numbers exercised a lasting influence upon his mode of thought ; and his
familiarity with the ideas and methods of the Higher Arithmetic continually
shows itself in his treatment of Geometry and Elliptic Functions. In the latter
subject the arithmetical tendency of his mind is especially evident in the point
of view from which the theory of Transformation is always regarded. Another
characteristic feature of his work is its completeness, both as regards attention
to details and accuracy of demonstration. He had a very strong dislike to
careless or slovenly work of any kind, and thought that it was nowhere so
much out of place as in Pure Mathematics. He was ready enough to pass over
the ground boldly and rapidly, without regard to ambiguities or details, when
he was seeking after new theorems, or merely endeavouring to decide upon the
truth of generalizations or guesses ; but he was of opinion that a mathematician
should refrain from publication until he had established his results by perfectly
rigorous demonstration. He had no sympathy with those who were contented
to give imperfect demonstrations, or to regard results as proved merely because
they had satisfied themselves of their truth. No task is more irksome to
a mathematician than that of working out in detail all the various particular
cases of a theorem, when the novelty of the investigation by which it was
discovered has long since worn off. The general result, too, of such examinations
is to produce modifications and limitations which at the same time add to the cum-
k
INTRODUCTION.
brousness of the demonstrations and detract from the simplicity of the theorems
themselves. But he held that any slurring-over of difficulties or ambiguities
was utterly repugnant to the nature of the subject, and that a mathematician
was bound to spare no amount of labour that was requisite in order to give to
his results the highest degree of precision of which they were susceptible. The
comparatively slow rate of progress of the memoir on the Theta and Omega
Functions was no doubt primarily due to the many other claims upon his time,
but it was also attributable, in no slight degree, to the extreme care taken to avoid
ambiguities of every kind, and to the attention bestowed upon the systematic
examination of all the special cases of the general theorems. His natural love
of precision in thought and expression was no doubt strengthened by his early
study of the writings of Gauss, for whom he always felt the most unbounded
admiration. The following notes, which he wrote for Mr. Tucker*, on the occasion
of the celebration of the centenary of Gauss's birth, find a fitting place here, as
they show, in his own words, not only his deep reverence for the great master
of the Higher Arithmetic, but also the extreme importance that he attached
to perfection of form in the presentation of mathematical results.
If we except the great name of Newton (and the exception is one which Gauss himself would have
been delighted to make) it is probable that no mathematician of any age or country has ever surpassed
Gauss in the combination of an abundant fertility of invention with an absolute rigorousness in
demonstration, which the ancient Greeks themselves might have envied. It may be admitted, without
any disparagement to the eminence of such great mathematicians as Euler and Cauchy, that they were
so overwhelmed with the exuberant wealth of their own creations, and so fascinated by the interest
attaching to the results at which they arrived, that they did not greatly care to expend their time in
arranging their ideas in a strictly logical order, or even in establishing by irrefragable proof propo-
sitions which they instinctively felt, and could almost see, to be true. With Gauss the case was other-
wise. It may seem paradoxical, but it is probably nevertheless true, that it is precisely the effort after
a logical perfection of form which has rendered the writings of Gauss open to the charge of obscurity
and unnecessary difficulty. The fact is that there is neither obscurity nor difficulty in his writings, as
long as we read them in the submissive spirit in which an intelligent schoolboy is made to read his
Euclid. Every assertion that is made is fully proved, and the assertions succeed one another in
a perfectly just analogical order ; there is nothing so fur of which we cau complain. But when we have
finished the perusal, we soon begin to feel that our work is but begun, that we are still standing on the
threshold of the temple, and that there is a secret which lies behind the veil and is as yet concealed
from us. No vestige appears of the process by which the result itself was obtained, perhaps riot even
a trace of the considerations which suggested the successive steps of the demonstration. Gauss says
more than once that, for brevity, he only gives the synthesis, and suppresses the analysis of his propositions.
' Pauea sed matura' were the words with which he delighted to describe the character which he
endeavoured to impress upon his mathematical writings. If, on the other hand, we turn to a memoir
of Euler's there is a sort of free and luxuriant gracefulness about the whole performance, which tells of
* ' Carl Friedrich Gauss,' by R. Tucker. Nature, vol. xv, p. 537 (April 19, 1877).
INTRODUCTION. Ixxv
the quiet pleasure which Euler must have taken in each step of his work ; but we are conscious
nevertheless that we are at an immense distance from the severe grandeur of design which is character-
istic of all Gauss's greater efforts. The preceding criticism, if just, ought not to appear wholly trivial ;
for though it is quite true that in any mathematical work the substance is immeasurably more important
than the form, yet it cannot be doubted that many mathematical memoirs of our own time suffer greatly
(if we may dare to say so) from a certain slovenliness in the mode of presentation ; and that (whatever
may be the value of their contents) they are stamped with a character of slightness and perishableness,
which contrasts strongly with the adamantine solidity and clear hard modelling, which (we may be
sure) will keep the writings of Gauss from being forgotten long after the chief results and methods
contained in them have been incorporated in treatises more easily read, and have come to form a part
of the common patrimony of all working mathematicians. And we must never forget (what in an
age so fertile of new mathematical conceptions as our own, we are only too apt to forget) that it is the
business of mathematical science not only to discover new truths and new methods, but also to establish
them, at whatever cost of time and labour, upon a basis of irrefragable reasoning.
The iiadrjpaTiKos m6avo\ay5>v has no more right to be listened to now than he had in the days of
Aristotle ; but it must be owned that since the invention of the ' royal roads ' of analysis, defective modes
of reasoning and of proof have had a chance of obtaining currency which they never had before. It is
not the greatest, but it is perhaps not the least, of Gauss's claims to the admiration of mathematicians,
that, while fully penetrated with a sense of the vastness of the science, he exacted the utmost rigorous-
ness in every part of it, never passed over a difficulty as if it did not exist, and never accepted
a theorem as true beyond the limits within which it could actually be demonstrated.
These words certainly express the ideal which Professor Smith had always
in his mind, and which has governed the character of his own work.
In passing the papers through the press I have corrected all the misprints and
errors that I detected, but no other alterations of any kind have been made in the
text. I have added notes only in those cases where they seemed to be absolutely
necessary. All additions, references, or notes which are not in the original
papers are enclosed in square brackets [ ]. The correction of misprints or slips
often involved matters of some delicacy, and occasioned frequent delays. There
were also other difficulties connected with the papers that were printed from
manuscript. The sheets containing the concluding portion of the Report on the
Theory of Numbers were passed through the press (during my absence abroad)
by Professor Cay ley, by whom the index to the Report was made. Professor
Cayley also kindly undertook the revision of the unconnected portion of the
Memoir on the Theta and Omega Functions.
Professor Smith did not leave many separate mathematical manuscripts,
most of his work being contained in note-books. These books, about forty in
number, cover the whole period of his mathematical career. Some contain his
early notes when making his first studies in Geometry, or reading the memoirs
upon which the Report on the Theory of Numbers was founded ; others relate
to his University lectures ; and rather more than a dozen are devoted to the
k2
INTRODUCTION.
records of original work, a very large portion of which has never been pub-
lished. I have repeatedly examined the note-books relating to the subjects
with which I was most familiar in hopes of being able to make extracts that
could have been included in the present volumes. But in this I have been
unsuccessful, for Professor Smith entered in these books not only the finished
theorems which he had demonstrated, but also results which he had arrived at by
rough explorations and inductions, as well as mere guesses sometimes ; and it is
certain that he would have published nothing himself from these books without
submitting it to the most careful examination and working out the demonstra-
tions afresh. Under these circumstances it was decided, but with great
reluctance, to confine the present work to the published writings, and make no
attempt to give an account of the varied contents of the note-books. The editing
of any considerable portion of the unpublished work would be a matter of great
difficulty, requiring much time and research, but it would not be so serious an
undertaking to prepare separately for publication some of the investigations
which he has left upon special subjects. In particular, it is very desirable
that his researches relating to the decomposition of numbers into seven squares
should be published ; and it would probably be found that the editing of this
application of the general formulae has been greatly simplified by his own treat-
ment (in the prize memoir) of the corresponding work on the five-square problem.
The principal subjects upon which he lectured in the University were
Modern Geometry, Analytical Geometry, Theory of Numbers, Calculus of Varia-
tions, and Differential Equations. With the exception of the Theory of Numbers,
his lecture-notes on these subjects are very fragmentary; but full and accurate
transcripts of the lectures themselves as delivered were kindly supplied by
Mr. Lazarus Fletcher, F.R.S., Mr. Thiselton Dyer, F.R.S., Mr. H. T. Gerrans, Mr.
Walter Larden, the late Mr. Arthur Buchheim, and other pupils. As no other
teaching on Modern Geometry was given in an English University, and as his
lectures on this subject exercised great influence upon the direction of mathe-
matical studies in Oxford, it was considered very important that they should be
published. The editorship was undertaken by Mr. H. M. Taylor, Fellow of
Trinity College, Cambridge, who after a careful comparison of the lectures as
delivered in different years wrote out for press a fair copy of what might be
regarded as the standard form of the course. It was, however, finally decided to
abandon the publication, partly because the same ground was more systematically
covered by foreign treatises, and partly because the extent of the lectures was
so limited (owing to the fact that students did not specialize in the subject) that
INTRODUCTION.
1
XXV11
the volume would be scarcely adequate to form an independent treatise on so im-
portant a branch of Mathematics. It may be mentioned that the courses delivered
in various years differed very much from one another, and it would appear as if
their nature and extent had to some degree depended upon the audience.
It is well known that Professor Smith intended to write an Introduction to
the Theory of Numbers, and regret was frequently expressed to him that the work
was still unpublished. Among his note-books there are several in which the
elements of the subjects are very clearly and succinctly explained in methodically
arranged paragraphs, and it cannot be doubted that these are successive editions
of the commencement of such a work, in which he was striving after greater
perfection. Other note-books contain carefully written articles which may have
been intended as chapters in such a work. The completed portion of the treatise,
however, is so small, only reaching to quadratic forms, that the idea at first
entertained of publishing it separately as a fragment was ultimately given up.
I hope that it will not be thought out of place for me to include in this
Introduction a few reminiscences of my own with respect to Professor Smith,
as he appeared to me, and to attempt a sketch, however slight, of his personality.
In the eleven years that have elapsed since his death many of those to whom
his presence was so familiar have passed away, and a new generation of mathe-
maticians has arisen to whom he is but a name, so that the time seems to have
already come when it is allowable to place on record matters which were
once of common knowledge or might have seemed too trivial for mention
in print.
I first saw Professor Smith at the British Association meeting at Nottingham
in 1866, when he was one of the secretaries of Section A (Mathematics and
Physics). I can perfectly remember his attitude and manner both when
as secretary he read the papers of others, and when standing by the black-
board, he explained, so simply and gracefully, the nature of his own communi-
cations to the section. His tall handsome figure, his commanding presence,
and the charm of his manners, stand out clearly before me, as I watched him
then ; and in no essential respect was there any change in him between the first
time I saw him and the last.
At this meeting he spoke upon the average frequency of prime numbers,
and I then for the first time heard of Legendre's approximate formula
2C
- for the number of primes inferior to x, a result which interested
log x l-Uodoo
Ixxviii INTRODUCTION.
me intensely, although I little thought that it was subsequently to occupy so
much of my own time *.
I was introduced to him in the committee-room of the section by my father,
and although I was not eighteen years of age, he welcomed me with as much
cordiality as if I had been a fellow-mathematician of equal standing with himself.
I was a shy and retiring schoolboy, but, in spite of the respect with which his
knowledge inspired me, his kind and friendly manner at once placed me at my
ease. I mention so particularly this experience of my own because it was very
characteristic of his gentle and considerate nature. I am sure that no one was
ever treated by him with less courtesy or attention on account of youth or junior
standing : on the contrary, I believe that in such cases he instinctively and
unconsciously showed even more consideration. I may perhaps mention that on
this occasion he gave me the first separate reprint of a mathematical paper
which I ever possessed : it was not a paper of his own, but one which had been
given to him, and seeing me interested in it he told me I might have it, as he
could procure another copy from the author.
I did not see him again till the meeting of the British Association at Brighton
in 1872, the year after that in which I took my degree at Cambridge. At that
meeting he spoke upon the circular transformation of Mobius. I was then able
for the first time to appreciate his wonderful power as an expositor of abstruse
mathematics. His winning manners and graceful delivery charmed me as before,
but I was even more struck with the skill with which he succeeded in giving, in
the simplest language, a correct idea of complicated theories to those to whom
they were entirely new.
* My memory is quite distinct that this account was given as a ' Report on the Theory of
Numbers,' and that he briefly explained to the section the nature of the subjects dealt with in the report.
This impression is confirmed by the fact that among the sectional papers there is no title under which
Legeudre's formula could have been introduced ; for the paper ' On the large prime numbers calculated
by Mr. Barrett Davis ' (which I also distinctly remember), was given on a different day and in a different
room. (Mr. Barrett Davis had communicated a manuscript list of large prime numbers, and Professor
Smith, in laying them before the section, merely called attention to the fact that, as in the case of the
smaller primes, they were sometimes clustered thickly together and sometimes widely separated.) It
would therefore appear, almost with ceiiainty, that Professor Smith had intended to write a seventh
part of the report, which should relate to the frequency of primes and other asymptotic formulae in the
Theory of Numbers. The early note-books written while the report was in preparation contain
references to Legendre's law, and a resumd of Lejeune Dirichlet's memoir on asymptotic formulae in the
Berlin Abhandlungen for 1849. Professor Pliicker was present at this meeting, and exhibited some
models of complexes to the section on the same morning as that on which Professor Smith spoke upon
the subject of his Report.
INTBODUCTION. Ixxix
All the papers of which he gave any verbal account in public after this
date were communicated either to the London Mathematical Society or to
Section A of the British Association, and I believe I was present on every
such occasion ; for I was a very regular attendant at the meetings of the
Mathematical Society; and was one of the secretaries of Section A from 1871
to 1880 inclusive. I was also present at the meetings in 1881 and 1882.
Several of these papers, like the one I have just referred to, related to subjects
with which I was quite unfamiliar ; but I never failed to derive some benefit
from his explanations or to feel a deeper interest in the theories of Pure
Mathematics in consequence of what he had said. In general I do not
readily gain an insight into new mathematical methods merely from verbal
explanations, but his papers had a wholly exceptional effect upon me in this
respect. He had the gift of fixing the complete attention of his audience,
and imparting valuable knowledge, no matter how remote or technical the
subject. Those who were present at the reading of any paper of his will
know that there is no exaggeration in this. Some mathematicians of our day
have regarded the reading of a technical paper before a society as a mere
formal preliminary to printing, which exists only as a survival from the past :
but the beautiful mathematical expositions by which Professor Smith could
gently lead on his audience into the remote intricacies of a difficult subject,
prove that it is possible even in Pure Mathematics to convey a true idea
of highly technical researches without being technical at all. He always began
at a point from which an ordinary mathematical listener could take up the
thread, and, laying down the main lines of his subject in a series of simple
and clear sentences, following each other in logical order, succeeded, apparently
with the greatest ease, in placing his audience in possession of sufficient general
knowledge to enable them to grasp the nature and scope of the new work
that he was bringing before them. He spoke slowly, with a marked emphasis
and a measured and almost rhythmical utterance, which were very distinctive
and attractive. His language was always peculiarly felicitous, both in formal
expositions and in private conversation ; and the elegance of his style may
be fairly judged by the papers printed in the Appendix, which, I think, those who
knew him could scarcely read without fancying that they heard in them the
cadence of his voice. Although dignified in words, manner, and bearing, he was
utterly free from any trace of formality : and indeed no small part of the charm
of his character was due to the way in which natural dignity was modified
by sweetness of disposition and gaiety of heart. Even when explaining
1XXX INTRODUCTION.
the most abstract theories with the severest logical accuracy, his liveliness
and wit would frequently peep out unexpectedly in parenthetical remarks.
He was always in touch with his surroundings, but never more perfectly so
than when addressing a mathematical audience, for his modesty and unselfishness
rendered it impossible for him ever to weary others by allowing himself to
be canned away by the interest which he felt himself in the researches he
was explaining. On no single occasion was he ever dull or tedious, and his
papers were always looked forward to with pleasure at the Mathematical Society
and by the habitue's of the mathematical Saturdays at the British Association
meetings. The power to render advanced researches intelligible and interesting
to a mixed audience is a rare gift ; and the only other brilliant expositor of
mathematics whom I have ever heard was Clifford, whose style however differed
widely in almost every respect from that of Professor Smith. Clifford spoke very
rapidly and fluently, in cleverly-worded sentences that were often startling
or paradoxical. The art with which he could invest familiar things with
a new interest, or connect them with novel ideas, and the facility with which
this was done, apparently on the spur of the moment, were truly surprising, but
it seemed to me that the effect produced was greatly dependent upon the exact
words which he used and upon his mode of delivery. In Professor Smith's expo-
sitions there was never anything paradoxical or artificial. The explanations
which he gave were perfectly matter-of-fact, his power being shown in the skill
with which he held the sustained attention of his hearers as he proceeded from
step to step.
It should be mentioned that very few of his papers were produced quite
spontaneously. Mr. Tucker, the secretary of the Mathematical Society, was always
anxious to have several communications announced for each meeting, and if he had
not received enough titles would write to those who were likely to have papers in
progress or suitable matter for verbal communication to the Society. Professor
Smith always responded willingly to such appeals, and would mention subjects upon
which 'he could say something, if required.' In the same way, at the meet-
ings of the British Association at which he was present, I always asked him for
papers, and he would give me a list of subjects which he could bring before the
section, sometimes offering me a choice and letting me select those which I pre-
ferred. In making his verbal communications he generally placed one of his quarto
note-books on the table, open at the place, and occasionally referred to it as he
proceeded with his explanation. These quarto note-books in their greyish covers
were well-known objects to all who attended mathematical meetings between
INTEODUCTION.
1873 and 1883. After laying before the Mathematical Society the results of
some researches of his own, probably carried out years before, great pressure
would be brought to bear to induce him to write out an account which would
be suitable for publication. This he did whenever he could find the time, but
unfortunately many of his most interesting communications remained unwritten
when death removed him. The communications to Section A were never intended
to be published in the volumes of the Association. One he wrote out for me
for the Messenger (No. xxviii, vol. ii.), and others which I had specially asked
for had been promised to me for the same journal.
The address which he delivered before the Mathematical Society on retiring
from his two years of oifice as President in 1876 possesses so much mathematical
interest that I felt justified in including it among the papers (No. xxxi, vol. ii.).
I think it would be admitted without question that this was by far the most
remarkable presidential address, both in substance and in mode of delivery,
which has been made to the Society.
I have been thus particular in trying to describe the characteristic features
of his method of exposition, partly because for some years before his death there
was no more conspicuous personal figure in English Mathematics, and partly
because in the severe style of the papers themselves there is no trace of the
bright and winning gaiety of manner with which their first introduction to
a mathematical audience was so often adorned.
I should despair of the possibility of myself conveying any adequate im-
pression of Professor Smith's position in University and general society, but
fortunately I am saved from the anxiety of any such attempt by the excellent
article in the Spectator from the accomplished pen of the late Lord Bowen, which
is reprinted on pp. xlvi-li. This tribute of affectionate appreciation, in which
Professor Smith's character is delineated with perfect justice and delicacy, enables
the reader to form a true idea of the unique place which he held in the larger
world in which he moved, while his special claims as a mathematician were
unknown to all except a few experts. His general attainments were so great
and varied, and his personal and social qualities so brilliant, that his mathematical
powers were completely overshadowed by other more conspicuous gifts. In an
article in the Times, published the day after his death, it was truly said : ' It is
probable that of the thousands of Englishmen who knew Henry Smith, scarcely
one in a hundred ever thought of him as a mathematician at all. . . . He was
a classical scholar of wide knowledge and exquisite taste, and there were few
who talked to him on English, French, German, or Italian literature, who were
1
Kxxii INTRODUCTION.
not struck by his extensive knowledge, his capacious memory, and his sound
and critical judgment.'
It always seemed to me very strange that it should have been possible
for him to have held so distinguished a position in the foremost rank of
mathematicians without his eminence, or his devotion to the subject, becoming
more widely recognized among his friends and colleagues. His official post in
the University was that of Professor of Geometry, and it was of course well
known that he was an accomplished mathematician of high reputation. But
I am sure that very few even of his intimate friends were aware that in his own
subjects he stood alone in England, and that his papers upon the Higher
Arithmetic held a place among the most important productions of the century
in abstract science. Even fewer still had any idea of the extent to which his
heart and mind were engrossed by his mathematical researches. This want
of recognition (if it may be so called) was no doubt partly due to his dis-
inclination to speak of his own work except occasionally to those whom he knew
to be interested in it, and his non-mathematical friends may be pardoned for
not discovering an enthusiasm which showed itself so little ; in fact it cannot be
doubted that he would have been spared much of the voluntary work which he
so unselfishly undertook at the solicitation of others, if the depth of his devotion
to his own subject had been generally known. But I think a truer explanation
is to be found in the fact that, as his whole time and powers were apparently
given up to other occupations, such as University work of all kinds and Royal
Commissions, it could scarcely be supposed that he would be much more than a dis-
tinguished amateur in so exacting a science. There was nothing that suggested
the specialist in his actions or conversation ; and it is indeed truly remarkable that,
in the midst of so many varied pursuits all requiring constant care and attention,
he should have been able to carry out original work which can compare in extent
and profundity with the researches of the ablest mathematicians, who have con-
centrated their whole lives upon their special subjects. Except in vacations he
seemed to have no time for mathematical investigation, and the amount that
he accomplished was always a mystery to me until I learned that after a hard day's
work, closing perhaps with a dinner party at which his lively wit and brilliant
conversation had made him seem the gayest and the brightest of the circle,
he would quietly settle down to work in his own room for some hours before
going to bed. What he then wrote related probably to matters that had been
more or less in his mind all day, and to which at intervals he had actively turned
his thoughts, making a few stray notes perhaps on slips of paper. The last thing
INTRODUCTION. Ixxxiii
of all at night he would enter the results of the day's work or thoughts in his
note book. Most of his mathematical work he did in his head, by sheer mental
effort, and he scarcely ever committed an investigation to paper in any detail
except when writing it out for publication. The notes which he made while
thinking out a subject were often written on scraps of paper or backs of
envelopes, which were destroyed as soon as he had made a definite advance
which would allow of an entry in his notes. The fact that he used pen and
paper so little, relying on his brain as it were, increased the mental strain of his
mathematical production, so that, as a rule, when struggling with difficulties or
exploring new fields, he did not work for long at a time. After an hour or two
he would leave the subject as it were to grow of itself in the background and
permeate his mind, while he was actively employed on something less exciting *.
I may here mention that the high standard of completeness which he exacted
from himself in his published writings, and which has been referred to on p. Ixxiv,
added considerably to the effort with which his finished work was produced.
The logical sequence of propositions, the absolute sufficiency of definitions, and
the rigour of demonstrations, were all matters that exactly suited the quality
of his mind ; but his mode of working did not readily adapt itself to the laborious
classification of the separate cases of a general theorem, or other details requiring
merely industry and attention.
As his attention was not specially directed to mathematics until after his
degree, he was in fact as regards its higher branches a self-made mathematician.
It was during the long period of isolated study in which he familiarized himself
with every formula in the greatest of the abstract Theories that his powers
were developed and that his interest in mathematics grew into the almost
passionate attachment of later years. Led on by pure fascination, under no
pressure, but without either assistance or encouragement, he slowly and surely
mastered everything that had been accomplished, and gained such an insight
into the principles of the subject, and such a command over its methods, as could
* In an article in the Fortnightly Review for May, 1883, I wrote: 'His victories were won by
the hardest of intellectual conflicts, in which for the time his whole heart and soul and powers were
entirely and absolutely absorbed. It was in his wide interests and sympathies, the pleasure of inter-
course with others, and the love of all that was good and cultivated, that he found relief from these
severe mental efforts. Had he not been gifted with a disposition that gave him the keenest sympathy
with every human interest, that attracted him to society and endeared him to his friends, that gave
him, in fact, his other noble life the life the world knew his fierce devotion to the subject he loved
would have ended his days long since.'
1 2
INTRODUCTION.
only have resulted from so long and complete a self-devotion. But one un-
fortunate result of his comparative isolation was that he allowed too much of his
own work to accumulate in manuscript, and that, the ' note ' of personal ambition
(as Lord Bowen described it) being wanting in his character, and no external
stimulus prompting him, he remained indifferent to the advantages of early
publication, and was too little sensible of the difficulties that would stand in the
way of preparing for the press any work which has been too long on hand. Thus,
when he was forty years of age, besides the Report, he had published only one
important memoir, although he was in possession of an immense amount of
original work relating to Quadratic Forms, Geometry, and Elliptic Functions.
The foundation of the London Mathematical Society in 1865 was an event
which exercised a marked influence upon the subsequent course of all his work.
He was a fairly regular attendant at the council and ordinary meetings, and
there met other mathematicians who appreciated his unique knowledge, and
urged him to bring papers before the Society. Wherever he was known he
was a persona grata, but nowhere more so than at the Society's rooms in
Albemarle Street. During his presidency he communicated nine papers to the
Society (besides the Address), only four of which however were written out.
As time went on, and engagements and duties thickened upon him, he became
more and more uneasy about the mass of work that lay unfinished in manu-
script. In declining to undertake a fresh piece of work he wrote : ' I have
twenty papers embedded in my note-books. I extricated and published seven
last year.' He found it impossible to obtain the amount of consecutive leisure
that was requisite to complete long and difficult investigations ; and he was
continually distracted between the fascination of new work and the desire to
publish portions of the old. He would often say, ' I must bind my sheaves ; ' and
only a few days before his death he said to his sister, ' My mind is teeming with
new ideas *.'
His power of reading rapidly the mathematical writings of others, seizing the
principles and grasping the methods as if by intuition, always struck me as very
remarkable. Up to the last, and in spite of the scanty allowance of time that
' Three months before his death, after the meeting at Cambridge for a memorial to the late
Professor F. M. Balfour, referring to the opinion expressed by one of the speaker, that a man's original
ideas came to him before he was thirty, he said to me that in his own case he was certain that not
only had his power of seeing and understanding increased without interruption all through his life,
but that his thoughts and ideas and invention had undergone a corresponding progression and
development
INTRODUCTION. IxXXV
he could devote to Mathematics, he continued to read new mathematical litera-
ture with the same ardour, and he never allowed the pursuit of his own work to
prevent him from keeping abreast of what was being done by others, not only in
his own departments of study, but also in other branches of the exact sciences.
I cannot refrain from devoting one brief paragraph to recording his admir-
able style and perfect taste when addressing a mixed scientific audience. Of
this I recollect three remarkable instances : the first when he proposed the late
Professor Tyndall as President of the British Association for the meeting at
Belfast in 1874 ; the second when, in reply to Lord Grimthorpe, he spoke on
the endowment of research, at a special meeting of the Royal Astronomical
Society, in 1881 ; and the third at the Balfour Memorial meeting, which has
just been alluded to. On the second occasion especially, I think that none who
heard the speech are likely to forget the power and brilliance of the speaker.
He spoke so lightly, and often with such whimsical disparagement of his
own attainments and performances, that even those who were conversant with
the nature of his published writings and the varied character of his pursuits
were frequently surprised to find how well acquainted he was with matters
and subjects which would not have been thought likely to be of special interest
to him. This was the case also in Mathematics ; and I can remember my own
astonishment when, long after I knew him well, I accidentally discovered how
familiar he was with every page of Jacobi's Fundamenta Nova. From the way
in which the subject of Elliptic Functions was treated in his writings, I had not
suspected that the Fundamenta Nova would have possessed so much attraction
for him.
The following extracts from a letter addressed to the late Mr. Todhunter
(in acknowledgment of some reprints of his papers) seem to be of sufficient
interest to deserve preservation :
I have been also reading, and with great interest too, your ' Conflict of Studies.' I am afraid
I am a shade less conservative than yourself. I have been led to entertain a somewhat higher
impression of the value of experimental science, at least when the pupil is made to experiment
himself. I am perhaps a little more willing than you are to consider favourably attempts to improve
Euclid, though I have a great dread of the Association for the Improvement of Geometrical Teaching.
Further (as I am a professor, and as there is nothing like leather), I am for having more professors, with
more work, and more pay. But I so heartily agree with much, or rather with most of your book, that
I should not have troubled you with this letter, if it were not that I cannot wholly subscribe to your
estimate of the present state of Mathematics. All that we have, one may say, comes to us from
Cambridge ; for Dublin has not of late quite kept up the promise she once gave. Further, I do not
think that we have anything to blush for in a comparison with France ; but France is at the lowest ebb,
is conscious that she is so, and is making great efforts to recover her lost place in Science.
INTRODUCTION.
Again, in Mixed Mathematics, I do not know whom we need fear : Adams, Stokes, Maxwell, Tait,
Thomson will do to put against any list, even though it may contain Helmholtz and Clausius.
But in Pure Mathematics I must say that I think we are beaten out of sight by Germany ; and
I have always felt that the Quarterly Journal is a miserable spectacle, as compared with Crelle, or
even Clelwch and Neumann. Cayley and Sylvester have had the lion's share of the modern Algebra
(but even in Algebra the whole of the modern theory of equations, substitutions, &c., is French
and German). But what has England done in Pure Geometry, in the Theory of Numbers, in
the Integral Calculus 1 What a trifle the symbolic methods, which have been developed in England,
are compared with such work as that of Riemann and Weierstrass !
But it is with the younger, or at least the less-known, people that I feel the difference most. Our
English papers are so often quite free from anything really new, whereas a German takes care to know
what is known before he begins to work, and besides generally takes care to work at some really im-
portant problem, and not at some trifling expression for the co-ordinates of the focus.
If I had room, I should vent my spleen (or perhaps my envy) by saying that I attribute the mis-
chief to the business of problem-making : ninety per cent, of the good problems in Pure Mathematics that
I see, are, if I mistake not, mere fragments of some great theory, of which the candidate is supposed to
be ignorant.
In the last paragraph but one he refers to the want of a sufficiently important
object in the papers of many English mathematicians. This was a subject which
was often in his mind, and I have heard him more than once express his regret
that so many writers, instead of attacking recognized difficulties or those parts
of their subject where real advances might be expected, should be content to
occupy themselves with developments of a comparatively trifling character.
In connexion with the reference to Cambridge problems, I may mention that
on one occasion, when I was telling him about a proposal to abolish the order of
merit in the Mathematical Tripos, he said that in his opinion a system which
was successful in extracting a great amount of hard work from the students
should not (in spite of many drawbacks) be lightly abandoned.
My own friendship with Professor Smith arose in connexion with the
interest I felt in some of the subjects in which he was an accomplished master,
but it was not until he began to write the Introduction to the Theta Tables for
me that I became intimate with him. The progress of this work naturally
brought us into closer and more frequent contact. I used to meet him at the
Mathematical and Astronomical Societies, often walking with him to the
Athenaeum Club at the close of the meetings, and we had long mathematical
conversations at Cambridge when he came to the dinners of the Ad Eundem
Club. When the memoir on the Theta Functions in its final form was passing
through the press, we both read the proof-sheets, and at the same time he
was sending me the Notes on Elliptic Transformation for the Messenger :
I also had occasion to consult him on several mathematical and other questions ;
INTRODUCTION. Ixxxvii
and all these causes combined to produce a rapid interchange of correspondence
during the last two years of his life.
It was not until I became really intimate with him that I had any idea
of the intensity and earnestness of his devotion to Mathematics. Even among
mathematicians he referred so gaily and with so light a heart to his own studies
and pursuits that I have been almost startled to find, when alone with him, how
engrossed he really was with mathematical researches, and how completely they
possessed his mind and affections. He derived intense pleasure both from
working at Mathematics and from the contemplation of its truths and processes ;
and although he was undoubtedly anxious in the latter part of his life that
what he had accomplished should not perish in his note-books, he seemed
quite indifferent to the amount of recognition that was accorded to his
published writings by his contemporaries * : in fact, the only word of impatience
that, so far as I know, ever escaped him with reference to the slight attention
that had been paid to his best work, was the sentence quoted in the private
letter to myself on p. Ixvi.
The last paper of which he gave a verbal account had for its title ' On
a property of a small geodesic triangle on any surface ' (p. Ixxiii), and was com-
municated to the Meeting of the British Association at York in 1881. The
object of this note was to point out that if a, b, c are the sides of a small
geodesic triangle, then the correction to be applied to the formula a* = & 2 + c 2
2 fee cos A is | (Area) 2 x curvature.
I have no word to express the admiration and affection with which I re-
garded him myself. As regards his qualities and abilities, if I had not known
him as I did it would have seemed to me incredible that such varied gifts
and powers could be combined in the same person. All the assistance that I
have ever received with respect to the direction of my own work, or the manner
of conducting research, came from him, and I have never ceased to miss
his advice and help : and more and more with each succeeding year. It will be
long indeed before his place in Mathematics can be held by another ; but in the
lives of those who were personally indebted to him the void can never be filled.
* In communicating a paper to the Mathematical Society he once had occasion to refer to some
results contained in one of his memoirs in the Philosophical Transactions, and he playfully apologized
for having ' to quote from a paper which he had no reason to think that any one had ever looked at.'
His indifference to personal prominence or display of any kind was frequently shown at the meetings
of the British Association, for whenever there was any pressure upon the limited time of the section,
he always waived his own claims in favour of those of others.
Ixxxviii INTRODUCTION.
It is always somewhat hazardous to quote from private letters (except for
the sake of facts), as they so often give to strangers a very different impression
from that conveyed to those who knew the writer personally. Still I am tempted
to close this Introduction with a few extracts from letters which, though too
trivial perhaps to deserve publication on their own account, are yet not without
a certain interest in connexion with the published papers. All the extracts are
from letters written to myself during the last two or three years of his life ; and
most of them have been selected because they relate to the progress of the
Introduction (or Memoir) on the Theta and Omega Functions and the Notes
on Elliptic Transformation, with which he was occupied to the very last.
Oxford, 2 November, 1880.
I enclose the penultimate copy of the four 0-functions. The Society is reprinting its early numbers,
and I have ordered fifty separate copies. There is an erratum in the note on p. 9, viz. it should
be, I think, j3 = i/ v not /3 = v v'. This I have altered in the reprint.
The trodden worm will turn; and I feel sure that even Cayley will admit any defender of
suffixes to all the privileges which appertain to the status of a worm. I therefore, speaking as a
worm, declare that I do not in the least care for suffixes, but that any one who does not admit that
a double notation is, for certain purposes, imperatively required by the circumstances of the case,
is not fit to be an annulated animal at all, but only a mere zoophyte. I will, seriously, quite as
willingly write ^ ( > x\ , or ^ 0*> v ; a;), as ^ (x) ; indeed to me it is a mere printer's question. But
if I am told that 3i i 3a> ^3> 3< (however convenient as abbreviations), or again 0, H, ,, H,, are as
handy for use in general formulae applying to all the four ^-functions, I am disposed to dissent. The
Germans, I perceive, are great lovers of suffixes ; and I confess that when I try to do without them,
I soon want another alphabet.
Of course you are most welcome to do what you please with my paper : it will be much honoured
by any use you may make of it. The ' Logic,' such as it is, you should have had long since, but that
I sit seven hours a day, day after day, with our Commission. ... It is my birthday and I am feeling
very old.
The paper referred to is No. xvi (vol. i). I had accused him of exulting
in the number and complication of the suffixes, and had said that the
criticism of Professor Cayley (who disliked suffixes and avoided their use
as much as possible) would be, ' Too many suffixes ! ' I was in the habit
of giving the principal theorem of the paper in my lectures, and had asked
for the separate copies, as the formulae were unsuited for writing on the black-
board. I had also said that when, in printing my 'Lectures,' I came to the
Theta Functions I wished to reproduce the whole of the paper just as it stood
as a separate chapter. The 'Logic' was the paper whose title appears as
No. 13 on p. Ixxii. It had been promised for the Messenger.
INTRODUCTION. Ixxxix
Oxford, 5 June, 1881.
Best thanks ; only I have not time to express them. [He then refers in detail to some misprints.]
It is very kind of you to take the trouble you have done about a wretched little paper, of which the
only interest, if any, is that it applies Liouville's theorem to a question of convergence. . . . When I sent you
the manuscript of my paper I had almost asked you to print (at the end of it) Riemann's proof of
Abel's ' little theorem.' There would then have been a good tail to a poor little thing ; because
Riemann's proof is a model of what such a proof should be. (I notice Todhunter in his 'Laplace's
Functions ' refers a little contemptuously to the ' little theorem ' this designation is mine, not his ; and
in this he is quite wrong, as I think the trigonometrical series at once shows.)
The paper referred to in this letter is No. xl (vol. ii.), ' On some dis-
continuous series considered by Eiemann.'
Eyde, 15 July, 1881.
Alas ! I am not yet at the Elliptic Functions. For three weeks I was tied to my sofa in Oxford
by a sprained thigh : and during that time I was exposed to continual interruptions, as in addition to
the usual Oxford business at the end of term I had become (just at that time) executor of my dear
friend Eolleston's will, and guardian of his children. . . Finding I could not be quiet enough to work
at my Introduction to your Tables, I took up a very different bit of work, the Introduction to Clifford'a
Collected Mathematical Works. This is three-parts done and must be finished next week early : indeed it
would have been done long ago except that thinking about it takes me into space of many dimensions, &c.
I grudge the time I am giving to it because I can say nothing on the one hand fit for mathematicians
to read, nor on the other fit for non-mathematicians. So I have to maunder a good deal, which is
neither acceptable to me nor suitable to my ideas of the right way of honouring Clifford's memory.
I long to be at the Elliptic Functions, I can tell you.
If you think the Messenger would like a note of three pages on one or two points in Riemann's
' Hypotheses which lie at the basis of Geometry ' (viz. on the only two results which he announces in
formulae), the said Messenger would be mobt welcome.
On another occasion, he said that this Introduction was inferior to the
similar work which he had done in connexion with the writings of the late
Professor Conington, and that it ' savoured of the sick couch on which it had
been written."
Oxford, 12 December, 1881.
I have stolen a few hours for the Elliptic Functions, chiefly to try and get my hand in again for
work immediately after Christmas. (Till then I am liable to many interruptions.) I mutt rewrite the
transformations of the second order ; I fear that nine of them must be given, viz. the nine which give
different transformations of the elliptic functions.
Oxford, 7 February, 1882.
I have not seen you for a long time, and am afraid I am not likely to see you very soon.
I am a close prisoner to my sofa with an inflamed vein in my thigh (gouty phlebitis, they call it).
I hope I am beginning to get slowly better, but it will be a good bit of time before I am able to move
about again. I have had to rewrite ' Transformations of the Second Order,' Art. 33, and while about it
I have also made many changes in Art. 31 (' Linear Transformations of Elliptic Functions'). All this
I could send to the printer if it were any good as yet for me to do to. I am not allowed to
111
xc INTRODUCTION.
work very much, and I find I can only do rather easy things. But I think I am up to doing what remains
to be done with the Memoir. It is horrible to me to think you should take any more trouble
over the thing. And so I hope you will do no more than look through the proof-sheets very
hastily indeed, if indeed you do as much as that.
I have been preparing a little ' paperlet ' to show (1) that the coefficients a, b of the general
elliptic transformation
~M i+
are rational in V and A*, not only in u and v as they appear in Jacob! and Cayley; (2) that when
X s is an equal root of the modular equation, they are not rational (in general) in tc* and X 2 (viz.
in this case Cayley's system of equations at the beginning of his memoir admits of more than one
solution); (3) giving a new (slightly new) process for determining them which shows that in all
cases (even of equal roots) they are rational in k* and X 2 and (in the equal-root case, need
not be rational in /t 2 and X"). I had thought of bringing this to the Mathematical Society on Thursday ;
but finding such a journey out of my reach I am thinking of inflicting it on you for the Messenger.
By the way, I will try and finish the little fragment of Logic for you. My difficulty is that I cannot
get upstairs to my study and no one can find my papers for me.
Oxford, 22 February, 1882.
I am putting several interesting little things together in the ' Notes on Transformation ' which
I am writing for you.
Another extract from this letter has been given on p. Ixvi.
Oxford, 9 March, 1882.
I am sorry to say that to-morrow I shall not be able to be at the Astronomical Society.
I shall however probably venture up to London in order to go to the Meteorological Office, under
a solemn promise to my doctors to be carried up and down stairs and do nothing else. So you see I am
getting on, and if I am only patient I may soon hope to be about again.
I have just finished going over the revise of sheets 2, 3, 4 ; and am sending them to you
at Cambridge. I am ashamed to have kept them so long. I find a few errata of my own, but
none (I hope) to give much trouble to the printer. I am putting together several ' Notes on
Transformation ' for you. The paper is getting rather larger than I expected, because I have found
two or three new (to me) little things while lying on my sofa.
Several of his friends were desirous that he should be nominated as the
President of the British Association. The following is an extract from a reply
to a letter of mine on this subject :
Oxford, 14 March, 1882.
I can tell you in a very few words what I feel about the Presidency of tLe British Association ;
indeed I do not know any one more likely to understand my feelings with regard to the matter than
yourself. I should esteem the office a most horrid nuisance ; at the same time I know my duty better
to the British Association, to the University here, and to myself, than to refuse it if it were offered to
me. For the honour (which I know to be a great one) I cannot bring myself to care (perhaps this is
INTRODUCTION. XC1
owing to a temporary weariness of the world, induced by lying on a sofa) ; but on the other hand I have
a great horror of the indolence which induces one to refuse a position because the duties of it are irk-
some ; and I think Dante was quite right to put the man in hell ' che il grau rifiuto fe ' (I forget who
he was, and what he declined). What makes me say that the position would be an unmixed nuisance,
is that I have (by this time), in the University and out of it, had my full share of the sort of work
which calls my mind away from the subjects which interest me most, and I am very anxious (before the
evening closes in) to concentrate myself as much as I can. If I had to be President of the British
Association, the best work of a year would have to be given to my address, and that is much more than
I can afford. It would certainly be a sad interruption to my plans of work, and I should have a per-
petual sense of unreality about it.
He afterwards said in conversation that the only scientific topics of
general public interest upon which he could usefully discourse in a Presidential
address were the motion of the atmosphere, the law of storms, &c.
Folkestone, 13 April, 1882.
I have been here for a fortnight, and can now limp about enough for purposes of business.
I hope to meet you on Friday, and to have a few words on Mathematics with you then.
A quotation from a letter written a few days afterwards has been given
on p. Ixvii. The prize memoir was completed and sent off by the end of May.
Oxford, 30 July, 1882.
Have you returned from the United States 1 and, if so, when and where can we have a conference ?
I have been absolutely idle for thirty days at Eoyat in Auvergne, and have returned, a good deal better,
I hope ; but I am totally demoralized, and I feel as if I was too sleepy ever to do anything like a day's
work again. However, I am now your slave, till I have accomplished my engagements with you
(Introduction that was and Messenger). But my mental forces are in complete disarray, and you
will have to use the whip severely to rally them.
Do you see that Lindemann has covered himself with immortal renown by proving the transcen-
dentality of T, ? Of course, nine-tenths of the discovery is really Hermite's : but then Lindemann has
the immense glory of having seen that Hermite's method could be applied to prove the transeendentality
of IT, when Hermite himself despaired of it. I have never examined Hermite's method closely, but
taking his results for granted, Lindernann's reasoning seems all right. It is difficult not to envy,
as well as admire, people who do such beautiful things : Lindemann's name is sure of a place in every
history of mathematics hereafter *.
* Nine years before (May 31, 1873) he had written to me: 'I am much pleased in particular
with the way in which you call attention to the question of arithmetical irrationality. So far as JT is
concerned, I do not believe that any one has ever proved even so much as that n- cannot be the root of
an affected quadratic equation. And I always maintain that, until geometers have done this, the}'
should not treat the problem of the rectification of the complete circumference as a demonstrated
impossibility. Perhaps, however, the proof of the quadratic equation theorem may be obtained by
Lambert's method. But this I have never tried.' Dr. Lindemanu was the guest of Professor Smith
(when I was so too) at the Oxford Commemoration in 1876.
m 2
xc ii INTRODUCTION.
Ryde, 20 August, 1882.
I have four of ray notes nearly ready for you, and hope to finish them before I leave. They will
make about eighty of my little pages ; will this fill a number for you ?
I have been led in Note II to your question about convergence of series like sin am u in powers of u.
The only one giving any trouble is -. - ; here the radius of convergence is the analytical modulus
sin am u
of K or iK' or KiK', whichever of these four is least ; and the question is to find the values of k* for
which each of these is least.
I have put headings to the Memoir, but have not sent it off, having been absorbed, so far as I had
time, in my ' Notes.' But I will send it before I leave.
On August 10 I had gone through the first seven sheets of the Memoir
with him at the Athenaeum Club.
Eyde, 23 August, 1882.
Can you let me have a figure in the Messenger ? Here it is *. ... It is one of the modular curves
of order 4 ; it divides the plane (as you see) into five regions. The least possible ' quarter periods ' of
sin am u are, if A? lies in 1, 2, . . . , 5 (i. e. if the extremity of the vector k* lies in 1, 2 5), 1 . K, $ iK' ;
2 . | f K', K ; 3 . $iK', K J iK'; 4 . \ iK', KiK';5.K iK', IK', the + sign being taken according
as # is below or above the axis. The absolutely least period is put first; of course K and K'
are the rectilinear integrals, and least refers to absolute magnitude, i. e. to analytical modulus. Of
course also there is a general theory relating to transformation to which this proposition belongs
(it is in fact the theory of a problem which Jacobi touches on in the Fundaments Nova, saying it
is very difficult).
A more complete account of these results is given on pp. 411-413 of vol. ii.
The first portion of the manuscript of the ' Notes ' was given to me at the
meeting of the British Association at Southampton on August 29.
Margate, 8 September, 1882.
I return the proof. I am heartily ashamed of the state it is in. ... My excuse is that I pressed
myself a little too much to deliver the manuscript to you at Southampton. I am very glad I did so,
however, for I think that it would have taken a longer time if I had tried to revise it thoroughly in
manuscript, even allowing for the time it will take the printer to go through it.
I think I have now made it hang together in an intelligible way. I confess that till I wrote out the
pages, which I sent you from Spottiswoode's, I had imagined that, when the modular equation has equal
roots, the multiplier might be a root (square or cube) of a rational function of A 2 and X 2 . But I found
that what really happens is that the multiplier (when the roots are equal) still continues to be a rational
function of k 1 and X s , but is a function of k 1 and X 2 with irrational coefficients, viz. the coefficients
contain an imaginary quadratic surd such as *J in, where m is a whole number ; whereas in all other
cases the coefficients are rational numbers. I had said nothing to contradict this ; but some of my
' The figure represented a symmetrical closed curve, consisting of four loops, each of which
included the next smaller one, and having three double points on the axis of x. The region ' 1 ' was
the interior of the smallest loop, the region ' 2 ' the space interior to the next loop but exterior to the
smallest, the region ' 3 ' the space interior to the next and exterior to the second loop, and similarly
for the region ' 4 ' ; the region ' 5 ' being the space exterior to the whole curve.
INTRODUCTION. XC111
present alterations are made with a view to lead up to it. More of them however are made simply to
make the meaning, and connexion, clearer. It all lies close to what is known, but I think it is full
enough of new little things to make it fit for the Messenger.
I send all that I have received from you so that it ends abruptly. I ought to have before me this
portion when I revise the remainder. Correcting this has taken me two and a half days of (for me) hard
work. I return at once to Note II : but would you not prefer to follow up Note I with something else,
and let Note II take its chance by and bye ? Notes I and II together would carry you nearly to the
end of a third number ; and this would be dreary for your subscribers.
Please send a card to say you have received this and have not gone mad with indignation at the
state of the proof.
London, 16 September, 1882.
I enclose the revise. Of course I need not see another revise, and I should think you need not, as
Metcalfe might well be trusted to make the corrections. On Monday morning you shall have the
manuscript of the remainder of Note I. Of course I could not resist the temptation of re-scribbling it.
Enough of Note II to fill up the September number, and more, shall, if I can possibly manage it,
be in your hands on Monday morning also.
London, 17 September, 1882.
I enclose the remainder of Note I, rewritten and made as tidy as I can.
As for Note II, a great part of it is nearly ready, but none of it quite. I will send you, very soon
indeed, as much as you are likely to want and more. I am sorry to tell you it will make more
than a number and a half. Now this is intolerable, and I must divide it, for I will not take up three
numbers running (even if you would let me, which, for your credit as an editor, I hope you would not).
I think I can manage to divide it, though Borne of the beginning part is written solely with a view to
the end. Till I get the September number of the Messenger safe in your hands I don't look at the
Memoir : alas !
Oxford, 24 September, 1882.
Here is some more copy for Metcalfe. It will take him a good bit on into the October number.
But now the worst of it is, that a lot more of Note II remains I think twenty -five slips at least and
this is after my cutting off all about the absolutely least periods (with the curly cue curves), which
I now propose to make into Note III (when you have got over the surfeit occasioned by Notes I and II).
So that you see Note II, if allowed to run on, will take up nearly all, perhaps quite all, the October
number. I cannot divide it into two Notes, because it really has a unity of its own, and the arithmetic
of Arts. 2 and 3 (especially Art. 3) would be unmeaning (in a note on Elliptic Functions) without what
follows. But there can be no objection to your dividing it in print, with a ' To be continued.' And
this I should advise you to do. But I put myself wholly in your hands, and will do what you please.
I think I could let you have the rest of Note II very soon. The Messenger must have as many lives as
a cat, if it survives my Notes. Still I am prepared to maintain that the stuff in them is reasonably
good, though by trying to be complete and exact I have become diffuse.
Brockham, 29 September, 1882.
I enclose the rest of Note II. There is not quite so much of it as it looks. Still I think it will
run on pretty far into the October number. I have (as I said) left out the parts that would require
a diagram or two.
If the London Mathematical Society are in want of food at their first meeting, I could give them
an account of these omitted portions, which are to be Note III (when you allow such a tiling to appear).
This would also give me an opportunity of saying briefly what Note II comes to ' when it comes to be
fired.'
XC1V INTRODUCTION.
The next thing that I shall do is to send you the revise of the Memoir, and to this I shall now
stick till it is done; I shall begin at it this very evening.
If Metcalfe could, without putting himself out, send me the whole of Note II together, it would save
time. But I have treated him abominably about Note I, and only hope that Note II will come out
decently straight ; there really are some things in it worth a moment's attention.
Brockham, 4 October, 1882.
I return the proofs. This time they are very clear, and Mr. Metcalfe will not be able to reproach
me. A couple of references to Gauss and to my own Report have to be inserted.
I send with the proof a little fragment which conies ill after the end of the August number, and
before the beginning of the slips now sent to me. I also return the copy for the October number. All
I shall want will be a revise (in pages) of the September number, and that will enable me to correct the
part that comes out in October. I do hope, and I think it likely, that I shall not run quite to the end
of the October number. I think I shall be more easily forgiven by your public, if they see that I really
liave come to an end, and that someone else is going ahead. . . .
All this interests me very much, because it turns on the theory of 'reduction' as applied to doubly
periodic functions, and seems to me to excuse the amount of space I have made you give to it in the
Messenger. I have still to make out whatever I can about the course of the curves P; but I fear this
will not be much. I shall try (whenever Note III comes into existence) to put all this stuff into it.
So Note III will want figures. My Jiexagon, curiously enough, had already been considered by
Dirichlet ; not, of course, in relation to Elliptic Functions, but in proving Gauss' famous theorems
about the minimum value of a ternary definite quadratic form.
The concluding paragraph relates to the limits of convergence of a series
for arg sn x, about which I had consulted him. He took great interest in
the question, and several letters were entirely devoted to it.
The portion that was written of Note III, referred to in the last three
letters, appears as No. X of the ' Notes ' (vol. ii, pp. 408-414).
Abergavenny, 7 October, 1882.
I enclose the revise ; I see there is one page over, to run on into the October number . . . Your
remark as to the complexity of the result in the case of the value of the series for arg sn x, has made
me begin to doubt whether I am really right in saying that one of the branches of the three-forked
curve of discontinuity does really enter the circle of convergence. If it does the nature of things is
a fool ; if it does not, I am a fool ; the latter hypothesis seems to me the more probable, and I gladly
embrace it. Besides, I begin to see dimly a weak point in my demonstration. If only the curve can
be coaxed into staying outside the circle the result will be the simplest possible, viz. that the series,
when convergent, always gives the least value possible.
It was on the 12th of October, that I went over all the manuscript of the
' Notes ' at the Athenaeum Club with him (p. Ixiv).
Oxford, rfl October, 1882.
At last I return the revise. I dare say it is full of blunders of mine, and is peppered over with
printer's errata, but I cannot find any more than I have marked.
If the alteration of the note on p. 88 and the rearrangement on p. 89 are troublesome, it would
not be ruinous if they were left alone. Item, on p. 96 the signs in lines 2 and 3 are not very wrong
INTRODUCTION. XCV
as they stand, and might be left as they are ; I have now made them correspond exactly with the
' elementary matrices of Art. 3 ' ; as they stand, they do not.
I am very sorry to have kept you waiting so long. I comfort myself by thinking that the number
has not been expected with great impatience by any one.
Oxford, 7 November, 1882.
I am almost sorry you took the trouble of sending me a revise. "When I returned the proof I had
intended to tell you that you might print it straight off. The small world that reads the Messenger
will give an audible sigh of relief when they come to p. 99, an,d find there is no more of me. How-
ever, you will have, within a year or so, to print Note III, and some figures with it. That done I absolve
you from all further Notes on Elliptic Functions, and if I ever write them, I will inflict them on
the London Mathematical Society, or the Quarterly Journal, or on the new Scandinavian journal, or
on Sylvester's Journal, or on any one but you.
London, 30 December, 1882.
Do you happen to have a copy of the sheet pp. 423-431 that you could send me? I have two,
but the printer could hardly make them out. I mean now to do nothing but proofs for a long time.
I have the two sheets which follow those now printed off practically ready, and there is nothing to
cause delay for a long time to come.
Oxford, 20 January, 1883.
I enclose four more sheets of the Memoir : the rest (as far as set up in 4to) will follow
immediately. I am sorry to say that the first two of the sheets I send have had to undergo great
alterations. This will not happen with any of the remaining sheets. I should be very glad to
have, as soon as you can, the manuscript which is in your hands set up. For the next three or
four months I can give a great deal of time to this work, and hope (D.V.) to bring it to a close.
Oxford, 1 February, 1883.
Best thanks for your letter. I cannot be at the R. A. S. to-morrow. ... I have returned to
the printers four sheets of the Memoir for revise but this includes the sheet which really has
to be set up again, and made, I should think, into two. I find the stuff (now that I have quite
forgotten it) more intelligible and hanging together better than I supposed. I find many little
slips of mine and some of the printers', but very few great blunders so far. It takes an enormous
amount of time to go through it. I must write to you before the week is over about figures, and
about completing, or rather shutting up, the whole thing: there are now 136 pp.; I think it will
run to about 170, or a little over. Please regard this letter as not needing any answer. I shall
see you on the ninth.
These were the last words I was to hear from him. The 9th of February
was the anniversary meeting of the Royal Astronomical Society, and I entered
the Society's rooms expecting to meet him, and go over some of the sheets
of the Memoir in the way that had become habitual to us ; but Mr. Stone,
who had just arrived from Oxford, told me that he had died at seven o'clock
that morning.
J. W. L. GLAISHER
ON SOME OF THE METHODS AT PRESENT IN USE IN
PURE GEOMETRY.
[Transactions of the Ashmolean Society, VoL II, No. xxv. Read December 1, 1851.]
J.HE principles of the Analytical Geometry introduced by Descartes effected
a change in the nature of the science, the importance of which it is impossible
to over-estimate. This change consisted in two things principally; first, in
the creation of a wholly new system of Geometry by the side of the old syn-
thetical methods ; and secondly, in the complete, though gradual, metamorphosis
which these methods themselves underwent. For a long time, it is true, this
second eifect did not manifest itself. No one, for example, would say that the
Pure Geometry of our great English mathematicians, of Newton, or Maclaurin,
or Matthew Stewart, exhibits so essential a difference from the Geometry of the
ancients, as that which strikes us in the works of far less original writers in
the present day. But the change, though long delayed, appears now to be
complete, and the geometrical methods, by long contact with analysis, seem
to have acquired much of its spirit, and of its peculiar power and facility, and
this without losing the intuitiveness proper to themselves.
This has been the work of the last sixty or seventy years ; but it is
historically interesting to observe that two of Descartes' contemporaries had
anticipated the change, and had introduced methods into Pure Geometry wholly
unlike its ancient resources. The theorems due to Desargues and Pascal are
still primary in the geometrical theory of the conic sections, and the methods
by which those results were obtained, so far as it is possible to judge, appear
to have partaken fully of the generality of the results themselves. In particular,
2 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
had the great work of Pascal upon Conic Sections been published, there
can be no doubt that geometrical theories, which we now owe to the writers
of the last half century, would have been in the possession of the world
for a much longer period. Unfortunately all that now remains of Pascal's
Conic Sections is comprised in a letter of Leibnitz to Pascal's executor, giving
an account of the work, which had been submitted to him for his inspection,
and earnestly recommending its immediate publication; and in a fragment of
three or four pages in length, entitled 'Essais pour les coniques,' and written
by Pascal when he was only sixteen years old. But even this fragment, though
printed and circulated by Pascal himself, was lost for nearly a hundred years
after his death, and the magnificent theorem contained in it remained fruitless
till comparatively recent times. Nor has Desargues been more fortunate. His
works are completely lost, and we are left to form our opinion of him from
the isolated expressions of Pascal and Descartes, and from the virulent attacks
of his enemies, of whom he appears to have had a great many. Had it not
been for the ill-fortune of their works, Pascal and Desargues would have been
the founders of modern Geometry; as it is, to Monge, before all others, this
honour justly belongs. Himself a great master of Analytical Geometry how
little, for example, has been added to the general analytical theory of surfaces
since his time he yet seems to have been the first who fully felt of how great
an extension the old Geometry was capable. He not only enriched it with a
method and a body of doctrine, which has rendered thoroughly rational the
relation of the science to the arts depending on it, but he also infused into
Geometry two qualities which had seemed peculiar to analysis method in its
processes and generality in its results. What gave, and what still gives,
analysis so immense an advantage over Geometry, is, that when we have once
expressed analytically a definition, say, or a theorem, the known laws of the
combination of symbols enable us to transform that expression in a thousand
different ways, and whenever we can interpret any such transformation we have
a new theorem. To take an instance, perhaps too simple, if we write x* + y 2 = a 2
we express merely the common definition of the circle ; if we write y' 1 = a?-x i
we express a theorem, deduced from the definition, and deduced, too, by
changing the place of a single letter in an equation. If we add to this copious
power of transformation, first, the generality of analysis, that is, its power of
expressing theorems, essentially the same, however they may differ accidentally,
in one and the same formula, and, secondly, the facility with which relations
dependent on the consideration of infinity may be algebraically expressed and
ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
transformed, we shall perhaps have stated the three principal prerogatives of
analysis over the old methods of Pure Geometry. Now it would be hazardous
to say that rational Geometry can at present compete with analysis in any one
of these respects, or even that it can ever hope to do so, but it is not too
much to say that it is in possession of extensive and powerful methods for the
transformation of theorems ; that it has attained to a generality of expression
unknown before ; and, finally, that it has learned to employ all the resources
of the theory of infinitesimals, and to employ them with facility and elegance
for the demonstration of theorems that seemed formerly to require the aid of
analysis. Nay, further, whereas since the time of Monge Analytical Geometry
has received three principal improvements, the introduction of symmetry, the
method of triliteral or quadriliteral coordinates, and the method of tangential
coordinates, it is a fact that for the two last we are indebted, in the first
instance, to the rapid development of Pure Geometry, and to the efforts suc-
cessfully made by analytical writers to reconquer the ground that seemed for
the moment lost to their favourite branch of the science.
Among the principal peculiarities by which the present Geometry dis-
tinguishes itself from the old we must reckon the theory of transversals, the
different theories of the transformation of figures, and the frequent use of the
geometrical method of infinitesimals. The object of the present paper is very
briefly to characterise the first two of these theories, and to illustrate the last
by a few examples of its application to the theory of geodesic lines.
But before doing so it will be well to allude to a question, very obscure
in itself, but which puts the spirit of the new Geometry in a clear light, I
mean the theory of imaginary quantities and imaginary figures. The analytical
writers on Geometry have adopted one of two courses in this matter. They
have either attempted to construct imaginary magnitudes geometrically, or else
they have asserted the impossibility of constructing them, and have thence
inferred the impossibility of getting any good at all out of them. With respect
to the attempted constructions of imaginaries, it cannot be denied that they
have been of great use ; but they have been of use not to Geometry, which they
have conducted to no new results, but to analysis, which they have enriched
by an important interpretation of symbolical expressions. In fact, the persons
by whom these constructions were introduced were much more familiar with
analysis than Geometry, and hence they were guided much rather by ideas of
analytical than of geometrical continuity. If, for example, in the equation
y 2 = a 2 - x 2 we assign to x values beyond the limits + a, and proceed to construct
B 2
4 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
the imaginary ordinates in a plane perpendicular to the plane of reference, we
preserve it is true the algebraic continuity of the function, and we give an
admissible interpretation of the symbol *J 1', but what geometer can persuade
himself that the equilateral hyperbola thus obtained stands in any but the
most arbitrary relation to the circle ? The continuity of the circle requires that
every line in its plane, without exception, should cut it twice, and that every
point in its plane, without exception, should be the intersection of a pair of
tangents to it ; does the equilateral hyperbola enable us to realise or to interpret
either of these properties in the cases in which they become unmeaning ? Again,
it is an obvious remark, though apparently either not made or not attended to,
that through every imaginary point there passes one, and but one, real line,
and that on every imaginary line there exists one, and but one, real point.
Can the principle of perpendicularity furnish any explanation of this fact ?
Apparently not; and yet this is only one out of many cases that might be
mentioned. Analytically considered, the theory is faultless ; but geometrically,
it introduces discontinuity, it is inadequate to explain the phenomena, and
(what is still worse in the eyes of a geometer) it is barren of results. Nor
should we forget that till the constructions in question can be extended to
loci in space, their use can never become general even in Plane Geometry, since
it is frequently requisite to consider plane curves as sections of surfaces. For
these reasons all attempts to construct imaginaries have been wholly abandoned
in Pure Geometry; but, by asserting once for all the principle of continuity,
as universally applicable to all the properties of figured space, geometers have
succeeded, if not in explaining the nature of imaginaries, yet, at least, in de-
riving from them great advantages. They consider it a consequence of the
law of continuity, that if we once demonstrate a property for any figure in any
one of its general states, and if we then suppose the figure to change its form,
subject of course to the conditions in accordance with which it was first traced,
the property we have proved, though it may become unmeaning, can never
become untrue, even if every point and every line, by means of which it was
originally proved, should wholly disappear. In this way geometers are enabled
not only to present theorems, in appearance the most dissimilar, as really
identical (which in a scientific point of view is of immense importance), but
also to make one easy demonstration serve where many dissimilar ones were
before required.
The practice of demonstrating real properties by means of imaginary ones
was first introduced by Monge, who employed it, though tacitly and without
ON SOME OF THE METHODS AT PEESENT IN USE IN PUBE GEOMETEY. 5
enunciating the general principle, in demonstrating what we should now call
the properties of poles and polars in surfaces of the second order. The principle
was first broadly stated, I believe, by Poncelet, in his ' Traite* des Proprie'te's
Projectives.' It was somewhat unfavourably criticised by the Commission which
was appointed by the Institute to report on that work, and which consisted
of Cauchy, Arago, and Poisson. It seems however to have more than held
its ground, and to be frequently appealed to by all the most eminent writers.
A single example of its application must suffice. It has been shown very
recently that if we draw two tangents to a conic from a point without it, and
inscribe a circle touching the two tangents and the curve, the arc of the curve
intercepted between the two tangents will be divided at its point of contact
with the circle into two parts such that their difference shall be geometrically
rectifiable. This theorem is of great importance : it enables us to construct
with extreme simplicity the principal formulae for the addition and multipli-
cation of elliptic functions of the two first species, and its demonstration depends
ultimately on two general properties of confocal conies, viz., that if from any
point on a conic A we draw two tangents to a confocal B, the angle between
these tangents is bisected by the tangent and normal to A ; and secondly, that
if from two points on A we draw pairs of tangents to B, these four lines shall
be tangents to one and the same circle. Now it is a known theorem, due to
Quetelet, that if two spheres be inscribed in a cone, so as to touch a plane
section, the two points of contact will be the two foci of the section. It is
also known that if a surface of the second order be inscribed in another surface
of the same order, the two sections determined on the two surfaces by any
plane will have double contact, the chord of contact being the intersection of
the plane of the sections and the plane of contact. We may therefore consider
the foci of a conic as two evanescent sections of a sphere, that is, as two
evanescent circles, having double contact with the conic. If we now observe
that an evanescent circle degenerates into a pair of imaginary lines, we shall
perceive that a system of confocal conies may be regarded as all inscribed in
one and the same imaginary quadrilateral. That is to say, they may be
regarded as possessing all properties incident to a system inscribed in a real
quadrilateral, and therefore as forming a system of curves in tangential
involution, and from this general property the two properties required to be
proved are at once deducible, though since the demonstration at this point
ceases to be imaginary the subsequent steps are here omitted.
Did time permit it might be shown how from the same imaginary property
6 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
of confocal conies all the known theorems respecting such a system might be
easily deduced, the geometrical theory of involution enabling us to transform
the imaginary relation in a multitude of ways, and with a facility that renders
the process almost mechanical. Nay, more, with merely verbal, or at least very
simple alterations, the same proofs would apply to spherical as well as to plane
conies ; so that, for example, we might extend the constructions before alluded
to for elliptic functions of the second order, so as to include those of the third
also. If any one will give himself the pains to examine any one of those
theories of modern Geometry in which frequent use is made of imaginaries
for example, Poncelet's theory of homological figures, or Chasles' construction
for the semi-axes of an ellipsoid he will rise perhaps with no clearer idea of
imaginary magnitude than when he began, but he will probably be satisfied
that this is one of those cases to which Dr. Woodhouse's remark applies with
all its force, that a method which leads to true results must have its logic.
In passing to speak of the theory of transversals, it is hardly necessary
to observe that only a very few of its most characteristic features can be alluded
to here. Under the term Theory of Transversals I mean to include (somewhat
improperly, though not without precedent) all those methods for the investi-
gation of the properties of curves and surfaces which rest upon metrical rather
than descriptive relations, and which operate on the figure as it stands, and not
on figures derived by transformation from it. Several of the isolated theories
belonging to this head appear to bear a special character, from the simple fact
of their having been first developed as parts of the theory of Conic Sections.
This is the case, for example, with the principles of harmonic and anharmonic
section, and especially of involution. But it is a mistake to suppose that this
special character is more than accidental. In fact, the harmonic properties of
all geometric curves have been known since the days of Cotes, and have in
recent times acquired great interest from the discoveries of Poncelet, and from
the still imperfect theory of polar curves. But the importance of the theory
of harmonic section cannot be fully understood till we reflect that of all con-
ceivable metrical relations among points on the same right line, not one can
by possibility belong to the sphere of linear Geometry, except it belong to the
class of harmonic properties. With the ruler alone we cannot bisect lines or
draw parallels, still less take proportionals, but we can always determine
harmonic means and construct harmonic progressions. This observation alone
would suffice to show that the harmonic relation, far from being special in its
application to curves of the second order, must meet us at every turn in the
ON SOME OF THE METHODS AT PKESENT IN USE IN PURE GEOMETRY. 7
science of space, and such in fact we find to be the case. For example, let a
geometric curve be traced on a sheet of paper, and let it be required to assign
its tangent at a given point, not singular, with the ruler alone. Let P be the
given point ; through it draw four transversals, which will cut the curve in
n 1 points apiece. On each of these transversals take the harmonic centre
of its n 1 points with respect to P, and consider the four points thus obtained
as determining a conic passing through P. Pascal's theorem will now assign
the tangent of this conic at P, and this tangent will be the tangent required.
We may add that were the curve of the nth class, instead of the nth order,
i. e. were it such that only n tangents could be drawn to it from a given point,
and if we supposed that we were given, or could construct, the tangents passing
through any point in the plane, we might, by a reciprocal construction,
determine with the ruler alone the point of contact on any tangent, supposed
not to be a double tangent to the curve. This solution of the general problem
of tangents, being purely linear, is so far simpler than any that has ever been
deduced from analysis, and it is worth while to ask why, in this particular case,
Geometry possesses an advantage over analysis. The answer plainly is, that
the analytical method introduces elements foreign to the real question, videlicet,
a pair of right lines termed axes, and standing in a purely arbitrary relation
to the curve and its tangent at the point P. When therefore we proceed to
construct the tangent by means of its relation to these axes (by constructing
its intercepts, for example), we lose sight for the moment of its immediate
connection with the curve, and substitute for that immediate connection a
mediate, and therefore a less simple relation. It will perhaps be found that
similar observations apply to almost all those cases in which the Cartesian
analysis is outstripped either by the present Geometry or by the newer methods
of triliteral and indeterminate coordinates.
Similarly, by properly generalising the definition of involution, we can
obtain without any trouble at all very many general properties of systems of
curves and surfaces, some of which would not be without interest, and would
admit of a multitude of corollaries.
Thus, if four surfaces have their complete intersection common, the an-
harmonic ratio of their four tangent planes is constant for every point on the
line of intersection, and reciprocally, if any number of surfaces of the nth order
be completely circumscribed by one and the same developable, the anharmonic
ratio of the points determined by four of the lines of contact on any edge of
the developable is constant for all the edges of the developable ; so that, in
8 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
particular, if the lines of contact be lines of curvature on three of the surfaces,
they are lines of curvature upon all of them.
But, after all, it is in the method of transversals, strictly so called, that
we find the principal resources of geometrical inquiry respecting general loci.
That method is comprised entirely in the general theorem known hy the name
of Carnot, and in the means that have been devised for transforming and
developing the equation supplied by it. Its importance in Pure Geometry is
so unquestionable that a few remarks upon the place which it holds in the
theory of curves, and on the geometrical considerations by which it may be
established, cannot be out of place here. What renders at all possible a purely
geometrical theory of curves, is precisely the introduction of considerations
involving imaginary points and lines into the principles of the science. It is
impossible to define geometrically a plane curve of the nth order, except by
saying that it is a curve such that every right line in its plane must of necessity
cut it in neither more nor less than n points, and of course this definition is
inadmissible so long as imaginaries are excluded from consideration. But this
definition once admitted, it is found possible to construct a purely geometrical
theory of curves. If, for instance, we wish to find the locus of a point subject
to certain conditions, these conditions give us the means of determining how
many points of the locus can lie on one line, and, since the continuity of the
locus implies that all lines in its plane cut it in the same number of points,
we can in general determine the order of any proposed locus ; and as soon as
this is done, we are in a condition to inquire still further into its properties,
to construct it by points, to determine its general form, &c. We will take
one or two very simple examples of this a priori determination of the order
of a locus. The first shall be Cotes' theorem already alluded to, namely, that
if we take a fixed point P in the plane of a geometrical curve, and draw
transversals through it, and then take the harmonic centres with respect to
P of the n points in which every transversal cuts the curve, the locus of these
centres for every position of the transversal shall be a right line. Now it is
clear that the distance of the harmonic centre from the fixed point, being a
symmetrical function of the distances of the n points of intersection from the
same point, can have but one value for one position of the transversal, and
therefore but one point of the locus can lie on each transversal through P.
Unless therefore P be itself upon the locus, either as an ordinary or as a
singular point, the locus is a right line. But that P is not on the locus, may
readily be verified in the case before us, so that the theorem is proved. As
ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
a second instance, we will take the following from solid geometry. If a body
be anyhow in motion, it is required to prove that the tangents to the trajectories
of all the points in it, which lie upon a given line, form at any instant a
hyperbolic paraboloid. To show this, we observe that no two tangents can
lie in the same plane (except in a particular case, when all of them lie in one
plane), and that consequently the complete section of the surface, formed by
any plane containing the given line, consists of that given line and of one
tangent only ; so that the locus is a surface of the second order. That it is
a paraboloid now follows from the fact that all the tangents are manifestly
parallel to one and the same plane.
It should be added, that precisely in the same way in which we define a
curve of the nth order by the number of its points which lie on a line, we define
a curve of the nth class by the number of tangents (real or imaginary) that can
be drawn to it from a given point, and this definition enables us to find
envelopes, just as the former enabled us to find loci : an obvious example will
suffice. ' One side of a constant angle passes through a fixed point, and the
vertex lies on a fixed line, the other side will envelope a parabola.' For here
the fixed line is itself one of the tangents, and therefore from each point upon
it two, and only two, tangents can be drawn to the envelope the envelope
is therefore a conic section and since the tangent line can in one position
remove to an infinite distance, it is a parabola.
We see then that when a locus is investigated geometrically, the most
general and the most important question we can ask respecting it is, in how
many points it can be out by a right line, and the preceding examples may
serve to show how this question may in many cases be answered, that is, how
the loci occurring in particular problems may be brought under the general
definition of geometric curves. The next step is, from the purely descriptive
relation asserted in the definition, to deduce an equally general metrical one ;
and this is exactly what is effected by Carnot's theorem. Carnot gives, in
the ' Geometric de Position,' two demonstrations of his theorem, one analytical
and one geometrical. The former is very simple and elegant, but the latter
is unsatisfactory; and though there can be no real objection to rest a general
geometrical theorem on an analytical proof, it was still a problem of some
interest to show that geometry could dispense with this assistance. To do
this, M. Poncelet first showed that Carnot's theorem passed into Newton's by
perspective, and then succeeded in demonstrating the latter by considerations
analogous to those which we have been just employing for the theorem of
c
10 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
Cotes. Newton's theorem is, that if we take a fixed axis of abscissas, and
draw ordinates to it, the ratio of the continued product of the abscissas to the
continued product of the ordinates is constant, at whatever point of the axis
of abscissas the ordinate is drawn. To prove this, we need only take upon
each ordinate a line proportional to the value of the ratio for that ordinate,
there is then no difficulty in establishing, first, that the locus of the extremities
of these lines is a right line, and secondly, that it is a right line parallel to
the axis of abscissas. And this is evidently equivalent to the required proof.
Carnot's theorem, notwithstanding its simplicity, is not very easily enun-
ciated. If we take a triangle ABC in the plane of a curve of the nth order, and
if its sides taken in order be cut in the 3 n points, p 1 p i . . p n , q^q^.qn, r^r t ..r n ;
and if we denote by (Aq) the continued product of Aq t Aq^ .... Aq n , we shall
have, by the theorem,
(Aq).(Br).(Cp) = (Ar).(Bp).(Cq).
We see that this equation establishes a relation between the 3 n points,
which must of necessity subsist in order that they may all lie on one and
the same curve of the nth order, and that when all these points are given,
except one, we are able to determine that remaining one. If, for example, all
the branches of a curve be completely described excepting one, and if two
points upon the remaining branch be given, the theorem enables us immediately
to describe it, or at least to determine as many points as we please upon it.
But the principal applications of the theorem depend mainly upon the facility
with which the fundamental equation may be modified and transformed. These
modifications are rendered possible, first by the absolutely arbitrary position of
the triangle in the plane of the curve, and secondly, by the facility with which
evanescent segments may be eliminated from the equation, by introducing new
transversals, and combining the equations supplied by them with the original
equation. Thus, if we take one of the vertices of the triangle of transversals
as A upon the curve, Carnot's equation will contain an evanescent segment on
either side ; and if we introduce a new transversal, passing through the ex-
tremities of the evanescent segments, and cutting the third side in a point t,
the equation connecting the segments determined by this line on the sides of
the triangle will immediately eliminate the evanescent segments, and give an
equation for determining the point t, that is, for determining the tangent at A.
By continuing the same process we might with the utmost facility determine
the position and magnitude of the circle of curvature at A, and with a little
more trouble might extend the investigation to the case of singular as well as
ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. 11
ordinary points. But our present limits preclude the possibility of our pursuing
this subject further. We will only add that it is possible so to transform
Carnot's equation as to render the relation given by it capable of linear con-
struction, and that hi this way an immense number of descriptive properties
of geometric curves may be obtained. For instance, we might demonstrate in
this way Cotes' theorem, or the linear construction before given for the tangent
at any point, which was exhibited as a consequence of the harmonic property
of curves. But we will confine ourselves to merely stating one very general
property due to Poncelet, ' If the points determined on the curve by a sufficient
number of transversals be given, it is possible to determine the intersections of
the curve with any other transversal by means of curves depending on the
intersections of right lines only.'
We must dismiss with a still more imperfect notice the theory of the
transformation of figures ; and this, not because the subject is less interesting,
but because, both in this country and on the continent, it has attracted so
much more attention than any other part of pure geometry, that it would be
no easy task to give a summary view of the whole system, while at the same
time it would be hard to present anything with respect to particular appli-
cations that should have the interest of novelty. A few general remarks is
all that will be attempted here. Figures may be transformed in two ways ;
either directly, that is, into others of the same kind, or inversely into reciprocal
ones. In the first case, to every point in the original figure a point corresponds
in the derived, and a line to every line. In the second case, this relation is
inverted, and a point of one figure corresponds to a line of the other, and vice
versa. It follows from this, that to the points of a curve line of the nth order
there will correspond in the first case the points of a curve of the same order,
but in the second case we shall have as the correlatives of the points of a curve
in the primitive figure the tangent lines of a curve, no longer now of the nth
order, but of the wth class. Consequently, the descriptive properties of a figure
derived directly will be precisely the same as those of its primitive, but the
descriptive properties of a figure derived inversely will be reciprocal to those
of its primitive. This will frequently enable us to extend a descriptive relation
from a particular to a general state of a figure, and from a descriptive relation
of one figure to deduce another belonging to a different figure. The use of
such processes in discovering new theorems, or in establishing a connection
between ones already known, is too obvious to be dwelt on. We also see that
any method of transformation, which satisfies the single condition, that to
C 2
12 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
the points of a right line there should correspond the points of a right line
in the first case, and a system of right lines passing through a point in the
second, will enable us to generalise or transform any purely descriptive property.
And it is possible to invent an unlimited number of methods of transformation,
which shall comply with this restriction, and which, so far, possess no advantage
one over another. But by a proper selection of the methods to be employed
it has been found that we are enabled to transform not only all descriptive
relations, but very many metrical ones also ; in fact, all such as can be enun-
ciated in a sufficiently general form. The two methods of projection and of
reciprocal polars, for both of which geometry is mainly indebted to M. Poncelet,
besides including in themselves almost all the methods for transformation that
had previously been proposed, possess this last property in a pre-eminent degree.
What is singular is, that though the principles upon which the two methods
rest are so widely different, exactly the same class of metrical theorems which
can be brought under the first are capable of being also transformed by the
second. All harmonic properties, and consequently the whole 'geometrie de
la regie,' all anharmonic and involutional relations, and, besides, all the general
theorems of the theory of transversals, can be operated on by either of the
two methods. Wherever it is applicable, the method of projection will enable
us to make the proof of a general theorem depend upon its simplest cases, and
on the other hand to explain and follow the modifications which a general
principle undergoes in its application to particular instances, while the method
of reciprocal polars unveils the singular duality which pervades so large a portion
of the science of space, and which now finds its analytical expression in the
method of tangential coordinates, but which at the time of M. Poncelet's
invention had hardly been observed at all. It was of course known that to
any triangle upon the surface of the sphere there always corresponded a second
triangle, the angles and sides of which answered to the sides and angles of
the first triangle, and were connected with them by an uniform and simple
metrical relation. But this remark had never been generalised so as to extend
to all spherical figures, still less had it been perceived that the property in
question was so far from being confined to figures on a sphere, that it was
only a particular case of a general property of a far more extensive class of
figures. Now, of course, it is well ascertained, that it is impossible to assert
any theorem respecting a figure on a sphere without, at the same time, asserting
a different property of a different figure on the same sphere, and this whether
the theorem be metrical or descriptive. Even any proposition respecting the
ON SOME OF THE METHODS AT PBESENT IN USE IN PURE GEOMETRY. 13
rectification of a spherical curve gives us at once a theorem respecting the
quadrature of the supplementary curve, so that if we could find all spherical
quadratures we could rectify all spherical curves, and vice versa. We can even
in this way obtain transformations of definite integrals; for example, of elliptic
functions of the third order. For if we express first the area of a spherical
conic, and then the length of the arc of its supplemental conic, we shall obtain
two elliptic functions of the third order, with different moduli and parameters,
and the supplementary relation of the two figures will at once establish an
equation between these two integrals. The duality, then, of spherical figures
is absolute, and extends to every conceivable case ; but as soon as we pass
to plane figures, or to figures in space, the case is different, and it is only in
certain definite, though still very extensive classes of properties, that we find
the principle manifesting itself, though this perhaps may be partly owing to
the imperfection of our means of investigation. For it is certain that in many
particular cases it is a matter of considerable difficulty to discover the reciprocal
relations between theorems, even where it can be shown to exist. Take, for
instance, the two theorems, ' A tangent to the interior of two similarly placed
and concentric conies cuts off a constant area from the exterior conic;' and
again, ' The sum of two tangents to an ellipse, which intersect on a confocal
ellipse, diminished by the arc intercepted between them, is constant.' No one
would have suspected, at first sight, that these two theorems are supplementary
to one another, in exactly the same sense in which the word is understood in
Spherical Trigonometry. But we should find, that if we were to imagine the
two figures to become infinitely small, and to be placed upon a sphere, they
would become supplementary, and the properties specified would follow the
one from the other. In the same way many properties of the asymptotes of
an hyperbola might be shown to be supplementary to the focal properties of
an ellipse. From the constancy of the sum of the radii vectores in the ellipse,
we might deduce the constancy of the triangle contained by the asymptotes
and any tangent to the hyperbola ; and from the equality of the angles made
by a tangent to the ellipse with the focal radii vectores, we might infer the
known theorem, that the intercept determined by the asymptotes on any
tangent to a hyperbola is bisected at the point of contact. These instances
may serve to show how cautious we should be in inferring that theorems, which
seem to give rise to no reciprocal property, are really incapable of assuming
this double character. It is therefore quite conceivable that future discoveries
in Geometry may render the application of the principle of duality to the
14 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
general properties of space quite as universal as it already is in the case of the
sphere. But in the present state of the science it would be hard to name a
case in which the existence of duality can be proved, and in which, nevertheless,
it cannot be brought to light by the method of reciprocal polars. This method,
therefore, still extends to the full extent of our present knowledge. It is
now equalled, but it has not yet been surpassed in this respect, by the analytical
method of tangential coordinates.
In forming the polar reciprocal of any proposed plane figure, we replace
every point by its polar, and every line by its pole, with respect to an auxiliary
conic taken in the plane of the figure. M. Poncelet has himself observed, that
it would be unfair to argue against the generality of his method on account -
of its reposing (as it thus is made to do) on a particular property of curves of
the second order. In fact, provided we once assure ourselves that the trans-
formation we are employing is capable of being applied to any proposed figure,
it will seldom signify whether the principles upon which the transformation
rests be general or not ; the only object is to obtain such a transformation as
will enable us to transform the greatest number of metrical relations possible.
It has been said, that any projective property may be transformed into a re-
ciprocal one ; but other relations, not projective, can nevertheless be made to
yield reciprocal properties, by employing a circle or a parabola as the trans-
forming conic. These particular applications, though of inferior interest with
respect to general Geometry, are of great importance in the case of curves and
surfaces of the second order, and in some physical applications of pure geometry
for example, in Professor Mac Cullagh's theory of apsidals, and his demon-
stration of Fresnel's construction of the wave surface.
Poncelet's account of Reciprocal Polars is to be found in a memoir in the
third volume of Crelle's Journal, but he has devoted a separate work (the Traitd
des Propri^te's Project! ves) to the use of central projection, that is to say, of
perspective in geometry. Few works, perhaps, could be named more calculated
to awaken a taste for Pure Geometry than this admirable treatise : though the
greater part of it is occupied with applications to the theory of conic sections,
the reader feels all along that the methods developed in it are perfectly general,
and that it only needs the genius of the author to apply them with equal
success in almost any investigation. It is so natural an idea to simplify a
diagram by forming a perspective representation of it, that it is surprising it
should not have been introduced long before into geometry, especially when we
remember for how long a time the conic sections were studied only on the cone.
ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. 15
Perhaps no better proof can be given of the rationality of the projective method
than that which is supplied by the fact, that it is to it mainly that we owe
the introduction of triliteral coordinates, a modification of the conception of
Descartes, which possesses, it is true, many undeniable advantages over the
purely geometrical perspective of Poncelet, but which cannot be properly under-
stood in its relation to space till it is regarded, if we may so express ourselves,
as a translation of perspective into the language of analysis. In fact, when
we express the equation of a curve in triliterals, we are merely putting it into
a form in which it becomes common to all possible perspective representations
of the curve we are considering ; we, as it were, divest the curve of all its non-
projective properties, for the purpose of exhibiting in a more palpable and ex-
plicit form those essential ones which still continue to characterise it.
We can immediately determine whether any proposed metrical relation be
projective or not, by merely examining whether it leads or does not lead to
a relation involving the angles that are at any point subtended by the lines
of the figure, and not involving the projecting lines. Applying this criterion,
we should find, for example, that Carnot's theorem is projective, or, on the other
hand, that it is impossible for a quadrature or a rectification to be so. But it
would not be easy to lay down any general formula for determining a priori
what properties are projective and what are not. Such a determination, though
practically of little use, would theoretically be of the greatest value. However,
the criterion we have given enables us now to see what was before observed,
that every projective property can be made to yield a reciprocal one. For let
the auxiliary conic be a circle, we shall have a relation between the angles
contained by the rays drawn from its centre to the points of the figure, and
since this same relation will subsist between the angles formed by the polars
of those points, the new figure will possess a property reciprocal to and derived
from that of the old.
Lastly, it may be remarked that the principle of perspective seems well
calculated to form the basis of classification for geometric curves of any given
order ; a splendid example of this is given by Newton's famous theorem, that
all curves of the third order may be generated by the shadows of five of them.
This surely is the first step towards a purely natural classification of these
curves, and it is much to be regretted that Pliicker in his enumeration of curves
of the third order, which is the most complete that has yet appeared, appears
to have paid so little attention to the fundamental distinction between curves
that can and that cannot be cut from the same cone, for there can be no doubt
16 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
that it would have enabled him to construct his classification of the 219 varieties
on a principle at once more general and more simple than that which he has
adopted.
We come last of all to the method of infinitesimals. Ever since the in-
vention of the Differential Calculus many of its resources have been at the
command of Pure Geometry; and yet, in the first instance, the effect of its
introduction upon that part of mathematical science was anything but favour-
able to it. For as soon as Newton and his immediate successors had passed
away, the geometrical methods ceased (with rare exceptions) to be cultivated ;
the attention of mathematicians was so engrossed by the brilliant successes of
the new calculus, that the era which is the most remarkable of all in the
history of analysis was almost wholly unproductive in Pure Geometry. But
the school of Monge, who delighted in finding geometrical solutions for all kinds
of problems, soon attempted to rival the Differential Calculus on its own ground.
And in particular cases they not only obtained very simple proofs of known
theorems, but succeeded in discovering new properties, to which analysis might
not have guided them so easily. Two striking instances of this might be men-
tioned, both from the works of Ch. Dupin. One is his celebrated theorem,
that three series of surfaces which cut one another orthogonally cut one another
in their lines of curvature. This he demonstrated by direct and purely geo-
metrical considerations ; and yet, perhaps, his proof is not the most simple that
might be given. The other is the proposition which is the most general yet
obtained in Dioptrics, ' That if a system of rays possess the property of being
all normal to some one and the same surface, they will still continue to possess
it after any number of refractions or reflexions at surfaces of absolutely arbitrary
form and position.' Malus had succeeded in showing that if a system of rays
emanate from a point they will form after a first reflexion two series of develop-
ables intersecting orthogonally, i.e. that they will all be normal to the same
surface. He then proceeded to inquire whether they would continue to possess
this property after a second reflexion, and, deceived by a slight error in his
analysis, he concluded that they would not. But Dupin showed, that if the
surface normal to the incident rays were imagined to envelope a system of
spheres of variable radius, but having their centres on the reflecting surface,
those spheres would determine a second envelope behind the reflecting surface,
and that every reflected ray would be normal to this the second sheet of the
complete envelope of the spheres, that is, that all the reflected rays would be
normal to one and the same surface. In the case of refraction, we have only
ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. 17
to substitute for the single sphere a pair of concentric spheres, having their
radii in the ratio of the refractive index to unity*.
Many other instances, some of them of even greater interest, might be
given ; but instead of doing so we will conclude this Paper with demonstrations,
simpler perhaps than those usually given, of some of the principal theorems
relating to geodesic lines ; first, upon surfaces in general, and then upon the
ellipsoid.
We will set out with the assumption, which is easily justified, that a plane
drawn parallel to the tangent plane at any point P of a curved surface, and at
an infinitely small distance from it, cuts the surface in a curve which for all
points indefinitely near the point of contact assumes the form and properties
of an evanescent conic section, having its centre upon the normal at the point P.
If we now consider a point Q situated on the circumference of the conic, it is
plain that the normal to the conic at' Q will be the orthogonal projection on its
plane of the normal to the surface at the same point Q. We can therefore find
the angle which the normal at Q makes with the normal section PQ, and this
angle, divided by the arc PQ, is equal to the reciprocal of the radius of torsion
of the geodesic line PQ, since the normal section is the osculating plane of that
curve at P, and the normal to the surface at Q is its principal normal at that
point. Transforming the expression thus obtained for the radius of torsion, we
obtain the still simpler one,
111
where R 1} R. 2 denote the principal radii of curvature, while p lt p 2 are the
radii of curvature of the normal sections tangent, and perpendicular to, the
geodesic line PQ. Several consequences may be deduced from this formula, a
few of which will be mentioned here.
First, if a geodesic line be tangent to a line of curvature, its torsion is
invariably suspended at the point of contact. If therefore a line, of curvature
become a geodesic line it must at the same time become plane, since every point
upon it will be a point of suspended torsion.
Secondly, if two geodesic lines intersect at right angles, their torsions at
the point of intersection are equal.
Thirdly, it is possible to trace upon a given surface lines of maximum
* For an analytical proof of this theorem see the first part of Sir William Hamilton's Essay
on Systems of Rays, or Prof. Minding in Poggendorff, 1847, p. 268.
D
18 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
geodesic torsion, i.e. curves such that at any point upon them the geodesic
tangent to the curve will have greater torsion than any geodesic line passing
through the point. Two of these curves will pass through every point on the
surface. They will be orthogonal trajectories upon one another, and will in-
tersect the lines of curvature at angles of 45.
Fourthly, if a point be found on a curve surface such that it is a point
of suspended torsion on eveiy geodesic line passing through it, it must be an
umbilic.
Consequently, if a surface have all its geodesic lines plane it must be
umbilical at every point. But Monge has shown that the sphere is the only
surface possessing this property; therefore the sphere is the only surface all
whose geodesic lines are plane curves.
Let us now consider any curve S traced on a given surface A. If P be
a point on S, and if we project S orthogonally on the tangent plane at P, the
projected curve will pass through P, and will possess at that point a definite
curvature. This curvature we shall term the tangential curvature of the curve
S at the point P; it is obviously equivalent to the curvature of S multiplied
by the cosine of the inclination of its osculating plane upon the tangent plane.
If <S be a geodesic line, its tangential curvature will be zero, and its tangential
projection will be inflected at P, so that any small arc in the immediate vicinity
of P may be regarded as rectilinear. It hence appears, that when S is not
a geodesic line its tangential curvature is equal to the angle between two
consecutive geodesic tangents, divided by the arc intercepted between the points
of contact ; or again, it is equal to an evanescent geodesic chord, divided by the
square of the sagitta bisecting it perpendicularly. Now if A' be a second surface
such that A can be developed on it without disruption or duplication, the
minimum property of geodesic lines shows us that every geodesic line on A
will determine a geodesic line on A', and that consequently the tangential
curvatures of corresponding curves are equal at corresponding points on the
two surfaces. In particular, we see that if A be a surface developable on a
plane, the tangential curvature of S is precisely the curvature of the plane curve
into which S is transformed, when A is developed on a plane. Or, when A
is any surface whatever, if we imagine it to be circumscribed by a developable
along S, since the tangential curvature will continue the same, whether we
regard S as traced on the developable or on A, we may define the tangential
curvature of S as the curvature of the plane curve into which S is transformed
by the complanation of the developable circumscribing A along S.
ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. 19
Gauss, in his celebrated memoir ' Disquisitiones generales circa superficies
curvas,' has introduced one or two expressions into Geometry which it is con-
venient to preserve. If we take a finite area upon a curved surface, bounded
by any closed contour whatever, and if through any fixed point we draw
parallels to the normals to the surface along the given contour, they will inter-
cept a spherical area on a concentric sphere of radius unity, which is termed
the ' spherical value ' of the given curved area. The spherical value divided
by the true value is the ' integral curvature ' of the area ; and if instead of a
finite area we take an evanescent one, including a given point P, the limiting
value of the integral curvature is the curvature of the surface at P. By the
' integral curvature ' of a finite arc of a plane curve we understand (it is hardly
necessary to observe) the angle between the extreme normals divided by the
arc, and the angle itself may be called the ' circular value ' of the arc.
The theorems which we shall now endeavour to establish geometrically are
the following :
I. The curvature of a surface at any point is equal to the product of the
reciprocals of the radii of curvature. (For simplicity, we consider only surfaces
doubly concave, but the demonstrations, mutatis mutandis, will apply to surfaces
having their curvatures of opposite signs.)
II. The spherical curve, which is supplementary on the auxiliary sphere,
to the spherical value of any proposed area, is equivalent to the integral of the
angle of tangential curvature extended over the whole contour of the area.
III. Corresponding areas on surfaces developable upon one another have
equal spherical values, and consequently equal 'integral curvatures.'
I. If we take an evanescent rectangle dS contained by four lines of
curvature, it is plain that if S<j>, Sty' denote the angles subtended by two
adjacent sides at their respective centres of curvature, we shall have the
equation dS = RR' <?< <S<'. But if d Q be the spherical value of dS, we also
have dQ = Sd>Sd>'. Therefore -7^7 = -jrjf., and the truth of the result is inde-
KK
pendent of the peculiar form we have assigned to the element dS. For,
whatever form we assign to that evanescent element, we can always imagine
it made up of an infinite number of such rectangles, for every one of which
the product T , will retain the same value within an infinitesimal, so that if
KK
da-, da-' etc. be the little rectangles, du>, d<a etc. their spherical values, we shall
always find dS = 2d<r, cZQ =
D 2
20 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
and, in the limit, when dS is itself rendered evanescent,
da> da>' , c
, = -j- = -T-. = . . . = ->-, as before.
RR* da- d</~ dS
II. Let S represent the contour of the given area, and 2 the contour of
the corresponding spherical area. The spherical curve supplementary to 2,
which we will term Z', is the envelope of great circles having their poles on 2,
and, consequently, having their planes parallel to the tangent planes of the
given surface along S. The consecutive intersections of the planes of the great
circles determine the sides of the cone subtended by 2' at the centre of the
sphere, and, in like manner, the tangent planes of the surface determine the
arfites of the developable circumscribing the surface along S. Therefore the
sides of the cone are respectively parallel to the aretes of the developable, so
that 2', which is obviously equal to the sum of the angles subtended by its
elements at the centre of the sphere, is equal to the sum of the angles con-
tained by the consecutive are"tes of the developable. Observing that every
ardte intersects S once, and once only, we see that, after complanation, the
sum of the consecutive angles will be precisely equal to the angle contained
by the two extreme ardtes, that is, to the angle contained by the two extreme
normals, or, finally, to the integral of the angle of tangential curvature extended
over S.
A particular case of this theorem deserves special attention.
Let S, instead of a continuous curve, form a polygon composed of geodesic
lines. The theorem will evidently still subsist, only the quantity we have
designated as the integral of the angle of tangential curvature will be simply
replaced by the sum of the external angles of the polygon. We have therefore
this theorem given by Gauss :
' The excess of the angles of any geodesic polygon, above the sum of the
angles of a plane polygon of the same number of sides, is equal to the spherical
value of the area of the polygon.'
This property will serve, in its turn, to establish another of Gauss' pro-
positions : ' If from any point two geodesic lines OP, OP' be drawn containing
an evanescent angle o, and if OP = p, PP' perpendicular to OP = P, the
quantity P will satisfy the differential equation of the second order,
<VP P
dp* ~ V RR' =
Let Q denote the spherical value of the triangle OFF. Then, since
ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY. 21
dP = dp cos OP'P, we find OP'P = I - -7 ; and therefore the preceding theorem
dp
gives at once -j |- Q w = ; or, differentiating, =-^ + ~^ = 0. But, by theo-
dp dp dp
dQ P d 2 P P
rem I, ^ = ^yr,, and, substituting, -j-j +
rfjO -run, a/)
We may also observe that this demonstration supplies us with a first
and second integral of Gauss' equation. In fact, we have not only
dP f
-=-+3 <0=0, but also P = po> / Qdp.
dp Jo
The first arbitrary constant is <a, the second has been put equal to zero in the
inferior limit of the definite integral in the expression of P.
In this form the equation suggests some interesting remarks, which our
limits compel us to omit.
III. The integral of the angle of tangential curvature, extended over any
arc of a curve, is constant for all developments of the surface on which the
curve is traced. For since each element of the integral is an angle contained
by two consecutive geodesic lines, it is apparent that neither the number nor
the magnitude of the elements will be affected by the transformation, and con-
sequently the sum will remain unaltered. If the arc become a closed contour,
it will follow that not only this integral of contingence, but also the quantity
supplementary to it, that is, the spherical value of the area, will remain constant
for all developments of the surface.
In particular, if we consider an evanescent area we shall find -^ =
But since di2 and dS are both constant, it follows that -j^f^ will be so too.
HK
This gives us the theorem which Gauss has demonstrated by a singularly
beautiful analysis :
' If two surfaces be developable one upon another, the product of the
principal radii of curvature is the same for any two corresponding points.'
The propositions we have been considering are of great importance in the
theory of surfaces. The properties of geodesic lines, or more generally those
properties of a surface which remain unchanged so long as the geodesic distances
of its points remain unaltered, are doubtless as yet but very imperfectly known,
notwithstanding the attention bestowed on them since the publication of Gauss'
memoir. The subject is one of great difficulty, as those who have tried it well
know, but it is at the same time of great interest, as it is certain that any
22 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
general results obtained here would find frequent and useful application.
Recently, too, the properties discovered by Mr. Roberts on the geodesic lines
of the ellipsoid have attracted increased attention to the general question, and
have themselves furnished fresh examples of the resources of Pure Geometry;
for though Mr. Roberts' results were obtained in the first instance analytically,
the geometrical proof of them since given is so direct that one is almost sur-
prised that they were not discovered sooner.
As it is possible to exhibit this proof in a very simple form, and one which
shows clearly its connection with the general theory of surfaces, we will allow
it to find a place here. We have to show that the sum of two geodesic lines
drawn from the umbilics of an ellipsoid to any point on a line of curvature,
including the two umbilics, is constant ; or, which comes to the same thing,
that the angles made by the two geodesic radii vectores with the line of
curvature are equal. Now it is a well-known property of confocal surfaces that
the cones which envelope them from any point in space are themselves confocal,
and consequently orthogonal, so that from whatever point in space two confocal
surfaces be viewed their apparent contours will intersect at right angles. If
we compare this property with the theory given by Monge, of the ' Surface of
Centres ' of any given surface, we shall perceive that any two confocal surfaces
may be regarded as forming the two sheets of the surface of centres of some
one and the same transcendental surface, and that every geodesic line existing
on either confocal surface, and touching their common intersection, will be the
cuspidal line of a developable circumscribing the second confocal. This pro-
position is the geometrical expression of Liouville's equation, M 2 cos 2 i-f i> 2 shW = p 2 ,
or Joachimsthal's, PD = const. As a particular case, we observe that every
tangent to an umbilical geodesic line on an ellipsoid will pass through the focal
hyperbola, so that the two tangents to the two geodesic radii vectores of any
point on a line of curvature will be sides of the cone which from that point
envelopes the focal hyperbola. But since these two sides lie in a principal plane
of the cone they will make equal angles with the principal axes of the cone, that
is, with the tangent and normal to the line of curvature.
The same principles would serve to demonstrate the other theorems dis-
covered by Mr. Roberts ; but the proof of this one may be enough to show,
that though any property of space is doubtless discoverable by analysis, yet it
sometimes happens that in particular cases it is more convenient to lay aside
for a moment our analytical formulae, and consider the questions that arise in
some more special but less artificial manner. This is especially requisite when
ON SOME OF THE METHODS AT PEESENT IN USE IN PURE GEOMETRY. 23
the subject of inquiry is one about which little is as yet known. For here
we cannot be sure a priori that the analysis of coordinates is the most natural
method, and therefoi-e the surest to lead to results. In such cases we find
ourselves obliged to adopt, though it be but tentatively, a more direct study
of the circumstances of the case in preference to a method which is apt at times
to conceal from us in a singular manner the real grounds of the results with
which it supplies us. In the words of M. Poinsot, ' Rien ne nous dispense
d'etudier les choses en elles-memes, et de nous bien rendre compte des ide*es qui
font 1'objet de nos speculations. Si le calcul seul peut quelquefois nous offrir
une ve"rite nouvelle, il faut songer que cette verite etant inde'pendante des
methodes ou des artifices qui ont pu nous y conduire, il existe certainement
quelque demonstration simple qui pourrait la porter a 1' evidence ; ce qui doit
etre le grand objet et le dernier re"sultat de la science mathe'matique.'
24 ON SOME OF THE METHODS AT PRESENT IN USE IN PURE GEOMETRY.
[The following abstract of the preceding paper was published in the Proceedings of the
Ashmolean Society, Vol. II. pp. 305, 306.]
The object of this paper was to show how, during the course of the last
fifty years, the geometrical methods had acquired that generality and facility
which had been, at an earlier period, regarded as exclusively characteristic
of analysis. This rapid development of the resources of Pure Geometry was
illustrated partly by general observations on the nature of some of its principal
theories, and partly by a series of more particular examples.
The use of imaginary magnitudes in Geometry was especially dwelt on,
and it was shown how by their aid we may sometimes comprehend in one
and the same statement theorems at first sight widely different, and exhibit
them as expressions of some one and the same general principle, assuming a
different form, under different accidental circumstances. Other illustrations
were taken from the application of the theory of transversals to the investi-
gation of the properties of algebraic curves, and, among other conclusions, a
linear solution of the direct and inverse problem of tangents (analogous to
that given by M. Poncelet) was deduced from the harmonic properties of
such curves.
The various methods for the transformation of figures, whether into other
of the same kind, or into reciprocal ones, were also alluded to ; and the law
of Geometric Duality, which manifests itself in these transformations, was
commented on, and an attempt made to fix the limits within which it is
applicable.
In order similarly to exemplify the use of infinitesimals in Pure Geometry,
some applications were made of this method to the theory of curved surfaces,
and outline demonstrations of some of the results of Gauss on geodesic lines,
and the mutual developability of surfaces, were given without the aid of
coordinate Geometry.
Lastly, a proof was proposed of Mr. Roberts' theorems respecting the
geodesic lines of ellipsoid, in which those results were exhibited as immediate
corollaries from two well-known theorems of Pure Geometry, due to Monge
and Jacobi.
II.
ON SOME GEOMETRICAL CONSTRUCTIONS.
[Cambridge and Dublin Mathematical Journal, vol. vii. pp. 118-126 ; May, 1852.]
IF a geometrical curve be completely traced on a sheet of paper, the
principles of the Theory of Transversals enable us to assign its tangent line
and radius of curvature at any point, without supposing its equation known,
and without employing any operations excluded from the sphere of elementary
geometry.
The construction given by M. Chasles for this purpose is the following.
Let m be the point on the curve, M any point assumed in the plane, mp, mq,
two transversals, and MP, MQ, two parallels to them. Let also p, q, P, Q,
denote the continued products of the segments on mp, mq, MP, MQ, respect-
ively, excepting the evanescent segments on mp, mq. Then if we take on
P Q
mp, mq, two lines respectively proportional to , , the line joining their
extremities shall be parallel to the tangent at m. To find the osculating circle,
let t denote the continued product of the segments on the tangent at m, except-
ing the two evanescent segments, T the continued product of the segments on
MT drawn parallel to mt ; then, if on any transversal mp we take me equal to
T p
T-'PJ the point c shall lie on the osculating circle. For the diameter of this
' JT
T n
circle we have the expression -== ; n, N denoting products of segments on the
v J*\
normal and on a parallel to it.
If the point be a double point, the preceding constructions fail ; but by
slightly modifying them, we may determine the two tangents and two radii
E
26 ON SOME GEOMETRICAL CONSTRUCTIONS.
of curvature, if the point be nodal : or, if it be conjugate, we can assign the
elements of an ellipse, whose imaginary asymptotes shall be the imaginary
tangents in question ; that is to say, an ellipse concentric, similar, and similarly
placed, with the evanescent conic formed by the conjugate point. In this case
the two radii of curvature are in general imaginary and therefore cannot be
constructed : but, since they are conjugate imaginary magnitudes, any rational
symmetrical functions of the two (for example, the rectangle under them, or
their harmonic or arithmetic mean), may readily be determined. The process to
be employed is as follows. Take three transversals mp, mq, mr, and three
parallels to them MP, MQ, MR, and let MR cut mp, mq in A, B, and the two
tangents in 6 l} 2 . We shall have
Now if the point m be conjugate, the products A0 l} A0 2 , B0 lt B6 2 , are essentially
positive ; and if it be nodal we can ensure their being so, by taking the three
transversals mp, mq, mr, in one and the same pair of vertically opposite angles.
Hence, if we put 7? n /? a
(*4ff-a, (m^f.^-*,
the lines a and 6 can always be constructed. Therefore to determine 6 lt 6 2 ,
describe two circles round A and B, with radii a and 6, respectively. The
radical axis of these circles will intersect MR at o the middle point of 6 lt 2 ; and
any circle of the system orthogonal to the two circles (A) and (B), (i.e. any circle
having its centre on the radical axis and its radius equal to the tangential
distance of its centre from either of those two circles), will intersect MR in 6 1} 6 2 .
If (A) and (B) intersect in real points, their radical axis is instantly found ; but
in this case 0j and 6 2 are always imaginary. Let s lt s 2 be the points in which the
radical axis is cut by any one of the orthogonal circles ; on mr take mo' a mean
proportional between osj and os 2 (i.e. equal to the tangential distance of o from
any one of the orthogonal circles): the ellipse having its centre at m, and
mo, mo' for semi-conjugate diameters, will have the two imaginary tangents
for its asymptotes. If (A) and (B) touch, the points 1} 6 2 coincide in o, and the
double point becomes a cusp, having mo for its tangent. Lastly, if (A) and (B)
intersect in imaginary points, the radical axis, though not immediately given,
can always be determined by the ruler alone, and in this case, 9 lt 2 being always
real, the tangents mQ l} m0 2 can be directly constructed.
The direction of the tangents once ascertained, the radii of curvature may
ON SOME GEOMETRICAL CONSTKUCTIONS. 27
be immediately found. In fact, if we denote by c 1} c 2 the chords intercepted on
mp by the two circles, and by 1 , 2 the two points in which the tangents are cut
by MP, parallel to mp, we have
and by making mp coincide successively with the two normals, we get the
values of the two diameters of curvature. Or we may first determine one, and
then obtain the second by the proportion, which is easily demonstrated,
T T
P # 1 2 .
-fa -fi'2 . '-j"
l \ l z
If the point be triple the determination of the directions of its tangents, which
in analysis depends on the solution of a cubic equation, is not in general possible
by the intersections of right lines and circles. The problem in its simplest form
is this : Given three points on a right line ABC, and given the products
Ae L .Ae 2 .Ae 3 , Be^Be^BO^, Cfy. C0 2 . Ce 3 , find e lt 2 , 6 3 . But whatever the order of
the point, if the direction of its tangents be once known, the construction of its
radii of curvature is very easy. If, for example, the order of the point be r, the
chord determined on mp, by the circle tangent to m0 1} is readily seen to be
given by the equation Q&.8&...6&
which chord is therefore imaginary for an imaginary tangent, as it ought to be.
Returning to the case of double points, we see from the formula
_ W* ?! P.
^'mO,' t, P'
that if the two tangents coincide, the osculating circles become simultaneously
evanescent, except a fourth segment on the tangent become evanescent also,
that is, except the tangent cut the curve in four coincident points at m. In this
case the point m is not cuspidal, but is a point of osculation, and possesses two
radii of curvature, for which we proceed to give a graphical construction. If
T n
D l9 D 2 be the two diameters, we find readily enough Z\ Z) 2 = "AT > ^ut ^ e
C -iV
theorem of Newton's, which has hitherto guided us, is perhaps insufficient
immediately to furnish a second relation. Such a relation, however, may be
obtained by the following considerations. It is well known that the polar conic
of a point of inflexion breaks up into two lines : one of these is the tangent at
the point of inflexion, the other will be found to be the locus of the harmonic
E 2
28 ON SOME GEOMETRICAL CONSTRUCTIONS.
c.-i it res of the n - 1 points in which the curve is cut by a transversal through the
point. Exactly in the same way the polar curve of the third order at a point m
of the nature here considered, resolves itself into the tangent line and into a
conic section ; this conic touches the tangent at m, and is the locus of harmonic
centres of the n - 2 points in which the curve is cut by a transversal through m ;
consequently its curvature at m, multiplied by n - 2, is precisely the sum of the
curvatures sought. Now the diameter of curvature in a conic is to the chord
intercepted on the normal, as the rectangle under the segments of a parallel to
the tangent is to the rectangle under the segments of the normal chord. Hence,
if mpi be any radius vector of the conic, p\p 2 a chord parallel to the tangent at
m, the radius of curvature is known as soon as the point p 2 has been constructed.
To effect this, take any circle tangent to the conic at m ; this circle and the
conic being homological, their axis of homology may be first found, and then the
line homologous to pip 2 ', this will give the point homologous to p 2 , and therefore
Pi itself ; in fact, the circle once described, p 2 may be found by the ruler alone.
We now know the rectangle under the two radii of curvature, and the harmonic
mean between them : the radii may therefore themselves be found by a simple
and well-known construction.
It may be observed that the theory of polar curves leads to a construction
for the tangent of a curve line, which is different from M. Chasles', and in fact
linear. Through the given point P draw four transversals ; each of these will
cut the curve in n 1 points. Take the harmonic centre of each of these four
groups with respect to P, and consider the four points thus obtained as deter-
mining a conic section passing through P. Pascal's theorem will then determine
the tangent to this conic at P ; that is, the tangent required. It is unnecessary
to give the reciprocal construction, which enables us, when a curve of the w th
class is given tangentially, to determine with the ruler alone the point of contact
on any one of its tangents, supposed not to be a double tangent. It should
however be added, that a method for the linear solution of these two problems
has been long since given in a different and less explicit form by M. Poncelet, in
his excellent memoirs on the Analysis of Transversals.
The radius of curvature of any point is, of course, by its nature, incapable of
linear construction ; but if we imagine ourselves to have constructed the normal at
any point, and to have determined on it the centre of curvature of the given curve
or of any one of its superior or inferior polar curves ; and if, in addition, a line
parallel to the normal be given, in order that the point at infinity on the normal
may be known ; we can find linearly the centre of curvature of every single curve
ON SOME GEOMETRICAL CONSTRUCTIONS. 29
of the polar system continued as far upwards as we please. This is a consequence
of the following theorem : The distances of the centres of curvature of the
successive polar curves from their common tangent form a harmonic progression,
commencing at infinity and having the given point for its point of evanescence.
All the preceding methods admit of an easy application to the theory of
surfaces. For example, to determine the tangent plane at m, we must draw
three transversals mp, mq, mr, not in one plane, and then proceed as the case
of plane curves. If the surface be of the n ib order, its tangent plane will
determine on it a curve of the same order ; the point m being a double point in
that curve nodal, cuspidal, or conjugate, according as the contact is hyperbolic,
parabolic, or elliptic. Taking the last case, we must determine two conjugate
semidiameters of an ellipse concentric, similar, and similarly placed with the
evanescent conic in the tangent plane, and therefore with the indicatrix of
the point m : the directions and magnitudes of the semiaxes of this ellipse
may now be deduced by a construction of extreme simplicity (vide Note xxv. on
M. Chasles' History of Geometry), and therefore the ratio of the principal curva-
tures, and the traces of the principal normal sections on the tangent plane are
known. If now we construct the radius of curvature in either of these normal
sections, the square of one semi-axis of the indicatrix is found, and therefore that
curve may be regarded as completely determined. It hence appears, that to find
the tangent plane and the indicatrix of any point, it is requisite to draw sixteen
transversals ; not that so many are absolutely essential, but the trouble is rather
increased than lessened by taking fewer.
From their connexion with the present subject the following geometrical
demonstrations of Meunier's and Euler's theorems on curvature may find a place
here. If we take a point P on a curve line, and if we consider an evanescent
chord drawn parallel to the tangent at P as an infinitesimal of the first order,
this chord will be bisected by the normal at P; that is to say, it will be divided
into two segments whose difference will be infinitesimal of the second order.
Moreover, if we take any sagitta perpendicular to the chord, and intersecting it
in a point distant only by an infinitesimal of the second order from its centre, it
is readily seen that the square of either segment of the chord, divided by the
sagitta, may be taken to represent the diameter of curvature of the evanescent
arc. Hence, if we take two plane sections of a surface intersecting in an evanes-
cent chord, the radii of curvature of the evanescent arcs are inversely as any two
sagittse perpendicular to the chord, and bisecting it approximately. If, now, one
of the sections be a normal one, we may take for the sagitta in that section the
30 ON SOME GEOMETRICAL CONSTRUCTIONS.
intercept on the normal to the surface. Consequently the triangle formed
by joining the extremities of the two sagittae will be right-angled ; and therefore
the radius of the oblique section is equal to the orthogonal projection of the
normal radius upon the plane of the oblique section, which is Meunier's theorem.
It follows also from what has been said, that if we draw a plane parallel to
a tangent plane, and distant from it by an infinitesimal of the second order, the
curve surface will, in general, determine upon this plane an evanescent hyperbolic
or elliptic oval ; and that the chords of the oval, being themselves infinitesimals
of the first order, will be bisected within an infinitesimal of the second order at
the point at which the normal meets the plane, and which we will call C.
We may therefore consider the oval as a central curve having its centre at C : it
only remains to shew that it is a conic section. This may be done as follows :
Every transversal passing through C and lying in the plane of the oval, will cut
the surface in two points belonging to the oval, and in n 2 points whose
distance from C is infinitely great compared with that of the two first points.
Now, if for a moment we consider the diameters of the oval to be finite, the
remaining n 2 points will lie at infinity, and therefore an infinitely magni-
fied representation of the section we are considering would consist of a finite
central conic, replacing the oval, and of the line at infinity n 2 times repeated,
replacing the n 2 branches which lie at a finite distance from C.. Since, then,
the radii of curvature of the normal sections vary as the squares of the diameters
of the evanescent oval, they vary as the squares of the central radii vectores
of a conic section.
If there be a double line upon the surface we can construct the two
tangent planes at any point m by taking two plane sections passing through
m and constructing the tangents of the double points. Each of these tangent
planes will cut the surface in a curve having a triple point at m ; but as
the direction of one of the three tangents is known a priori, being the
intersection of the tangent planes, the directions of the remaining two may
be found by the construction used for double points ; consequently the directions
of the tangents to the principal sections on each sheet of the surface are known,
and the principal radii of curvature may be determined by the construction
before given for finding either radius of curvature at a double point. The
two indicatrices at the point m may therefore be considered as ascertained
in magnitude and position.
If the osculating plane and radius of curvature of the double line itself
be required, they may be obtained very simply by a method to be given below.
ON SOME GEOMETRICAL CONSTRUCTIONS. 31
If the singular line be of the r th order (r > 2), a little consideration will
shew that though we cannot determine the tangent planes by any elementary
construction, yet, if we assume these planes as known, the indicatrices upon each
sheet may be found as easily as if the point were not singular. This is the more
remarkable, since the expressions given by analysis for the principal radii of
curvature at such points appear to be of great complexity.
Let us now take a point m on a curve surface where two sheets of the
surface meet and have a common tangent plane. This tangent plane will
intersect the surface in a curve having a quadruple point at m ; but the direc-
tions of the four tangents may always be ascertained by a quadratic construction.
For at such a point the polar surface of the third order will resolve itself into
the tangent plane and into a surface of the second order. And it may be shewn
that the two generatrices (real or imaginary) of that surface which lie in the
tangent plane are in involution with the two pair of asymptotes of the two
indicatrices ; that is, with the four tangents before mentioned. Now these two
generatrices may be determined by means of the theory of homological figures,
since that theory enables us to assign a pair of semi-conjugate diameters of
a section of the surface of the second order parallel to the tangent plane at m,
whence the directions of the asymptotes of that section become known, and
therefore the two generatrices required. The problem now will be : Given four
points in a line PQRS, and the four products P0 1 .P6 2 .Pd s .Pd t , &c., determine
flu #2) #3 j #4) a pah* of points G lt G 2 being also given which form an involution with
the two pairs 6 l , 2 and 6 a> O t . It is plain that, A being any point whatever, any
symmetrical function of the distances A0 1 , A6 2 , A6 3 , Ad t , may be constructed.
Hence H 1} H 2 , the harmonic centres of the four points 8 lt 2 , 6 3 , 4 with respect to
GI, G 2 , are known. But H lf H 2 form a pair of points in involution with the two
pair sought and therefore H lt H 2 together with G 1} G 2 completely determine the
involution. Therefore the centre and foci of the system are known, and con-
sequently Q l} 2 and 6 3 , 4 may be now quadratically determined. Points of the
nature here considered may exist isolated on a curve surface ; but if there be a
continuous series of them, we shall have a line along which two sheets of the
surface envelope one another (not a cuspidal line, for any transversal plane will
determine a section having not cusps, but points of osculation at its intersections
with the singular line), and at any point on such a line the two generatrices
before mentioned will be found to coincide : and consequently the surface of the
second order will degenerate into a cone. The side of this cone, existing in the
tangent plane at m, may be determined by proceeding as in the general case :
32 ON SOME GEOMETRICAL CONSTRUCTIONS.
and then, instead of a pair of points in involution with 6 lt 2 , 6 3 , 9 4 , we shall have
one focus of that involution given. The second focus may next be constructed
(being the harmonic centre of 6 1} 2 , 3 , 4 with respect to the given focus), and
the problem becomes quadratic as before.
Lastly, let there be a point on a curve surface having a tangent cone of the
second order. It will be possible to determine three conjugate diameters of that
cone. For, take any two planes passing through the vertex, and having con-
structed the tangent lines of the double points in those planes, take the harmonic
conjugates of the line of intersection of the two planes with respect to each pair
of tangents. This will give the plane conjugate to the line of intersection ; and
by taking any two lines harmonically conjugate with respect to the two tangent
lines existing in that plane, we shall obtain the directions of the three semi-
diameters required. Likewise, their ratios, or rather the ratios of their squares,
may be found, since the two sides of the cone in each conjugate plane may be
constructed. Hence, we may deduce the directions and the ratios of the squares
of the principal semiaxes of the cone. But this determination, involving the
solution of a cubic equation, requires the construction of a conic, and is con-
sequently not within the limits of elementary geometry. (Vide the Note on
M. Chasles' History, already quoted, and a paper by Mr. Townsend in this
Journal.)
If a curve of double curvature be given in space the principles of the theory
of transversals are not immediately applicable : but if we regard it as the inter-
section of two geometrical surfaces completely given, we may immediately find
its tangent, osculating plane, and radius of curvature. The tangent at m is of
course determined by the intersection of the two tangent planes ; and if we take
the two normal sections containing that tangent, and, having constructed their
radii of curvature, let fall a perpendicular from m on the line joining the two
centres, this perpendicular will represent in magnitude and direction the radius
of curvature of the given curve. This (it will be seen) follows at once from
Meunier's theorem, or from that known as Hachette's.
III.
DE COMPOSITIONE NUMERORUM PRIMORUM
FORMAE 4A+1 EX DUOBUS QUADRATIS.
[Crelle's Journal, vol. L. pp. 91, 92; 1855.]
s
IT
.
H --
fractio continua, cujus numerator, qui determinant!
qi, 1, 0, 0, ...
-1, q,, 1, 0, ...
0, -1, fc,l, . . .
0, 0, -1, &, ...
....... 1
0, 0, 0, 0, . -l,q n
aequalis est, per hujusmodi formulam (qiq z q 3 ... <?_!<?) exprimatur. Erit ergo
[gi g g<-i qi\ =
et [g, ... 2 B ] = [3ig 2 ... 5 f ] . b <+ i .
quae aequationes pendent ab ilia forma determinantali, ambae autem L. Eulei-o
debentur.
Itaque, si quantitatum q par sumatur numerus, ipsaeque ita serie sym-
metrica disponantur, ut binae inter se aequales fiant, elucet, quantitatem
34 DE COMPOSITIONS NUMERORUM PRTMORUM FORMAE 4X + 1, ETC.
[7, q t ... g, 5,- ... q a qt] summam fore duorum quadratorum inter se primorum ;
fit enim [9,9., ... 7,?, ... q a g,] = [qi q, . . . q t ] 2 + fa q, ... g,--,] 2 ....
Contra in numero quotientium impari, erit
unde colligis, numerum [7, . . . q { . . . 5,] primum esse non posse, nee duplicem
mimeri primi ; si quidem casus excipis, in quibus, aut i unitati aequatur, aut
i binario, q unitati.
Slip numerus integer datus ; /u t , /u 2 , .../*, series numerorum, qui ad p primi
sunt, ipsiusque p dimidio minores.
P P P
Formentur fractiones continuae , , ... ; quae omnes ita terminentur,
Mi Ms M,
ut is quotiens qui in extreme loco ponatur unitatem superet. Hinc patet,
quanta fuerit numerorum n lt n 2 , ...n, multitudo, tantum fore numerum determi-
nantium [q t ... 5,,], qui dato numero p aequales erunt, neque praeter illos ullum
dare ejusdem formae determinantem, cujus et primus et extremus quotiens
unitate major sit, quique numero p aequalis esse possit.
Jam vero, quum duo determinantes [qi-..q n ] et [<?...<?!] aequales sint,
quumque ipsum q n unitate majus sit, apparet [q n ... #1] ex una aliqua fractionum
- oriri. Unde sequitur, data quavis fractione -, inveniri posse aliam in eadem
serie, quae quotientes eosdem, ordine inverse, repraesentet.
Sit p primus, formae 4 X + 1 ; ut numerus determinantium ipsi p aequalium
par existat. Quum ipse p unus e determinantium serie fiat, unus certo alius
inveniri poterit in quo quotientium ordo invertendo non mutatur. Cum sit ergo
p = [q l q 2 ...q i q i ...q 2 q l ~\
erit denique p = [q, q 2 . . . g,.]-' + [q 1 q > ... q^] 2 .
Quam theorematis Fermatiani demonstrationem maxime elementarem esse
patet, quum pendeat a conversione fractionum vulgarium in fractiones continuas.
Singulos autem formae 1 + x 2 divisores ex duobus quadratis conflari, eodem
modo demonstrare in promptu est. Sit enim
ftv = 1 + X 2 ,
apparet fore M = [^ q 2 . . . q { q { ... q 2 q^
OXFORD, Maio 1854.
IV.
ON THE HISTORY OF THE RESEARCHES OF
MATHEMATICIANS ON THE SUBJECT OF THE SERIES
OF PRIME NUMBERS.
[Proceedings of the Ashmolean Society, Vol. III. No. xxxv. pp. 128-131. Read March 2, 1857.]
IT is probable that the Pythagorean school was acquainted with the definition
and nature of prime numbers ; nevertheless the arithmetical books of the
elements of Euclid contain the oldest extant investigations respecting them ;
and, in particular, the celebrated, yet simple, demonstration that the number
of the primes is infinite. To Eratosthenes of Alexandria, who is for so many
other reasons entitled to a place in the history of the sciences, is attributed the
invention of the method by which the primes may successively be determined in
order of magnitude. It is termed, after him, 'the sieve of Eratosthenes ;' and is
essentially a method of exclusion, by which all composite numbers are suc-
cessively erased from the series of natural numbers, and the primes alone are
left remaining. It requires only one kind of arithmetical operation ; that is
to say, the formation of the successive multiples of given numbers, or, in other-
words, addition only. Indeed it may be said to require no arithmetical operation
whatever ; for if the natural series of numbers be represented by points set off at
equal distances along a line, by using a geometrical compass we can determine
without calculation the multiples of any given number. And it was in fact by a
mechanical contrivance of this nature, that M. Burckhardt calculated his table of
the least divisors of the first three millions of numbers. But simple as this
process is, the questions to which it gives rise are among the most obscure of the
theory of numbers.
F 2
X ON THE HISTORY OF THE RESEARCHES OF MATHEMATICIANS
Adopting (with a slight variation in its meaning) an expression introduced
by M. Polignac, we may call the series of numbers left unerased, after the
erasure of the multiples of any given primes, the diatomic series of those primes.
Thus the diatomic of 2.3.5 is 1, 7, 11, 13, 17, 19, 23, 29 : and it is unnecessary
to continue the series further, because if a denote any one of the eight numbers
we have written down, the remaining terms of the series are included in the
eight arithmetical progressions 30m + a. In general, if O denote the product of
any given primes, in forming their diatomic we need only attend to the terms
less than Q. A few of the properties of this finite series are very easily seen.
In the first place, the number of the diatomic terms is that of the numbers less
than Q and prime to it, or II . (P 1), if P be any one of the given primes and II
denote a continued product. Secondly, the diatomic terms are distributed sym-
metrically ; that is to say, if a be a diatomic term, Q a is one too. The sum of
the diatomic terms is therefore iQII(P 1) or ^ILP(.P 1). It is not difficult
similarly to form expressions giving the sums of any positive and integral powers
of the diatomic numbers ; or series giving expressions convergent up to a certain
point, for their product or the powers of their reciprocals. It is therefore
possible to form an equation of which the coefficients are functions of the given
primes, and the roots are the diatomic numbers. But it does not appear that
this equation throws much light on the nature of its roots.
A remark of greater interest is due to Legendre. Let us denote by (r) the
greatest whole number not surpassing a given positive numerical quantity r ; let
j 2 3 ... be the diatomic terms of any given primes, and let a be a numerical
quantity inferior to u k + l , but not inferior to u k ; then
The series is to be continued till it stops of itself, and the signs of summation 2
extend to every possible combination of the given primes P 1 P 2 ... taken one by
one, two by two, &c. The principle on which the demonstration of this equation
(and of many resembling it, which occur in the theory of numbers) is founded,
may be termed the principle of cross classification, and may be enunciated thus.
If 0*1 o> s ... be a system of cross-classifying classes, and if (o^) denote the number
of things in the class a- lt (o-j o- 2 ) the number of things common to the two classes
o-, and <r 2 , (o-j o-jj <7 3 ) the number of things common to the three classes o- 1; cr 2 , <r 3 :
then S^-S^or^ + Z^o-ijo-a) ... will express the whole number of things
present in the classes o-,, <r 2 ... .
Legendre's formula, it is readily seen, assigns the index k of any given
ON THE SUBJECT OF THE SERIES OF PRIME NUMBERS. 37
diatomic term u k . Conversely we can express u k in terms of k. Let us denote
the function -a\ a
and let us form the series of terms
till we arrive at last (as we shall certainly do) at two consecutive terms equal to
one another, say a n and a n + l ; we may then stop, for we should find
The expression for u k will then be
u k = k + a n , or u k =
It must be confessed that this result is one which, for diatomics derived from a
numerous group of primes, would involve far too much labour to be of any use.
But it is of some slight theoretical interest. For if the given primes P 1 P 2 . . . P x
be the x first primes in order of magnitude, it is clear that their 2 d , 3 d , ...
diatomic terms will be P x + i P x + 2 , and that the first diatomic term which will
not be a prime is P 2 x + i. Given therefore the first x primes, the formula
prescribes a direct method for the calculation of the primes intermediate between
P x and P* X + I. Thus let the given primes be 2, 3, 5, and let it be required
to determine the next prime after 5, we find
We have therefore
It may be added, that the calculation of the functions <f> does not absolutely
require the knowledge of the primes P 1 ...P X ; only the arithmetical operations
which would be requisite for determining their value would involve more trouble
than the determination of the primes P^.^.P,..
V.
REPORT ON THE THEORY OF NUMBERS.
PABT I.
[Report of the British Association for 1859, pp. 228-267.]
1. 1HE ' Disquisitiones Arithmeticae ' of Karl Friedrich Gauss (Lipsiae, 1801
{ed. 1. 1798}*) and the 'Theorie des Nombres' of Adrien Marie Legendre (Paris,
1830, ed. 3) are still the classical works on the Theory of Numbers. Nevertheless,
the actual state of this part of mathematical analysis is but imperfectly repre-
sented in those celebrated treatises. The arithmetical memoirs of Gauss himself,
subsequent to the publication of the ' Disquisitiones Arithmeticae ;' those of
Cauchy, Jacobi, Lejeune Dirichlet, Eisenstein, Poinsot, and, among still living
mathematicians, of MM. Kummer, Kronecker, and Hermite, have served to
simplify as well as to extend the science. From the labours of these and other
eminent writers, the Theory of Numbers has acquired a great and increasing
claim to the attention of mathematicians. It is equally remarkable for the
number and importance of its results, for the precision and rigorousness of its
demonstrations, for the variety of its methods, for the intimate relations between
truths apparently isolated which it sometimes discloses, and for the numerous
applications of which it is susceptible in other parts of analysis. ' The higher
arithmetic,' observes Gauss f, confessedly the great master of the science, 'presents
us with an inexhaustible store of interesting truths, of truths, too, which are
not isolated, but stand in a close internal connexion, and between which, as our
knowledge increases, we are continually discovering new and sometimes wholly
' The additions enclosed in { } are taken from manuscript notes in the author's interleaved
copy ; they are all in his own handwriting.
t Preface to Eisenstein'a ' Mathematische Abhandlungen,' Berlin, 1847.
Art. 3.] BEPORT ON THE THEORY OF NUMBERS. 39
unexpected ties. A great part of its theories derives an additional charm from
the peculiarity that important propositions, with the impress of simplicity upon
them, are often easily discoverable by induction, and yet are of so profound
a character that we cannot find their demonstration till after many vain
attempts ; and even then, when we do succeed, it is often by some tedious
and artificial process, while the simpler methods may long remain concealed.'
2. It is the object of the present report to exhibit an outline of the results
of these later investigations, and to trace (so far as is possible) their connexion
with one another and with earlier researches. An attempt will also occasionally
be made to point out the lacunae which still exist in the arithmetical theories
that come before us ; and to indicate those regions of inquiry in which there
seems most hope of accessions to our present knowledge. In order, however,
to render this report intelligible to persons who have not occupied themselves
specially with the Theory of Numbers, it will be occasionally necessary to in-
troduce a brief and summary indication of principles and results which are to
be found in the works of Gauss and Legendre. It is hardly necessary to add
that we must confine ourselves to what we may term the great highways of
the science ; and that we must wholly pass by many outlying researches of
great interest and importance, as we propose rather to exhibit in a clear light
the most fundamental and indispensable theories, than to embarrass the treat-
ment of a subject, already sufficiently complex, with a multitude of details,
which, however important in themselves, are not essential to the comprehension
of the whole.
3. There are two principal branches of the higher arithmetic : the Theory
of Congruences, and the Theory of Homogeneous Forms. The first of these
theories relates to the solution of indeterminate equations, of the form
in which a n _! . . . a l and P are given integral numbers, and x and y are
numbers which it is required to determine. The second relates to the solution
of indeterminate equations of the form
F(x l x 2 ...x m ) = M,
in which M denotes a given integral number, and F a homogeneous function
of any order with integral coefficients. In this general point of view, these
two theories are hardly more distinct from one another than are in algebra
the two theories to which they respectively correspond, the Theory of Equa-
tions, and that of Homogeneous Functions ; and it might, at first sight, appear
as if there was not sufficient foundation for the distinction. But, in the present
40 REPORT ON THE THEORY OF NUMBERS. [Art. 4.
state of our knowledge, the methods applicable to, and the researches suggested
by these two problems, are sufficiently distinct to justify their separation from
one another. We shall therefore classify the researches we have to consider
here under these two heads ; those miscellaneous investigations, which do not
properly come under either of them, we shall place in a third division by them-
selves.
(A) Theory of Congruences.
4. Definition of a Congruence. If the difference between A and B be
divisible by a number P, A is said to be congruous to B for the modulus P ;
so that, in particular, if A be divisible by P, A is congruous to zero for the
modulus P. The symbolic expressions of these congruences are respectively
A=B, mod P,
A=0, mod P.
Thus 7 = 2, mod 6 ; 13 = - 3, mod 8.
It will be seen that the definition of a congruence involves only one of
the most elementary arithmetical conceptions, that of the divisibility of one
number by another. But it expresses that conception in a form so suggestive
of analogies with other parts of analysis, so easily available in calculation, and
so fertile in new results, that its introduction into arithmetic (by Gauss) has
proved a most important contribution to the progress of the science. It will
be at once evident, from the definition, that congruences possess .many of the
properties of equations. Thus, congruences in which the modulus is the same
may be added to one another ; a congruence may be multiplied by any number ;
each side of it may be raised to any power whatever, and even may be divided
by any number prime to the modulus.
5. Solution of a Congruence. If < (x) denote a rational and integral func-
tion of x with integral coefficients (we shall, throughout this report, attach this
meaning to the functional symbols F, f, <p, &c., except when the contrary is
expressly stated) ; the congruence (f) (x) = 0, mod P, is said to be solved, when
all the integral values of x are assigned which make the left-hand number of
the congruence divisible by P ; i. e. which satisfy the indeterminate equation
(x) = Py. It is evident that if x = a be a solution of the congruence <p (x) = 0,
every number included in the formula x = a + nP is also a solution of the con-
gruence. But the solutions included in that formula are all congruous to one
another and to a. It is proper, therefore, to consider all these congruous solu-
tions as identical, and in speaking of the number of solutions of a congruence
Art. 7.] REPORT ON THE THEORY OF NUMBERS. 41
to understand the number of sets of incongruous solutions of which it is sus-
ceptible. To assign, by a direct method, all the solutions of which a proposed
congruence is capable, is the general problem which, in the Theory of Numbers,
corresponds to the problem of the solution of numerical equations in ordinary
algebra. But the solution of the arithmetical problem is attended with even
greater difficulties than that of the algebraical one ; and the attention of geo-
meters has been turned with more success to the improvement of the indirect
or tentative methods of solution, and to the discovery of criteria of possibility
or impossibility for congruential formulae, than to their direct solution. It is
to be observed that, by virtue of a remark already made, the tentative solution
of a congruence involves no theoretical difficulty. For if x a be a solution,
every number included in the formula x = a + nP is also a solution, and among
these numbers there is always one, and only one, comprised within the limits
and P 1 inclusively. By substituting, therefore, for x all numbers in suc-
cession less than the modulus, and rejecting those which do not satisfy the
congruence, we shall obtain its complete solution. But the interminable labour
attending this operation, notwithstanding all the abbreviations in it suggested
by the Calculus of Finite Differences, renders its application impossible, except
when the modulus is a low number.
6. Systems of Residues. The set of numbers 0, 1, 2 ... P-l (or any set
of P numbers respectively congruous for the modulus P to those numbers) is
termed a complete system of residues for the modulus P. By a system of residues
prime to P, we are to understand a complete system, from which every residue
has been omitted which has any common divisor with P. Thus 1, 5, 7, 11,
or 1, 5, 5, 1, are the terms of a system of residues prime to 12. The word
Residue is employed instead of Remainder, because the word Remainder would
suggest the idea of a positive number less than the modulus or divisor ; whereas
it is frequently convenient to consider residues differing from those positive
remainders by any multiples of the modulus whatever.
7. Linear Congruences. The general form of a linear congruence is
ax + b =0, mod P ;
a, b, and P denoting given numbers, and x a number to be determined.
The theory of these congruences may be considered to be complete, both
as regards the determination of the solutions or roots themselves and of their
number. If a be prime to the modulus, there is always one solution, and one
only; if a have a common divisor with the modulus which does not also divide
b, the congruence is irresoluble ; if S be the greatest common divisor of a
G
42 REPORT ON THE THEORY OF NUMBERS. [Art. 8.
and P, and if 3 also divide b, the congruence has S solutions. In every case
when the congruence is resoluble, the direct determination of its roots may
be made to depend on the solution of a congruence of the form ax = 1, mod P,
in which a is prime to P. This congruence coincides with the indeterminate
equation ax = 1 + Py, methods for the solution of which were known to the
ancient Indian geometers*, and have been given in Europe by Bachet de
Meziriacf, EulerJ, and Lagrange^. The methods of these writers ultimately
depend on the conversion of a vulgar fraction into a continued fraction, and
in one form or another have passed into every book on algebra. Nor would
it have been proper to allude to them here, were it not that they serve to
supply us with a clear conception of what we have a right to expect in the
solution of an arithmetical problem. In such problems, we cannot expect to
express the quaesita as (discontinuous) analytical functions of the data. Such
expressions may indeed, in many cases, be obtained (by the use of the roots
of unity or by other methods) ; but the results of the kind which have hitherto
been given, though sometimes of use in calculation, may be said, with few
exceptions, to conceal rather than to express the real connexion between the
numbers required and the numbers given. The arithmetical solution of a
problem should consist in prescribing a finite number of purely arithmetical
operations (exempt from all tentative processes), by which all the numbers
satisfying the conditions of the problem, and those only, are obtained. It is
clear that this description exactly applies to the methods on which the solution
of linear congruences depends ; but, unfortunately, the higher arithmetic pre-
sents but few examples of solutions of equal perfection.
8. Besides the older methods for the solution of the equation ax = 1 + Py,
others have, in very recent tunes, been suggested. Of these the following may
serve as examples :
A. In the equation ax 1 + Py, or the congruence ax = 1, mod P, form
* See the Arithmetic of Bhascara, cap. xii, and the Algebra of Brahmegupta, cap. i, in
Mr. Colebrooke's translation, London, 1817.
t Problemes plaisans et delectables, qui se font par les nombres. Seconde edition. Par Claude
Caspar Bachet, Sieur de Meziriac, Lyon, 1624. (See Props, xv to xxv.)
J Comment. Acad. Petropol. torn. vii. p. 46, or in the Collection of Euler's Arithmetical Memoirs
(L. Euleri Commentationes Arithmeticae Collectae, Petropoli, 1849), vol. i. p. 2 ; and in hia Elements
of Algebra, part ii. cap. 1.
Sur la B&iolution des Problemes Indetermines du seconde degre". Hist, de 1' Acad. de Berlin,
1767, p. 165. (See Arts. 7, 8, and 29 of the Memoir.) Also in the Additions to Euler's Algebra,
eects. i and iii. (Lyon, an. in.)
Art, 9 ]
REPORT ON THE THEORY OF NUMBERS.
43
the residues of the successive powers of a for the modulus P. If a be prime
to P, we shall at last arrive at a power which has +1 for its remainder or
residue. The residue of the power immediately inferior to this power is the
value of x in the congruence ax = l, mod P. This solution is evidently an
application of Fermat's Theorem*.
B. Let there be P points A lt A 2 , ... A p , arranged at equal distances on the
circumference of a circle. Join A^ to A a + l , A a + 1 to A 2a + l ... arid so on con-
tinually. It can be proved that if a be prime to P, we shall not return again to
A lt until we have passed through every one of the P points, and have formed a
polygon of P sides. Let X l} X 2 , ... X p be the vertices of this polygon, taken in
order, and let A z = X m + x ; then x=m is the value of x in the congruence
ax = 1, mod Pf.
C. Let an origin and a pair of axes be assumed in a plane, and let all the
points be constructed whose coordinates are integral multiples of the linear unit ;
call these points unit points. Join the origin to the point (a, P}. If a be prime
to P, no unit point can lie on the joining line, but on each side of the joining
line there will be a point lying nearer to it than any other. Let ( t ft), ( 2 1? 2 ) be
the coordinates of these points, and let & : ft < 2 : 7 2 ; then j, ft, and 2) iz are
the least positive numbers satisfying the equations
aft-jPi = l, ai/ 2 -P 2 = -1.
The late M. Crelle, of Berlin, in the 45th volume of his Journal (p. 299),
has given a very useful table, containing the least positive numbers x l and x z
which satisfy the equation a 1 x 1 a 2 x a = l, for all values of a^ up to 120, and for
all values of a 2 prime to a x and less than it.
9. Systems of Linear Congruences. The theory of these systems is left
imperfect in the work of Gauss (see 'Disq. Arith.' art. 37); but, by the aid of
a few subsidiary propositions relating to determinants, we may, in every case,
obtain directly all possible solutions of any proposed system ; and (what is
frequently of more importance) we can decide a priori whether a given system
of linear congruences be resoluble or not, and if it be resoluble we can assign the
* Binet, sur la Resolution des Equations du premier degr6 en Nombres entiers. (Journal de
1'Ecole Polytechnique, cahier xx. p. 289.)
Libri, M6moires de Math&natique et Physique (Florence, 1829), pp. 65-67.
Poinsot, Reflexions sur lea Principes Fondamentals de la Th6orie des Nombres (Paris, 1845),
cap. iii. nos. 19 and 20. For another solution by M. Binet, see Comptes Rendus, xiii. p. 349. See
also Cauchy, Comptes Rendus, xii. p. 813. {Exer. d'Anal. et de Pbys. Math., vol. ii. p. 1.}
t Poinsot, Reflexions, &c., cap. iii. nos. 17 and 18.
O 2
44 REPORT ON THE THEORY OF NUMBERS. [Art. 9.
number of its solutions. The following theorems by which the determination of
the number of solutions is, in every case, effected, will sufficiently indicate the
nature of these investigations.
Let the proposed system of congruences be represented by
(1, 1) aj, + (l, 2) ^ + (1, 3) Xs + ... + (1, n)x n =u l ,
(2, 1) x, + (2, 2) ^ + (2, 3) a%+...+(2, n)x n =u 2 , (A)
(n, 1 ) Xj + (n, 2) a^ + (n, 3) x 3 + . . . + (n, n) x n =u n ;
let the modulus be q, and the determinant 2 + (1 , 1) (2, 2) . . . (n, n) = D. If the
determinant be prime to the modulus, these congruences will always admit of
one, and only one, system of solutions, namely, that supplied by the system of
congruences &= B ^j^
Dx r = 2 - -u k .
But if D be not prime to q, let q =p 1 m i . p 2 m * ... where p lt p 2 , &c. denote
different primes. In order that the proposed system should be resoluble for the
modulus q, it must be separately resoluble for each of the modules p^i , p^ , &c. ;
and, conversely, if it be resoluble for each of those modules, and admit P 1
solutions when taken with respect to the modulus p^ , P z solutions when taken
with respect to the modulus p*, and so on, it will be also resoluble for the
modulus q, and will admit P l xP 2 xP s ... solutions for that modulus. It is,
therefore, only necessary to assign the number of solutions of the congruences
(A), for a modulus p m which is the power of a prime. Let / be the index of the
highest power of p which divides D ; and similarly, let I r denote the index of
the highest power of p which divides all the minors of D which are of order r ;
then if / /_] S m, the system (A) (if resoluble at all) admits of p 1 " solutions ;
but if I n >m + I n _ l , it will always be possible, in the series of differences
-*n~-'-l> 'n-l~-*n-2> >
to assign a pair of consecutive terms I r+1 I r , I T I r _- i , satisfying the in-
equalities r _/>,.>/_/ .
J r + l J r - W4 = J- r -L r -l>
and then the number of solutions (supposing always that the congruences are
resoluble) is expressed by the formula p / - +( "~ r) ' B .
The analogy of this theory with the corresponding algebraic theory of
systems of linear equations is in particular cases very striking. For example,
we have in Algebra the theorem :
Art. 10.]
REPORT ON THE THEORY OF NUMBERS.
45
' The system of n linear equations
(I, 1) x, + (l, 2) x 2 + (l, 3) x 3 + ... +(1, w) ce n = 0,
(2, 1) ^ + (2, 2) * 2 + (2, 3) ar, + . . . + (2, n) 03 n = 0,
, 2) Cjj + (w, 3) x 3 + ... +(n, n) x n = 0,
implies either that D = 2 + (1, 1) (2, 2) . . . (w, w) = 0, or else that o^, x 2 , . . . x n are
separately equal to zero.'
In the Theory of Numbers we have the corresponding theorem :
' If n linear and homogeneous functions of an equal number of indetermi-
nates be congruous to zero for a prime modulus, either the determinant of the
system is congruous to zero for that modulus, or else every one of the indetermi-
nates is separately congruous to zero.'
10. Fermat's Theorem. The theory of congruences of the higher orders
is so essentially connected with Fermat's Theorem, that it will be proper before
proceeding further to introduce a few considerations relating to that celebrated
proposition.
It may be considered from two different (though closely connected) points
of view, each of which has proved equally fertile in consequences. First, it may
be regarded as asserting that, if p be a prime number, and x any number prime
to p, the remainder left by the power x v ~ l when divided by p is unity. It is
thus the fundamental proposition in the arithmetical theory of the residues
of powers, or, which is the same thing, of binomial congruences. Or, secondly,
it may be regarded as asserting that the congruence x p ~ 1 = l, mod p, has
precisely p 1 roots ; and that these roots are the terms of a system of residues
prime to p. It is in this latter point of view that the theorem is the basis of
the general theory of congruences.
We may observe that the demonstrations of Fermat's Theorem point to this
twofold aspect.
The proof, which is found in most English treatises of Algebra (it is the
first of those given by Euler*), and which depends on the property of the
binomial or multinomial coefficient, would naturally lead us to regard the
Theorem in the first point of view. The same may be said of Euler's second
* Comment. Acad. Petropol., vol. viii. p. 141, or Comment. Arith., vol. i. p. 21. This is the
first demonstration of the Theorem discovered, since the time of Fermat. The memoir containing
it was presented to the Academy of St. Petersburg, Aug. 2, 1 736.
4 g REPORT ON THE THEORY OF NUMBERS. [Art. 10.
demonstration*, which consists in showing that the index of the lowest power
of x in the series 1, x, x*, x\ Ac., which leaves unity for its remainder when
divided by p, is either p-1, or some submultiple of p-l ; or again of
demonstration of MM. Dirichlett, BinetJ, and Poinsotf, which depend*
the observation that the terms of a system of residues prime to any modulus,
being multiplied by any residue prime to the modulus, still form a sysl
residues prime to the modulus.
But a remarkable proof of the theorem, in the second expression we
given to it, occurs in a memoir of Lagran g e||. As this proof (though very
elementary) has not been copied by subsequent writers, and is consequently but
little known, its nature may be indicated here.
Let the product (z + 1) (x + 2) (as + 3) ... (x + p-1) be represented by
x denoting an absolutely indeterminate quantity. Writing x + l for x, and
multiplying by x + l, we obtain the identity
whence, by equating the coefficients of like powers of x, we find
p(p-l)
Al ~ 1.2
--
A *~
1.2.3 1.2
_, + .
3= 1.2.3.4 1.2.3
* Novi Commentarii Petropol., vol. vii. p. 49, or Comment. Arith., voL i. p. 260. From the
point of view in which Fermat presents his theorem, it is not improbable that the demonstration
he had found of it was no other than this of Euler's. (See Fermati Opera Mathematics, Tolosae,
1679, p. 163.) It has been adopted by Gauss in the Disquisitiones, Art. 49.
t Crelle's Journal, vol. iii. p. 390.
J Journal de l'cole Polytechnique, Cahier xx. p. 289.
Reflexions sur la Theorie des Nombres, p. 32. But the principle of this demonstration i.
employed by Gauss in a memoir published in the Comm. Soc. Getting, vol. xvi. p. 69, to whicl
we shall have again to refer. (See Art. 19 of this Report.)
|| Demonstration d'un TWoreme nouveau concernant les Nombres Premiers (Nouveaux Memoires
de 1'Academie Eoyale de Berlin, 1771, p. 125). The 'new theorem' is that known as Sir. J. Wilson's.
Art. 11.] REPORT ON THE THEORY OF NUMBERS. 47
From these equations we successively infer the congruences A l = 0, A 2 = 0,
A 3 = 0, ... A p _ z = 0, and lastly, A p _ l = 1, mod_p. We have, therefore, the
indeterminate congruence
(x + l)(x + 2)(x + 3)...(x+p-l) = x p - l -l, modp,
which is evidently identical, i. e. it subsists for all values of x. And since, if
Oi, a 2 ,'... a p _ l be the terms of any system of residues prime to p, the factors
x a 1} x a 2 , x a 3 , ... x a p _ l are one by one congruous to the factors x + 1,
x + 2, x + 3, ... x+p 1 taken in a certain order, the products
(x ttj) (x a 2 ) ... (x a p _^) and (x + 1) (x + 2) ... (x +p 1)
are also identically congruous for the modulus p, so that we may write
(x Oj) (x a 2 ) ... (x a p _j) = x p ~ 1 l, modp.
This congruence exhibits in the clearest manner possible what the real
nature of the function x p ~ l 1 is when considered with respect to the modulus p,
and explains to us why it assumes a value divisible by p, when we assign to
x any integral value not divisible by p.
It will be observed that the last of the p 1 congruences included in the
congruence
(x-l) (x-2) (x-3) ... (x-p-l) = x p - l -l, modp,
(which is a particular case of that last written), namely, the congruence
1.2.3 ...p-l = -l, modp,
is the symbolic expression of Sir J. Wilson's Theorem.
11. Lagrange's Limit of the Number of Roots of a Congruence. The full
development of the consequences of Fermat's Theorem requires the aid of the
following proposition, which was first given, in a slightly different form, by
Lagrange *.
' If F (x) be a function of x of n dimensions, such that F (a) = 0, mod p,
then a function of as of n 1 dimensions, F 1 (x), can always be assigned such
that we shall have the identical congruence F(x} = (x a)F- i (x), modp.'
Hence we may infer that no congruence, of which the modulus is prime,
can have more incongruous roots than it has dimensions ; and, if a congru-
ence have congruous roots, we obtain a definition of their multiplicity; viz.,
if F(x) = (x-a) r F l (x), mod p, then we may say that F(x) = 0, modp, has
* Nouvelle M6thode pour resoudre les Problemes Ind6termin6s en Nombres entiers. (See Hist.
Ac. Berl. 1 768, p. 1 92.) The case of binomial congruences of the form x n = 1 had already been
treated by Euler. (See Nov. Comment. Petropol. vol. xviii. p. 85, or Comment. Arith. vol. i. p. 516,
Art. 28 of the Memoir.)
48 REPORT ON THE THEORY OP NUMBERS. [Art. 12.
r roots congruous to a. We may also observe that this theorem enables us
at once to infer Lagrange's indeterminate congruence from the first expression
of Fermat's Theorem. For since x~ l -l is = for the values x = l, x = 2,
...x = p-l, we may, by successive applications of the preceding theorem,
show that a;"- 1 -! = (x-l)(x-2) ... (x-p + 1), mod p.
12. Theory of the Residues of Powers. The principal elementary theorems
relating to the Residues of Powers are the following. They are all due to
Euler *, who was the first to demonstrate Fermat's Theorem, and to develope
the numerous arithmetical truths connected with it.
I. If e and f be conjugate divisors of p 1 so that p \-ef; the con-
gruence a/ = 1, mod p, always admits of f incongruous roots. Let these roots
be denoted by a^, a. z , ... a f . Then each of the / congruences x e = a r admits of
e solutions, and the ef roots of these f congruences exhaust completely the
p l residues prime to p. It appears, therefore, that if we raise the residues
of p to the power e, they will divide themselves into f groups of e numbers
apiece ; the e numbers of each group giving, when raised to the power e, the
same residue for the modulus p. The numbers a 1} a 2 , ...ay are termed the
quadratic, cubic, biquadratic, quintic, &c. residues of p, according as e = 2,
(! = 3, e = 4, e = 5, &c., because they are each of them congruous to an e th power
(and indeed to an e th power of e different numbers), and because no other
number beside them can be congruous to such a power. Thus every uneven
prime has h(p 1) quadratic, and as many non-quadratic residues; every prime
of the form 4w + l has \(p 1) biquadratic residues, and three times as many
non-biquadratic residues, &c.
II. It is readily seen that if the same number x satisfy the two congruences
o/i = 1, and o/s = 1, it also satisfies the congruence x d = l, modjD; where d
is the greatest common divisor ofyi and /a- If therefore/ 1 be the lowest index
for which the number x satisfies the congruence a/= 1, mod 2^, f is a divisor
* Euler's memoirs on this Theory are :
(i.) Theorematum quorundam ad numeros primos spectantium demonstratio. Comment. Arith.
Tol. i. p. 21.
(ii.) Theoremata circa residua ex divisione potestatum relicta. Ibid. p. 260.
(iii.) Theoremata arithmetica novo methodo demonstrata. Ibid. p. 274.
(iv.) Disquisitio accuratior circa residua ex divisione quadratorum aliarumque potestatum per
numeros primos relicta. Ibid. p. 487.
(v.) Demonstrationes circa residua ex divisione potestatum per numeros primos resultantia.
Ibid. p. 516.
Art. 13] REPORT ON THE THEORY OF NUMBERS. 49
of p 1 ; as indeed appears directly from Euler's second demonstration of
Format's Theorem. Let -^ (f) denote the number of numbers less than f and
prime to it; then there are always ^(/) roots of the congruence o/ = 1, mod^>,
which cannot satisfy any other congruence of lower index and similar form.
These are called primitive roots of the congruence a/ = 1, modj) ; they are also
said to appertain to the exponent/. Iff=p l, the ^(p 1) primitive roots
of the congruence x p ~ l = 1, mod^>, are termed for brevity (though the de-
signation is somewhat improper) the primitive roots of p. There are therefore
^(p 1) primitive roots of p.
13. Primitive Roots. The problem of the direct determination of the
primitive roots of a prune number is one of the 'cruces' of the Theory of
Numbers. Euler, who first observed the peculiarity of these numbers, has yet
left us no rigorous proof of their existence * ; though, assuming their existence,
he succeeded in accurately determining their number. The defect in his
demonstration was first supplied by Gauss t, who has also proposed an indirect
method for finding a primitive root. This method J consists in taking any
residue a of p, and determining (by the successive formation of its powers)
the exponent f to which it appertains. JS f=p 1, a is itself a primitive
root of p ; if not, let b be a second residue of p, not contained in the period
of a, (i.e. not congruous for the modulus p to any one of the numbers a,
a, a 2 , ...a/~ 1 ,) and let the exponent to which b appertains be determined.
This exponent cannot (as is shown by Gauss) be identical with, nor yet a
divisor of, the exponent to which a appertains ; but it is always possible by
a comparison of the values of a and b to determine a third number, c, which
shall appertain to an exponent divisible by each of the exponents to which
a and b appertain. By proceeding in this way we shall evidently obtain num-
bers appertaining to exponents continually higher, till at last we come to a
number appertaining to the exponent p l; i.e. to a primitive root of p.
M. Poinsot proposes the following method. If 2, q lt q 2 , ... &c. be all the
prime divisors of p 1, raise the numbers
1, +2, +3,... l(p-l),
which form a system of residues prime to p, to the powers of which the
* See the memoir (i) of the preceding note ; and Gauss's criticism on it ; Disq. Arith. Art. 56.
t Disq. Arith. Art. 52-55.
J Ibid. Art. 73-74.
Reflexions sur la Th6orie des Nombres, cap. iv. art. 3.
H
50 REPORT ON THE THEORY OF NUMBERS. [Art. 13-
indices are 2, q lt q t , &c. ; so as to determine all the quadratic residues of p,
and its residues of the powers q lt q 2> &c. If from the system of residues
1, 2, 3, ...p 1, we successively exclude these residues of squares and higher
powers, we shall have -^(p-1) numbers left, which cannot be congruous to
any power having an index that divides p1, and which are consequently
(as may easily be shown) the primitive roots of p.
This method is very symmetrical ; and if the problem proposed be to find
all the primitive roots of p, it is sufficiently direct. But it is (like many other
direct methods in the Theory of Numbers) of interminable prolixity; and
becomes absolutely impracticable if p be a number even of moderate size,
as it requires us to form the residues of the successive powers of the numbers
1, 2, 3, ... %(p 1). Of course, in performing this operation, the multiples of p
are to be rejected as fast as they arise ; but, notwithstanding this abbreviation,
and others which a little experience will readily suggest, Gauss's method is, for
any practical purpose, greatly preferable.
In a memoir by M. Oltramare in Crelle's Journal (vol. xlix. p. 161), several
considerations are offered for facilitating the determination of the primitive roots
of primes in numerous special cases. Some, however, of the general results of
this memoir are erroneous, at least in expression, and the demonstrations of the
more particular conclusions contained in it involve no new principle, but may be
obtained by combining the definition of primitive roots with the criteria by
which (as we shall hereafter see) we are enabled to decide on the quadratic
or cubic characters of the residues of given primes. The following may
serve as examples of the very interesting results which are thus obtained
by M. Oltramare :
' If a be a prime number and 2 a + 1 be also a prime, 2 or o is a primitive
root of 2a + l, according as a is of the form 4w + l or 4n + 3.' Thus 2 is a
primitive root of 83, 11 is a primitive root of 23, 83 of 167, &c.
' If o be a prime number, other than 3, and if p = 2 m a + 1, where m is > 1,
be also a prime, 3 is a primitive root of p, unless the congruence
3 2m - l + l=Q, modp,
be satisfied.' Thus 3 is a primitive root of 89, and of 137.
Theorems of the same character will be found in the ' The"orie des Nombres * '
of M. Desmarest. By their aid M. Desmarest has constructed a table giving a
primitive root for every prime less than 10,000.
* Paris, 1852. See pp. 275-279.
Art. 14.] REPORT ON THE THEORY OF NUMBERS. 51
14. Indices. If 7 be a primitive root of p, the least positive residues of the
p 1 successive powers of 7, yi } ~ ~s yp-2 yp-i
which we may denote by y lt ?, y 3 , ... y p _ 2 , 1,
are all incongruous for the modulus p. These residues, therefore, irrespective
of the order in which they occur, coincide with the numbers 1, 2, 3, ...p 1,
i.e. they represent the terms of a complete system of residues prime to p.
If 7" = a, mod p, then /c, or any number congruous to K for the modulus p l, is
termed the index * of a for the primitive root or base 7 ; and this is expressed
symbolically by writing
K = Inda, mod(^> 1), or K = Ind^a, mod(p 1).
The principal properties of these indices, which it is clear are a kind of
arithmetical logarithm, are as follows :
(1) Ind (AB) = Ind A + Ind B, mod (p - 1).
(2) Ind (A") = <r Ind A, mod (p - 1).
(3) Ind , mod p) = Ind A - Ind B, mod (p - 1).
/A
[The symbol ( -^ , mod p\ is used to denote the value of x deduced from
the congruence Bx = A mod p.~\
(4) Ind T A = Ind 7 7'. Ind y A, mod (p - 1).
(5) 1{A = B, modp, Ind ^ = Ind B, modp- 1.
In these congruences yl and B represent numbers prime to p, a- any integral
number, and 7 and j two different primitive roots.
The great importance of these indices in arithmetical researches has induced
the Academy of Berlin to publish a volume containing tables of the numbers
corresponding to given indices, and of the indices corresponding to given
numbers for all primes less than 1000. This volume, the ' Canon Arithmeticust,'
was edited by C. G. J. Jacobi, and contains, besides the Tables, a preface
* The reader must be careful to distinguish between the index of a number and the exponent to
which the number appertains. The exponent does not depend on the choice of the primitive root : for
j 1
a given number it has but one value, a, which is such that is the greatest common divisor of the
index and of p 1. The index may have any one of \\r (a) different values; which of these it
Las depends on the particular primitive root chosen,
t Berlin, 1839.
H 2
52
REPORT ON THE THEORY OF NUMBERS.
[Art. 14.
explaining the methods which he adopted in their construction. The annexed
specimen will serve to exemplify the arrangement of the Tables :
Numeri.
Indices.
1
I.
1
2
3
4
6
6
7
8
9
10
13
14
24
8
22
17
25
18
1
6
2
20
26
28
19
16
15
5
21
2
7
12
4
11
23
27
9
3
1
N.
1
2
3
4
5
6
7
8
9
28
11
27
22
11
10
20
5
26
1
1
23
21
2
3
17
16
7
9
15
2
12
19
6
24
4
8
13
25
14
M. Burckhardt, to whom arithmetic is indebted for an excellent Table of
the divisors of numbers from 1 to 3,036,000*, has inserted in his work, and
apparently only to fill up a blank-page at the end of the first million, a table
stating the number of figures in the decimal period of the fraction -, for every
prime number p less than 2500. It is evident that the number of terms in the
decimal period of - : is nothing else than the exponent to which 10 appertains
for the modulus p. M. Burckhardt's Table, therefore, at once apprises us that
out of the 365 primes inferior to 2500 (2 and 5 are not counted in this enume-
ration, as being divisors of 10), 10 is a primitive root of 148 ; because there are
148 primes p below 2500, the reciprocals of which have decimal periods con-
sisting of p 1 figures. Again, for 108 of the remaining primes below 2500, the
exponent to which 10 appertains is ^(p 1). Of these 108 primes, 73 are of
the form 4n + 3, from which it may be inferred that -10 is a primitive root
of those 73 numbers. M. Burckhardt's Table supplies us, therefore, with a
primitive root (and that root the most convenient for the purposes of compu-
tation) of 148 + 73 = 221 out of the 365 primes inferior to 2500. Nor is this the
limit to its usefulness ; for when the exponent to which 10 appertains is as high
as ^(p-1) or $(p 1) or %(p l\ it is possible by methods which Jacob! has
indicated to construct the Table of Indices with very little labour.
Jacobi says that had it not been for this table of Burckhardt's he should
hardly have ventured on the construction of the 'Canon Arithmeticus,' on
* Paris, 1814-1817. A Table containing the exponents to which 10 appertains, for every prime
less than 10,000, has since been given by M. Desmarest. (See p. 308 of his 'Theorie dee Nombres.')
Art. 14.] REPORT ON THE THEORY OF NUMBERS. 53
account of the prolixity and uncertainty of the tentative methods for the in-
vestigation of primitive roots. But, while endeavouring to avail himself of
the results of M. Burckhardt's Table, for the computation of his own Tables
of Indices, in other cases besides those in which that Table immediately fur-
nishes a primitive root, he was led to the invention of a general method of
procedure, which, as he says, would have enabled him to dispense with the
assistance of Burckhardt's Table altogether, or to extend his Canon to any
higher limit which the expense of printing would have admitted. This method
is not in principle very different from Gauss's process for finding primitive roots,
but the form which Jacobi has given to it possesses great advantages, for the
purpose to which he has applied it. He first of all takes a number a (not quite
at hap-hazard, for quadratic residues can at any rate be excluded by the law of
reciprocity; see inf. Art. 16); and determines its period of residues, and the
exponent a to which it appertains. Let aa = p 1, and let the residues of
a, a 2 , a 3 , ... a be entered in a Table of which the arguments are the indices
1, 2, 3, ...p 1, opposite to the indices, a, 2 a', 3 a', ... aa', respectively. It has
been shown by Gauss that there are always 'j. . - primitive roots for which
this assignment is true. A number b is then taken, not contained in the period
of a, and the residues of its successive powers are formed till we come to the
lowest power of it that is congruous to any power of a ; so that b s = a A , mod p.
Let j8 be the exponent to which 6 appertains, 6 the greatest common divisor of
o
a and /3, and \ = ~ their least common multiple ; let also /3/3' = p 1. It may
u
be proved that B = ; A = ; where k is some number less than 6 and prime
c/ 6
to it, so that -r is the greatest common divisor of A and a. These relations
show, that when we know the numbers a, A, and B, we can immediately find
6, k, and ft, without having to raise b to any power higher than b s . We may
then assign to b any index of the form 1(3', where I is prime to /3, and congruous
to k for the modulus 9. The number of such values of I (incongruous for the
modulus (8) is ,;..{ ; and, whichever of them we take, there will be -- --
primitive roots, for which b will have the index 1/3', while a retains the index a'.
We must next form the residues of the A a products included in the formula
a x b"; where x has any value from 1 to a inclusive, and y any value from 1 to
B \. These residues are all incongruous ; the indices of all of them are known ;
54 REPORT ON THE THEORY OF NUMBERS. [Art. 14.
and, together with the a powers of a already entered in the table, they exhaust
p 1
all the numbers which have indices divisible by *-
In practice, it will almost always happen that X is equal to p 1. When
this is so, nothing remains to complete the operation but to enter in the Table
the residues of the numbers a x b" opposite to the indices corresponding to them.
f> 1
But, if X <p- 1, we may take that residue which has ^-r : for its index, and
use it to replace a in the preceding operation, while b is replaced by some other
residue not yet entered in the Table. In this way we shall ultimately (and in
practice very speedily) obtain a complete Table of Residues corresponding to
"given indices, which, of course, immediately supplies us with the inverse Table
of Indices corresponding to given residues. It will be seen (as has been already
observed) that the process is not dissimilar to Gauss's method for determining a
number appertaining to the exponent X when we already know two numbers
a and b appertaining to the exponents a and /3 respectively. But it is so
arranged by Jacobi that hardly a single figure is wasted, the primitive root,
instead of being found by a preliminary investigation, presenting itself at the
end of the operation, and being recognized by its standing opposite to the
index 1.
To calculate with rapidity the residues of the powers of a number, Jacobi
employs a method proposed by M. Crelle in his Journal, vol. ix. p. 30, and which
is most easily explained by an example.
Let p^ll, and let it be required to determine the residues of the powers
of 3 ; and the residues of those powers multiplied by 7.
Column I. 1,2, 3, 4, 5, 6, 7, 8, 9, 10 ;
II. 3, 6, 9, 1, 4, 7, 10, 2,5, 8 ;
III. 3, 9, 5, 4, 1 ;
IV. 10, 8, 2, 6, 7.
The first column contains the numbers 1, 2, 3, ... p 1. The second column
begins with 3 (the number the powers of which we are considering), and consists
of numbers formed by successive additions of 3, multiples of 11 being rejected as
fast as they arise. The third column also commences with 3, and is so formed
that any number r in it is followed by the number which in column II. stands
under r in column I. This column contains the residues of the powers of 3
taken in order, and stops at 3 5 because after that the same residues recur.
Art. 16.] REPORT ON THE THEORY OF NUMBERS. 55
Lastly, column IV. begins with 10 (the number which in column II. stands under
7 in column I.), and is formed in the same way as column III. It represents the
residues of 7.3, 7.3 2 , &c
15. Quadratic Residues. It appears from the theorems cited in Art. 12,
that the numbers 1, 2, 3, ...p 1 divide themselves into two classes of Qua-
dratic Residues and Quadratic non- Residues, comprising \(p 1) numbers each.
Every quadratic residue a satisfies the congruence o;z (p ~ 1) = 1, mod p ; every qua-
dratic non-residue b satisfies, instead, the congruence zz^- 1 ' = 1, mod p. Again,
for every quadratic residue the congruence x 2 = a, mod p, is resoluble ; for every
non-quadratic residue the congruence x 2 = b, mod p, is irresoluble. The solution
of almost every problem relating to the indeterminate analysis of quadratic
functions involves a congruence of the simple form x 2 = A, modp. It is there-
fore of great importance to obtain a criterion which shall enable us to determine
a priori whether a given number is or is not a quadratic residue of a given
prime. If we have a Table of Indices for the given prime, we have only to see
whether the index of the given number is even or uneven ; if even, it is a
quadratic residue ; if uneven, it is a quadratic non-residue. Or, again, we may
raise the given number a (by M. Crelle's method, or any other) to the power
i(_p 1), and see whether the residue is +1 or 1. It is usual to denote the
positive or negative unit which is the remainder of a? (p ~ l \ modp, by the symbol
- j, which is known as ' Legendre's Symbol ' ; so that in every case
a |(p-i) = ( _V modp, and (-)= +1 or = 1,
\p/ 1 \p/
according as a is or is not a quadratic residue of p. It will be seen that we also
have in every case the equation
Vp/ Vp/ \ p )
If a instead of being prime to p be divisible by p, it is convenient to attribute
to (-) the value zero.
16. Legendre's Law of Reciprocity. The two methods alluded to for the
discrimination of quadratic and non-quadratic residues, or, which is the same
thing, for the determination of the value of the symbol (-), are not satisfactory,
the first because it supposes a reference to a Table of Indices (i.e. to a
r l( ; REPORT ON THE THEORY OF NUMBERS. [Art. 16.
recorded solution of the problem it is proposed to solve), the second on account
of its inapplicability to high numbers. A very different solution of the problem
is supplied by a theorem which is known as ' Legendre's Law of Quadratic
Reciprocity,' and which is, without question, the most important general truth
in the science of integral numbers which has been discovered since the time of
Fermat. It has been called by Gauss* ' the gem of the higher arithmetic,' and is
equally remarkable whether we consider the simplicity of its enunciation, the
difficulties which for a long time attended its demonstration, or the number and
variety of the results which have been obtained by its means. The theorem is
as follows :
' If p and q be two uneven prime numbers,
-(-!)**- <-"/ 0):
to which we must add the complementary propositions relating to the resi-
dues -1 and 2,
In (ii), p is supposed to be positive ; in (i), p and q are supposed not to be
simultaneously negative.
The equation (^) = (~ I)*"-" ^
may be expressed in words by saying that ' if p and q be two primes, the
quadratic character of p in regard to q is the same as the quadratic character
of q in regard to p ; except both p and q be of the form 4 + 3, in which case
the two characters are opposite instead of identical.'
Gauss, who attributes the first enunciation of this theorem to Legendre,
while he justly claims the first demonstratidn of it for himself f, appears to have
considered that Euler was unacquainted with the theorem, at least in its simple
* {Jacobi, Crelle, vol. xix. p. 314.}
t 'Pro primo hujus elegantissimi Theorematis iuventore ill. Legendre absque dubio habendus
est, postquam longe antea summi geometrae Euler et Lagrange plures ejua casus speciales jam per
inductionem detexerant In ipsum theorema proprio marte incideram anno 1795, dum omnium,
qu in arithmetica sublimiori jam elaborata fuerant, penitus ignarus, et a subsidiis literariis omiimo
prsecluaus essem. Sed per integrum annum me torsit, operamque emxissimam effugit, etc.' Comm.
Soc. Gott. vol. xvi. p. 69.
Art. 16.] REPORT ON THE THEORY OF NUMBERS. 57
form. (See Disq. Arith., Art. 151.) Nevertheless, we find in the ' Opuscula
Analytica' of Euler, vol. i. p. 64, a memoir* the concluding paragraph of which
contains a general and very elegant theorem, from which the Law of Reciprocity
is immediately deducible, and which is, vice versd, deducible from that law.
But Euler (loc. cit.) expressly observes that the theorem is undemonstrated ;
and this would seem to be the only place in which he mentions it in connexion
with the theory of the Residues of Powers ; though in other researches he has
frequently developed results which are consequences of the theorem, and which
relate to the linear forms of the divisors of quadratic formulae. But here also
his conclusions repose on induction only ; though in one memoir he seems to
have imagined (for his language is not very precise) that he had obtained a
satisfactory demonstration. The theorem, in a form precisely equivalent to that
in which we have cited it, was first given by Legendre, in a Memoir contained in
the 'Histoire de 1'Acaddmie des Sciences' for 1785. (See pp. 516, 517.) But
the demonstration with which he has accompanied it is invalid for several
reasons. (See Gauss, Disq. Arith., Arts. 151, 296, 297, and the Additamenta.)
[Addition^. Legendre's investigation of the law of reciprocity (as presented
in the ' Thdorie des Nombres,' vol. i. p. 230, or in the ' Essai,' ed. 2, p. 198) is
invalid only because it assumes, without a satisfactory proof, that if a be a given
prime of the form 4n + l, a prime 6 of the form 4n + 3 can always be assigned,
satisfying the equation (r) = 1- M. Kummer (in the Memoirs of the Academy
of Berlin for 1859, pp. 19, 20) says that this postulate is easily deducible from
the theorem demonstrated by Dirichlet, that every arithmetical progression, the
terms of which have no common divisor, contains prime numbers. It would
follow from this, that the demonstration of Legendre (which depends on a very
elegant criterion for the resolubility or irresolubility of equations of the form
ax 2 + by 2 + cz 2 = 0) must be regarded as rigorously exact (see, however, the
' Additamenta' to arts. 151, 296, 297 of the Disq. Arith.). In the introduction
to the memoir to which we have just referred, the reader will find some valuable
observations by M. Kummer on the principal investigations relating to laws of
reciprocity.]
* Observationes circa divisionem quadratorum per numeros primes (Comment. Arith. vol. i.
p. 477).
t The additions to Arts. 16, 20, 22, 24, 25, 36, 37, and 38 were published at the end of Part II.
of the Report (1860).
58 REPORT ON THE THEORY OF NUMBERS. [Art. 17.
17. JacoU's extension of Legendre's Symbol The symbol (|), the introduc-
tion of which has greatly contributed to simplify the theories of the higher
arithmetic, does not appear in Legendre's Memoir of 1785. It first occurs in the
' Essai sur la Thdorie des Nombres ' ; the first edition of which appeared at
Paris in 1798, and the second in 1808.
Jacobi, in a note communicated to the Academy of Berlin in 1837*, has
extended the notation of Legendre. If P=pip 2 p 3 , where PI, p t , p 3 denote
(equal or unequal) uneven prime numbers, Jacobi defines the symbol \-p} by the
equation , i. ,. x t N />. .,
i\ t \ / i f
\p) " \ PI ) \pj \p s
and observes that we then have the equations
P and Q denoting any two uneven numbers relatively prime, the signs of which
are subject to the same restrictions as the signs of p and q in the corresponding
formula of Art. 16. The theorems expressed by these formulae of Jacobi are
very easily deducible from the formulae of Legendre, and will be found in the
Disq. Arith. (Art. 133). To prevent misconception, however, it is proper to
observe that, while Legendre's equation (\ = 1 is a necessary and sufficient
condition for the resolubility of the congruence x 2 = k, mod p, Jacobi's equation
-75) = 1, where P is not a prime number, though a necessary, is not a sufficient
condition for the resolubility of the corresponding congruence x 2 = k, mod P.
That congruence requires for its resolubility that the conditions
should separately be satisfied ; p ly p a , ... denoting the unequal prime factors of P.
Gauss (who had in the course of his own early researches arrived inde-
* Ueber die Kreistheilung und ihre Anwendung auf die Zahlentheorie. See the Monats-Bericht
of the Berlin Academy, vol. ii. p. 127 (Oct. 16, 1857), or Crelle's Journal, vol. xxx. p. 166.
Art. 18.] REPORT ON THE THEORY OF NUMBERS. 59
pendently at the Law of Quadratic Reciprocity), before finally abandoning the
theory, succeeded in obtaining no fewer than six demonstrations of this funda-
mental proposition. The first two are contained in the Disq. Arith. (Arts.
125-145, and Art. 262) ; the third and fourth in two memoirs presented in
1808 to the Society of Gottingen (Comm. Soc. Gott. vol. xvi. p. 69, Jan. 15, and
Comm. Recentiores, vol. i. Aug. 24), of which the latter bears the title ' Sum-
matio serierum quarundam singularium.' The fifth and sixth appeared nine years
later in the memoir entitled ' Theorematis Fundamentalis in doctrina de Residuis
quadraticis demonstrationes et ampliationes novae ' (Comm. Recent, vol. iv. p. 3,
Feb. 10, 1817). The fourth of these demonstrations is probably that which is
promised in the Disq. Arith., Art. 151, but which does not appear in that work,
because (as it would seem) Gauss had not yet succeeded in overcoming the diffi-
culties connected with it.
Independently of the fundamental importance of Legendre's Law of Reci-
procity, these demonstrations of Gauss possess such intrinsic interest, and have
contributed so much to the progress of the science, that we shall briefly review
them here.
18. Gauss's First Demonstration. -The first demonstration (Disq. Arith.,
Arts. 125-145), which is presented by Gauss in a form very repulsive to any
but the most laborious students, has been resumed by Lejeune Dirichlet in
a memoir hi Crelle's Journal (vol. xlvii. p. 139), and has been developed by
him with that luminous perspicuity by which his mathematical writings are
distinguished.
Let X represent any uneven prime. The single observation that
(s)- *-()
shows that the theorem of reciprocity is true for primes inferior to 7. To
establish its universal truth, it is, consequently, sufficient to show that, if true
for all primes up to X exclusively, it is also true for all primes up to X inclusively.
Let the theorem therefore be assumed to be true for all primes inferior to X ;
let p be any one of those primes ; and let the eight cases [2 x 2 x 2 = 8] be con-
sidered separately, which arise from every possible combination of the hypotheses
(a), (^)= +1, or = -1; (/3), X=l, or = 3, mod 4 ; ( 7 },p = l, or =3, mod 4.
It has to be shown that, in each of these eight cases, the symbol () actually
has the value which the Law of Reciprocity assigns to it. The nature of the
I 2
60 REPORT ON THE THEORY OF NUMBERS. [Art. 18.
proof in the four cases in which (-)= + 1 w ^ be rendered intelligible by a
single example.
Let (*-} = 1 and let \ =p = 1, mod 4. By virtue of the symbolic equation
() = !, we can establish the congruence x" 1 = p, mod X, or (which is the same
A '
thing) the equation x*=p + \y ; in which we may suppose x even and less than
X, y positive, less than X and of the form 4n + 3. From this equation it appears
that (--) = 1, and fj = 1, the symbol () being here used with the meaning
JT %y y
Jacobi has assigned to it. But every prime divisor of y is less than X ; and,
therefore, by Jacobi's formula of reciprocity (which is valid for all uneven num-
bers less than X, since by hypothesis Legendre's law is valid for all primes less
than X), (i) .().!. But (*) _!_() (i) ; so that, tolly. () - 1
in conformity with Legendre's law. We have here assumed that x is prime
to p ; a slight modification in the proof will adapt it to the contrary sup-
position.
Again, the two cases in which ( ) = 1, and X = 3, mod 4, admit of simi-
lar treatment. For the equation ( ) = 1 involves also the equation (-r^-) = + 1 ,
because X = 3, mod 4. We have therefore the congruence x 2 = p, mod X, which
will serve to replace the congruence x 2 =p, mod X, which presents itself in the
four cases first mentioned.
But the two remaining cases, in which ( ) = -1, X = l, mod 4, require
^ X '
a different mode of treatment. By a singularly profound analysis, Gauss has
succeeded in showing that every prime of the form in + 1 is a non-quadratic
residue of some prime less than itself. Assume, therefore, the existence of a
prime *r, less than X, and satisfying the condition ( \ 1. This condition
implies that (^) = - * ; for if (^) were equal to + 1, we should have ( ) = + 1,
by one of the first four cases. Hence we may infer that (^) = +!> and may
establish the congruence x z = vrp, mod X, which, treated as in the preceding
cases, will lead us to the conclusion that (} () = 1, i.e. that (-} = - 1.
vp/ VCT/ Vj>/
Art. 19.] REPORT ON THE THEORY OF NUMBERS. 61
19. Gauss's Second, Third, and Fifth Demonstrations. The second demon-
stration (Disq. Arith. 262) depends on the theory of quadratic forms, and will
be referred to in its proper place in this Report [see Art. 115].
The third and fifth (which are in principle very similar to one another)
depend on much simpler considerations.
A half-system of Residues for a prime modulus p is a system of ^(p 1)
numbers r lt r 2 , ...ri (p _^, such that the p 1 numbers r lt r 2 , ... rn p _- l)
constitute a system of residues prime to p. We might take for the numbers
t\, r 2 , &c., the even numbers less than p (as Eisenstein has done: see Crelle's
Journal, vol. xxviii. p. 246), but Gauss has preferred to take the numbers
1,2, 3, ...1(^-1).
Let q be any number prime to p, and let k be the number of the numbers,
<? r i <7 r 2> 9 r 3) <? r i(p-n> which are congruous, not to numbers in the series
T lt r%, ... r^ p _ l ), but to numbers in the series r lt r 2 , ... r^.^. It may be
shown (by a method similar to that employed in Dirichlet's proof of Fermat's
Theorem) that gi<- = ( - 1 )*, mod p ; so that ( ?) = ( - 1 ) fc . * Hence if q be a
prime as well as p, and k' denote the number which replaces k, when p and q are
interchanged in the preceding considerations, we find that
It has, therefore, to be shown that k + k' = %(p 1) (q 1), mod 2. The way in
which this is done is different in each of the two demonstrations, and is a little
complicated in each of them ; but by the aid of a diagram the congruence may
be demonstrated intuitively (compare Eisenstein: Crelle, xxviii. p. 246 {trans-
lated by Cayley in the Quart. Jour, of Math. vol. i. p. 186}). With a
pair of axes Ox and Oy construct a system of unit-points in a plane : only let
no such points be constructed on the axes themselves. If S be any geometrical
figure, let (S) stand for the number of unit-points contained inside it or on its
contour. On Ox and Oy respectively take OA=^q, OB = \p. Complete the
parallelogram OACB, and draw its diagonals, OQC, AQB. It is then easily
seen that
* {Mr. Morgan Jenkins in a paper read to the London Mathematical Society [vol. ii. p. 29, 1867]
shows that (-^) = (1)*, Q and P uneven, and (-^) being Jacobi's symbol.}
82 REPORT ON THE THEORY OF NUMBERS. [Art. 20.
k = (QCA) - (QBO),
k'=(QBC) - (QOA),
k + k'=(ABC) - (AOB),
= (OABC)- 2 (AOB),
= (OABC], mod 2.
But (OABC} = (p - 1) (q - 1) ; therefore, finally,
k + V = $(p-l)(q-l), mod 2.
These demonstrations (the 1st, 3rd, and 5th) introduce no heterogeneous
elements into the inquiry (the geometrical method of the present article is to
be regarded only as an abbreviation of an equivalent and purely arithmetical
process) ; they are based on the principles of the two theories with which the
Law of Reciprocity is most intimately connected, those of the residues of
powers, and of quadratic congruences. The third, in particular, appears to have
commended itself above the rest to Gauss's judgment*.
20. Gauss's Fourth Demonstration. The fourth and sixth demonstrations,
though somewhat different from one another, are both intimately connected
with the theory of the division of the circle. They must, therefore, be regarded
as less direct than the earlier proofs, but they have contributed even more to
the methods and resources of the higher arithmetic.
The fourth depends on the formula
in which i represents (as throughout this Report) an imaginary square root
of 1 ; n is any uneven number, ^/n its positive square root,
2lT 2-JT
T J , , r = cos - - + t sin ---
Let the series n n
1+r fc + r 4i + r o* + >+r (n-i)*t be denoted by ^(k,n);
in the particular case in which n is a prime number, it is easy to see that
^(k, n) = f-j ^(1, n). Further, p and q denoting two prime numbers, it is
' Sed omnes hae demonstrationes,' (he is speaking, apparently, of the 1st, 2nd, 4th, and 6th,)
' etiamsi respectu rigoris nihil desiderandum relinquere videantur, e principiis nimis heterogeneis
derivatw sunt ; prima forsan excepta, qu* tamen per ratiocinia magis laboriosa procedit, opera-
tiunibusque prolixioribus premitur. Demonstrationem itaque genuinam hactenus haud affuisse non
dubito pronunciare; esto jam penes peritos judicium, an ea, quam nuper detegere successit,'
(the 3rd,) 'hoc nomine decorari mereatur.' Comm. Soc. Qott. vol. xvi. p. 70.
Art. 20.] REPORT ON THE THEORY OF NUMBERS. 63
found by actual multiplication of the two series ^(p, q) and ^(q,p) that
If we substitute for the functions ^ their values given by the equation (A),
we find / rt\ / n \
(JL\ (JL) = fci(P4-D 2 -i(2>-i) 2 -i(9-i) 2
V g / Vj9/
an equation which gives a relation between ( ) and () coincident with that
assigned in Legendre's Law of Reciprocity.
The equation (A) is not easy to demonstrate. It is not indeed difficult
to show that the sum of the series on the left-hand side is ^/n when
n = l, mod 4; and +i x / when n = 3, mod 4. But the determination of the
ambiguous sign in these values appears to have long occupied Gauss. He has
effected it in his memoir (the ' Summatio Serierum &c.' ) by establishing the
equality
l+r + r* + r 9 +...+r (n - l) * = (r-r- l )(r 3 -r- s ) ... (r n - 2 -r~ n + 2 ) ..., (B)
which he obtains by writing r for x, and n 1 for m, in the series
l-3* (1 - x m ) (1 - af- 1 ) (1 - x m ) (1 - a:"*- 1 ) (1 - x m ~ 2 )
'
This series when m is a positive integer becomes an integral algebraical function,
and is proved by Gauss to be zero if m be uneven ; and if m be even, to be equal
to the product (1 x) (1 a; 2 ) ... (1 as" 1 " 1 ). From this last observation, the
demonstration of the formula (B) naturally flows. If n be an even number,
the formula (A) becomes
l+r + r 4 + r 9 + ...+r ( "- 1) ' = (l+t') N /n or =0, (A')
according as n is evenly or unevenly even.
A very different, but a simpler demonstration of these formulae (A) and (A'),
depending on the properties of the definite integrals
/+ 00 /*+ 00 /*+ 00
cosx 2 dx, I sinx*dx, or / e ix *dx,
QO J QO */ OO
has been given by Dirichlet in his memoir, ' Application de 1' Analyse Infinite'si-
male & la The"orie des Nombres' (Crelle, vol. xxi. p. 135).
The same formulae have also been deduced by Cauchy from the equation
or
64 REPORT ON THE THEORY OF NUMBERS. [Art. 20.
in which ab = *, {a* and 6 2 denoting real positive quantities, or imaginary quan-
tities the real parts of which are positive ; the real parts of a, b have the
same sign as a* 6* = ^/TT\ *. This equation Cauchy obtained, as early as 1817,
by the principles of his theory of reciprocal functions ; but it is also deducible
from known elliptic formulae. (See a note by M. Lebesgue in Liouville's
Journal, vol. v. p. 186. {See also Crelle, xvii. p. 57, and the Berlin Transac-
tions for 1835 ; the former Memoir contains the criticism of M. Libri's proof.})
If in it we write
a 2_f^ f or and 0* + - for b\
n *
a and j9 being two evanescent quantities connected by the relation a = 2#, the
two series
and
become respectively
^(1, n)x I e~ x * dx, and (1 + e~i n '*) x / e~ x *dx;
Jo Jo
whence, dividing by the definite integral, and observing that
a = A/ e-* iw ,
v n
we obtain finally, in accordance with the formulae of Gauss,
For the case in which n is a prime number, the equality (B) has been
* {Put a? = ittta ; we get the formula in Lacroix, vol. ii. p. 408 ; and the condition that a is
positive is the same as that the real part of Sito is positive. See Liouville (II.) vol. iii. p. 30 for a
general formula.}
t See M. Cauchy's ' Me'moire sur la Theorie des Nombres ' in the Me'moires de ' I'Acade'mie
de France, vol. xvii, notes ix, x, and xi. See also the Comptes Rendus for April 1840, or Liouville's
Journal, vol. v. p. 154 ; and compare (besides the note of II. Lebesgue quoted in the text) a
memoir by the same author in Liouville, vol. v. p. 42.
{Writing a' = a 1 - ^^, 6 = ff + 5!I, I find +(m, n) = J A/ (1 + *(-, 4m).
n 6tViii v wfc
If m 1, this is right. If n = 4f, since ty( 4i>, 4m) = Ity ( v, m), we have
^ (m,4v)= 2 /\y ~ (1 -H) ^(-".
making the case when n is even depend on the case when n is uneven, and agreeing with Art. 104,
note.}
Art. 21.] REPORT ON THE THEORY OF NUMBERS. 65
established in a very simple manner by M. Cauchy* and M. Kroneckerf. But,
as these latter methods have not been extended to the case in which n is a
composite number, they cannot be used to replace Gauss's analysis in this
demonstration of the law of reciprocity.
From the formula (A) combined with the equation ^(k, p) = (-} ^(l,p),
p denoting a prime number, we may infer "
(-)\/P = 2. coss 2 2 sins 2 = :
V = <> P 8 = P
or (-)<\/P= 2 sins 2 ; V coss 2 - = 0;
W .-o P ..o P
according as p = 1, or =3, mod 4.
These formulae serve to express the value of the symbol (-) by means of a
finite trigonometrical series, and are, therefore, of very great importance. Con-
versely, the circumstance that a trigonometrical summation should depend on
the quadratic characters of integral numbers, may serve of itself to show the use
of abstract arithmetical speculations in other parts of analysis.
[Addition. Dirichlet's demonstration of the formulae (A) and (A') first
appeared in Crelle's Journal, vol. xvii. p. 57. Some observations in this paper
on a supposed proof of the same formulae by M. Libri (Crelle, vol. ix. p. 187)
were inserted by M. Liouville in his Journal, vol. iii. p. 3, and gave rise to a
controversy (in the Comptes Rendus, vol. x) between MM. Liouville and Libri.
The concluding paragraphs of Dirichlet's paper contain the application of the
formulae (A) and (A') to the law of reciprocity (Gauss's fourth demonstration).] '
21. Gauss's Sixth Demonstration. This demonstration depends on an
investigation of certain properties of the algebraical function
8 =
in which p is a prime number, 7 a primitive root of p, k any number prime to p,
and x an absolutely indeterminate symbol. These properties are as follows :
(1) & - ( - l)i<- p is divisible by \
1
X
(2) -(-) & is divisible by 1 - x',
* In the M^moire sur la Thtorie des Nombres, Note xi, or Liouville, vol. v. p. 161.
t Liouville, New Series, vol. i. p. 392.
K
( ;i;
REPORT ON THE THEORY OF NUMBERS. [Art. 21.
(3) If k = q be a prime number,
, - , is divisible by q.
From (1) we may infer that ,'-' - ( - 1)* 1 *-" b-pW- is divisible by -
and, by combining this inference with (1) and (2), we may conclude that
x
is also divisible by [ ; that is to say,
J. * 3C
1 x p
is the remainder left in the division of the function & (^ - ,) by - - But
J. *"~ *C
every term in that function is divisible by q ; the remainder is therefore itself
divisible by q. We thus obtain the congruence
which involves the equation
Gauss has given a purely algebraical proof of the theorems (1), (2), and (3),
on which this demonstration depends. The third is a simple consequence of the
arithmetical property of the multinomial coefficient, already referred to in Art.
10 of this Report ; to establish the first two, it is sufficient to observe that
/k\
t 2 -(-l)i (p - 1) ^>and 4 -(-) 1 vanish, the first, if x be any imaginary root, the
second, if x be any root whatever, of the equation x p 1 = 0. If, for example, in
o o_
the function t we put x = r = cos --- 1- i sin - , we obtain the function -^(k,p),
which satisfies, as we have seen, the two equations [$(k,p)'] 2 = ( l)^*" 1 ^, and
/k\
'b(k,p) = (-)^'(l,p). It is, indeed, simplest to suppose x = r throughout the
whole demonstration, which is thus seen to depend wholly on the properties of
the same trigonometrical function -^, which presents itself in the fourth demon-
stration ; only it will be observed that here no necessity arises for the considera-
tion of composite values of n in the function \J/(&, TO) ; nor for the determination
of the ambiguous sign hi the fonnula (A). In this specialized form, Gauss's sixth
proof has been given by Jacobi (in the 3rd edit, of Legendre's ' Theorie des
Art. 21.] REPORT ON THE THEORY OF NUMBERS. 67
Nombres,' vol. ii. p. 391), Eisenstein (Crelle, vol. xxviii. p. 41), and Cauchy
(Bulletin de Ferussac, Sept. 1829, and more fully Me"m. de 1'Institut, vol. xviii.
p. 451, note iv. of the Me"moire), quite independently of one another, but
apparently without its being at the time perceived by any of those eminent
geometers that they were closely following Gauss's method. (See Cauchy 's
Postscript at the end of the notes to his Mdmoire ; also a memoir by M. Lebesgue
in Liouville, vol. xii. p. 457 ; and a foot-note by Jacobi, Crelle, vol. xxx. p. 172,
with Eisenstein's reply to it, Crelle, vol. xxxv. p. 273.)
MM. Lebesgue* and Eisenstein f have even exhibited a proof essentially the
same in a purely arithmetical form, from which the root of unity again disap-
pears, and is replaced by unity itself. Eisenstein considers the sum
in which k l} k 2 , ... k q denote q terms (equal or unequal) of a system of residues
prime to p, the sign of summation extending to every combination of the
numbers k lt k 2 , ... k q) that satisfies the congruential condition
This sum is, in fact, the coefficient of r in the development of the qih power of
*~p~ 1 /k\ r p 1
the function 2 (~)**f {reduced by the equation r p 1 = 0; not - ^ = 0},
which is equivalent in value to Gauss's function ^(l,p). From the equation
[k = p 1 T,
z (
it follows that
k=p 1 1. -iq
2 (*V | = (-
fc-i V J
whence
* See Liouville's Journal, vol. ii. p. 253, and vol. iii. p. 113. (The proof of the law of
reciprocity will be found in sect. i. art. 5, and sect. iii. art. 2, of the memoir). See also the
memoir referred to in the text, Liouville, vol. xii. p. 457.
t Crelle's Journal, vol. xxvii. p. 322.
J {This assumes that (7 = 0. Adding the p 1 equations
2 C a r a = ( 1)*(P-D (-D ^ite-i) 2 ( ) r a
to the equation 2 p ~ 1 C' a =0, we obtain C Q =0; then the equation follows from the irreducibility of
K 2
(58 REPORT ON THE THEORY OF NUMBERS. [Art 22.
And again, since
we have the congruence n _ / tt \ /q\ j
^ a = \p)\p)' q '
But these results, which, taken together, establish the law of reciprocity, are
obtained by Eisenstein from his arithmetical definition of C a , without any
reference to the trigonometrical function -^(l,p). If we write that function in
the form 2 i* 1 , instead of the form 2 (-)r fc , we obtain from its qih power
i-o *=i P
the coefficient C' a considered by M. Lebesgue. This coefficient, which is con-
nected with C a by the equation C' a =p q ~ 1 + C a , represents the number of solu-
tions of the congruence x^ + x. 2 2 + x s 2 + . . . + x q z = a, mod q. From this definition
M. Lebesgue deduces the equation (7 =^- 1 + (-l)* (p ~ 1)(lz ~ 1) (-)p^- 1 >, and the
congruence C" = ! + (-) (~}> m d 2> by processes which, though different from
those of Eisenstein, involve, like them, the consideration of integral num-
bers only.
22. Other proofs of the Theorem of Reciprocity have been suggested to
subsequent writers by a comparison of the different methods of Gauss. The
symbol r denoting a root of the equation - - = 0, it is very easily shown that
It is natural therefore to employ this equation to replace the equation
which presents itself in the 4th and 6th methods of Gauss. It is also found
that the product t-Kp-i)^ r~ k <J
II - is equal to ("V (D)
t-i r-r~ \p/
This is an immediate consequence of the property of a half-system of Residues
(see Art. 19 supra) on which Gauss's 3rd and 5th methods depend. From a
combination of the equations (C) and (D), the law of reciprocity is immediately
deducible. (See a note by M. Liouville, Compt. Rend. vol. xxiv., or Liouville's
Journal, vol. xii. p. 95, and especially a memoir by Eisenstein, entitled ' Appli-
Art. 23.] REPORT ON THE THEORY OF NUMBERS. 69
cation de 1'Algebre a FArithme'tique transcendante,' Crelle, vol. xxix. p. 177.
The proof by the same author in vol. xxxv. p. 257, is the same as that in the
earlier memoir, only that the properties of the circular functions, which here
replace the roots of unity, are in the later memoir deduced immediately from the
definition of the sine as the product of an infinite number of factors.)
[Addition. From a general theorem of M. Kummer's (see Arts. 43, 44 of
this Report), it appears that the congruence r 2 = ( l)^" 1 ^, mod q, is or is not
resoluble, according as qrz^-^s +1, or = 1, mod X, a result which implies
the theorem of quadratic reciprocity. This very simple demonstration (which
is, however, only a transformation of Gauss's sixth) appears first to have
occurred to M. Liouville (see a note by M. Lebesgue in the Comptes Rendus,
vol. li. pp. 12, 13).]
23. Algorithm, for the Determination of the Value of the Symbol (p
Gauss has shown in the memoir ' Demonstrationes et ampliationes novse,' already
quoted, that, if p be a prime number, the value of the symbol ( J may be
O
obtained by developing the vulgar fraction - : in a continued fraction, and con-
sidering the evenness or unevenness of a certain function of the quotients and
remainders which present themselves in the development. Jacobi has observed
(see Crelle, vol. xxx. p. 173) that a much simpler rule may be obtained, by the
use of his extension of Legendre's symbol to the case when p is not a prime.
The following is the form in which the rule has been exhibited by Eisenstein
(see Crelle, vol. xxvii. p. 319). LetjJj, J9| be two uneven numbers prime to one
another, and let us form by division the series of equations
in which * 2 > f s> e M + i denote positive or negative units, and^>], p 2 , p 3 , ... which
are all positive and uneven, form a descending series. Let a- denote the number
of the quantities 2k r _ 1 p r + e r p r + l , in which both p r and e r p r + l are of the form
4n + 3 ; then ( ^ = ( - IK The demonstration of this result flows immediately
^Pi'
from the definition of Jacobi's symbol of reciprocity.
A numerical example is added (see Disq. Arith., Art. 328) from which the
70 REPORT ON THE THEORY OF NUMBERS. [Art. 24.
reader will perceive the utility of these researches in their practical application
to congruences.
Let the proposed congruence be x* = -286, mod 4272943, where 4272943
is a prime number. _ ^oo
We have to investigate the value of the symbol (- ), in which p is
written for 4272943. Now
) * (f ) <T> " <T
because (--)=- 1, and /-) = + 1, p being of the form 8- 1. To find the
value of ( -- ) , we have
P 143 = x 4272943 + 143 f,
4272943 = 29880 x 143 + 103 f,
143 = 2x103-63,
103 = 2x63-23,
63 = 2x23 + 17,
23 = 2x17-11,
17 = 2xll-5t,
11 = 2x5 + 1.
The obelisk (f) denotes that the equation to which it is affixed is one of
those enumerated in <r. Hence
or the proposed congruence is resoluble. Its roots (as determined by Gauss) are
1493445.
24. Biquadratic Residues. Reverting to the general theory alluded to in
Art. 12, we see that, when p is a prime of the form 4w + l, the congruence
x 4 1 = 0, mod p, admits four incongruous solutions ; these are +1, 1, and the
two roots of the congruence x 2 + 1 = 0, mod p, which we shall denote by +/and
f, or by /and/ 3 , so that the four roots of a; 4 -! = are 1, / -1, and/ 3 .
Further, if k be any number prime to p, k satisfies one or other of the four con-
gruences
(i.) &*<*-= 1, mod 2?. (iii.) *<-= - 1, mod_p.
(ii.) k* ( '-=f,modp. (iv.) jfe*<*- ] > = /. mod p.
We see therefore that the p 1 residues of p divide themselves into four
classes, comprising each J(>- 1) numbers, according as they satisfy the 1st, 2nd,
3rd, or 4th of these congruences. The first class comprises those numbers a for
Art 24.] REPORT ON THE THEORY OF NUMBERS. 71
which the congruence x l = a, mod p, is resoluble ; that is, the biquadratic
residues of p ; the third comprises those numbers which are quadratic, but not
biquadratic, residues of p ; the second and fourth classes divide equally between
them the non-quadratic residues.
We owe to Gauss two memoirs* on the Theory of Biquadratic Residues,
which, while themselves replete with results of great interest, are yet more
remarkable for the impulse they have given to the study of arithmetic in a new
direction. Gauss found by induction that a law of reciprocity (similar to that
of Legendre) exists for biquadratic residues. But he also discovered that, to
demonstrate or even to express this law, we must take into consideration the
imaginary factors of which prime numbers of the form 4w + l are composed. By
thus introducing the conception of imaginary quantity into arithmetic, its
domain, as Gauss observes, is indefinitely extended ; nor is this extension an
arbitrary addition to the science, but is essential to the comprehension of many
phenomena presented by real integral numbers themselves.
Gauss's first memoir (besides the elementary theorems on the subject) con-
tains a complete investigation of the biquadratic character of the number 2 with
respect to any prime p = 4 + 1. The result arrived at is that if p be resolved
into the sum of an even and uneven square (a resolution which is always possible
in one way, and one only), so that p = a 2 + b 2 (where we may suppose a and b
taken with such signs that a = 1, mod 4 ; b = af, mod p), 2 belongs to the first,
second, third, or fourth class, according as ^b is of the form 4, 4n + l, 4n + 2,
or 4 n + 3. The considerations by which this conclusion is obtained are founded
(see Art. 22 of the memoir) on the theory of the division of the circle, and we
shall again have occasion to refer to them. In the second memoir Gauss
developes the general theory already referred to, by which the determination of
the biquadratic character of any residue of p may in every case be effected.
The equation p = a 2 + b 2 shows that p = (a + bi) (a bi), or that p, being the
product of two conjugate imaginary factors, is in a certain sense not a prime
number. Gauss was thus led to introduce as modulus instead of p one of its
imaginary factors : an innovation which necessitated the construction of an
arithmetical theory of complex imaginary numbers of the form A + Bi. The
* Theoria Residuorutn Biquadraticorum. Commentatio prima et secunda. (Gottingse, 1828
and 1832, and in the Comm. Recent. Soc. Gott., vol. vi. p. 27 and vol. vii. p. 89.) The articles
in the two memoirs are numbered continuously. The dates of presentation to the Society are
April 5, 1825, and April 15, 1831.
72
REPORT ON THE THEORY OF NUMBERS.
[Art. 25.
elementary principles of this theory are contained in the memoir in question ;
they have also been developed by Lejeune Dirichlet with great clearness and
simplicity in vol. xxiv. of Crelle's Journal (pp. 295-319, sect 1-9)*. The
following is an outline of the definitions and theorems which serve to constitute
this new part of arithmetic.
[Addition. A note of Dirichlet's, in Crelle, vol. Ivii. p. 187, contains an
elementary demonstration of Gauss's criterion for the biquadratic character of 2.
From the equation p = a* + b*, we have (a + b) 2 = 2ab, mod^>, and hence
... (A)
, or, observing
or, which is the same thing,
/a + i
(-,
But (-} = (} = 1, because p = b 2 , mod a ; and ( -} = f *-
vjx va/ V>/MI +
that 2p = (a + b)- + (a - 6) 2 ,
2
p
since f' 2 +l =0, mod p. Substituting these values in the equation (A), we find
2i<p-i) =^"2*, mod p, which is in fact Gauss's criterion.]
25. Theory of Complex Numbers. The product of a number a + bi by its
conjugate a bi is called its norm ; so that the norm of a + bi is a 2 + b- ; the
norm of a (which is its own conjugate) is a 2 . This is expressed by writing
A r (a + bi} = N(a bi) = a 2 + 6 2 ; N(a) = a 2 . If a and /3 be two complex numbers,
we have evidently N(a) x N(@) = N(a/3). There are in this theory four units,
l,i, 1, i, which have each of them a positive unit for their norm. The four
numbers a + bi, ia b, a ib, ia + b (which are obtained by multiplying any
one of them by the four units in succession, and which consequently stand to one
another in a relation similar to that of + a and a in the real theory) are said
to be associated numbers. These four associated numbers with the numbers
respectively conjugate to them form a group of eight numbers (in general
' The death of this eminent geometer in the present year (May 5, 1859) is an irreparable
loss to the science of arithmetic. His original investigations have probably contributed more to
its advancement than those of any other writer since the time of Gauss; if, at least, we estimate
results rather by their importance than by their number. He lias also applied himself (in several
of his memoirs) to give an elementary character to arithmetical theories which, as they appear
in the work of Gauss, are tedious and obscure; and he has thus done much to popularize the
theory of numbers among mathematicians a service which it is impossible to appreciate too highly.
Art. 25.] REPORT ON THE THEORY OF NUMBERS. 73
different), all of which have the same norm. These definitions are applicable
whatever be the nature of the real quantities a and b. If a and b are both
rational, the complex number is said to be rational ; if they are both integers,
a + bi is a complex integral number. One complex integer a is said to be
divisible by another /3, when a third y can be found such that a = f$y. Adopting
these definitions, we can show that Euclid's process for investigating the
greatest common divisor of two numbers is equally applicable to complex
numbers ; for it may be proved that, when we divide one complex number
by another, we may always so choose the quotient as to render the norm of the
remainder not greater than one-half of the norm of the divisor *. If, therefore,
we apply Euclid's process for finding the greatest common divisor to two
complex numbers, we shall obtain remainders with norms continually less and
less, thus at last arriving at a remainder equal to zero ; and the last divisor will
be, as in common arithmetic, the greatest common divisor of the two complex
numbers. Similarly the fundamental propositions deducible in the case of
ordinary integers from Euclid's theory are equally deducible from the correspond-
ing process in the case of complex integral numbers. Thus, ' if a complex
number be prime to each of two complex numbers, it is prime to their product. '
' If a complex number divide the product of two factors, and be prime to one of
them, it must divide the other.' 'The equation ax by=l, where a and b are
complex numbers prime to one another, is always resoluble with complex
numbers x and y, and admits an infinite number of solutions,' &c.
A prime complex number is one which admits no divisors besides itself, its
associates, and the four units.
There are three distinct classes of primes in the complex theory :
1. Real prime numbers of the form 4w + 3 (with their associates).
2. Those complex numbers whose norms are real primes of the form 4n + l.
3. The number 1 + i and its associates, the norm of which is 2.
Instead of dividing numbers into even and uneven, we must here divide
them into three classes, uneven, semi-even, and even, according as they are
(1) not divisible by (1 -I- i) ; (2) divisible by 1 + i, but not by (1 + i) 2 ; (3) divisible
by (1 +i) z = 2{, or, which is the same thing, by 2.
a ac + bd bcad. ac+bd ,
Since ,. = - = ft; if be the integral number nearest to -= TJ, and q that
e + di 2 2 2 '
. . be ad
nearest to -, ^", p + qt is the quotient required.
74 REPORT ON THE THEORY OF NUMBERS. [Art. 25.
Of four associated uneven numbers, there is always one, and only one, such
that b is even and a + b - 1 evenly even. This is distinguished from the others
as primary. Thus 7 and 5 + 2z are primary numbers. A primary number
is congruous to +1 for the modulus 2(1 + {); whence it appears that the product
of any number of primary numbers is itself a primary number. The conjugate of
a primary is also primary. In speaking of uneven numbers, unless the contrary
is expressed, we, shall suppose them to be primary. This definition of a primary
number is that adopted by Gauss (I.e. Art. 36), and after him by Eisenstein, and
we shall adhere to it in this Report. But Gauss has also suggested a second
definition (which is for some purposes slightly more convenient), and which has
been adopted by Dirichlet, who defines a primary uneven number to be one in
which b is even, and a= 1, mod 4. The object of singling out one of the four
associated numbers is merely that it serves to give definiteness to many theorems.
For example, the theorem that 'every real number may be expressed as the
product of powers of real primes in one way, and in one only,' may be now
transferred in an equally definite form to the complex theory, ' Every complex
number can be expressed in one way only in the form
where m, n, a, (3, y, &c. are real integral numbers, A, B, C ... primary com-
plex primes.'
If a + bi be a complex number, and N=N(a + bi) = a 2 + b 2 , and if h be the
greatest common divisor of a and b, it can be shown that every number is con-
gruous, for the modulus a + bi, to one, and one only, of the numbers x + iy, where
These numbers therefore (or any set of numbers congruous to them) form a
complete system of residues for the modulus a + bi. The number of the numbers
x + iy is evidently N, so that the norm of the modulus represents the number of
residues in a complete system. In particular, therefore, if the modulus a + bi be
a prime of the second kind, having p for its norm, the numbers 0, 1, 2, ...p 1
represent a complete system of residues ; and if the modulus be a prime of the
first kind, as q, the numbers included in the formula x + iy, where x and y may
have any values from to q 1 inclusive, will represent a complete system of
residues.
[Addition. Although the second definition has been adopted by Dirichlet
in his memoir in Crelle's Journal, vol. xxiv (see p. 301), yet in the memoir
Art. 27.] REPORT ON THE THEORY OF NUMBERS. 75
' Untersuchungen ilber die complexen Zahlen' (see the Berlin Memoirs for 1841),
sect. 1, he has preferred to follow Gauss.]
26. Fermat's Theorem for Complex Numbers. Dirichlet's proof of this
theorem for ordinary integers is equally applicable to complex numbers, and
leads us to the following result :
' If p be a prime in the complex theory, and k any complex number not
divisible by p, then k N P~^ = 1, mod p.'
Again, the demonstration of the theorem of Lagrange (see Art. 11) is equally
applicable here (see Gauss, Theor. Res. Biq., Art. 50), and therefore the general
theorems mentioned in Art. 12 may be extended, mutatis mutandis, to the com-
plex theory. In particular, the number of primitive roots will be "^\_N(p) 1],
or the number of numbers less than N(p) 1, and prime to it. It will follow
from this that, if the modulus be an imaginary prime p, every primitive root of
Np in the real theory will be a primitive root both of p and its conjugate.
Those Tables of Indices, therefore, in the ' Canon Arithmeticus,' which refer to
primes of the form 4n + l will continue to hold, if for the real modules we
substitute either of the imaginary factors of which they are composed. For
primes of the form 4n + 3 (considered as modules in the complex theory), it
would be requisite to construct new tables, a labour which no one as yet
appears to have undertaken.
27. Law of Quadratic Reciprocity for Complex Numbers. If p and q be any
two uneven primes (not necessarily primary, but subject to the condition that
their imaginary parts are even), and if we denote by the unit -residue of the
power _p JC-Mz- 1 ], mod q ; so that = + 1, or = 1, according as p is or is not a
quadratic residue of q : then a law of reciprocity exists, which is expressed by
the equation =
If p and q are both real primes, it is easily seen that either of them is
a quadratic residue of the other in the complex theory, or = = 1.
But, as p may or may not be a quadratic residue of q in the theory of real
integers, we see that the values of the symbols and | j are not neces-
i-i L<7-> \qs
sarily identical.
This theorem is only enunciated in Gauss's memoir (Art. 60), and, as he
speaks of it as a special case of the corresponding theorem for biquadratic
L 2
76
REPORT ON THE THEORY OF NUMBERS. [Art. 28.
residues, it is probable that his demonstration of it was of the same nature
with that which he had found of the law of biquadratic reciprocity. How-
ever, a simple proof of it, depending on Legendre's law of reciprocity, has
been given by Dirichlet in Crelle's Journal *. He shows that, if q be a prune
of the first kind, f 1 = (- ^-); and that, if a + bi be any prime of the
second kind in which b is even, [** ,. 1 = ( a a . fr2 ) ^he ^ aw ^ reciprocity
is easily deducible from these transformations. If, for example, a + bi, a + fti,
be prunes of the second species in which both b and /3 are even, we have
simultaneously
where p = N (a + b i) ; * = N (a + pi). But (- -) = (~r^g) b J Jacobi's
formula (see Art. 17 supra) ; and (- -) = (- r-r-sj- Also
\ & / Vaa + op/
per = (aa + 0/3) 2 + (aft - 6a) 2 ;
whence we infer / or \ , . , . , ,, .
( r ) = 1 or, which is the same thing,
Vaa + Op/
\ / * \ f G -\-\ r a
;-?.) = ( - -T-al ; and theretore finally,
+ fe/3/ \aa + bp/ J 'La
The complementary theorems which have to be united with this formula are
'
(see Dirichlet, Crelle, vol. xxx. p. 312) ; and they, as well as the formula of
reciprocity itself, admit of an extension similar to that which Jacobi has given
to the corresponding formulae of Legendre.
28. Reciprocity of Biquadratic Residues. We now come to the theorem
which first suggested the introduction of complex numbers.
If p be any (complex) prime, and k be any residue not divisible by p, we
denote by (} the power i e of i, which satisfies the congruence &4(*i>-i) = {. It
* Crelle, vol. ix. p. 379.
Art. 28.] EEPOET ON THE THEORY OF NUMBERS. 77
will be observed that when p is a prime of the second species, the quadripartite
classification of the real residues of p which we thus obtain is identical with
that which we obtain for Np in the real theory (see Art. 24 supra) ; for the
numbers /and /being the roots of the congruence tc 2 + l =0, mod Np, satisfy
the same congruence for either of the complex factors of Np, and are therefore
congruous to +i and i, for one of those factors, and to i and +i for the
/k\
other. Admitting this definition of the symbol ( ) , Gauss's law of biquadratic
reciprocity is expressed by the equation
a and /8 denoting two primary uneven primes, and A and B being their
norms.
The complementary theorems relating to the unit i and the semi-even prime
1+i are
(ii.) _ _ =i-J(-i) ; (iii.)
' VotrtV* v ' \
ill which a + ia' denotes a primary uneven prime. These formulae, like those
of the last article, are susceptible of the same generalization which Jacob! has
applied to Legendre's symbol ; and we may suppose in the first that a and /3
are any two primary uneven numbers, prime to one another ; and in the second
and third that a + ia is any primary uneven number.
If, in the formula (i.) which expresses the law of reciprocity, a = a + ia,
fi = l> + ib', it may be easily seen that the unit (-.l)i^-JJ-$(fl-iJ j s equal to
( i)i<-i)'i*-i). This gives us a second expression of the theorem. (See Eisen-
stein, 'Math. Abhandl.' p. 137, or Crelle, vol. xxx. p. 193.)
Further, if we observe that every primary number is either = 1, mod 4, or
else =3 + 2i, mod 4 ; and that %(A - 1).%(B- 1) and (a- 1) .%(!>- 1) are even
numbers, unless both a and /3 satisfy the latter congruence, we may enunciate
the law of biquadratic reciprocity by saying
' The biquadratic characters of two primary uneven prime numbers with
respect to one another are identical, if either of the prunes be =1, mod 4 ; but
if neither of them satisfy that congruence, the two biquadratic characters are
opposite.'
This theorem is only enunciated by Gauss, who never published his demon-
stration of it. ' Non obstante,' he observes, ' summit huius theorematis sim-
plicitate ipsius demonstratio inter mysteria arithmeticae sublimioris maxime
78 REPORT ON THE THEORY OF NUMBERS. [Art. 29.
recondite referenda est, ita ut, saltern ut nunc res est, per subtilissimas tantum
niodo investigationes enodari possit, quae limites praesentis commentationis
longe transgrederentur.' Thevr. Res. Biq. Art. 67.
Soon after the publication of the theorem, its demonstration was obtained
by Jacobi, and communicated by him to his pupils in his lectures at Konigsberg
in the winter of 1836-37 (see his note to the Berlin Academy, already cited
in Art. 17). These lectures have unfortunately never been published ; but
Jacobi's demonstration, from his criticism (see ibid.) on the first of those given
ten years later by Eisenstein, appears to have been very similar to it.
It is to Eisenstein that we are indebted for the only published proofs of
the theorem in question. That great geometer (so early lost to arithmetical
science a victim, it is said, to his devotion to his favourite pursuit) has left
us as many as five demonstrations of it ; the two earlier based on the theory
of the division of the circle ; the last three, on that of the lemniscate. We
proceed to explain the principles on which each of these two classes of proofs
depends :
29. Biquadratic Residues Researches of Eisenstein. It is possible, as we
have seen, to obtain a proof of Legendre's law of Reciprocity by considera-
k=p ~ l /k\
tions relating to the function 2 ( ) a*, p denoting a real prime, and x a
x p 1
root of the equation - = 0. This function is a particular case of the well-
30 -L
known function (introduced by Gauss and Lagrange into the theory of the
t=p 2
division of the circle) F (0, x) = 2 6' x"*', where is any root of the equation
=o
0p-i_l
71---: = 0, 7 a primitive root of the congruence x p ~ l = l, mod^>, and x a root
y.P J
of the equation - = 0. In the quadratic theory we assign to 9 the value
x ~~ l
1 ; in the theory of Biquadratic Residues we put = i, and are thus led to
=p 2
consider another particular form of the same function, viz. F(i, x) = 2 i' &',
=o
p denoting a prime of the form 4 + 1.
30. The function F (Q, x) or F(6) is characterized by the following general
properties; which have been given by Jacobi, Cauchy, and Eisenstein. (See
Jacobi, Crelle, vol. xxx. p. 166 ; Cauchy, Mdmoire sur la The"orie des Nombres
in the Mdm. de 1'Acad. de 1'Institut de France, vol. xviii ; Eisenstein, Crelle,
vol. xxvii. p. 269. (Also M. Lebesgue, Liouville, vol. xix}.)
Art. 30.] REPORT ON THE THEORY OF NUMBERS. 79
II.
F(6)F(0-)
F(Q-(m+nr) ^'
where -vj/ (0) does not involve x, and is an integral function of with integral
coefficients *. The function -^ (0) satisfies the equation
IV.
x p ~ 1 1
Lastly, let 6 be a primitive root of - - = 0, and in the function
let 7 be written for ; then if m and n be positive and less than p 1,'
IIi denoting the continued product 1.2.3 ...m.
Applying these equations to the particular form of the function F which
we have to consider here, we find
if tfo-'W, and m = n = !0>- 1).
IF ( - or =
^( 7 t<*-i))=0, mod_p.
Let ^f(i) = a + bi = p 1 ; ^f( i) = a bi = p z , so that p l p 2 =p.
The congruence
x/,[ r i<i>-i>] = 0, modja, or a + feyi^-^^O, mod_p,
involves also the congruence
a-H&-y* c - |) =0, mod jJiJ i.e. y$ (p - l) =i, mod ^ ;
so that T ) =i fc . Hence we have, putting y= k, modjo,
t-p-l
= 2
t-p-l J, 3
-{= 2
* In this equation &~ m and ^~" are supposed not to be reciprocals.
gO REPORT ON THE THEORY OF NUMBERS. [Art. 30.
From these formulae two cases of the law of Reciprocity are directly
deducible.
a. Let q be a real prime of the form 4n + 3. Raising S to the power q,
we have
Multiplying by 5, we find
, modq;
or, observing that p z = pf, mod q, and p=pip 2 ,
,*-=(- !)*<- (.-) , mod q;
^Pl *
that is to say (*)- () ......... (A.)
which is in accordance with the law of Reciprocity.
B. Again, let q be a prime of the form 4+ 1 ;
then S 9 = (~- ) S, mod q ; that is, S q ~ 1 = (^,modq,
or pita- 1)^4(9-0= (-!) , mod g ;
But, by changing i into - i,
^'=^, and W 3 =^,
so that C^) = (-i) (B.)
The symbolic equations (A.) and (B.) lead immediately to the conclusion
that if a and b be any two primary uneven numbers, one, at least, of which is
real, we have (r) = (-) ; and that if a and b be both real, the common value
^O ' i \Q/' i
of these symbols is + 1. By combining with these results the supplementary
(7 \
:-/) = i~l (a ~ l) , in which a + ia' denotes any primary uneven
Ct "f" W^t ' ^
number, and also the self-evident equations,
Art. 32.] REPORT ON THE THEORY OF NUMBERS. 81
c (a + bi) = (ac + bd) + bi(c + di),
a(c + di) = (ac + bd) + di(a + bi),
/a + bi\ /a bi\ _ 1
\c + dih \c-di/i~ '
Eisenstein* investigates a relation between the symbols (- yr) and
/c "4" di\
- =-. ) , which, when a + bi and c + di are primary, coincides with that ex-
i'
pressed by the law of reciprocity.
31. The proof in Eisenstein's second memoir f is identical in its essential
character with that in the first ; but he has given it a purely arithmetical form,
independent of the theory of the division of the circle. Instead of the sum
* =p ~ l /k\ x p l
S = 2 ( ) a* in which a; is a root of the equation - = 0, he considers
fc=i Vp/4 x-l
i=? ~ 1 //fc\
the powers of the series 2 ( J , and arrives by a process purely arithmetical
at the equations (A.) and (B.) of the preceding article. Thus the two forms in
which he has exhibited his demonstration are precisely analogous to the two
expressions which he has given to Gauss's sixth demonstration of Legendre's
law (see above, Art. 21).
32. The proofs of the Law of Biquadratic Reciprocity, which are taken from
the theory of elliptic functions, no less than those which we have just considered,
depend in great measure on a generalization of the principles introduced by
Gauss into his demonstrations of Legendre's law. Indeed, Gauss himself tells
us | that his object in multiplying demonstrations of Legendre's law, was that
he might at last discover principles equally applicable to the Biquadratic Theo-
rem. It would be interesting to know whether the proof which he ultimately
obtained of this theorem depended only on the division of the circle, or on
elliptic transcendents. Jacobi appears to have believed the latter ; for he
expresses his opinion that his own demonstration of the Biquadratic Theorem
See the memoir entitled ' Lois de Reciprocity,' in Crelle, vol. xxviii. pp. 53-67.
t ' Einfacher Beweiss und Verallgemeinerung des Fundamental-Theorems fur die biquadratischen
Reate,' in Crelle, vol. xxviii, p. 223.
J See the memoir, ' Theorematis Fundamental Demonstrationes et Ampliationes Novse,' p. 4 :
' Hoc ipsum incitamentum erat ut demonstrationibus jam cognitis circa residua quadratica alias
aliasque addere tantopere studerem, spe fultus, ut ex multis methodia diversis una vel altera ad
illustrandum argumentum affine aliquid conferre posset.'
M
82 REPORT ON THE THEORY OF NUMBERS. [Art. 33.
was widely different from that of Gauss* ; and he further conjectures that what
induced Gauss to introduce complex numbers, as modules, into the theory of
numbers, was not the study of any purely arithmetical question, but that of
/dx
/(I- *\^' ^" s
opinion of Jacobi's will not appear improbable, when we remember that in the
' Disquisitiones Arithmetic* ' (Art. 335) Gauss promises an ' amplum opus ' on
these transcendents ; and that a casual remark of his in relation to them renders
it perfectly certain (as Dirichlet has observed) J that he was at that early period
in possession of the principle of the double periodicity of elliptic functions thus
anticipating by twenty-five years the discoveries of Abel and Jacobi. Never-
theless the close analogy we have endeavoured to point out between Gauss's
sixth proof of the quadratic theorem, and the trigonometric demonstration of the
biquadratic one, may perhaps incline us to the opposite opinion. Nor is the
introduction of complex numbers, as modules, an idea unlikely to have suggested
itself, when once complex numbers were admitted ; though it is remarkable that
Jacobi, in the first printed memoir in which complex numbers appear, and to
which we shall presently refer, seems not to have thought of this extension of
his theory $.
33. Application of the Lemniscate Functions to the Biquadratic Theorem^.
Let pi be a complex prime (real or imaginary), p its norm ; and let the p 1
residues, prime to p lt be divided into four groups of %(p 1) terms, after
the following scheme :
(1) MI, tr,,
(2) - r lt - r 2 ,
(3) -ir lt -ir 2 ,
* ' Ueber die Kreistheilung,' Crelle, vol. xxx. p. 171.
t Crelle, vol. xix. p. 314, or in the ' Monntsbericht ' of the Berlin Academy for May 16, 1839.
t In his ' Gedachtnissrede fiber Karl Gustav Jacob Jacobi,' Mem. de I'Acad&nie de Berlin, 1852.
This remarkable 61oge is also inserted in Crelle's Journal, vol. lii, and in a French translation in
Liouville's Journal, vol. ii, 2nd series.
{Gauss's demonstration seems after all to have been more nearly comparable to the second and
fifth of the Quadratic Theorems. See his Works, vol. ii : but I have not yet examined the paper
carefully. }
II See Eisenstein's memoir, ' Applications de 1'Algebre a I'Arithmeiique transcendante,' in Crelle's
Journal, vol. xxx. p. 189, or in Eisenstein's ' Mathematische Abhandlungen,' p. 121.
Art. 33.]
REPORT ON THE THEORY OF NUMBERS.
83
so that of any four associated numbers one, and only one, appears in each group.
Let jj be any residue prime to p l ; h, k 2 , k s , ... the numbers of the residues
which belong to the groups (1), (2), (3), respectively; then
or
(3l\ = t t
w
(See Gauss, Theor. Res. Biq., Art. 71.)
The expression on the right-hand side of this equation may now be trans-
formed by means of the Lemniscate function <p, defined by the equations
dx
ft, X = <f>(v).
The function <j> (v) is doubly periodic, the arguments of the periods being
2 to and (1 + i) <a, or, more simply, (1 + i) ca and (1 i) a>, where
dx
so that we have (f>(v + 2k(a) = (f>(v), k denoting any complex integer whatever.
From this it appears that the relation of the Lemniscate functions to the theory
of complex numbers is the same as the relation of circular functions to the
arithmetic of real integers. The function <p (v) also satisfies the equation *
<t> (i k v) = i k < (v), whence
(i)
PI
the sign of multiplication II extending to every residue r included in the group
(0). Similarly, if q t , like j> 1; be a prime,
i w\
(2)
s denoting the general term of a quarter-system of Residues for the modulus <.
{This, which is evident from the definition of </>(), is also readily verified by applying the
transformation u> = ; we find the multiplier = i, &c.}
M 2
84 REPORT ON THE THEORY OF NUMBERS. [Art. 34.
<b (kv)
By an elementary * theorem in the calculus of Elliptic Functions, -^Vy is
for every uneven value of k a rational and fractional function of x = (f> (v). If
/ ( ?:1'U>\
p l be primary, as we shall now suppose, and if we put a r = <$> {- -), we have,
by the principles of that calculus,
(fl.*0 n(g-a)
<j,(v) ' 11(1 -a 4 * 4 )'
the sign II extending to all the different values of o ; and similarly,
(gi*) n(s 4 -/3 4 )
0(t>) "II(l-/3*o*)'
if & = < (- -Y Combining the equations (3) and (4) with (1) and (2), we find
* II(/3 4 -a 4 ) .
the sign of multiplication extending to the i(p 1)(? 1) combinations of the
values of a and /8 ; whence, evidently,
The priority of Eisenstein in this singularly beautiful investigation is in-
disputable.
34. In a later memoir ('Beitrage zur Theorie der EUiptischen Functionen,'
Crelle, xxx. p. 185, or Math. Abhandl. p. 129), Eisenstein has put this proof
into a slightly different form. He shows, by a peculiar method, that if j^ be
an imaginary and primary complex prime, every coefficient in II (a; 4 a 4 ) except
the first is divisible by p l , and that for every primary uneven value of p t
(whether prime or not) the last coefficient is p l} so that ( l)i (p ~ 1) ^> 1 = IIa 4 .
Representing therefore by p t an imaginary and primary prime, by q l any complex
prime, the equation
*CT' ,*'-
== 1A ; 7-7;
* {It is not the ordinary theorem of multiplication, for k is complex. Doubtless equations
(3) and (4) may be immediately proved by the general method of comparing the zeros and infinities
of either aide.}
Art. 35.]
EEPORT ON THE THEORY OF NUMBERS.
85
ass
umes the form () = (- l)*^-^- 1 ) *<*->, modp,
V'/
or
which establishes the law of Reciprocity for every case except that of two real
primes, when the value of the symbols ( ) = ( ) = 1 is at once apparent
V 9'l / 4 \Pl'4
from their definition and from Fermat's Theorem.
35. A third, and no less interesting application of the theory of elliptic
functions to the formula of Biquadratic Reciprocity, occurs in the memoir,
' Genaue Untersuchung der Unendlichen Doppel-Producte, aus welchen die
Elliptische Functionen als Quotienten zusammengesetzt sind' (Mathematische
Abhandl. p. 213, or Crelle's Journal, vol. xxxv. p. 249). The function
n= +00 =+oo
F(x} = n n (i
=_ ^-o
tx
which is considered in this memoir, and in which the factor 1 is to be
replaced by tx, coincides (if we disregard a constant factor) with the numerator
of (j>(v), when that function is expressed as the quotient of one infinitely con-
tinued product divided by another. This may be seen by comparing F(x) with
the expression of the general elliptic function < (a) given by Abel, viz.
x n n
1 +
(a + mto) 2
1 +
1 +
1 +
(M -I)'-"
1 +
1 +
(See Abel, (Euvres, vol. i. p. 213, equat. 178.)
If we particularize this expression, by putting = (which changes ^> (a)
into the Lemniscate-function) and then write to t x for a, we shall find that the
function of x which appears in the numerator is precisely Eisenstein's func-
tion F (x). This function (which is, consequently, a particular case of Jacobi's
function H in his ' Fundamenta Nova ') is only singly periodic ; so that
F (x) = F (x + -~P), if ft denote any real integer ; but F (x + -y*) is equal to the
product of F(x) by an exponential function, if M be an imaginary complex
86
REPORT ON THE THEORY OF NUMBERS. [Art. 36.
number. (Compare the formulae of sect. 61 of the ' Fundamenta Nova.') The
difficulty occasioned by this imperfect periodicity of F (x) Eisenstein has over-
come by the introduction of the number t, which is supposed to represent a real
even indeterminate integer. The formulae on which his proof depends, are
(i) F(x + k) = e lt F(x),
(ii) F(ix) = i
(BO
The symbol w which depends on x, but is independent of t, is different in
each of these equations : in the first, k is any complex integer ; in the third,
c is a numerical constant independent of x and p l ; p l a primary number prime
to t ; p its norm ; and r the general term of the p- 1 residues of p 1} the sign
of multiplication n extending to every value of r. These equations, the first
two of which depend on the most elementary properties of the function F(x)
or H (see ' Fundamenta Nova,' loc. cit.), while the third is of a more abstruse
character, Eisenstein has established by methods which are peculiar to himself,
and which it would take us too far from our present subject to describe. They
serve to replace the formulae
in Eisenstein's earlier demonstration ; and lead to the conclusion
w still denoting some quantity independent of t. And since in this formula
t may have any even value prime to p^ and q ls it is impossible that e" should
have any value but that of one of the fourth roots of unity, so that we have
e wf _ i . wn i c h gives the law of Reciprocity.
36. An algorithm has been given by Eisenstein * for calculating the value
of the symbol (^ r^-) by means of the development of -j - rp- , in a continued
* Crelle's Journal, vol. xxviii. p. 243. But the first invention of this algorithm, and of the
similar one which exists in the Theory of Cubic Residues, is due to Jacobi. (See the note, ' Ueber die
Kreistheilung &c.,' so often cited in this report.)
Art. 36.] REPORT ON THE THEORY OF NUMBERS. 87
fraction. This algorithm, in a slightly simplified form, is as follows : Let
a + ia'= p , b + ib'= p lt and form the series of equations
The numbers p and p l are supposed to be uneven, and prime to one another ;
_pj is primary; the quotients k , J^, Tc z , ...Tc n are all divisible by l + i, and are so
chosen that the norms of p 2 , p 3 ,... form a continually decreasing series (as is
always possible) ; lastly, the units i* are so chosen as to render p 2 , p s ,... primary.
Let p s = a e + ia t ; let ^(a a l) = g , mod 4; and in the series 6 lt Q 2 , ... O n + 1 , let
p be the number of sequences of uneven terms. Then ( ) = i 2 f + ' se ^.
Example. Let it be required to determine whether the congruence
x *= -3381, mod 11981
be possible or impossible.
Since 11981 = 109 2 + 10 2 , and is a prime number, the resolubility of this
congruence depends on that of the congruence oc*= 3381, mod ( 109 + 1(H).
OO Q~f
We have therefore to investigate the value of the symbol ( :- T/r 5 ) This
gives us the series of equations
- 3381 = (31 + 3^) ( - 109 + 1(H) + i 3 ( - 17 + 28 1),
- 17 + 28* = -2i (- 19-12i) + i(+ 7 - lOi),
- 19-12i= -2i ( 7-Wi) + i*(- 1 - 2i),
7-10i = ( 3 + 5i)(- 1- 2i) + f.
Here ^=-1, 2 =+1, 3 = 2, 4 = 1, 6 = 1; so that p = 2, 2 M = 0, and
OO O "|
/ = ^' or ^ e P r P se( ^ congruence is resoluble. Its four roots are
+ 87, +2646, as may be found by any of the indirect methods for the solution
of Quadratic congruences.
[Addition. In the algorithm given in the text, the remainders p 2 ,p 3 ...
are all uneven ; and the computation of the value of the symbol ( ) i g thus
rendered independent of the formula (iii) of Art. 28. The algorithm given by
Eisenstein is, however, preferable, although the rule to which it leads cannot be
expressed with the same conciseness, because the continued fraction equivalent
gg REPORT ON THE THEORY OF NUMBERS. [Art. 37.
to terminates more rapidly when the remainders are the least possible, and
Pi
not necessarily uneven.]
37. Cubic Residues. The Theory of Cubic Residues is less complex than
that of Biquadratic Residues, and is at the same time so similar to it, that it
will not be necessary to treat it with the same detail.
If p be a real prime of the form 3n + 1, and if 1, f, f 2 denote the roots of
the congruence 3 -l = 0, mod^j, the p-1 residues k l} k^, ... &_! of p divide
themselves into three classes according as kl (p - l) =l, or =f, or =/ 2 , modp;
the first class comprising the cubic residues, the two other classes comprising
the cubic non-residues. Now it can be proved that every prime number of
the form 3n + l may be represented by the quadratic form A 2 AB + B 2 ; i.e.
it may be regarded as the product of two conjugate complex numbers of the
forms A+Bp, A+Bp 2 , where p and p 2 are the two imaginary cube roots of
unity ; just as the theory of biquadratic residues involves the consideration of
the quadratic form A 2 + B 2 , and of complex numbers of the type A + Bi. The
real integer A 2 AB + B 2 is the norm of the complex numbers A + Bp and
A + Bp 2 , and expresses the number of terms in a complete system of residues
for either of those modules.
The theory of these complex numbers has not been treated of in detail by
any writer (see Eisenstein, Crelle, vol. xxvii. p. 290) ; but the methods of Gauss
or Dirichlet are as applicable to them as to complex numbers involving i*.
Thus it will be found that every fraction of the form -^ ~- can be developed
in a finite continued fraction, having for its quotients complex integers ; that
Euclid's process for finding the greatest common divisor is applicable in this
case also, and that the same arithmetical consequences may be deduced from
it as in the case of ordinary integers. The prime numbers to be considered
in this theory are
(1) Real primes, as 2, 5, 11, 17, &c. of the form Bn + 2.
(2) Imaginary primes of the form A + Bp, having for their norms real prunes
of the form 3 n + 1.
(3) The primes 1 - p, 1 - /> 2 , having 3 for their norm.
The units are 1, p, and + p 2 .
If A + Bp be any complex number not divisible by 1 p, it may be seen
* {There is a note by Gauss on this subject in vol. ii. of his Works.}
Art. 37.] REPORT ON THE THEORY OF NUMBERS. 89
that of the three pairs of numbers, + (A + Bp), p(A+ Bp), p 2 (A + Bp), there
is always one, and one only, which, when reduced to the form a + bp, satisfies the
congruences a= +1, & = 0, mod 3. Such a number is called a primary number.
The product of two primary numbers, taken {positively or} negatively, is itself
primary.
If a be any prime of this theory, and k any number not divisible by a,
Fermat's Theorem is here represented by the congruence k Na ~ 1 = 1, mod a.
Denoting by (- ) that power p* of p which satisfies the congruence
- 1 ^>=p', the law of cubic reciprocity is contained in the formula
a and /8 denoting any two primary complex primes.
The demonstration of this theorem follows quite naturally from the formulae
cited in Art. 30. Applying them to this particular case, we have, if p denote a
real prime of the form 3 n + 1,
(i) F(p).F(p*)=p,
(ii)
(iii)
(iv) >fr(7*<- '>)=(), modp;]
from which we may infer that y^ (9 ~ l) =p, mod ^(/o). (Compare Art. 29.) In the
equation (iii), ^(p) and ^(/o 2 ) are primary; for from the equation [f 1 (p)] 3 =p-^(p),
it appears that ^(p) = 1, mod 3. The congruence y* (l '- 1) =|0, mod ^(p), implies
that ( . , . ) = p', whence if y' = k, mod p,
v > y
k = p 1 , I-
2
k=l
where p^ = ^(p). By these formulae the several cases of the theorem of reciprocity
may be proved, as follows* :
* Eisenstein in Crelle's Journal, vol. xxvii. p. 289. But in this, as in many of his earlier
researches, Eisenstein had been anticipated more than ten years by Jacobi.
N
90 REPORT ON THE THEORY OF NUMBERS.
First, let q be a prime of the form 3n + 2. Then
-) x qk , mod q,
i, mod q,
._ / o \
or
But also
so that
or raising each side of this congruence to the power q 1,
ft *W-i)=(JL) or (^) -()
Secondly, let q be a real prime of the form 3n + 1 ; we find
T-I/ \ / Q \ 2
or f (p) q - 1 = (--) , mod q ;
[Art. 37.
and also
Hence ( } ( ) = ( ) > where q t is either of the complex factors of q ; or,
/Pi\ 2 /f>t\ /<7\ 2 /<7\
observing that ( i = ( } , and ( ) = f ) , we may write
<2V
It is clear from this, that if we denote the four symbols
by a i &n &z. <*2 respectively, and the reciprocal symbols by a/, &/, &/, f/.
we have the equations
Oz 6, = 02' 6j', 02 & 2 = a z ' 6/, 6j 6 2 = 6/ 6 2 ' = 1 .
which imply that a 2 = a/, & 2 = 6 2 ', &c., or, since a, a', 6, b', ... are cubic roots of
unity>
Art. 38.] REPORT ON THE THEORY OF NUMBERS. 91
If Pi and q l be conjugate primes, the preceding proof fails ; but it is easily seen
that in this case also
Lastly, if p and q are both of the form 3, + 2, it follows from the definition
of the symbols, and from Format's Theorem, that
The complementary theorems* relating to the unit p and the prime 1
(which are not included in the preceding investigation) are
(JL\ =
Ws
Pi
where p^ is a primary prime, and a and /3 are defined by the equality
Eisenstein has observed! that a demonstration of the law of cubic reci-
procity, precisely similar to that analysed in Art. 33 of this Report, may be
/dx
-j-rr gr and its inverse function, instead
/(I a; )
of the Lemniscate integral and Lemniscate function. He has not, however,
entered into any details on this interesting subject (which is the more to be
regretted, because there appears to be no published memoir treating specially
/dx
-7-7- -- JTT); although his latest proof of the Biquadratic law
/(L x)
(see Art. 35) has been exhibited by him in such a form as to extend equally
to Cubic Residues, and even to residues of the sixth power.
[Addition. In the definition of a primary number, for '=+!,' read
'a = 1.' But, for the purposes of the theory of cubic residues, it is simpler
to consider the two numbers + (a + bp) as both alike primary (see Arts. 52
and 57).]
38. The first enunciation of the law of Cubic Reciprocity is due to Jacobi,
and the demonstration of it which we have inserted in the preceding article
* Eisenstein, Crelle's Journal, vol. xxviii. p. 28 (the continuation of the memoir cited in the
preceding note).
t In the memoir, ' Application de 1'Algebre ' &c., already referred to.
X 2
92 REPORT ON THE THEORY OF NUMBERS. [Art. 38.
is doubtless the same with that which he gave in his Konigsberg Lectures.
In one of his earliest memoirs (' De residuis cubicis commentatio numerosa,'
Crelle, voL ii. p. 66), which was composed after the announcement, but before
the publication, of Gauss's memoirs on Biquadratic Residues, Jacobi had
already arrived at two theorems relating to Cubic Residues, which involve the
law of Reciprocity, and which he seems to have deduced from his formulae
for the division of the circle. But, as it had not occurred to Jacobi, at the
time when this memoir was written, to introduce, as modules, instead of the
prime numbers themselves, the complex factors of which they are composed,
the law of Cubic Reciprocity in its simplest form does not appear in the
memoir.
To complete the present account of the Theory of the Residues of Powers,
or of Binomial congruences, we should have in the next place to review the
recent investigations of M. Kummer on complex numbers, and on the reci-
procity of the residues of powers of which the index is a prime. But the
consideration of these investigations, as well as of the other researches be-
longing to our present subject, our limits compel us to postpone to the second
part of this Report.
[Addition. Jacobi's two theorems cannot properly be said to involve the
cubic law of reciprocity. If () = 1, it will follow from those theorems that
^Pz a
f\ = I. But if () p, or p 2 , they do not determine whether () = p,
or p*. It is remarkable that these theorems, ' form& genuina qu inventa sunt,'
may be obtained by applying the criteria for the resolubility or irresolubility of
cubic congruences (Art. 67) to the congruence r 3 3\r \M =0, modj (Art. 43),
which, by virtue of M. Rummer's theorem (Art. 44), is resoluble or irresoluble
according as q is or is not a cubic residue of X.]
VI
REPORT ON THE THEORY OF NUMBERS.
PART II.
[Report of the British Association for 1860, pp. 120-169.]
39. JLlESIDUES of the Higher Powers. Researches of Jacobi. The prin-
ciples which have sufficed for the determination of the laws of reciprocity
affecting quadratic, cubic, biquadratic, and sextic residues, are found to be
inadequate when we come to residues of the 5th, 7th, or higher powers. This
was early observed by Jacobi, when, after his investigations of the cubic and
biquadratic theorems, he turned his attention to residues of the 5th, 8th, and
12th powers*. It was evident, from a comparison of the cubic and biquadratic
theories, that in the investigation of the laws of reciprocity the ordinary prime
numbers of arithmetic must be replaced by certain factors of those prime num-
bers composed of roots of unity ; and Jacobi, in the note just referred to, has
indicated very clearly the nature of those factors in the case of the 5th, 8th, and
12th powers respectively. He ascertained that the two complex factors com-
posed of 5th roots of unity into which every prime number of the form 5n + 1 is
resoluble by virtue of Theorem IV. of art. 30 of this Report, are not prime num-
bers, i.e. are each capable of decomposition into the product of two similar com-
plex numbers; so that every (real) prime number of the form 5+l is to be
regarded as the product of four conjugate complex factors ; and these factors
are precisely the complex primes which we have to consider in the theory of
* See a note communicated by him to the Berlin Academy, on May 16, 1839, in the Monats-
berichte for that year, or in Crelle, vol. xix. p. 314, or Liouville, vol. viii. p. 268, in which, however,
he implies that he had not as yet obtained a definitive result ; nor does he seem at any subsequent
period to have succeeded in completing this investigation.
04 REPORT ON THE THEORY OF NUMBERS. [Art. 40.
fpiintic residues, in the place of the real primes they divide. To this we may
odd that primes of the forms 5n2 continue primes in the complex theory;
while those of the form 5n 1 resolve themselves into two complex prime factors.
Thus
7 = 7; Il = (2 + a)(2 + a')(2 + a 3 )(2 + a<); 13 = 13;
19 = (4 - 3(a + a*)) (4 - 3(a 2 + a 3 )) ; 29 = (5 - (a + a*)) (5 - (a 2 + a 3 )) ;
31 = (2 - a) (2 - a 2 ) (2 - a 3 ) (2 - a), &C.,
where a is an imaginary 5th root of unity. Precisely similar remarks apply
to the theories of residues of 8th and 12th powers, real primes of the forms
8n+l, 12n + l, resolving themselves into four factors composed of 8th and 12th
roots of unity respectively. By considerations similar to those previously
employed by him in the case of biquadratic and cubic residues, Jacobi succeeded
in demonstrating (though he has not enunciated) the formulae of reciprocity
affecting those powers for the particular case in which one of the two primes
compared is a real number. But it would seem that he never obtained the law
of reciprocity for the general case of any two complex primes ; and indeed, for a
reason which will afterwards appear, it was hardly possible that he should do so,
so long as he confined himself to the consideration of those complex numbers
which present themselves in the theory of the division of the circle. No less
unsuccessful were the efforts of Eisenstein to obtain the formulas relating to 8th
powers, by an extension of the elliptical properties employed by him in his later
proofs of the biquadratic theorem*. It does not appear that any subsequent
writer has occupied himself with these special theories ; while, on the other hand,
the theory of complex numbers composed with roots of unity of which the
exponent is any prime, has been the subject of an important series of investi-
gations by MM. Dirichlet and Kummer, and has led the latter eminent mathe-
matician to the discovery and demonstration of the law of reciprocity, which
holds for all powers of which the exponent is a prime number not included in a
certain exceptional class.
40. Necessity for the Introduction of Ideal Primes. The fundamental pro-
position of ordinary arithmetic, that if two numbers have each of them no
common divisor with a third number, their product has no common divisor with
that third number, is, as we have seen, applicable to complex numbers formed
See M. Kummer, ' Ueber die allgemeinen Reciprocitatsgesetze,' p. 27, in the Memoirs of the
Berlin Academy for 1859.
Art. 4\]
BEPOBT ON THE THEORY OF NUMBERS.
95
with 3rd or 4th roots of unity, because it is demonstrable that Euclid's theory of
the greatest common divisor is applicable in -each of those cases. With complex
numbers of higher orders this is no longer the case* ; and it is accordingly found
that the arithmetical consequences of Euclid's process, which are of so much
importance in the simpler cases, cease to exist in the general theory. In
particular, the elementary theorem, that a number can be decomposed into prime
factors in one way only, ceases to exist for complex numbers composed of 23rd f
or higher roots of unity if, at least (in the case of complex as of real numbers),
we understand by a prime factor, a factor which cannot itself be decomposed into
simpler factors J. It appears, therefore, that in the higher complex theories, a
number is not necessarily a prime number simply because it cannot be resolved
into complex factors. But by the introduction of a new arithmetical conception
that of ideal prime factors M. Kummer has shown that the analogy with
the arithmetic of common numbers is completely restored. Some preliminary
observations are, however, necessary to explain clearly in what this conception
consists.
41. Elementary Definitions relating to Complex Numbers. Let X be a prime
a x 1
number, and a a root of the equation - = ; then any expression of the form
F(a) = a + a 1 a + a 2 a 5! +...+a A _ 2 a x - 2 , (A)
in which a , c^, a 2 , ... a x _ 2 denote real integers, is called a complex integral
number. To this form every rational and integral function of a can always be
reduced ; and it follows, from the irreducibility of the equation
-1
a-1
= 0, that
the same complex number cannot be expressed in this reduced form in two
different ways. The norm of -^(a) is the real integer obtained by forming the
product of all the X 1 values of F (a), so that
' {See Gauss, vol. ii, ' Zur Theorie der complexen Zahlen,' and Schering's Note.}
t For complex numbers composed with 5th or 7th roots of unity, the theorem still exists ; for
23 and higher primes it certainly fails; whether it exists or not for 11, 13, 17, and 19, has not been
definitely stated by M. Kummer (see below, Art. 50).
% ' Maxime dolendum videtur' (so said M. Kummer in 1844) 'quod haec numerorum realium
virtus, nt in factores primos dissolvi possint, qui pro eodem numero semper iidem sint, non eadem
est numerorum complexorum, quae si esset, tota hsec doctrina, quee magnia adhuc difficultatibus
premitur, facile absolvi et ad finem perduci posset.' (See his Dissertation in Liouville's Journal,
vol. xii. p. 202.) In the following year he was already able to withdraw this expression of regret.
96 REPORT ON THE THEORY OF NUMBERS. [Art. 41.
The operations of addition, subtraction and multiplication present no peculiarity
in the case of these complex numbers ; by the introduction of the norm, the
division of one complex number by another is reduced to the case in which the
divisor is a real integer. Thus
/(a) /
N.F(a)
and /(a) is said to be divisible by F(a) when every coefficient in the product
/(a) F(a 2 ) F(o 3 ) ... F(a x ~ l ), developed and reduced to the form (A), is divisible by
N. F(a). When /(a) is not divisible by ^(a), it is not, in general, possible to
render the norm of the remainder less than the norm of the divisor ; and it is
owing to this circumstance that the common rule for finding the greatest common
divisor is not generally applicable to complex numbers. If, in the expression (A),
we consider the numbers a , a 1} ..., a x _ 2 as indeterminates, the norm is a certain
homogeneous function of order X 1, and of \ 1 indeterminates ; so that the
inquiry whether a given real number is or is not resoluble into the product of
X 1 conjugate complex factors, is identical with the inquiry whether it is or is
not capable of representation by a certain homogeneous form, which is, in fact,
the resultant of the two forms
and a^
The problem is considered in the former aspect by M. Kummer, in the latter by
Dirichlet. The methods of Dirichlet appear to have been of extreme generality,
and are as applicable to complex numbers, composed with the powers of a root of
any irreducible equation having integral coefficients, as to the complex numbers
which we have to consider here. Nevertheless, in the outline of this theory
which we propose to give, we prefer to follow the course taken by M. Kummer :
for Dirichlet's results have been indicated by him, for the most part, only in a
very summary manner* ; nor is it in any case difficult to assign to them their
proper place in M. Rummer's theory; while, on the other hand, it would, perhaps,
be impossible to express adequately, in any other form than that which M. Kum-
mer has adopted, the numerous and important results (including the law of
* See his notes in the Monatsberichte of the Berlin Academy for 1841, Oct. 11, p. 280; 1842,
April 14, p. 93; and 1846, March 30; also a Letter to M. Liouville, in Liouville's Journal, vol. v.
p. 72; a note in the Comptes Rendus of the Paris Academy for 1840, vol. x. p. 286; and another in
the Monatsberichte for 1847, April 15, p. 139.
Art. 41.] REPORT ON THE THEORY OF NUMBERS. 97
reciprocity itself) contained in the elaborate series of memoirs which he has
devoted to this subject*.
* The following is a list of M. Kummer's memoirs on complex numbers :
1. De numeris complexis qui radicibus unitatis et numeris realibus constant, Breslau, 1844.
This is an academical dissertation, addressed by the University of Breslau to that of Kb'nigsberg,
on the tercentenary anniversary of the latter. It has been inserted by M. Liouville in his Journal,
vol. xii. p. 185.
2. Ueber die Divisoren gewisser Formen der Zahlen, welche aus der Theorie der Kreistheilung
entstehen. Crelle, vol. xxx. p. 107.
3. Zur Theorie der complexen Zahlen, in the Monatsberichte for March 1845, or in Crelle,
vol. xxxv. p. 319.
4. Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Prim-
factoren. Crelle, vol. xxxv. p. 327. The date is Sept. 1846.
5. A note addressed to M. Liouville (April 28, 1847), in Liouville's Journal, vol. xii. p. 136.
6. Bestimmung der Anzahl nicht aquivalenter Klassen fur die aus A.ten Wurzeln der Einheit
gebildeten complexen Zahlen, und die idealen Factoren derselben. Crelle, vol. xl. p. 93.
7. Zwei besondere Untersuchungen iiber die Classen-Anzahl, und fiber die Einheiten der aus
Xten Wurzeln der Einheit gebildeten complexen Zahlen. Crelle, vol. xl. p. 117. (See also the
Monatsberichte of the Berlin Academy for 1847, Oct. 14, p. 305.)
8. Allgemeiner Beweis des Fermat'schen Satzes, dass die Gleichnng x* + y^ = z*- unlb'sbar ist,
fur alle diejenigen Potenz-Exponenten A, welche ungerade Primzahlen sind, und in den Zahlern der
ersten J(A 3) Bernouillischen Zahlen als Factoren nicht vorkommen. Crelle, vol. xl. p. 131. (See
also the Monatsberichte for 1847, April 15, p. 132.) This and the two preceding memoirs are dated
June 1849.
9. Kecherches sur les Nombres Complexes. Liouville, vol. xvi. p. 377. This memoir contains
a very full resume of the whole theory, and may be read by any one acquainted with the elements
of the theory of numbers.
10. A note in the Monatsberichte of the Berlin Academy for May 27, 1850, p. 154, which con-
tains the first enunciation of the law of reciprocity.
11. Ueber die Erganzungssatze zu den allgemeinen Reciprocitatsgesetzen. Crelle, vol. xliv.
p. 93 (Nov. 30, 1851), and vol. Ivi. p. 270 (Dec. 1858).
12. A note on the irregularity of determinants, in the Berlin Monatsberichte for 1853, March 14,
p. 194.
13. Ueber eine besondere Art aus complexen Einheiten gebildeter Ausdriicke. Crelle, vol. I.
p. 212 (Aug. 31, 1854).
14. Ueber die den Gaussischen Perioden der Kreistheilung entsprechenden Congruenzwurzeln.
Crelle, vol. liii. p. 142 (June 5, 1856).
15. Einige Satze Uber die aus den Wurzeln der Gleichung a x = 1 gebildeten complexen Zahlen
fur den Fall dass die Klassenanzahl durch A. theilbar ist, nebst Anwendung derselben auf einen weiteren
Beweis des letzten Fermat'schen Lehrsatzes. Memoirs of the Berlin Academy for 1857, p. 41. An
abstract of this memoir will be found in the Monatsberichte for 1857, May 4, p. 275.
16. Theorie der idealen Primfactoren der complexen Zahlen, welche aus den Wurzeln der Gleichung
O
98 REPORT ON THE THEORY OF NUMBERS. [Art. 42.
42. Complex Units. A complex unit is a complex number of which the
norm is unity. If X = 3, there is only a finite number [six] of units included
in the formula + a*. But for all higher values of X, the number of units is
infinite. Nevertheless it is always possible to assign a system of M 1 units
(putting, for brevity, (X 1) =M) such that all units are included in the formula
a*ttJ l ttJ 1 ...t!l7 1 ; in which u lt u a , u 3 , ... u ll _ 1 are the assigned units, and
k, H!, n,, ...W M _! are real (positive or negative) integral numbers. A system
of units, capable of thus representing all units whatsoever, is called a funda-
mental system. The existence, for every value of X, of fundamental systems of
M 1 units may 'be established by means of a general proposition due to Dirichlet
and relating to any irreducible equation having unity for its first coefficient, and
all its coefficients integral. If, in such an equation, R be the number of real,
and 2 / of imaginary roots, there always exist systems of R + I 1 fundamental
units, by means of which all other units can be expressed ; or, in other words,
the indeterminate equation ' Norm = 1 ' is always resoluble hi an infinite number
of ways, and all its solutions can be expressed by means of R + 1 1 fundamental
solutions *. The demonstration of the actual existence, in every case, of these
w" = 1 gebildet sind, wenn n eine zusammengesetzte Zahl 1st. Memoirs of the Berlin Academy for
1856, p. 1.
1 7. Ueber die allgemeinen Reciprocitatsgesetze unter den Resten und Nicht-resten der Potenzen,
deren Grad eine Primzahl ist. Memoirs of the Berlin Academy for 1859, p. 20. It was read on
Feb. 18, 1858, and May 5, 1859. An abstract will be found in the Monateberichte of the former year.
A memoir by M. Kronecker (De unitatibus complexis, Berlin, 1845; it is his inaugural dis-
sertation on taking his doctorate) connects itself naturally with the earlier memoirs of the preceding
series.
' To enunciate Dirichlet's theorem with precision, let f(x) = be the proposed equation; let
a,, a lt ... a B be its roots, and ^(flj), ^( a 2 )> ^( a n) a system of n conjugate units. If the analytical
modulus of every one of the quantities ^(aj), i/f(a 2 ), ... \^(a n ) be unity, the system of units is an
isolated or singular system. The number of singular systems (if any such exist) is always finite,
whence it is easy to infer that the units they comprise are simply roots of unity. For if \}r (a) be
a singular unit, its powers are evidently also singular units, and therefore cannot be all different from
one another; i.e. ^(a) is a root of unity. If f(x) be of an uneven order, there are no singular
units; if /(a;) be of an even order, 1 is a singular unit; and if f(x) = have any real roots,
it is the only singular unit ; whereas if all the roots of / (a;) = be imaginary, other singular units
may in special cases exist. Thus the equation - = has 2 (A 1) singular units included in
3G ~ 1
the formula a*. Admitting this definition of singular units, we may enunciate Dirichlet's theorem
as follows: A system of h units [h = 7+^-1}, ,(<,), e t (a), ... e h (a), composed with any root a,
can always be assigned such that every unit composed with the same root can be represented (and
in one way only) by the formula
ft> . , (a) . ,* (a) . e s j (a) .... / (a),
Art. 42.] REPORT ON THE THEORY OP NUMBERS. 99
systems of fundamental units (a theorem which is, as Jacobi has said*, 'un
des plus importants, mais aussi un des plus dpineux de la science des nombres')
is of essential importance in the theory of complex numbers, and has the same
relation to that theory which the solution of the Pellian equation x 2 Dy z = 1
has to the theory of quadratic forms of determinant D. It may be observed,
however, that in the case which we have to consider here, that of the equation
r- = 0, the existence of fundamental systems of n 1 units has been demon-
a 1
strated independently of Dirichlet's general theory by MM. Kronecker and
Kummer f.
IfX = 5, a + a" 1 is the only fundamental unit; so that every unit is in-
cluded in the formula + ^(a + a" 1 )".
If X = 7, the complex units are included in the formula
But for higher primes the actual calculation of a system of fundamental units
involves great labour ; and a method practically available for the purpose has
not yet been given. It is remarkable that every unit can be rendered real
(i.e. a function of the binary sums or periods o 1 + o~ 1 , &c.) by multiplying it
by a properly assumed power of a. We shall therefore suppose, in what follows,
that the units of which we speak have been thus reduced to a real form.
For all values of X greater than 5, the number of systems of fundamental
units is infinite. For if u 1} u z , ...u^^ still represent a system of fundamental
units, it is evident that the system E lt E z , ... E^^, defined by the equations
E-i = w ( 1 1 ' 1 .tt ( a 1>2) ^-i" 1 ' 1
JL 9 =U,' 'U a ' ' U.. ' , , . / A\
is also a fundamental system, if the indices (1, 1), &c. be integral numbers, and
if the determinant 2 + (1, 1) (2, 2) ... (/* !, /* !) be equal to unity. And
where n t , n t , . . . n h are positive or negative integral numbers and o> is unity, or some one of the
singular units composed with a.
The principles on which the demonstration of this theorem depends are very briefly indicated
in the notes presented by Dirichlet to the Berlin Academy in 1841, 1842, and 1846. {See also
Liouville, vol. v. p. 72. }
* Crelle's Journal, vol. xl. p. 312.
t See Kronecker, De unitatibus complexis, pars altera; and Kummer, in Liouville's Journal,
vol. xvi. p. 323.
O 2
ion
REPORT ON THE THEORY OF NUMBERS.
[Art. 43.
conversely, every system of fundamental units will be represented by the equa-
tions (A), if in them we assign to the indices (1,1), (2, 2), &c. all systems of
integral values in succession consistent with the condition
so that a single system of fundamental units represents to us all possible
systems.
We shall also have occasion to allude to independent systems of units. A
system of n 1 units, u 1} u 2 , ... M _!, is said to be independent when it is im-
possible to satisfy the equation
?/"* 1/" 5 I/" 3 1/V-l
M-, tt<~ (*o *_.
whatever integral values are assigned to the indices %, n^, n 3 , ...n lt _ l . The
equations (A) will represent all possible systems of independent units, if we
suppose that in them the indices (1, 1), (2, 2), (3, 3) ... receive all positive and
negative integral values, subject only to the condition that the determinant
A = 2 + (1, 1) (2, 2) ... (/u. 1, fj. 1) does not vanish. Every system of funda-
mental units is also independent ; but not conversely. Every unit can be
represented as a product of the powers of the units of an independent system ;
but if the system be not also fundamental, the indices of the powers are not
in general integral, but are fractions having denominators which divide A.
Lastly, if Cj (a), c 2 (a), ... c^.^a) be a system of independent units, the log-
arithmic determinant
r.Cj(a) , L.C 2 (a) , ,.C M _!(a)
in which y denotes a primitive root of X, is different from zero ; and conversely,
if the determinant be different from zero, the system of units is independent.
For all systems of fundamental units, the absolute value of the logarithmic
determinant is the same ; for any other independent system, its value is A
times that least value. The quantities denoted by the symbols L.c 1 (a),
L . c 2 (a), &c. are the arithmetical logarithms of the real units Cj (a), &c. taken
positively.
43. Gauss's Equations of the Periods. In Gauss's theory of the division
of the circle, it is shown that if X be a prime number, and if ef = X - 1, the
e periods of/ roots each, that is the quantities i , ii, 12, i e -i, defined by the
equations
Art. 43.] REPORT ON THE THEORY OP NUMBERS. 101
i; =cP +a"f e +aT 26 + + a? ~ l ",
<ul I y64~l I jy26 + l i
tj 1 = a i + a ' + a '
2e 1 . 3e 1 .Y/ 1
+ aT + ...... +a>
(7 still denoting a primitive root of X), are the roots of an irreducible equation
of order e having integral coefficients, which we shall symbolize by
(see Disq. Arith., Art. 346). This equation is of the kind called Abelian ; that
is to say, each of the e periods is a rational function of any other, in such a
manner that we may establish the equations
where it is to be observed that the coefficients of the function (j) are not in
general integral. The determination of the coefficients of the equation F (y) =
may be effected, for any given prime X, and any given divisor e of X 1, by
methods which, however tedious, present no theoretical difficulty. Every rational
and integral function of the periods can be reduced to the form
o 1o + <*l 1l + a 2 12+ ' + a e-l 1e-l-
If we combine the equation
1 + no + 1l + 12 + + n e -i =
with the e - 2 equations, by which ^, ^, ... n e ~ l are expressed in that linear form,
we may eliminate >? 2 , ti 3 , ... n e -\, and shall thus obtain an equation of order e,
satisfied by / , i.e. the equation of the periods, or F(y) = 0. This is the method
proposed by Gauss (Disq. Arith., Art. 346) ; M. Kummer, instead, forms the
system of equations
, 0*o + , l, + , 2,+...+ , e-l_ 1;
2 , 0), + ( 2 , !),, + ( 2 , 2) % +... + ( 2 , e -
and eliminates 1/1, ? 2 > f-i from them. The symbol (k, K) represents the number
of solutions of the congruence y 6 "** = 1 + y ex + k } mod X, x and y denoting any two
terms of a complete system of residues for the modulus f: % is zero for all
values of k, excepting that n = l if /be even, and n^ e = l if f be uneven*.
* Liouville's Journal, vol. xvi. p. 404.
102 REPORT ON THE THEORY OF NUMBERS. [Art. 43.
The systems of equations corresponding to the particular cases e = 3, e = 4, have
been given by Gauss, who has succeeded in expressing the values of the coeffi-
cients (k, h) in each of those cases by means of numbers depending on the
representation of X by certain simple quadratic forms ; and has employed these
expressions to demonstrate the criterion already mentioned in this Report for
the biquadratic character of the number 2 *. A third method has been given
by M. Libri f : he establishes the formula
in which N k represents the number of solutions of the congruence
x
If Si, S a , S 3 ... denote the sums of the powers of the roots of the equation
F(y) = 0, this formula may be written thus,
or, solving for Si, S 2 , ...,
From this equation, when the values of N 1} N 2 , &c., have been determined,
S l , S 2 , ... may be calculated, and thence by known methods the values of the
coefficients of the equation F(y) = 0. Lastly, M. Lebesgue has shown that, if
we denote by a- k the number of ways in which numbers divisible by X can be
formed by adding together k terms of the series 7, y l , ..., 7 X ~ 2 , subject to the
condition that no two powers of 7 be added the indices of which are congruous
for the modulus e, the function (X 1) F(y) assumes the form
^[y-ijr- 1 +r,3r-*-...+(-i)'<r.]-(y-/)'f
But the practical application of any of these methods is very laborious when
* Disq. Arith., Art. 358, and Theor. Res. Biq., Arts. 14-22.
t See the memoir ' Sur la Theorie des Nombres," in his ' M6moires de Math^matique et de
Physique,' pp. 121, 122. The notation of the memoir has been altered in the text. See also M. Le-
besgue, in Liouville's Journal, vol. ii. p. 287, and vol. iii. p. 113.
% In this congruence a;,, x t ,...x t are k terms (the same or different) of a complete system of
residues for the modulus A ; and in counting the number of solutions, two solutions are to be con-
sidered as different in which the same places are not occupied by the same numbers. A simpler
formula for 5 t+1 may be obtained by considering a;,, x t , . . . x k to represent terms of a system of residue
prime to \, and denoting by e* y k the number of solutions of M. Libri's congruence on this hypothesis.
We thus find S M = Xy-/* (Liouville, vol. iii. p. 116).
Liouville, vol. iii. p. 119.
Art. 44] EEPOBT ON THE THEORY OF NUMBERS. 103
X is a large number, chiefly on account of the determinations which they all
require of the numbers of solutions of which certain congruences are susceptible.
For e = 2 the equation is
or, putting r = 2?/ + l, r 2 -(- l^X = 0. The cubic and biquadratic equations
corresponding to the cases e = 3 and e = 4 are also known from Gauss's investiga-
tions. The results assume the simplest forms if we put r = ey + l. We then
have
(1) e = 3, 4X = Jlf 2 + 27JV 2 , Jtf=l,mod3; r 3 -3Xr-X M = 0.
(2) e = 4; \ = A
Though these determinations are not required in M. Kummer's theory, we have
nevertheless given them here, in order to facilitate arithmetical verifications of
his results. The forms of the period-equations for the cases e = 8 and e 12 can
(it may be added) be elicited from the results given by Jacobi in his note on the
division of the circle (Crelle, vol. xxx. pp. 167, 168).
44. The Period- Equations considered as Congruences. An arithmetical
property of the equation F(y~) = 0, which renders it of fundamental importance
in the theory of complex numbers, is expressed in the following theorem :
' If q be a prime number satisfying the congruence qf = 1 , mod X, the
congruence F(y) = Q, mod q, is completely resoluble, i.e. it is possible to establish
an indeterminate congruence of the form
F (y) = d/- O (y-ui).:(y- ._i), mod 2 ,
u , u lt ... u e _! denoting integral numbers, congruous or incongruous, mod jf.'
* M. Lebesgue, Comptes Rendus, vol. li. p. 9. Gauss has not exhibited this last equation in its
explicit form. See Theor. Ees. Biq. /. c.
t This theorem was first given by Schoenemann (Crelle, vol. xix. p. 306) ; his demonstration,
however, supposes that q ^ e, a limitation to which the theorem itself is not subject. The following
proof is, with a slight modification, that given by M. Kumrner (Crelle, vol. xxx. p. 107, or Liouville,
vol. xvi. p. 408). {See also Liouville (II), vol. v. p. 369.} From the indeterminate congruence of
Lagrange (see Art. 1 of this Report),
x(x \}(x 2) (x q+1) = x"x, mod q,
it follows that
(y-it) (y-->h- l ) (2/-i*- 2 ) (y-'?t-2+ 1) = (y-^) 9 -(y-^)
= y g - r tk g (y'nk) = y"y< mod ?>
observing that T// = *) 4+ i n d 9 > an d that, if Ind q be divisible by e (or, which is the same thing, if q
104 REPORT ON THE THEORY OF NUMBERS. [Art. 44.
X* 1
A particular case of this theorem, relating to the equation - = (which
may of course be regarded as the equation of the X 1 periods, consisting each of
a single root), is due to Euler, and is included in his theory of the Residues of
Powers ; for it follows from that theory (see Art. 12 of this Report), that the
o^ l
binomial congruence xt - 1 = (and therefore also the congruence -- = 0, mod q)
X ^ J-
is completely resoluble for every prime of the form m\ + 1.
A remarkable relation subsists between the periods /, ^ ... ri e _^ of the
equation F(y) = Q, and the roots u , u l} u z , ...,_! of the congruence F(y) = 0,
mod q. This relation is expressed in the following theorem :
' Every equation which subsists between any two functions of the periods,
will subsist as a congruence for the modulus q when we substitute for the
periods the roots of the congruence F(y) = taken in a certain order.'
It is immaterial which root of the congruence we take to correspond to any
given root of the equation. But when this correspondence has once been esta-
blished in a single case, we must attend to the sequence which exists among the
roots of the congruence corresponding to the sequence of the periods. When
MO, !, ..., u e _ l are all incongruous, their order of sequence is determined by the
congruences
tti=0(o), M 2 =0(i), ...... , W =0K-i)> mod ?>
which correspond to the equations
12 = <>ll, ...... ,1o = <>le-l,
and which are always significant, although the coefficients of (f> are fractional,
because it may be proved that their denominators are prime to the modulus q.
When u , u l} ...,u e _ l are not all incongruous [an exceptional case which implies
that q divides the discriminant of F(y}], a precisely similar relation subsists,
though it cannot be fixed in the same manner, and though the number of
incongruous solutions of the congruence is not equal to the number of the
satisfy the congruence g^= 1, mod X), J? t+ i n d = */* Multiplying together the e congruences obtained
by giving to k the e values of which it is susceptible in the formula
(y-n) (y-i- V (y-i*-2) . . . (y-n-+ 1) = y"-y, mod q ,
we find f(y) F(y-l) F(y-2) ... F(y-q+ 1) = (y-y), mod q ;
whence, by a principle to which we shall have occasion to refer subsequently (see Art. 69), it appears
that F(y) is congruous for the modulus q to a product of the form
Art. 45.]
REPORT ON THE THEORY OP NUMBERS.
105
periods. (See a paper by M. Kummer in Crelle's Journal, vol. liii. p. 142, in
which he has established this fundamental proposition on a satisfactory basis*.)
45. Conditions for the Divisibility of the Norm of a Complex Number by
a Reed Prime f. Instead of the complex number
/(a) = a + Oi a + a 2 a 2 + . . . + a A _ 2 a x - 2 ,
let us now, for a moment, consider the complex number
which, with its conjugates
... +C e _ l r, e _ 2 ,
is a function of the periods only, and is therefore a specialized form of the
general complex number f(d) and let q still denote a real prime, satisfying
the congruence q f =l, mod A. By means of the relation subsisting between the
equation-roots >? , ti l} ..., rj e _ 1 , and the congruence-roots u , u 1} ..., u e _ lt M. Kum-
mer has demonstrated the two following theorems :
(i.) ' The necessary and sufficient condition that \J<- (?) should be divisible
by q (i.e. that the coefficients c , c 1} ..., c e _ 1 should be all separately divisible
by q) is that the e congruences
e _ l u e _ 1 = 0, mod 2,
e _ 1 tt =0, mod 2,
>;), taken
should be
_0 = c w e _! + Cj u a + c 2 u-i + . . . + c e _j u e _ 2 = 0, mod q,
should be simultaneously satisfied.'
(ii.) ' The necessary and sufficient condition that the norm of
with respect to the periods, i.e. the number ^(lo)-^ (ii)
divisible by q, is that one of the e congruences
^ (o) =0, ^ (u^ =0, ... ^ (_!) =0, mod g-,
should be satisfied.'
" {If q does not divide the discriminant, it is true, conversely, that if F (y) = for any value
of y, mod q, q/=l, mod A. For we readily find (;ind~ r ?) e = 0. m <>d q: that is g 1 divides the
discriminant, which is contrary to the hypothesis.
M. Kummer shows that if q divides F(y), and yf is not = 1, modf), then q is a residue of
a power having with e some common divisor other than unity; therefore if e is a prime, q is a
residue of an e-th power.}
t The outline of the theory of complex numbers contained in this and the subsequent Articles is
chiefly derived from M. Rummer's me'moire in Liouville, vol. xvi. p. 411.
106 REPORT ON THE THEORY OF NUMBERS. [Art. 45.
These results may be extended to any complex number /(a), by first
reducing it to the form
This is always possible ; for, since the / roots which compose any one period,
e.g. 7 , are the roots of an equation x ( a ) = f order /, the coefficients of which
are complex integers involving the periods only*, we may simply divide /(a)
by x( a ) and the remainder will give us the expression of f(a) hi the required
form. Further, let q now denote a prime appertaining to the exponent f (not
merely satisfying the congruence q f = l, mod A, but also satisfying no congruence
of lower index and of the same form). The two preceding theorems are then
replaced by the two following, which are analogous to them, and include them.
(i.) ' The necessary and sufficient condition that /() should be divisible
by q, is that the congruences
x^ (u k ) = 0, ^! (%) = 0, . . . , xJ 7 _ 1 (u k ) = 0, mod q,
should be simultaneously satisfied for every value of k.'
(ii.) ' And the condition that the norm of f (a) should be divisible by q, is
that the same congruences should be satisfied for some one value of k.'
When the congruences \|/- (u k ) = 0, ^ (u k ) =0, ... >!//_! (u k ) = 0, mod q, are
simultaneously satisfied, f (a) is said to be congruous to zero (mod q), for the
substitution q = u k . These f congruences may be replaced by a single congru-
ence in either of two different ways. Thus, if we denote by F (i? ) the complex
number involving the periods only, which we obtain by multiplying together
the f complex numbers
/w. /(-n /v 2 v-.yv (/ - 1)e ),
it may be proved that the single congruence F (%) = 0, mod q, is precisely
equivalent to the f congruences
^ (u k ) = 0, ^ (u k ) = 0, . . . , ^^ ( 4 ) = 0.
Or, again, if we denote by ^ (i/ ) a complex number congruous to zero for every
one of the substitutions no = u i> ^o = u 2> r io = u e _ l , but not congruous to zero
for the substitution >i = u (such complex numbers, involving the periods only,
can hi every case be assigned) t, it is readily seen that the same / congruences
are comprehended in the single formula
* Disq. Arith., Art. 348.
t Crelle, vol. liii. p. 145. The number * (rf) of this memoir possesses the property in question.
Art. 46.] REPORT ON THE THEORY OF NUMBERS. 107
The utility of this latter mode of expressing the f congruences will appear in
the sequel ; the formula F (u k ) = 0, mod q, is of importance, because it supplies
an immediate demonstration of the important proposition, that if a product
of two factors be congruous to zero for the substitution % = u k> one or other of
the factors must be congruous to zero for that substitution.
46. Definition of Ideal Prime Factors. To develope the consequences of
the preceding theorems, let us consider a prime number q appertaining to the
exponent f\ and let us first suppose that it is capable of being expressed as the
norm (taken with respect to the periods) of a complex number \^ (>;), which
contains the periods of /terms only ; so that
2 = 4" (%) ^ (ll) -"^(le-l)-
If the substitution of u in \J<- render ^ ( ) = 0, mod q, we may distinguish the
e factors of q by means of the substitutions which respectively render them
congruous to zero ; so that, for example, ^ (i e -k) is the factor appertaining to
the substitution / = u k .
We thus obtain the theorem that if /(a) be congruous to zero, mod q, for
any substitution ri = u , f(a) is divisible by the factor of q appertaining to that
substitution. For if \f/- (? ) be that factor of q,
2
but f(a) xff (^j) -^ (i7 2 ) ... ^f (i e -i) is congruous to zero, mod q, for every one of the
substitutions i = u , i u iy"- t ia u e-iJ it is consequently divisible by q; i.e.
f(a) is divisible by -^ (>? ). A useful particular case of this theorem is that
u k -tj k = 0, mod ^ (; ), if ^ (u ) = 0, mod q.
Again, it may be shown that these complex factors of q are primes in the
most proper sense of the word : i. e., first, that they are incapable of resolution
into any two complex factors, unless one of those factors be a complex unit ;
and secondly, that if any one of them divide the product of two factors, it
necessarily divides one or other of the two factors separately. That -^ (>? )
possesses the first property is evident, because its norm is a real prime, and
that it possesses the second is a consequence of the last theorem of Art. 45.
For if ^ (rj ) divide / (a) xf 2 (a), either f (a) or /% (a), by virtue of that theorem,
is congruous to zero (mod q) for the substitution r; = u ; that is to say, either
fi (a) or / 2 (a) is divisible by ^ (*).
Now, if every prime q which appertains to the exponent f were actually
capable of resolution into e complex factors composed of the e periods of f roots,
these factors would represent to us all the true primes to be considered in the
P 2
108 REPORT ON THE THEORY OF NUMBERS. [Art. 47.
theory of the residues of X-th powers. And for values of X inferior to 11, perhaps
to 23, this is, in fact, the case. But for higher values of X, the real primes
appertaining to the exponent f divide themselves into two different groups,
according as they are or are not susceptible of resolution into e conjugate factors.
Let, then, q represent any prime appertaining to the exponent/ whether sus-
ceptible or not of this resolution, and let f(a) still denote a complex number
which is rendered congruous to zero by the substitution *i = u ; f(a) is said
by M. Kummer to contain the ideal factor of q appertaining to the substitution
>] ti = u . This definition is admissible, because it is verified, as we have just seen,
when q is actually resoluble into e conjugate factors ; and its introduction is
justified, as M. Kummer observes, by its utility. To obtain a definition of the
multiplicity of an ideal factor, we may employ a complex number ^ (17) possessing
the property indicated in the last article. If of the two congruences
] n /() = 0, mod 2",
] n + 1 /() = 0, mod 2 + ',
the former be satisfied, and the latter not, f(a) is said to contain n times pre-
cisely the ideal factor of q which appertains to the substitution ^ = u .
47. Elementary Theorems relating to Ideal Factors. The following pro-
positions are partly restatements (in conformity with the definitions now intro-
duced) of results to which we have already referred, and partly simple corollaries
from them. They will serve to show that the elementary properties of ordinary
integers may now be transferred to complex numbers.
(1.) A complex number is divisible by q when it contains all the ideal
factors of q. If it contain all of those factors n times but not all of them n + 1
times, it is divisible by q n but not by q n + J .
(2.) The norm of a complex number is divisible by q when the complex
number contains one of the ideal factors of q. If (counting multiple factors)
it contain, in all, Ic of the ideal factors of q, the norm is divisible by (ff, but by
no higher power of q (/denoting the exponent to which q appertains).
(3.) A product of two or more factors contains the same ideal divisors as its
factors taken together.
(4.) The necessary and sufficient condition that one complex number should
be divisible by another is, that the dividend should contain all the ideal factors
of the divisor at least as often as the divisor.
(5.) Two complex numbers which contain the same ideal factors are iden-
tical, or else differ only by a unit factor.
(6.) Every complex number contains a finite number of ideal prime factors.
Art. 47.] EEPOET ON THE THEOEY OF NUMBERS. 109
These ideal prime factors (as well as the multiplicity of each of them) are per-
fectly determinate.
The prime number X is the only real prime excluded from the preceding
considerations. Since \ = (1 a) (1 a 2 ) ... (1 a*- 1 ), it appears that the norm
of 1 a is a real prime, and therefore I a cannot be resolved into the product
of two factors, unless one of them be a unit. Again, because the necessary and
sufficient condition for the divisibility of a complex number by 1 a is that the
sum of the coefficients of the complex number should be congruous to zero for
the modulus X, and because the sum of the coefficients of a product of complex
numbers is congruous, for the modulus X, to the product of the sums of the
coefficients of the factors, it appears that if the norm of a complex number is
divisible by X, the complex number is itself divisible by 1 a ; and also that,
if the product of two complex numbers be divisible by 1 a, one or other of
the factors separately must be divisible by 1 a. Hence 1 a is a true complex
prime, and is the only prime factor of X ; in fact,
if e (a) denote the complex unit
1-a 2 1-a 3 1-a*- 1
1 a 1 a 1 a
The theorems which have preceded enable us to give a definition of the
norm of an ideal complex number. If the ideal number contain the factor
I am times, and if it besides contain k, k', k", . . . prime factors of the primes
q, q', q",... appertaining to the exponents f, f, f",... respectively, we are to
understand by its norm the positive integral number
a definition which, by virtue of the second proposition of this article, is exact
in the case of an actually existing number.
It will be observed that the number of actual or ideal prime factors (com-
pound of X-th roots of unity) into which a given real prime can be decomposed,
depends exclusively on the exponent to which the prime appertains for the
modulus X. If the exponent is f, the number of ideal factors is ^ = e. Thus,
if q be a primitive root of X, q continues a prime in the complex theory ; if it
be a primitive root of the congruence x$ (K ~ l) = 1, mod X, it is only resoluble into
two conjugate prime factors. This dependence of the number of ideal prime
HO REPORT ON THE THEORY OF NUMBERS. [Art. 48.
factors of a given prime upon the exponent to which it appertains is a remark-
able instance of an intimate and simple connexion between two properties of
the same prime number, which appear at first sight to have no immediate con-
nexion with one another.
It may be convenient to remark that the word Ideal is sometimes used so
as to include, and sometimes so as to exclude, actually existent complex num-
bers ; but it is not apprehended that any confusion can arise from this ambiguity,
which it is not worth while to remove at the expense of introducing a new
technical term.
48. Classification of Ideal Numbers. An ideal number (using the term in
its restricted sense) is incapable of being exhibited hi an isolated form as a
complex integer ; as far as has yet appeared, it has no quantitative existence ;
and the assertion that a given complex number contains an ideal factor is only
a convenient mode of expressing a certain set of congruential conditions which
are satisfied by the coefficients of the complex number. Nevertheless we may,
without fear of error, represent ideal numbers by the same symbols, /(), ^(a),
<f) (a), ..., which we have employed to denote actually existing complex numbers,
if we are only careful to remember that these symbols, when the numbers which
they represent are ideal, admit of combination by multiplication or division,
but not by addition or subtraction. Thus f(a) x /J (a), f(a)-^-fi (a), [/(a)]" 1 , are
significant symbols, and their interpretation is contained in what has preceded ;
but we have no general interpretation of a combination such as f(<*)+fi (),
or f(a) f^ (a) *. This symbolic representation of ideal numbers is very con-
venient, and tends to abbreviate many demonstrations.
Every ideal number is a divisor of an actual number, and, indeed, of an
infinite number of actual numbers. Also, if the ideal number < (a) be a divisor
of the actual number -^(a), the quotient (p 1 (a) = F(a)-^-(f) (a) is always ideal;
for if 0! (a) were an actual number, <p (a), which is the quotient of F (a) divided
by 0! (a), ought also to be an actual number. It appears, therefore, that there
exists an infinite number of different ideal multipliers, which all render actual
the same ideal number. It has, however, been shown by M. Kummer that a
finite number of ideal multipliers are sufficient to render actual all ideal numbers
whatever; so that it is possible (and that in an infinite number of different
* These symbols are, however, interpretable when /(a) and /, (a) belong to the same class.
Thus, if (j>(a) X/(a) and $(a) X /, (a) be both actual, f(a)+f l (a) is the ideal quotient obtained
by dividing j> (a) x /(a) + <f> (a) x /, (a) by <j> (a).
Art. 49.] REPOET ON THE THEORY OF NUMBERS. Ill
ways) to assign a system of ideal multipliers, such that every ideal number
is rendered actual by one of them, and one only. Ideal numbers are thus dis-
tributed into a certain finite number of classes, a class comprehending those
numbers which are rendered actual by the same multiplier ; and this distribution
into classes is independent of the particular system of multipliers by which it
is effected, inasmuch as it is found that if two ideal numbers be rendered actual
by the same multiplier, every other multiplier which renders one of them actual
will also render the other actual. Ideal numbers which belong to the same class
are said to be equivalent ; so that two ideal numbers, which are each of them
equivalent to a third, are equivalent to one another. We may regard actual
numbers (which need no ideal multiplier) as forming the first or principal class
in the distribution, and, consequently, as all equivalent to one another. If
f(a) be equivalent to f^ (a), and (f> (a) to fa (a), f(a) x < (a) is equivalent to
/! (a) x fa (a), a result which is expressed by saying that ' equivalent ideal
numbers multiplied by equivalent numbers give equivalent products ; ' and the
class of the product is said to be the class compounded of the classes of the
factors.
49. Representation of Ideal Numbers as the roots of Actual Numbers. An
important conclusion is deducible from the theorem that the number of classes of
ideal numbers is finite. Lety(a) be any ideal number ; and let us consider the
series of ideal numbers /"(a), f(a) 2 ,f(<*) 3 , These numbers cannot all belong to
different classes ; we can therefore find two different powers of /(a), for example
[/(")] m an d [/( a )] m ~ l " n > which are equivalent to one another. But the equivalence
of these numbers implies that [/(a)] is equivalent to the actual number + 1 ;
i.e. that [/()]" is itself an actual number. We may therefore enunciate the
theorem, ' Every ideal number, raised to a certain power, becomes an actual
number.'
The index of this power is the same for all ideal numbers of the same class,
but may be different for different classes. By reasoning precisely similar to that
employed by Euler in his 2nd proof of Fermat's Theorem *, it may be proved
that the index of the first term in the series /(a), [/(a)] 2 , [/(a)] 3 ..., which is an
actual number, is either equal to the whole number of classes, or to a sub-
multiple of that number. This least index is said to be the exponent to which the
class of ideal numbers containing f (a) appertains. It would seem that for certain
values of the prime X, there exist classes of ideal numbers appertaining to the
* See Art. 10 of this Report.
112 REPORT ON THE THEORY OF NUMBERS. [Art. 50.
exponent H, if // denote the number of classes of ideal numbers *. Such classes
(when they exist) possess a property similar to that of the primitive roots of
prime numbers; i.e., by compounding such a class continually with itself we
obtain all possible classes, just as by continually multiplying a primitive root by
itself we obtain all residues prime to the prime of which it is a primitive root.
It has, however, been ascertained by M. Kummer that these primitive classes do
not in all cases, or even in general, exist.
The theorem of this article enables us to express ideal numbers as roots of
actually existing complex numbers. Thus, if q be a prime appertaining to the
exponent/ for the modulus X, and resoluble into the product of e conjugate ideal
factors <(i7 ), ^(^i), <P(i2)> ) <t>(ie-i)> these ideal numbers, which will not in general
belong to the same class, will nevertheless appertain to the same exponent h ; so
that [<H'7o)]*> \jp(ii)] h > ' w iH a H be actual numbers. The power j* is therefore
resoluble into the product of e actually existing complex factors. If we effect
this resolution, and represent the factors of 5* by $ (?<,)> (ii)> the ideal numbers
(/>(i7 ), (f>(ii), ... may be represented by the formulae
50. The Number of Classes of Ideal Numbers. The number of classes of
ideal numbers was first determined by Dirichlet. He effected this determination
by methods which he had previously introduced into the higher arithmetic, and
which had already led him to a demonstration of the celebrated theorem, that
every arithmetical progression, the terms of which are prime to their common
difference, contains an infinite number of prime numbers ; and to the determina-
tion of the number of non-equivalent classes of quadratic forms of a given
determinant f. Dirichlet's investigation of the problem which we are here
* See on this subject M. Rummer's note ' on the Irregularity of Determinants ' in the Monats-
berichte of the Berlin Academy for 1853, p. 194. M. Kummer's investigation, however, is restricted
to classes containing ideal numbers /(a) such that /(a) x /(a" 1 ) is an actual number.
t See his memoirs on Arithmetical Progressions, in the Transactions of the Berlin Academy
for the years 1837 (p. 45) and 1841 (p. 141), or in Liouville, vol. iv. p. 393, ix. p. 255. The first
of these papers relates to progressions of real integers, the second to progressions of complex numbers
of the form a + bi. In the memoir 'Recherches sur diverses applications de 1'analyse infmit&imalo
4 la Theorie des Nombres' (Crelle, vol. xix. p. 324, xxi. pp. 1 & 134), Dirichlet has applied his
method to quadratic forms having real and integral coefficients ; and in a subsequent memoir (Crelle,
vol. xxiv. p. 291) he has extended this application to quadratic forms, of which the coefficients are
complex numbers containing f. See also Crelle, vol. xviii. p. 259, xxi. p. 98 (or the Monatsberichte
for 1840, p. 49), xxii. p. 375 (Monatsberichte for 1841, p. 190). "We shall have occasion, in a later
part of this Report, to give an abstract of the contents of this invaluable series of memoirs.
Art. 50.]
REPORT ON THE THEORY OF NUMBERS.
113
considering has never been published ; but that since given by M. Kummer is
probably in all essential respects the same, as it reposes on an extension of the
principles developed in Dirichlet's earlier memoirs. Our limits compel us to omit
the details of M. Kummer's analysis ; the final result, however, is that, if H
denote the number of non-equivalent classes of ideal numbers, H =
In this formula P is a quantity defined by the equations
P D
(2X)"- 1 X A '
/3 representing a primitive root of the equation ft*' 1 = 1, 7 a primitive root of the
congruence 7 X ~ 1 = 1, mod X, and y 1} y 2 , y 3 , ... the least positive residues of
7> 7 2 , 7 3 , ... for the modulus X ; A is the logarithmic determinant (see Art. 42 of
this Report) of any system of / 1 fundamental units, and D the logarithmic
determinant of a particular system of independent but not fundamental units,
e(a), e(aT), e(a"* 2 ), ...,e(a'^~ 2 \ defined by the equation
SO that
D =
(1- )(l--i)
L.e(a) , L.e(a)
L.e(at) , L.
a
. 7T
sin
\
Each of the two factors
(2X)
and -.r, of which the value of H is com-
D .
posed, is separately an integral number. That -r is integral is a consequence of
the relation which exists between the logarithmic determinant of a system of
fundamental units, and that of any system of independent units ; that P is
divisible by (2X)''~ 1 may be rendered evident from the nature of the expression
P itself*. The factor -r-, taken by itself, represents the number of classes that
contain ideal numbers composed with the periods of two terms a + a -1 , a 2 + a~ 2 , ...
* See the investigation in the next article.
Q
114 REPORT ON THE THEORY OF NUMBERS. [Art. 51.
only; or, which is the same thing, it represents the number of classes each of
which contains the reciprocal /(a" 1 ) of every ideal number /(a) comprehended in
p
it ; , r - , on the other hand, is the number of classes of those ideal numbers
\ )
which become actual by multiplication with their own reciprocals *. The actual
calculation of the factor -r is extremely laborious, as it requires the preliminary
investigation of a system of fundamental units. For the cases X = 5, X = 7, the
trigonometrical units e(a), e(a"t), e(ai' 2 ) ... are themselves a fundamental system,
so that in these two cases D = A, and = + 1. The computation of the first
p
factor -7^^ presents somewhat less difficulty; and M. Kummer (though not
(2Xf
without great labour) has assigned its value for all primes inferior to 100. For
the primes 3, 5, 7, 11, 13, 17, 19, that value is unity; for 23 it is 3, and then
increases with extraordinary rapidity; so that for 97 it already amounts to
411322823001 =3457 x 118982593. The asymptotic law of this increase is ex-
pressed by the formula
p
when X increases without limit f. It will be seen that the number of classes of
ideal numbers for X = 3, X = 5, X = 7, is unity; i.e., for those values of X every
complex prime is actual. In the absence of any determination of a system of
fundamental units for X = 11, X = 13, X = 17, and X = 19, it is not possible to say
whether this is or is not the case for these values also. But from and after the
p
limit X = 23, the value of the factor , . indicates that a complex number is
not necessarily a complex prime because it is irresoluble into factors.
51. Criterion of the Divisibility of H by X. The number of classes of ideal
numbers, which we have symbolized by H, is not in general divisible by X ; but
in certain cases it may happen that it is so. The quotient is never divisible
p
by X, except when the other factor . . _ 1 is also divisible by X. And it has
been found by M. Kummer that the necessary and sufficient condition for the
* See the note already cited, ' on the Irregularity of Determinants,' in the Monatsberichte for
1853, p. 195.
t Liouville, vol. xvi. p. 473. The formula is given without demonstration.
Art. 51.] REPORT ON THE THEORY OP NUMBERS. 115
p
divisibility of , > _ 1 by X is that the numerator of one of the first /w 1 functions
of Bernoulli should be divisible by X. The investigation of this singular criterion
depends on a transformation of the function <(/3) which enters into the expression
of P. If we represent the product
(7/3 - 1) 0(0) = (77 X _ 2 - 1) + (7 - y00 + (yyi - 7s)/3 2
in which every coefficient is divisible by X, by
2 +...&A-2|8 x - 2 ], or
(b m denoting the quotient <yy "'~ 1 , or J^"*" 1 , if /represent the greatest in-
X X
teger contained in the fraction before which it is placed), we obtain by multipli-
cation the equality
or, since 7** + 1 is divisible by X, and may be supposed not divisible by X 2 *,
p
C denoting a coefficient prime to X. The congruence . . _ x = 0, mod X, is there-
fore equivalent to the congruence
-2) = , modX,
which may, in its turn, be replaced by the following,
^ (?) ^ (r 3 ) 4" (r*" 2 ) = 0, mod X.
For, if there be an equation which, considered as a congruence for a given
modulus X, is completely resoluble for that modulus, any symmetrical function
of the roots of the congruence is congruous, for the modulus X, to the cor-
responding function of the roots of the equation. The function
which is a symmetric function of /3, /3 3 , ... /6*~ 2 , the roots of the equation
2^ + 1 = 0, is therefore congruous to ^ (7) . >!/ (7") ... -^ (7*~ 2 ), which is the same
function of 7, y 3 , 7 s , ... y*" 2 , the roots of the congruence x<* + l=Q, mod X.
* For y+1 and (y + A^+l are both of them divisible by \; but only one of them can be-
divisible by X 2 , since their difference is not divisible by A 2 . We can therefore, without changing
Xo> y\i > yA._ 2 , determine y in accordance with the supposition in the text.
Q2
116
REPORT ON THE THEORY OF NUMBERS.
[Art. 51.
m-f
Hence the necessary and sufficient condition for the divisibility of v by X
is that one of the M congruences included in the formula
(a)
, = 1, 2,3,..., X
should be satisfied. Now
_
or, observing that y , -y lf 7 2 , ... 7 X _ 2 are the numbers 1, 2, 3, ... X - 1, taken in a
certain order, and introducing the values of b , b lt b 2 , ...
"- 1 /^?, mod X.
X
z-1
This last expression may be further transformed as follows. If /() denote any
x=x
function of x, and F(x)= 2 /(a), we have the identical equation
a=l
7 and X being any two numbers prime to one another. To verify this equation,
we may construct a system of unit points in a plane ; then the right-hand
member is the sum of the values off(x) for all unit points in the interior of the
parallelogram (0, 0), (X, 0), (X, 7), (0, 7) ; while the two terms of the left-hand
member represent similar sums for the two triangles into which the parallelogram
is divided by its diagonal yx \y = 0. Writing then in this identity x 2n ~ l for
x = x
f(x), and employing the symbol F Zn _ l (x) to represent the sum 2 x 2n ~ l , or
rather the function
-or
_
-i n (2)n(2n-2)
in which B lt J5 2 , .-.,B n are the functions of Bernoulli, and which, when a; is an
integral number, coincides with that sum, we find
1=^1 ,,/. 3=7 1
2 x /*?+ 2 J P 2n
ae=l ft x = l
But ^ 2n _i (X - 1) = ^ 2n _! (X) - X 2 "- 1 is evidently divisible by X ; so that
x=X 1 -,~, z=ir-l r \~-i
2 aji.-ijE? + 2 ^T/- 1 = 0, mod X.
z=l X x=1 L 7 J
Art. 52.] REPORT ON THE THEORY OF NUMBERS. 117
The congruences (a) may therefore be replaced by the congruences
3=1
which may be written in the simpler form
if we observe that (\ being prime to 7) the numbers / , / , ..., I - '
77 7
are congruous (mod X) to the fractions , , ... -- , taken in a certain
order. But, by a curious property of the function F 2n _ l} demonstrated for the
first time by M. Kummer,
x
The condition for the divisibility of H by X is therefore that one of the /u con-
gruences included in the formula B n (y 2n 1) = 0, mod X, should be satisfied.
The last of these congruences, or B li (y z>l 1) = 0, is never satisfied; for it is
easily proved that the denominator of B^ contains X as a factor, while
though divisible by X, is not divisible by X 2 . And since, if n< /u, y zn 1 is prime
to X, that factor may be omitted in the remaining M 1 congruences ; so that
the condition at which we have arrived coincides with that enunciated at the
commencement of this article.
We have exhibited M. Rummer's analysis of this problem with more fulness
of detail than might seem warranted by the nature of this Report, not only on
account of its elegance, but also because it exemplifies transformations and pro-
cesses which are of frequent occurrence in arithmetical investigation *.
52. 'Exceptional' Primes. A prime number X, which, like 37, 59, and 67
in the first hundred, divides the numerator of one of the first ^(X 3) functions of
* In Liouville, vol. i. (New Series) p. 396, M. Kronecker has given a very simple demonstration
of the congruence
2nAi/A (y 1 " 1 - 1 ) = (y 2B - 1) [I 2n + 2 2 " + ... + (\- I)""], mod A 2 ,
which, combined with another easily demonstrated formula, viz.,
leads immediately to the theorem of M. Kummer.
118 REPORT ON THE THEORY OF NUMBERS. [Art. 53.
Bernoulli, and which consequently divides the number of classes of ideal numbers
composed with X-th roots of unity, is termed by M. Kummer an exceptional
prime. Such primes have to be excluded from the enunciation of several
important propositions ; and their theory presents difficulties which have not
yet been overcome. Thus the following propositions are true for all primes
other than the exceptional prunes, but are not true for the exceptional primes.
(1.) The exponent to which any class of ideal numbers appertains (see
Art. 49) is prime to X.
(2.) The index of the lowest power of any unit which can be expressed as a
product of integral powers of the trigonometric units is prune to X. For that
index is a divisor of -r- (see Art. 42).
(3.) Every complex unit which is congruous to a real integer for the
modulus X is a perfect X-th power. (Whether X be an exceptional prime or not,
the X-th power of any complex number is congruous, for the modulus X, to a real
integer, viz. to the sum of the coefficients of the complex number.)
(4.) If f(a) denote any (actual) complex number prime to X (i. e. not
divisible by 1 a), a complex unit e (a) can always be assigned, such that the
product F(a) = e (u)f(a) shall satisfy the two congruences
A complex number satisfying these two congruential conditions is called a
primary complex number; the product of two primary numbers is therefore
itself primary. This definition, in the particular case X = 3, includes the primary
numbers of Art. 37, taken either positively or negatively.
53. Fermat's Theorem for Complex Primes. Let < (a) be an actual or ideal
complex prime, and let N=N.<j> (a) represent its norm. A system of N actual
numbers can always be assigned such that every complex number shall be con-
gruous to one and only to one of them for the modulus <j> (a). These 2V" numbers
may be said therefore to form a complete system of residues for the modulus
q> (a) ; and by omitting the term divisible by < (a), we obtain a system of N 1
residues prune to (j> (a).
Let q be a prime appertaining to the exponent/, so that N=qf, and let
< (a) or <, (? ) be the prime factor of q which appertains to the substitution
io = w ! the formula
1 af- 1 (A)
Art. 53.] RETORT ON THE THEORY OF NUMBERS. 119
will represent a complete system of residues for the modulus fa (%), if we assign
to the coefficients a , a 1} a 2 , ... the values 0, 1, 2, ..., q- 1 in succession. For if
be any complex number, /(a) is congruous for the modulus fa (>? ) to
x^o (o) + ^i (MO) + . . . + a/- 1 x^ (tt ),
because w >? = 0, mod fa (%) : that is, /(a) is congruous to one of the complex
numbers included in (A) ; nor can any two numbers
a + a 1 a + a 2 a 2 + ...+a f _ 1 a f - 1 and & + 6 1 a + & 2 a 2 + ... + b f _ 1 a f ~ 1
included in that formula be congruous to one another ; for the congruence
( - 6 ) + a (! - 6j) + a 2 (a 2 - & 2 ) + . . . + of- 1 (c^-i - &/_i) = 0, mod fa (i, ),
involves, by M. Rummer's theory (see Art. 45), the coexistence of the /
congruences
a -b = 0, m d 2J i-&i = 0, mod <?; ...... ; a/_i-&/_i = 0, mod q;
i.e. the identity of the complex numbers
a + aa 1 + a 2 a 2 +. ..+c/~ 1 a / _ 1 , and b + ab 1 + a 2 b 2 + ... +a f ~ 1 b f _ 1 .
It is worth while to notice that, if q be a prime appertaining to the exponent 1,
for the modulus X, that is if q be of the linear form raX + 1, the real numbers
0, 1, 2, 3, ..., q 1 will represent the terms of a complete system of residues for
the modulus (f> (a) ; but if <p (a) be a factor of a prime appertaining to any higher
exponent than unity, a complete system will contain complex as well as real
integral residues.
By applying the principle (see Art. 10) that a system of residues prime to
the modulus, multiplied by a residue prime to the modulus, produces a system
of residues prime to the modulus, we obtain the theorem, which here replaces
Fermat's Theorem, that if ^ (a) be any actual number prime to (f> (a),
If we combine with this theorem the principle of Lagrange (cited in Art. 11)
which is valid for complex no less than for real prime modules, we may extend,
mutatis mutandis, to the general complex theory the elementary propositions
relating to the Residues of Powers, Primitive Roots, and Indices, which, as we
have seen, exist in the case of complex primes formed with cubic or biquadratic
roots of unity. In fact, these propositions are of a character of even greater
generality, and may be extended, not only to complex numbers formed with
roots of unity whose index is a composite number, but also to all complex
120 REPORT ON THE THEORY OF NUMBERS. [Art. 54.
numbers formed with the roots of equations having integral coefficients, as soon
as the prime factors of those complex numbers are properly defined.
54. M. Rummer's Law of Reciprocity. We can now enunciate M. Rummer's
law of reciprocity. It appears, from the last article, or it may be proved
JV-1
immediately by dividing the Nl residues of < (a) into X groups of
terms, after the following scheme,
(0) r 1} r z , , r N _y
K
(1) ar n ar 2 , , ary_ lt
x
(2) ar lf a*r 2 , , a*r N _ lt
(X-l)
'!> " '2 J u ' NI>
A.
and proceeding as in Art. 33 of this Report, that if ^ (a) be any actual complex
N-l
number prime to (j> (a), \J<- (a) x is congruous for the modulus < (a) to a certain
power a* of a. This power of a may be denoted by the symbol | ^4-r ] ; so that
we have the congruence
The symbol | T/xl , which we may term the X-tic character of ^ (a) with regard
L<p (a)-U.
to <f) (a), is evidently of the same nature as the corresponding symbols with
which we have already met in the quadratic, cubic, and biquadratic theories, and
admits of an extension of meaning similar to that of which they are susceptible.
Availing himself of this symbol, M. Kummer has expressed his law of reciprocity
by the formula
~
<f> (a) and \fr (a) denoting real or ideal primes. But, to interpret this equation
rightly, it is important to attend to the following observations.
(1) When xj' (a) and $ (a) are both actual numbers, the formula supposes
that they are both primary prime numbers. The prime 1 a is therefore
excluded.
(2) The definition that we have given of the symbol rrr I becomes un-
Art. 55.]
REPORT ON THE THEORY OF NUMBERS.
121
meaning when <p (a) is ideal, because no signification can be assigned to an ideal
number which presents itself, not as a modulus or divisor, but as a residue.
Let, therefore, h denote the index of the lowest power of <f) (a) which is an actual
number ; i.e., let h be the exponent to which the class of <p (a) appertains ; and
let [$ (a)]* represent the actually existing primary complex number which con-
tains the factor ^> (a) h times, but contains no other prime factor ; then the
[(h (CL\^~\
*j v . ; . has by the preceding definition a perfectly definite meaning.
Let then , , ', = a? ; we may define the value of the symbol -7-4-c by means
L\p(a)Jx Ly(a]JA.
of the equation f A / a ) -.* f $ /)-,
I = I = a
L>l r (a) JA L ^ (a) J\
which, if h be prime to X, always gives a determinate value a fc for . . . ,
k being defined by the congruence hk = k', mod X. For the symbol . \ { so
defined, the law of reciprocity still subsists, subject however to the condition
that [<p (a)] 6 is primary.
It will be seen, therefore, that the exceptional primes of Art. 52 are ex-
cluded from M. Kummer's law of reciprocity, for a twofold reason : first, because
if X be one of those numbers, the definition of a primary number is not in
general applicable ; and secondly, because, on the same supposition, the symbol
r<H a )i
. . ( may become unmeaning.
55. The Theorems complementary to M. Kummer's Law of Reciprocity.
The prime I a, and its conjugate primes, as well as the complex units, are
excluded from the law of reciprocity ; but complementary theorems by which
the X-tic characters of these numbers may be determined have been given by
M. Kummer. For a simple unit a k , we have the formula
r a '
,ff-i
n A.
With regard to X, which is the norm of 1 a, it may be observed that if (p (a) be
a prime factor of a real prime q appertaining, for the modulus X, to any exponent
/different from unity, i.e. if q be not of the linear form mX + 1, the character
of every real integer, and therefore of X, with respect to (f) (a) is + 1, because,
qf-l
is divisible by q 1. But whatever be the linear form of q, the
122 REPORT ON THE THEORY OF NUMBERS. [Art. 55.
characteristic of X or x (*) (f r 8 we sna M ^ or brevity term the index of a in
the equation ftr-Trl = a *) ^ determined by the congruence
= -Z> A , modX,
D^ being the value (for v = 0) of the differential coefficient - , if 9 (a)
1 d K log [d> (e"\] h
be an actually existent number, or of j- ^ ^ ^ ^e ideal. To obtain
the characteristics of the units, M. Kummer considers the system of independent
defined by the formula
E k (a) =e (a) e (a^' 2fc e (a^ 4 * ...... e (,"-7- 2( *- 1)4 ,
in which e (a) represents the trigonometrical unit of Art. 50, and 7 is the same
primitive root of X which occurs in the expression of e (a). We have then, for
X [E k (a")] and %.(!- <**), the formulae
X^^a'Ol^C-l)^? 24 -!)^-^-^ modX,
and
jy representing the norm of 9 (a), J5 1? jB 2 , ..., -B M _j the functions of Bernoulli,
and D m the value of the differential coefficient
d m log 9 (e v ~) / d m log [d> (e)1
r 1 ( or - LJ m
dv m V hdv m
These formulae do not in general hold for the exceptional prime numbers X,
which divide the numerator of one of the first n 1 functions of Bernoulli. This
is evident from the occurrence in them of the coefficients D m , which if 9() be
ideal, and h be divisible by X, may acquire denominators divisible by X, thus
rendering the congruences nugatory. It is sufficient to have determined the
characteristics of the particular system of units Ei(a), E 2 (a), ..., E fl _ l (a), be-
cause, as that system is independent, every other unit e (a) is included in the
formula e (a) = ^ (),, ^ (a)n , 2 ...... E ^ ^^ .
so that x [ e ()] may be found from the congruence
X [e(a)]= 2 m kX [E k (a)], mod X,
4 = 1
which cannot become unmeaning, except in the case of the exceptional primes ;
Art. 56.]
REPORT ON THE THEORY OF NUMBERS.
123
because if D' be the logarithmic determinant of the system of units E x (a),
E 2 (a), ..., E ll _ l (a), D and A retaining the meanings assigned to them in Art. 50,
it may be shown that -~- is prime to \, and therefore = -=- x -r- is also
prime to X ; i.e., the denominators of the fractions
m
z ,
m^^ are prme
to X (see Art. 42). But M. Kummer has also given a formula which assigns
directly the characteristic of any unit e (a) whatsoever. If A k denote the value
of the differential coefficient - ^ '
, .
a-zr
- "2
, for v = 0. we have
^.rt, modX*.
56. We have already observed (see Art. 39) that it is impossible to deduce
a proof of the highest laws of reciprocity from the formulae which present
themselves in the theory of the division of the circle. It is true (as we shall
presently see) that the formulae IV. and V. of Art. 30 determine the decom-
position of the real prime p (supposed to be of the form &X + 1) into its X 1
complex prime factors ; but it will be perceived that these complex factors occur,
not isolated, but combined in a particular manner. From equation IV. of the
article cited we infer that p = \^ (a) \^ (a" 1 ) ; let then
a lf a 2 , ..., a^ being /u different roots (of which no two are reciprocals) of the
equation - = 1 ; so that /("i), /(a 2 ), ,f( a i^ are one-half of the complex
primes of which p is composed ; if e (a) be any real unit, satisfying the equation
e (a) = e (a" 1 ), it is plain that
e (aj* e (a 2 ) 2 ...e (atf = 1, or ^ (a) = + e (a 1 )/(a 1 ) x e ( 2 )/( 2 ) x . . . x e (<*)/(<*).
The consideration, therefore, of the number >// (a) cannot supply us with any
determination of the X-tic character of /(c^) which will not equally apply to
/( a i) x e (i)- But for all values of X greater than 3, the number of real complex
units is, as we have seen, infinite ; and the character of any complex prime f(a)
with respect to any other complex prime evidently changes when f(a) is mul-
tiplied by a unit of which the X-tic character is not unity. The inapplicability
of the formulae of Art. 30 to any general demonstration of the law of reciprocity
* The formulae of this article are taken from M. Rummer's second memoir on the complementary
theorems (Crelle, vol. Ivi. p. 270).
R 2
124 REPORT ON THE THEORY OF NUMBERS. [Art. 57.
is thus apparent. The only equation of reciprocity that has been elicited from
them is the following :
r * r *
...
L ft JA L ft J* L </ e
in which < (a) is a complex prime factor of a prime number p of the form raX + 1,
and q lt q t , ..., q e are the e conjugate factors of a prime number q appertaining
to the exponent / for the modulus X. This equation, which, if we adopt the
generalised meaning of the symbol of reciprocity, may be written more briefly
thus, I 1 = I y-r- , was first obtained by Eisenstein, who inferred it from
L 9 J x L#(a)J A
M. Rummer's investigation of the ideal prime divisors of ^ (a) (see a note ad-
dressed by Eisenstein to Jacobi, and communicated by Jacobi to the Berlin
Academy, in the Monatsberichte for 1850, May 30, p. 189). In a later memoir
(Crelle's Journal, vol. xxxix. p. 351), Eisenstein proposes an ingenious method-
reposing, however, on an undemonstrated principle for the discovery of the
higher laws of reciprocity; but it would seem that the application of this
method failed to lead him to any definite result ; and it is unquestionably to
M. Kummer alone that we are indebted for the enunciation as well as for the
demonstration of the theorem.
57. M. Kummer appears to have waited until he had developed the theory
of complex numbers with a certain approximation to completeness, before pro-
ceeding to apply the principles he had discovered to the purpose which he
had in view throughout, the investigation of the law of reciprocity. He suc-
ceeded in discovering the law which we have enunciated, in the year 1847,
and, after verifying it by calculated tables of some extent, he communicated
it to Dirichlet and Jacobi in January 1848, and subsequently, in 1850, to the
Berlin Academy, in a note which also contained the demonstration of the com-
plementary theorems relating to the units, and the prime divisors of X. From
the analogy of the cubic theorem, it was natural to conjecture that the law
of reciprocity would assume the simple form for primes p and p z
<-P^\ L^>jJ x
reduced, by multiplication with proper complex units, to a form satisfying
certain congruential conditions. But to determine properly these conditions,
i.e. to assign the true definition of a primary complex prime, was no doubt
the principal difficulty that M. Kummer had to overcome in the discovery of
his theorem. If X = 3, the single congruence /(a) =/(!), mod(l-a) 2 , suffi-
ciently characterises a primary number; and since, whatever prime be repre-
Art. 58.]
REPORT ON THE THEORY OF NUMBERS.
125
sen ted by \, that congruence is satisfied by one, and one only, of the numbers
included in the formula a fc /(a), it was probable that it ought to form one
of the congruential conditions included in the definition of a primary complex
prime. In determining the second condition, M. Kummer appears to have been
guided by a method which depends on the arithmetical properties of the log-
f( \
arithmic expansion of a complex number. If we develope log 'j. in ascending
powers of a ' f . ^ - and represent by L~ the finite number of terms
which remain in this expansion after rejecting those which are congruous to zero
for the modulus X, we are led, after some transformations, to the congruence
where X k (a) represents the function
the value (for v = 0) of the differential coefficient
=X-2
2 y-' k a*', and D k denotes, as in Art. 55,
=o
d k log/(e")
dv k
In this con-
gruence the first coefficient alone is altered when f(a) is multiplied by a
simple unit ; and only the even coefficients are altered when f(a) is multi-
plied by a real unit. Now D l is rendered congruous to zero by the condition
f(a) = f(l), mod (1 a) 2 ; and M. Kummer has shown that, by multiplying
f(a) by a properly chosen real unit, D 2 , D 4 , ..., Z> A _ 3 may be similarly made
to disappear, so that we obtain
a congruence which is proved to involve the second congruence of condition
satisfied by a primary number, i.e. f(<*)f(a~ l ) =/(l) 2 > m d X *.
58. The methods to which M. Kummer at first had recourse in order to
obtain a demonstration of his theorem, consisted in extensions of the theory of
the division of the circle. By such extensions he demonstrated the comple-
mentary theorems, and even a particular case of the law of reciprocity itself
that hi which the two complex primes compared are conjugate. But, after
repeated efforts, he found himself compelled to abandon these methods, and to
seek elsewhere for more fertile principles. ' I turned my attention,' he says,
'to Gauss's second demonstration of the law of quadratic reciprocity, which
depends on the theory of quadratic forms. Though the method of this demon-
* Crelle, vol. xliv. pp. 130-140.
126 REPORT ON THE THEORY OF NUMBERS. [Art. 59.
stration had never been extended to any other than quadratic residues, yet its
principles appeared to me to be characterised by such generality as led me to
hope that they might be successfully applied to residues of higher powers ; and
in this expectation I was not disappointed *.'
M. Rummer's demonstration of the law of reciprocity was communicated to
the Academy of Berlin in the year 1858, ten years after the date of his first
discovery of it. An outline of the demonstration is contained in the Monatsbe-
richte for that year ; and it is exhibited with great clearness and fulness of
detail in a memoir published in the Berlin Transactions for 1859, which con-
tains what is for the present the latest result of science on a problem which,
if we date from the first enunciation of the quadratic theorem by Euler, has
been studied by so many eminent geometers for nearly a century. It would,
however, be impossible, without exceeding the limits within which this Report
is confined, to give an account of its contents, which should be intelligible to
persons not already familiar with the subject to which it refers. Taken by itself
the demonstration of the theorem is, indeed, sufficiently simple ; but it is based
on a long series of preliminary researches relating to the complex numbers that
can be formed with the roots of the equation w^ = D (a), in which D (a) itself
denotes a complex number composed of Xth roots of unity. To those researches,
and to the demonstration of the law of reciprocity founded on them, we shall
again very briefly refer, when we come to speak of the corresponding investiga-
tions in the theory of quadratic forms, an acquaintance with which is essential to
a comprehension of the method adopted by M. Kummer in his memoir. We may
add that M. Kummer has intimated that he has already obtained two other
demonstrations of his law of reciprocity, which, though they also depend on the
consideration of complex numbers containing w, yet do not require the same
complicated preliminary considerations.
59. Complex Numbers composed of Roots of Unity, of^vhich the Index is not
a Prime. In a special memoir (see the list in Art. 41, note, No. 16), M. Kummer
has considered the theory of complex numbers composed with a root of the
equation w n = 1, in which n denotes a composite number. The primitive roots of
this equation are the roots of an irreducible equation of the form
See the Berlin Transactions for 1859, p. 29.
Art. 59.] REPOET ON THE THEORY OF NUMBERS. 127
Pi, P2, Pa, denoting the different prime divisors of n*. If \^ (n) be the
number of numbers less than n and prime to it, F(u>] is of the order ^(n), and
every complex number containing o> can be reduced (and that in one way only)
to the form y( w ) = + x + a 2 a, 2 + . . . + a m _ t
The numbers conjugate to /(<>) are the ^ (n) numbers obtained by writing in
succession for a> the -^ (n) primitive roots of a>" = 1 ; and the norm of /() is the
real and positive integer produced by multiplying together the ^ () conjugates.
If q be a prime number not dividing n, the sum
W^O^ + to^ + W* "+...,
in which the series of terms is to be continued until it begins to repeat itself, is
termed a period. The n periods w 1} in- 2) ..., & remain unchanged if for we
write afl, w ?2 , etc. Hence, if q appertain to the exponent t for the modulus n
(i.e. if q satisfy the congruence g* = 1, mod n, but no congruence of a lower order
and similar form), the number of different numbers conjugate to a given complex
number containing the periods only is at most For brevity, a complex
t
number containing the periods only for example, the number
C + C 1
may be symbolised by/(i), so that
If 1, r 1} r 2 , ... are a set of j-* numbers prime to n and such that the quotient
6
of no two of them (considered as a congruential fraction f) is congruous for the
modulus n to any power of q, the numbers conjugate iof(a) may be represented
* The irreducibility of the equation = when n is prime was first established by Gauss
(Disq. Arith., Art. 341). For other and simpler demonstrations of the same theorem, see the memoirs
of MM. Kronecker (Crelle, xxix. p. 280, and Liouville, 2nd series, vol. i. p. 399), Schonemann (Crelle,
vol. xxxi. p. 323, vol. xxxii. p. 100, <fc vol. xl. p. 188), Eisenstein (Crelle, vol. xxxix. p. 166), and
Serret (Liouville, vol. xv. p. 296). The principles on which these demonstrations depend suffice to
establish the irreducibility of the equation ' m _ 1 = 0, but they fail, as M. Kronecker has observed,
to furnish the corresponding demonstration when n, as in the text, is a product of powers of different
primes. This demonstration was first given by M. Kronecker (Liouville, vol. xix. p. 177), who has
been followed by M. Dedekind (Crelle, vol. liv. p. 27), and by M. Arndt (ib. Ivi. p. 178).
t For the definition of a congruential fraction see Art. 14.
128 REPORT ON THE THEORY OF NUMBERS. [Art. 59.
by /(,), /(O /(**-,) The periods are the roots of certain irreducible
equations, each of which is completely resoluble when considered as a congru-
ence for the modulus q ; and the roots u lt u z , ... of the congruences are connected
with the roots w u w 2 , ... of the equations, by a relation precisely similar to that
enunciated in Art. 44. This relation M. Rummer has established by introducing
certain conjugate complex numbers* *(wi), *(^ n ), * 0*r s )> involving the
* These complex numbers are defined as follows (see the memoir cited at the commencement of
this article, sect. 3, and that in Crelle, vol. liii. p. 142) : Let r t be a period satisfying the irreducible
equation <<>(w t ) = 0, and let <*,, a,, . . . be the incongruous roots of <f> (y) = 0, mod q ; 6,, b t , . . . the
remaining terms of a complete system of residues, mod q, so that <f> (&,), </> (6 2 ), . . . are prime to q.
Since vr k q = or*,, mod q, and ro> 4 = w t , we have, by Lagrange's indeterminate congruence (see
Art. 10 of this Keport),
(r t -a,) K-o 2 ) . . . (w 4 -6,) (i-6) . . . = 0, mod q,
or, since w t -&, divides <f> (&,) etc.,
K^^j) K~i) (^i-a.,) ... = 0, mod q;
i. e. (aT ll a l ) (iff t a 2 ) . . . = 0, mod g;.
We may now consider the series of factors
si k a l , sr i a a , ,, ....,
corresponding to the n values of k [the numbers a lt a,, ... are of course the same for two periods
which satisfy the same irreducible equation, but not in general the same for any two periods], and,
retaining among these factors only those which are different, we may take for ^(or,) the complex
number formed by combining as many of them as possible, in such a manner as to give a product
which is not divisible by q, but which is rendered divisible by q by the accession of any one factor not
already contained in it. It is evident that * (or,) cannot contain all the factors
cr k a lt Tsr t o a ;
let us then denote by TB^ u k a factor which is not contained in * (orj) ; we thus obtain the relation
* ( CT i) ( OT *-%) = > mod 1'
or, changing the primitive root o> into o> r ,
*(or r ) (sfrku k ) = 0, mod g.
The conjugates of *(oj- 1 ) are all complex numbers formed according to the same law as *(or,)
itself; and, besides ^(tsfj) and its conjugates, no other complex number can be formed according to
that law. Also the number w t which corresponds to a given period w k is absolutely determined as
soon as we have selected the multiplier ^(in-j); for if two of the factors ar k a l , w t a 2l ...were
absent from * (IT,) we should have
* fa) ( w t ~ i) = . * K) fat - a ) = 0, mod ? ;
and thence (i~ 2 ) * (>) = 0, mod q,
contrary to the hypothesis that a, and a a are incongruous, and that * (wj is not divisible by q. The
correspondence of the numbers w,, u , u n , with the periods or,, in- 2 , . . . , sr n , can thus be fixed in us
many ways as there are numbers conjugate to * (srj), t. e. in - different ways.
Art. 60.]
REPORT ON THE THEORY OF NUMBERS.
129
periods only, not themselves divisible by q, but each satisfying the n con-
gruences included in the formula
* (v) far ~ *) = 0, mod q,
&=1, 2, 3, ...,n.
From these congruences it is easy to infer that, if f(& r , w 2r , ..., w nr ) = be
any identical relation subsisting for the periods, a similar relation
/(!, u 2 , ..., u n ) = 0, mod g,
will subsist for the numbers u lt u 2 , ... , u n ; for we find
i.e.f(u lt u 2 , ...) = 0, mod*?. Another important property of the complex num-
ber Sk (OTJ) is that it is congruous to zero, mod q, for every one of the sub-
stitutions vr l = u 1 , vr 1 = u ri , -a 1 = u ri , ... except the first: thus the congruences
^ (u ri ) = 0, *" (u r ^) = are satisfied, . . . but not ^ (wj) = 0, mod q. If, then,
f(u>) be any complex number satisfying the congruence
Sk fa } m f(ta) = 0, mod q m ,
but not the congruence
Sk (w r ) m + l f((e) = 0, mod 3 m + 1 ,
f(<a) is said to contain m times precisely the ideal factor of q corresponding to
the substitution Tir kr = u k . Since it can be shown that the numbers conjugate
to ^(rai) are all different from one another, it follows from the definition,
represents the number of conjugate ideal prime factors
that the quotient
i
contained in the real prime q, appertaining to the exponent t. If q be a
divisor of n, the definition of its ideal factors requires a certain modification,
which we cannot here particularise. (See sect. 6 of M. Kummer's Memoir.)
The two definitions, corresponding to the cases of q prime to n, and q a divisor
of n, enable us, when they are taken together, to transfer to the general case
when n is composite, the elementary theorems already shown to exist when
n is prime (see Art. 47). We may add that it is easy to prove, in the general
as in the special case (see Art. 48), that the number of classes of ideal
numbers is finite.
60. Application to the Theory of the Division of the Circle. We cannot
quit the subject of complex numbers without mentioning certain important
investigations in which they have been successfully employed. The first
relates to the problem of the division of the circle. In this problem the
130 REPORT ON THE THEORY OF NUMBERS. [Art. 60.
*=p 2
resolvent function of Lagrange 2 6'xT (see Art. 30) is, as is well known,
o
of primary importance. Retaining, with a slight modification, the notation
of Art. 30, and still representing by X a prime divisor of p 1, and by a a
o x 1
root of the equation - = 0, let us consider the function F(a, x), which
is a particular case of the resolvent, and let us represent the quotient
"**><> We thus find
and in particular, observing that ^(a, x) F(a*~ l , x) =p,
[^(,z)]^vM)-^()-"^- 2 (), ....... (2)
a result which is in accordance with the known theorem that [^(a, x)] x is
independent of x and is an integral function of a only. The resolution of
the auxiliary equation of order X, the roots of which are the X periods of
p 1 x p 1
roots of the equation - r- = 0, depends solely on the determination
A 3C -1
of the complex numbers ^i(a), ^ 2 ( a )> > " v f f \-2( a )- For when these complex
numbers are known, we may equate F(a, x) to any X-th roots of the expression
jp^i(a).>fr s (a) ... ^x-a( a )> from the value of F(a,x), thus obtained, those of
F(a 2 ,x), F(a 3 ,x), ... may be inferred by means of equation (1); and, lastly,
from the values of F(l,x), F(a,x), ..., F(a*' l ,x), the values of the periods
themselves are deducible by the solution of a system of linear equations. To
determine the numbers >K(a), ^2 ()/ M. Kummer assigns the ideal prime
factors of which they are composed, employing for this purpose the results
cited in Art. 30. The equation -^ k (a) ^ k (a~ 1 )=p shows that 4* (a) contains
precisely ^(p 1) ideal prime divisors of p, and no other complex prime. To
distinguish the prime factors of p contained in ^ k (a) from those contained
in ^(a- 1 ) M. Kummer avails himself of the congruence V. of Art. 30, viz.,
JO 1
Let X *- , and u = y*', mod p, so that u, u z , ..., x-1 are the roots of
A
y*_ I
- = 0, mod p ; also, to adapt the formulae of Art. 30 to our present
&> X
purpose, let Q~ K ' = a, m = \', n = k\'; it will result from these substitutions,
that \ta(u-*) = 0, modjp, if k and h satisfy the inequality [A] + [A]>X, where
Art. 61.] REPORT ON THE THEORY OF NUMBERS. 131
[h] and [M] are positive numbers less than X, and congruous, mod X, to h
and kh respectively. If we represent by /(a) the ideal prune factor of p
which appertains to the substitution a = u, this may be expressed by saying
that vj'j. (a) contains the factor f(a -''), if U- + U- > X, the symbols and j-
denoting the least positive numbers satisfying the congruences hx=l, mod X,
and hx = k, mod X. Assigning, therefore, to the number h every positive value
less than X compatible with this condition, we may write
+ a 8 being a simple unit which may be determined by the congruence
>h(a)= -1, mod(l-a) 2 *:
it is not necessary to add a real complex unit, for a reason which has already
appeared (see Art. 56, supra). From the expression for ^(a) a still simpler
formula for F(a, x) x may be obtained, viz.
61. Application to the Last Theorem of Fermat. The second investigation
to which we shall advert relates to the celebrated proposition known as the
'Last Theorem of Fermat,' viz. that the equation x n + y n = z n is irresoluble,
in integral numbers, for all values of n greater than 2 J . As Fermat himself
* The numbers ^ t (o) are primary according to M. Kummer's definition (Art. 52) ; for
_F(a,x)F(cf,x)_
the summation extending to every pair of values of y^ and y 2 that satisfy the congruence
yi + ya = 1, mod p,
in which y represents the same primitive root of p that occurs in the expression F (a, x). Hence
^k( l )=P-2= - 1 ' mod A, and ^(a) ^(cT 1 ) = p = 1 = [^(l)] 2 , mod A.
Also ^(a) 1/^(1) is divisible by (1 a) 2 ; for
observing that y l and y t each receive all the values 1, 2, . . . , p 2 in succession. We have, therefore,
the congruence ^//' 4 (1) = 0, mod A, from which it follows (see a note on the next article) that
\lr k (d) = \lr k (l),mod(l a.y, or ^(a) = 1, mod(l a) 2 , as in the text.
t Liouville, vol. xvi. p. 448. M. Rummer has also extended his solution of this problem to the
case in which n is any divisor of p 1. See the memoir quoted in the last article, sect. 11.
% Format's enunciation of this celebrated theorem is contained in the first of the MS. notes
placed by him on the margin of his copy of Sachet's edition of Diophantus. It would seem that this
copy is now lost ; but in the year 1670 an edition of Bachet's Diophantus was published at Toulouse,
8 2
132 REPORT ON THE THEORY OF NUMBERS. [Art. 61.
has left us a proof of the impossibility of this equation in the case of n 4, by a
method which Euler has extended to the case of n - 3, we may suppose, without
by Samuel de Fermat (the son of the great geometer), in which these notes are preserved (Diophanti
Alexaudriui Arithmeticorum libri sex, et de Numeris Multangulis liber unus, cum commentariis
C. O. Bacheti V. C. et observationibus D. P. de Fermat senatoris Tolosani. Tolosse 1670). The
theorems contained in them are, with a few exceptions, enunciated without proof; and it may be
inferred from the preface of S. Fermat that he found no demonstration of them among his father's
papers. Nevertheless, in the case of several of these propositions, we have the assertion of Fermat
himself that he was in possession of their demonstration; and although, when we consider the
imperfect state of analysis in his time, it is surprising that he should have succeeded in creating
methods which subsequent mathematicians have failed to rediscover, yet there is no ground for the
suspicion that he was guilty of an untruth, or that he mistook an apparent for a real proof. In fact
these suspicious are refuted, not only by the reputation for honour and veracity which he enjoyed
among his contemporaries, and by the evidence of singular clearness of insight which his extant
writings supply, but also by the facts of the case itself. {Gauss, vol. ii. p. 160, expresses himself
unfavourably to Fermat: see especially p. 152.} It would be inexplicable, if his conclusions reposed
on induction only, that he should never have adopted an erroneous generalization ; and yet, with the
exception of the ' Last Theorem * (the demonstration of which, after two centuries, is still incomplete),
every proposition of Fermat's has been verified by the labours of his successors. There is, indeed, one
other exception to this statement ; but it is an exception which proves the rule. In the letter to
Sir Kenelm Digby which concludes the ' Commercium Epistolicum etc.' edited by Wallis (Oxford,
1658), Fermat enunciates the proposition that the numbers contained in the formula 2 2 "+l are all
primes, acknowledging, however, that, though convinced of its truth, he had not succeeded in obtaining
its demonstration. This letter, which is undated, was written in 1658 ; but it appears, from a letter
of Fermat's to M. de * * *, dated October 18, 1640, that even at that earlier date he was acquainted
with the proposition, and had convinced himself of its truth (D. Petri de Fermat Varia Opera
Mathematica, Tolosse, 1679, p. 162). It was, however, subsequently observed by Euler that
2'*+ 1 = 4294967297 = 641 x 6700417, i.e. that the undemonstrated proposition is untrue (Op.
Arith. collecta, vol. i. p. 356). The error, if it is an error, is a fortunate one for Fermat ; it exemplifies
his candour and veracity, and it shows that he did not mistake inductive probability for rigorous
demonstration : ' Mais je vous advoue tout net,' are his words in the letter last referred to, ' (car par
advance je vous advertis que comme je ne suis pas capable de m'attribuer plus que je ne S9ay, je dis
avec mime franchise ce que je ne scay pas), que je n'ay peu encore demonstrer 1'exclusion de tous divi-
seurs en cette belle proposition que je vous avois envoy^e, et que vous m'avez conference touchant les
nombres 3, 5, 17, 257, 6553, &c. Car bien que je reduise 1'exclusion & la pluspart des nombres, et
qne j'aye mfime des raisous probables pour le reste, je n'ay peu encore demonstrer n^cessairemeut la
v^ritd de cette proposition, de laquelle pourtant je ne doute non plus a cette heure que je faisoia
auparavant. Si vous en avez la preuve assuree, vous in'obligerez de me la communiquer : car apres
cela rien ne m'arrestera en ces matieres.'
The ' Last Theorem ' is enunciated by Fermat as follows :
' Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter
nullam in infinitum ultri quadratum potestatem in duos ejusdem nominis fas est dividere ; cujus rei
demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.' (Fermat's
Diophantus, p. 51.)
Fermat has also asserted that neither the sum (ibid. p. 258) nor the difference (ibid. p. 338) of
Art. 61.] REPORT ON THE THEORY OF NUMBERS. 133
loss of generality, that n is an uneven prime number X greater than 3, and we
may write the equation in the symmetrical form x* + y* + z* = Q. The impos-
two biquadrates can be a square. Each of these propositions comprehends the theorem that the sum
of two biquadrates cannot be a biquadrate ; and of the second, we possess a very remarkable demonstra-
tion by Fennat himself (ibid, p. 338; and compare Euler, E16mens d'Algebre, vol. ii. sect. 13;
Legendre, Th6orie des Nombres, vol. ii. p. 1). The essential part of this demonstration consists in
showing that, from any supposed solution of the Diophantine equation x* y* = a square, another
solution may be deduced in which the values of the indeterminates are not equal to zero, and yet are
absolutely less than in the proposed solution, from which it immediately follows that the Diophantine
equation is impossible. This method has been successively employed by Euler (toe. cit.) to demonstrate
several negative Diophantine propositions, and in particular the theorem that the sum of two cubes
cannot be a cube. The only arithmetical principles (not included in the first elements of the science)
which are employed by Euler and Fermat in their applications of this method, relate to certain simple
properties of the quadratic forms a: 2 + y 2 , y? + 2 y 2 , a? + 3 y 2 ; and as these principles seem inadequate to
overcome the difficulties presented by the equation x" + y n + z n = 0, when n is > 4, it is probable that
Fermat's ' demonstratio mirabilis sane ' of the general theorem was entirely different from that which
he has incidentally given of the particular case.
The impossibility of the equation x" + y" + z" = for n = 5 was first demonstrated by Legendre
(Memoires de 1' Academic des Sciences, 1823, vol. vi. p. 1, or Th^orie des Nombres, vol. ii. p. 361. See
also an earlier paper by Lejeune Dirichlet, Crelle, vol. iii. p. 354, with the addition at p. 368, and a
later one by M. Lebesgue, Liouville, vol. viii. p. 49); for n = 14, by Dirichlet (Crelle, vol. ix. p. 390);
and for n = 7, by M. Lam6 (Memoires des Savans Etrangers, vol. viii. p. 421, or Liouville, vol. v.
p. 195. See also the Comptes Kendus, vol. ix. p. 359, and a paper by M. Lebesgue, Liouville,
vol. v. pp. 276 and 348). But the methods employed in these researches are specially adapted to the
particular exponents considered, and do not seem likely to supply a general demonstration. The
proof in Barlow's Theory of Numbers, pp. 160-169, is erroneous, as it reposes (see p. 168) on an
elementary proposition (cor. 2, p. 20) which is untrue. A memoir by M. Kummer on the equation
a^+y 2 * = 2 2A , in which complex numbers are not employed, and in which no single case of the
theorem is demonstrated (Crelle, vol. xvii. p. 203), is nevertheless of great interest for the number of
auxiliary propositions contained in it. Of the same character are the notes by MM. Lebesgue and
Liouville, in Liouville's Journal, vol. v. pp. 184 and 360, and a few theorems given without demon-
stration by Abel, (Euvres, vol. ii. p. 264.
In the year 1847, M. Lam6 presented to the Academy at Paris a memoir containing a general
demonstration of Fermat's Theorem, based on the properties of complex numbers (Comptes Rendus,
vol. xxiv. p. 310; Liouville, vol. xii. pp. 137 and 172). It was, however, observed by M. Liouville
(Comptes Rendus, vol. xxiv. p. 315), that this demonstration is defective, as it assumes, without proof,
the proposition that a complex number can be represented, and in one way only, as the product of
powers of complex primes a proposition which, as we have seen, is untrue, unless we admit ideal as
well as actual complex primes. The discussion on M. Lamp's memoir attracted Cauchy's attention to
Fermat's Theorem; and the 24th and 25th volumes of the Comptes Eendus contain several communi-
cations from him on the subject of complex numbers [or polynomes radicaux, as he has preferred to
term them]. In the earlier papers of this series, Cauchy attempts to prove a proposition which, as we
have already observed (see Art. 41), is untrue for complex numbers considered generally, viz. that the
norm of the remainder in the division of one complex number by another can be rendered less than the
norm of the divisor (see Comptes Rendus, vol. xxiv. pp. 517, 633, and 661). Elsewhere (ibid. p. 579) he
134 REPORT ON THE THEORY OF NUMBERS. [Art. 61.
sibility of solving this equation has been demonstrated by M. Kummer, first,
for all values of X not included among the exceptional primes * ; and secondly,
for all exceptional primes which satisfy the three following conditions :
(1) That the first factor of H, though divisible by X, is not divisible by
X* (see Art. 50).
(2) That a complex modulus can be assigned, for which a certain definite
complex unit is not congruous to a perfect X-th power.
(3) That B KK is not divisible by X 3 , B K representing that Bernoullian
number [<c ^ ft 1] which is divisible by X f.
Three numbers below 100, viz. 37, 59, 67, are, as we have seen, exceptional
primes. But it has been ascertained by M. Kummer that the three conditions
just given are satisfied in the case of each of those numbers ; so that the
impossibility of Format's equation has been demonstrated for all values of
the exponent up to 100. Indeed, it would probably be difficult to find an
exceptional prime not satisfying the three conditions, and consequently excluded
from M. Rummer's demonstration.
We must confine ourselves here to an indication of the principles on which
the demonstration rests in the case of the non-exceptional primes \.
assumes the proposition as a hypothesis, and deduces from it conclusions which are erroneous (pp. 581,
582). But at p. 1029 he recognises and demonstrates its inaccuracy. The results at which he arrives
in his subsequent papers on the same subject are, for the most part, comprehended in M. Rummer's
general theory (Comptes Kendus, vol. xxv. pp. 37, 46, 93, 132, 177). In one place, however (p. 181),
he enunciates, though without demonstrating, the following important result :
'If the equation x^ + y^+z*- = be resoluble, x, y, z denoting integral numbers prime to A,
the sum
is divisible by A."
(Compare M. Rummer's memoir in the Berlin Transactions for 1857, p. 64.)
The investigation of the Last Theorem of Fennat has been twice proposed as a prize-question by
the Academy of Paris first at some time previous to 1823 (see Legendre's memoir already cited, in
vol. vi. of the H&noires de I'Acad&nie des Sciences, p. 2), and again in 1850 (Comptes Kendus,
vol. xxx. p. 263) : at neither time was the prize adjudged to any of the memoirs received. On the
last occasion, after several postponements of the date originally fixed for the award, the prize was
ultimately, in 1857 (ib. vol. xliv. p. 158), conferred on M. Rummer, who had not been a competitor,
for his researches on complex numbers.
' Liouville, vol. xvi. p. 488, or Crelle, vol. xl. p. 131.
t See the memoir No. 15 in the list of Art. 41.
I When A is not an exceptional prime, the equation as* + y* + 2 A = is irresoluble not only
in ordinary integral numbers, but also in any complex integers composed of A-th roots of unity. The
demonstration does not possess the same generality when A is an exceptional prime satisfying the
three conditions cited in the text. In this case M. Rummer has only shown that the equation
Art. 61.] REPORT ON THE THEORY OF NUMBERS. 135
We may suppose that X is greater than 3, and that no two of the numbers
x, y, z admit any common divisor. And first, let none of them be divisible by
1 - a, a still representing a root of the equation - - = 0. Since for x we may
a 1
write a'x, we may assume that x, y, z are of the form
z=c+(i- a yz,
a, b, c denoting integral numbers prime to X, which evidently satisfy the con-
gruence a + b + c = Q, mod X. The equation x*- + y K + Z K = may then be written
tnus
No two of the factors of which the left-hand member is composed can have
any common divisor; each of them is therefore the product of a perfect X-th
power by a unit; so that we may write, x + a'y = a<'e(a) v^, e(a) denoting a
real unit. Since v*- is an actual number, it follows (remembering that X is not
an exceptional prime) that v is also actual ; hence v^ is congruous, mod X, to a
certain integral number ra. Eliminating m x e(a) between the two congruences
x + a'y = mai'e(a), and x + a~ s y = ma~Pe(a), mod X,
p j
a~l'(x + a e y) ai > (x + a- s y) = 0, mod X.
For the modulus (1 a) this congruence is identically satisfied *. That it should
be satisfied, mod (1 a) 2 , we must have the relation (a + b) p = bs, mod X ;
whence, putting - j- = k, mod X, we have p = ks, mod X. Substituting this
d/ "T"
value for p, we find that the congruence
is identically satisfied, mod (1 a) 3 ; but in order that it should be satisfied,
mod (1 a) 4 , we have the condition
s 3 b(2k-l)(k-l)-3s{(k-l)y" + kx"} = 0, modX,
x^ + y^ + z*- = is irresoluble when we suppose that x, y, z are ordinary integral numbers prime
to A, or else complex numbers containing the binary periods a + o~*, one of which has a common
divisor with A.
* Since A is divisible by (1 a)*" 1 , and since
it is readily seen that, if r ^ A 1, the conditions for the divisibility of $ (a) by (1 a) r are
4> (1) = 0, c'(l) = 0, . . . , ^ r - J ) (1) = 0, mod X
130 REPOKT ON THE THEORY OF NUMBERS. [Art. 61.
where x" and y" are the values (for a = 1) of the second derived functions
of x and y with respect to a. This conditional congruence must be satisfied
for every value of s; either therefore k=l, mod X, or 2k =1, mod X. The
supposition k = 1 is inadmissible ; for it implies that a = 0, mod X, contrary
to the hypothesis. Hence we must have 2=1, and a = b, or, by parity of
reasoning, a = b = c, mod X. But also a + b + c = 0, mod X, whence we again
infer the inadmissible conclusion a = b = c = 0, mod X.
Secondly, let one of the numbers x, y, z (for example, z) be divisible by
1 a ; it will be convenient to consider the equation in the generalised form
y* + y* = E(a)(l-a} n *z\ ........ (1)
in which x, y, and z are all prime to 1 a, and E (a) is any unit. We may
assume that the values of x and y are of the form
a and b being prime to X, but satisfying the relation a + b = 0, mod X. In the
first place, m must be greater than 1. For since
a^ = ct\ and y x = 6 X , mod (1 a) x + 1 ,
if x* + y* be divisible by (1 a) x , a x + 6 x is divisible by X 2 , and therefore x* + y K
by (1 a) x+1 . Again, each of the factors x + ay, x + a 2 y, ..., x + a*~ l y is
divisible once, and once only, by 1 a ; whence it follows that x + y is divisible
by (1 a) mx ~ x + 1 , and that no two of the X factors of x x + y x have any other
common divisor than I a. Hence the X factors
x + y x + ay
_ a - -
are relatively prime, and may be represented by expressions of the form
e<>()&>\ ^(a)^, ...... , e^.^a)^-!,
e o( a ), ^(a), ... representing units, and < X , (pf, ... X-th powers prime to Ia.
Eliminating x and y from the three equations
X+ 3/ =
we obtain a result of the form
e(a) and E- i (a) denoting two units. But, as in the former case, it may
be shown that < r x and <, x are congruous, mod X, to real integers, and
Art. 62.] REPOET ON THE THEORY OF NUMBERS. 137
(1 ) (m )K = 0, mod X, because m>l. Hence e(a) is also congruous to a real
integer for the modulus X, and is therefore a perfect X-th power by a property
of every non-exceptional prime (see Art. 52). The equation (2) therefore
assumes the form x x + y K = j^ ( a ) z x ^ _ a )(.-i)x
If, therefore, the proposed equation (1) be possible, it will follow, by successive
applications of this reduction, that the equation
is also possible. But this equation has been shown to be impossible ; the
equation (1) is therefore also impossible.
62. Application to the Theory of Numerical Equations. In the Monats-
berichte for June 20, 1853 (see also the Monatsberichte for 1856, p. 203),
M. Kronecker has enunciated the following theorem :
' The roots of any Abelian equation, the coefficients of which are integral
numbers, are rational functions of roots of unity.' The demonstration of this
theorem (Monatsberichte for 1853, pp. 371-373) depends on a comparison of a
certain form, of which the resolvent function of any Abelian equation is
susceptible, with M. Kummer's expression for the resolvent function in the
case of the equation of the division of the circle (see Art. 60). It thus involves
considerations relating to ideal numbers.
Two propositions of a more special character, and closely connected with
one another, have also been given by M. Kronecker (Crelle, vol. liii. p. 173).
Their demonstration is immediately deducible from the principles of Dirichlet's
theory of complex units :
' If unity be the analytical modulus of every root of an equation, of which
the first coefficient is unity and all the coefficients are integral numbers, the
roots of the equation are roots of unity.'
' If all the roots of an equation (having its first coefficient unity and all
its coefficients integral) be real and inferior in absolute magnitude to 2, so
that they can be represented by expressions of the form 2 cos a, 2 cos /3, 2 cos % . . .
the arcs a, /3, 7 are commensurable with the complete circumference.'
In the following proposition M. Kronecker has extended a theorem of
M. Kummer's (Art. 42) relating to complex units composed with roots of unity
of which the index is a prime, to complex units composed with any roots of
unity (Crelle, vol. liii. p. 176) :
' Every complex unit composed with the roots of the equation <o" = 1, can
be rendered real by multiplication with a 4-th root of unity. If n be even,
T
138 REPORT ON THE THEORY OF NUMBERS. [Art. 63.
a 2-th root will always suffice ; and if n be a power of a prime, an n-th root
will suffice.'
The demonstration of this proposition is also deducible from Dirichlet's
principles.
63. Tables of Complex Primes. In M. Kummer's earliest memoir on com-
plex numbers (Liouville, vol. xii. p. 206) he has given a table of the complex
factors, composed of X-th roots of unity, which are contained in real primes of
the form mX + 1 inferior to 1000, \ representing one of the primes 5, 7, 11, 13,
17, 19, 23. This memoir was written before M. Kummer had considered the
complex factors of primes of linear forms other than raX + 1, and before he had
introduced the conception of ideal numbers. The complex prime factors of
real primes of those other linear forms are, therefore, not exhibited in the
Table; and the five numbers of the form 23m+l, 47, 139, 277, 461, 967, each
of which contains 22 ideal factors composed of 23rd roots of unity, are repre-
sented as products of 11 actual factors (each of which contains two reciprocal
ideal factors). The tentative methods by which the complex factors were dis-
covered are explained in sect. 9 of the memoir cited. Since the full develop-
ment of M. Kummer's theory, Dr. Reuschle has undertaken to complete and
extend the Table. He has already given tables containing the complex prime
factors of all real primes less than 1000, composed of 5th, 7th, llth, 13th,
17th, 23rd, and 29th roots of unity, together with the complete solution of
the congruences corresponding to the equations of the periods (see the Mo-
natsberichte for 1859, pp. 488 and 694, and for 1860, pp. 150 and 714). For
5, 7, 11, 13, 17, the complex primes are exhibited in a primary form; for 19,
23, and 29 they are exhibited in a form which satisfies the condition
/() =/(!), mod (!-)',
but not the condition /( a )/(a~ 1 ) = [/(1)] ! , mod X.
The ideal factors Dr. Reuschle represents by their lowest actual powers ; for
23 this power is the cube, for 29 it is the square; for 11, 13, 17, 19, as well
as for 5 and 7, all complex prime factors of real primes less than 1000 are
actual. It appears from the Table (and it has indeed been proved by M. Kum-
mer), that 29 is an ' irregular determinant ' (see Art. 49, note) ; for the number
of classes is 8, while the square of every ideal number (occurring as a factor
of a real prime inferior to 1000) is actual. The methods employed by Dr.
Reuschle in the calculation of his tables have not yet been published by him.
In some instances, as M. Kummer has observed, they have not led him to the
simplest possible forms of the ideal primes.
Art. 64.]
REPORT ON THE THEORY OF NUMBERS.
139
A particular investigation relating to the ideal factors of 47, composed of
23rd roots of unity, has been given by Mr. Cay ley (Crelle, vol. Iv. p. 192,
and Ivi. p. 186).
64. The investigations relating to Laws of Reciprocity, which have so long
occupied us in this report, have introduced us to considerations apparently
so remote from the theory of the residues of powers of integral numbers, that
it requires a certain effort to bear in mind their connexion with that theory.
It will be remembered that the complex numbers to which our attention has
been directed are not of that general kind to which we have referred in Art. 41,
but are exclusively those which are composed of roots of unity. The theory
of complex numbers, in the widest sense of that term, does indeed present to
us an important generalisation of the theory of the residues of powers ; for
the theorem of Fermat (see Art. 53) subsists alike for every species of com-
plex numbers. But the complex numbers of Gauss, of Jacobi, and of M. Kum-
mer force themselves upon our consideration, not because their properties are
generalisations of the properties of ordinary integers, but because certain of the
properties of integral numbers can only be explained by a reference to them.
The law of quadratic reciprocity does not, as we have seen, necessarily require
for its demonstration any considerations other than those relating to ordinary
integers ; the real prime numbers of arithmetic are here the ultimate elements
that enter into the problem. But when we come to binomial congruences of
higher orders, we find that the true elements of the question are no longer
real primes, but certain complex factors, composed of roots of unity, which are,
or may be conceived to be, contained in real primes. For we find that the
law which expresses the mutual relation (with respect to the particular kind
of congruences considered) of two of these complex factors is a primary and
simple one ; while the corresponding relations between the real primes them-
selves are composite and derivative, and, in consequence, complicated. It thus
becomes indispensable, for the investigation of the properties of real numbers,
to construct an arithmetic of complex integers ; and this is what has been
accomplished by the researches, of which an account has been given in the
preceding articles.
The higher laws of reciprocity (like that of quadratic residues) may be
considered as furnishing a criterion for the resolubility or irresolubility of
binomial congruences ; and this, though not the only application of which they
are susceptible, is that which most naturally suggests itself. When the binomial
congruence is cubic or biquadratic, it is easy to resolve the real prime modulus
T 2
1 40 REPORT ON THE THEORY OF NUMBERS. [Art. 65.
into factors of the form a + bp, or a + bi (Arts. 37 and 24), and equally easy
to determine the value of the critical symbol of reciprocity by a uniform and
elementary process (see Art. 36). For these, therefore, as well as for quadratic
congruences, the criterion deducible from the laws of reciprocity is all that
can be desired. But for binomial congruences of higher orders this criterion
is not a satisfactory one, because of the difficulty of obtaining the resolution
of a real prime into its complex factors, and also because of the impossibility
of determining the value of the critical symbol by the conversion of an ordinary
fraction into a continued fraction.
The only known criterion applicable to such congruences is the following,
the demonstration of which is deducible from the elements of the theory of
the residues of powers : Let x" = A, mod p, represent the proposed congruence ;
it will be resoluble or irresoluble according as the index of A is or is not divisible
by d, the greatest common divisor of n and p 1, i.e. according as the exponent
pl
to which A appertains is or is not a divisor of -7 (see Arts. 14 and 15).
65. Solution of Binomial Congruences. We now come to the problem of
the actual solution of binomial congruences a subject upon which our know-
ledge is confined within very narrow limits.
When a table of indices for the prime p has been constructed, the reso-
lution of every binomial congruence, if it be resoluble, or, if not, the demon-
stration of its irresolubility, is implicitly contained in it. But to use a table
of indices for the solution of a binomial congruence is, as we have already
observed in a similar case (Art. 16), to solve a problem by means of a recorded
solution of it. When the congruence x n = A, mod p, is resoluble, its solution
may always be made to depend on that of a congruence of the form x d = a,
mod p, where d is the greatest common divisor of n and p 1, and where
a = A', mod p, and ns = d, mod p 1. We may therefore suppose that, in the
congruence x n = A, mod p, n is a divisor of p 1. This congruence (if re-
soluble at all) will have as many roots as it has dimensions ; if be any one
of them, and 1, 1} 6 2 , ..., 6 n _ l be the roots of the congruence x" = l, mod p,
the roots of x n = A, mod p, will be , 1} 2 , ..., #_! ; so that the complete
resolution of the congruence x n =A, mod p, requires, first, the determination
of a single root of that congruence itself, and, secondly, the complete resolution
of the congruence x" = l, mod p. With regard to the first of these requisites,
in the important case in which the exponent t to which A appertains is prime
to n, a value of x satisfying the congruence x n =A, mod^>, can be determined
Art. 66.] REPORT ON THE THEORY OF NUMBERS. 141
by a direct method (Disq. Arith., Arts. 66, 67). For, in this case, it will
always happen that one value of x is a certain power A k of A, where k is
determined by the congruence kn = l, mod t. Nor is it necessary, in order
to determine k, to know the exponent t to which A appertains ; it is sufficient
to have ascertained that it is prime to n ; for, if we resolve p 1 into two
factors prime to one another, and such that one of them is divisible by n
and contains no prime not contained in n, the other will be divisible by t,
and may be employed as modulus instead of t in the congruence kn = l, mod t.
When this method is inapplicable, we can only investigate a root of the
congruence x" = A, mod p (where A is different from 1), by tentative processes,
which, however, admit of certain abbreviations (Disq. Arith., Arts. 67, 68).
The work of Poinsot (Reflexions sur la Theorie des Nombres, cap. iv. p. 60)
contains a very full and elegant exposition of the theory of binomial congru-
ences ; but neither he nor any other writer subsequent to Gauss has been able
to add any other direct method to that which we have just mentioned.
66. Solution of tJie Congruence x n = l, mod p. When a single root of the
congruence x n = A is known, we may, as we have seen, complete its resolution
by obtaining all the roots of the congruence x n = l, mod p. The methods of
Gauss, Lagrange, and Abel for the solution of the binomial equation x n 1 =
are in a certain sense applicable to binomial congruences of this special form.
It is evident, from a comparison of several passages in the Disquisitiones
Arithmetics *, that Gauss himself contemplated this arithmetical application
of his theory of the division of the circle, and that he intended to include
it in the 8th section of his work, which, however, has never been given to
the world. In fact, the method of Abel t which comprehends that of Gauss,
and which gives the solution of any Abelian equation, is equally applicable
to any Abelian congruence ;' i.e. to any completely resoluble congruence of order
TO, the m roots of which (considered with regard to the prime modulus p)
may be represented by the series of terms
r, <j>(r), < 2 (r), ..., <p- l (r),
the symbol ( denoting a given rational [fractional or integral] function. And
as we can always express the roots of an Abelian equation by radicals (i.e. by
* See Disq. Arith., Arts. 61, 73, and especially Art. 335.
t See Abel's memoir, ' Sur une classe particuliere d'equations resolubles algebriquement,' sect. 3.
((Euvres, vol. i. p. 114, or Crelle, vol. iv. p. 131), and M. Serret's Algebre Superieure, 26th and 27tU
lessons.
142 REPORT ON THE THEORY OF NUMBERS. [Art. 66.
the roots of equations of two terms), so also the solution of an Abelian con-
gruence depends ultimately on the solution of binomial congruences. When,
for any prime modulus, an Abelian equation admits of being considered as an
Abelian congruence, so precise is the correspondence of the equation and the
congruence, that (as Poinsot has observed in a memoir in which he has
occupied himself with the comparative analysis of the equation x n = l, and
the congruence x n = l, mod p*) we may consider the analytical expression of
the roots of the equation as also containing an expression of the roots of
the congruence ; and by giving a congruential interpretation f to the radical
signs which occur in that expression, we may elicit from it the actual values
of the roots of the congruence. An example taken from Poinsot's memoir
x 1 1
will render this intelligible!. The six roots of the equation * r =
*C ~ JL
comprised in the formula
where the signs + and are to be successively attributed to *J 7, and
where the product of the two cube roots is + ^/ 7, or *J 7, according
to the sign attributed to ^ 7. Considering the equation as a congruence
with regard to the modulus 43, and observing that
,y^7 = + 6, mod 43, ^21 = + 8, mod 43,
we obtain in the first place
5 1 ,
x= - + - 4/16 + ^4/ -8, mod 43,
and ^=- + ,3/22 + -2, mod 43,
the product of the two cube roots being congruous to + 6 in the first formula,
and to 6 in the second ; and finally, observing that
* ' Sur 1'Application de 1'Algebre u la Theorie des. Nombres,' Memoires de 1'Academie des
Sciences, vol. iv. p. 99.
. _
t Gauss employs the symbol \j A, mod p, to denote ai root of the congruence x" = A, mod p,
n
just as he employs the symbol , mod p, to denote the root of the congruence Ax = B, mod p.
A
The congruential radical \/A, mod p, has of course as many values as the congruence x" = A, mod p,
has solutions ; if that congruence be irresoluble, the symbol is impossible.
1 See the memoir cited above, p. 125.
Art 66.]
REPORT ON THE THEORY OF NUMBERS.
143
4/16 = 21, 3, - 18, mod 43,
4/~^~8 = 14, 2, - 12, mod 43,
4/22 = - 15, 4, 19, mod 43,
4/^~2s + 9, -20, +11, mod 43,
and attending to the limitation to which the cube roots are subject,
x=-8, + 11, +21, or, -2, + 4, +16; mod 43.
Thus the complete solution of a congruence of the sixth order is obtained by
means of binomial congruences of the second and third orders only.
An essential limitation to the usefulness of this method arises from the
circumstance that it does not always (or even in general) happen that (as in
the example just given) each surd entering into the expression of the root
becomes separately rational. For that expression may itself acquire a rational
value, while certain surds contained in it continue irrational, precisely as, in
the irreducible case of cubic equations, a real quantity is represented by an
imaginary formula. To illustrate this point by an example, let us consider
x 7 1
the same congruence - = with respect to the modulus 29 *. Here in
the expression
where p denotes a cube root of unity, we have, putting ^/ 7 = + 14, and p = 1,
the irrational cube roots disappearing of themselves. Again, putting
\ ,
we find
x =
'7T/'7\4 '
= 7 + (7)=7 + 16=-6 or -9,
where every radical becomes rational of itself. Similarly taking the values
we find x = 5 or 13. But lastly, putting ^/ 7 = 14, p = 1, we find
z = 12 + J[14 + 7, v /2]* + J[14-'
* Hid. p. 132.
144 REPORT ON THE THEORY OF NUMBERS. [Art. 6ti.
To rationalise this expression, we have to observe that 14 + 7^/2, relatively
to the modulus 29, is the cube of a complex number of similar form ; in fact,
we have (147 N /2) = (5 + ll N /2) 1 , mod 29, whence x= -4. To elicit, therefore,
the value of this root from the irrational formula, we are obliged to solve
the cubic congruence x s = 14 + 7^2, which, although of lower dimensions than
the proposed congruence, is probably less easy to solve tentatively, because
29 has 29" - 1 = 840 residues of the form a + 6 v /2, and only 29-1=28 ordinary
integral residues ; so that practically the method fails. Theoretically, however,
the relation between the analytical expression of the equation-roots and the
values of the congruence-roots is of considerable importance, and the subject
would certainly repay a closer examination than it has yet received. We may
add that, if m be a divisor of p 1, the complete solution of an Abelian con-
gruence of order m requires only two things, 1st, the complete solution of
the congruence of" ' 1 = 0, mod p, and, 2ndly, the determination of a single
root of a certain congruence of the form x m a = 0, mod^>, in which a is an
ordinary integer; so that in this case (which is that of the congruence
x 1 1
-r = 0, mod 43) we obtain a real, and not only an apparent reduction of
3C ~~ -I.
the proposed congruence *.
It should also be observed that the primitive roots of the equation
= furnish, when rationalised, the primitive roots of the congruence
x-l
x n 1
- = 0, mod p. This, the only direct method that has ever been suggested
for the determination of a primitive root, appears to be the same as that
referred to by Gauss in the Disq. Arith. (Art. 73).
Poinsot expresses the conviction that this method of rationalisation is
applicable to any congruence corresponding to an equation, the roots of which
can be expressed by radicals f. With regard to equations of the second, third,
and fourth orders this is certainly true. If, for example, the biquadratic
equation F t (x) = Q be completely resoluble when considered as a congruence
' This will be at once evident, if we observe that when the congruence x m = 1, mod p,
is completely resoluble, its roots may be employed to replace, in Abel's method, the roots of the
equation x m 1 = 0.
t See the memoir cited above, p. 107, and M. Libri, Memoires de Mathdmatique et Phy-
sique, p. 63.
Art. 67.] REPORT ON THE THEORY OF NUMBERS. 145
for the modulus p, so that
F< (x) = (x- a t ) (x - a 2 ) (x - a 3 ) (x - a 4 ), mod p,
it is plain that the four roots of F(x) = 0, and the four numbers a ls 2 , a 3 , a 4
may be obtained by substituting, in the general formula which expresses the
root of any biquadratic equation as an irrational function of its coefficients,
the values of the coefficients of the functions F(x) and
(x ttj) (x 2 ) (x 3 ) (x a^
respectively. But these two sets of coefficients differ only by multiples of p ;
i.e. the values of a 1} a. 2 , a 3 , a 4 can be deduced from the expressions of the
roots of F(x) = by adding multiples of p to the numbers which enter into
those expressions. But this reasoning ceases to be applicable to equations
of an order higher than the fourth, because no general formula exists repre-
senting the roots of an equation of the fifth or any higher order. If, therefore,
F(x) = be an equation of the nth order, the roots of which can be expressed
by a radical formula, and which is also completely resoluble when considered
as a congruence for the modulus p, so that
F(x) = (x ctj) (x a 2 ) ... (x ), mod p,
it will not necessarily follow that the formula which gives the roots of F(x) =
is also capable (when we add multiples of p to the numbers contained in it)
of giving the roots of
(x-a l )(x-a 2 ) ... (x-a n ) = Q,
i.e. the roots of the congruence F(x) = 0, mod p; and thus the principle
enunciated by M. Poinsot is, it would seem, not rigorously demonstrated.
67. Cubic and Biquadratic Congruences. The reduction of cubic con-
gruences to binomial ones has been treated of by Cauchy (Exercices de Mathe-
matiques, vol. iv. p. 279), and more completely by M. Oltramare (Crelle, vol. xlv.
p. 314). Some cases of biquadratic congruences are also considered by Cauchy
in the memoir cited, p. 286. The following criteria for the resolubility or
irresolubility of cubic congruences include the results obtained by M. Oltramare,
I. c., and appear sufficiently simple to deserve insertion here :
Let the given cubic congruence be
a6 3 + 3b8* + 3ce + d = 0, mod p,
p denoting a prime greater than 3, which does not divide the discriminant of
the congruence ; i.e., the number
D= -a 2 <Z 2 + 6a&cd
TJ
146 REPORT ON THE THEORY OF NUMBERS. [Art. 68.
and, in connection with the congruence, consider the allied system of functions *
U = (a,b,c,d)(x,y)>,
H= (ac - b\ \(ad - be), U - c 2 ) (x, y)\
which are connected by the equation
let also u and (/> denote the values of U and $ corresponding to any given
values of x and y, which do not render H=Q, mod p. Then, if (* ~) = - 1,
the congruence has always one and only one real root ; if (^ -) = + 1, it has
either three real roots, or none : viz., if (?Ai_ -') = + 1, it has three ;
if {*W - .) =p } or = p 2 , it has none. The interpretation of the cubic
symbol of reciprocity will present no difficulty if we observe that V D, modp,
A
is a real integer if p = 3n + l, i.e. if ( -- ) = 1, and that, if p = 3n~-l, i.e. if
f- - j = 1, we have
> modp,
so that \/ D, mod^>, is a complex integer involving p. It will however be
observed that the application of the criterion requires in either case the solu-
tion of a quadratic congruence, r 2 = D, modp, or r 2 = ^D, mod^>.
Similar, but of course less simple, criteria for the resolubility or irresolu-
bility of biquadratic congruences may be deduced from the known formulae
for the solution of biquadratic equations.
68. Quadratic Congruences Indirect Methods of Solution. The general
form of a quadratic congruence is ax 2 + 2 bx + c = 0, mod p, where p denotes an
uneven prime modulus, and a is a number prime to p. It may be immediately
reduced to the binomial form r 2 = D, mod p, by putting
r = ax + b, D=b 2 ac, modjp.
The number of its solutions is 2, 0, or 1, according as D is a quadratic residue
* See a note by Mr. Cayley in Crelle's Journal, vol. 1. p. 285.
Art. 68.] REPORT ON THE THEORY OF NUMBERS. 147
or non-residue of p, or is divisible by p, and is therefore in every case expressed
by the formula 1 + ( )
If p = 4n + 3, and () = !, the congruence r 2 D=0, mod p, is satisfied
by r = D n + l , and r= D n + 1 , and is in fact resoluble by the direct method
of Art. 6). But no direct method, applicable to the case when p = 4:n + l,
is at present known. Two tentative methods are proposed in the sixth section
of the Disquisitiones Arithmetic*. They are both applicable to congruences
with composite as well as with prime modules. This circumstance is important,
because, when the modulus is a very great number, we may not be able to
tell whether it is prime or composite, and, if composite, what the primes are
of which it is composed, although, when the prime divisors of a composite
modulus are known, it is simplest first to solve the congruence for each of
them separately, and afterwards (by a method to which we shall hereafter
refer) to deduce from these solutions the solution for the given composite
modulus. To apply the first of Gauss's methods, the congruence is written
in the form r 2 = D + Py, P denoting the modulus. If in the formula V= D + Py
we substitute for y in succession all integral values which satisfy the inequality
-p<y<^P-^p, and select those values of V which are perfect squares,
their roots (taken positively and negatively) will give us all the solutions of
the congruence. We should thus have I(\P] or 1+I(^P) trials to make, /
denoting the greatest integer contained in the fraction before which it is
placed. If, however, we take any number E, greater than 2, and prime to P
(it is simplest to take for E a prime, or power of a prime), of which the
quadratic non-residues are a,b,c, ..., and then determine the values of a, /3, 7, ...
in the congruences a = D + aP, mod E, b = D + /3P, mod E, &c., we shall find
that every value of y contained in one of the linear forms mE + a, mE+/3, &c.,
gives rise to a value of V which is a quadratic non-residue of E, and which
cannot, therefore, be a perfect square ; so that we may at once exclude these
values of y from the series of numbers to be tried. A second excludent E'
may then be taken, and by its aid another set of linear forms may be
determined, such that no value of y contained in them can satisfy the con-
gruence. Thus the number of trials may be diminished as far as we please.
The application of this method is still further facilitated by the circumstance
that it is not necessary actually to solve the congruences a = D + aP, mod E, ...
u 2
148 RETORT ON THE THEORY OF NUMBERS. [.\rt.68-
but only the single congruence D + Py=Q, mod E (Disq. Arith., Art. 322).
Gauss's second method depends on the theory of quadratic forms ; it supposes
that the congruence is written in the form ?- 2 + Z)=0, mod P. By a tenta-
tive process (abbreviated, as in the first method, by the use of excludents)
Gauss obtains all possible primitive representations of P by the quadratic
forms of determinant D ; whence the complete solution of the congruence
r 2 + D = 0, mod P, is immediately deduced. This method involves the con-
struction of a complete system of quadratic forms of determinant -D, or, if
the prime factors of D be known, of one genus of forms of that system ; it
becomes therefore more difficult of application as D increases, whereas the
first method is not affected by the increase of D. The second method, how-
ever, especially recommends itself when P is a very great number; in fact,
if we do not employ any excludent, the number of trials required by the
first method varies (approximately, and when P is a great number) as P,
whereas, on the same supposition, the number of trials required by the second
method varies as v D x x/P.
M. Desmarest (in his Thdorie des Nombres) has proposed a method less
scientific in its character than those of Gauss, but sometimes easily applicable
in practice. He has shown that if the congruence r 2 + D = 0, mod P, be
resoluble, we can always satisfy the equation mP = x 2 + Dy 2 with a value. of
p
m inferior to rr; + 3, and of y not superior to 3. The demonstration of this
theorem is not very satisfactory, and the number of trials that it still leaves
is very great, viz. 3 (I ( ) + 3j
The application of Gauss's second method is rendered somewhat more
uniform, and at the same time the necessity for constructing a system of
quadratic forms of determinant D is avoided by the following modification
of it : By a known property of quadratic forms, whenever the congruence
r 2 + D = 0, mod P, is resoluble, the equation mP = x 2 + Dy 2 is resoluble for
some value of m< 2,^/1 .D. By assigning, therefore, to m all values in suc-
cession which are inferior to that limit, and which satisfy the condition
, and then obtaining (by Gauss's method) all prime representations
of the resulting products by the form a; 2 + Dy 2 , we shall have
x' x"
Art. 69.] REPORT ON THE THEORY OF NUMBERS. 149
x, y', x", y", etc. denoting the different pairs of values of x and y in the
equation mP = x 2 + Dy 2 .
69. General Theory of Congruences. We may infer from several passages
in the Disquisitiones Arithmeticse, that Gauss intended to give a general
theory of congruences of every order in the 8th section of his work, and
that, at the time of its publication, he was already in possession of the prin-
cipal theorems relating to the subject *. These theorems were, however, first
given by Evariste Galois t, in a note published in the Bulletin de Fe"russac
for June, 1830 (vol. xiii. p. 438), and reprinted in Liouville's Journal, vol. xi.
p. 398. An account of Galois's method (completed and extended in some
respects) will be found in M. Serret's Cours d'Algebre SupeYieure, lecon 25.
The theory has also been independently investigated by M. Schonemann, who
seems to have been unacquainted with the earlier researches of Galois (see
Crelle's Journal, vol. xxxi. p. 269, and vol. xxxii. p. 93). In several of Cauchy's
arithmetical memoirs (see in particular Exercices de Mathematiques, vol. i.
p. 160, vol. iv. p. 217 ; Comptes Rendus, vol. xxiv. p. 1117 ; Exercices d'Analyse
et de Physique Mathe"matique, vol. iv. p. 87) we find observations and theorems
relating to it. Lastly, in a memoir in Crelle's Journal (vol. liv. p. 1) M. Dede-
kind has given (with important accessions) an excellent and lucid resumd of
the results obtained by his predecessors.
In the following account of the principles of this theory, the functional
symbols F, <f>, -^, ... will represent (as in general throughout this Report)
rational and integral functions having integral coefficients ; we shall use p
to denote a prime modulus, and x an absolutely indeterminate quantity. As
we shall have to consider the functions F (x), f(x), ^(x), etc., only in relation
to the modulus p, we shall consider two functions F (x) and F z (x), which
differ only by multiples of p, as identical, and we shall represent their identity
by the congruence F l (x) = F 2 (x), mod p, which is equivalent to an identical
equation of the form F l (x) = F 2 (x) +p$> (x). The designation ' modular function,'
* See Disq. Arith., Arts. 11 and 43.
t Galois was born October 26, 1811, and lost his life in a duel, May 30, 1832. He was
consequently eighteen at the time of the publication of the note referred to in the text. His
mathematical works are collected in Liouville's Journal, vol. xi. p. 381. Obscure and fragmentary
as some of these papers are, they nevertheless evince an extraordinary genius, unparalleled, perhaps,
for its early maturity, except by that of Pascal. It is impossible to read without emotion the letter
in which, on the day before his death and in anticipation of it, Galois endeavours to rescue from
oblivion the unfinished researches which have given him a place for ever in the history of mathe-
matical science.
150 REPORT ON THE THEORY OF NUMBERS. [Art. 69.
which has been introduced by Cauchy (Comptes Rendus, vol. xxiv. p. 1118),
will serve (though, perhaps, not in itself very appropriate) to indicate that
the function to which it is applied is thus considered in relation to a prime
modulus. Since in any modular function we may omit those terms the co-
efficients of which are multiples of p, we shall always suppose that the
coefficient of t ie hig-i est power of a; in the function is prime to p.
If F ()=/i (x)xfi(x), mod p, fi(x) and f 2 (x) are each of them said to
be divisors of F(x) for the modulus p, or, more briefly, modular divisors of
F(x), or even simply divisors of F(x) when no ambiguity can arise from this
elliptical mode of expression. If a be a function of order zero, i. e. an integral
number prime to p, a is a divisor, for the modulus p, of every other modular
function; so that we may consider the p 1 terms a lf 2 > 3> .*-!> of
a system of residues prune to p, as the units of this theory, and, in any
set of p 1 associated functions
a l F(x), a 2 F(x), ..., a p _ l F(x),
we may distinguish that one as primary in which the highest coefficient is
congruous to unity (mod p).
If F(x) be a function which is divisible (mod p) by no other function
(except the units and its own associates), F(x) is said to be a prime or irre-
ducible function for the modulus p. And it is a fundamental proposition in
this theory, that every modular function can be expressed in one way, and
one way only, as the product of a unit by the powers of primary irreducible
modular functions. The demonstration of this theorem depends (precisely
as in the case of ordinary integral numbers) on Euclid's process for finding
the greatest common divisor, which, it is easy to show, is applicable to the
modular functions we are considering here. For, if fa (x) and fa (x) be two
such functions [the degree of fa (x) being not higher than that of fa (a;)],
we can always form the series of congruences
fa (x) = q l (x) fa (x) + T! fa (x), mod p,
fa (x) = q 2 (x) fa (x) + r 2 fa (x), mod p,
in which r lt r 2 , ... denote integral numbers, qi(x), q 2 (x), ... modular functions,
and fa (x~), fa (x), . . . primary modular functions, the orders of which are suc-
cessively lower and lower, until we arrive at a congruence
<Pk (x) = q k (x) fa +i(x) + r k fa + 2 (x), mod p,
in which r k =0, mod p. The function fa +l (x) is then the greatest common
Art. 69.] REPORT ON THE THEORY OF NUMBERS. 151
divisor (mod p) of the given functions fa (x) and fa (x) ; and, in particular, if
fa + l (x) be of order zero, those two functions are relatively prime. We may
add that, if R be the Resultant of fa (x) and fa (x), the necessary and sufficient
condition that these functions should have a common modular divisor of an
order higher than zero is contained in the congruence R = Q, mod p * a theorem
exactly corresponding to an important algebraical proposition. From the nature
of the process by which the greatest common divisor is determined, we may
infer the fundamental proposition enunciated above, by precisely the same
reasoning which establishes the corresponding theorem in common arithmetic.
Similarly, we may obtain the solution of the following useful problem :
'Given two relatively prime modular functions A m and A n , of the orders
m and n, to find two other functions, of the orders m 1 arid n 1 respectively,
which satisfy the congruence
The assertion, that f(x) is a divisor of -^(a;) for the modulus p, is for
brevity expressed by the congruential formula
which represents an equation of the form
F(x)=p$(x} +f(x
Similarly the congruence
F 1 (x) = F t (x
is equivalent to the equation
F l (x) = F s (x
If f(x) be a function of order m, it is evident that any given function is
congruous, for the compound modulus [p, f(x)~\ to one, and one only, of
the p m functions contained in the formula a + a l x+ ... +a m _ 1 x m ~ 1 , in which
, a,, ..., a m _, may have any values from zero to p 1 inclusive. These p m
functions, therefore, represent a complete system of residues for the modulus
[ft/Ml
A congruence F(X) = Q, mod [p, /()], is said to be solved when a
functional value is assigned to X which renders the left-hand member divisible
* See Cauchy, Exercices de Mathematiques, vol. i. p. 160, or M. Libri, Meinoires de Math6-
matique et de Physique, pp. 73, 74. But a proof of this proposition is really contained in Lagrange's
Additions to Eider's Algebra (sect. 4).
152 REPORT ON THE THEORY OF NUMBERS. [Art 70.
by f(x) for the modulus p ; and the number of solutions of the congruence
is the number of functional values (incongruous mod [.p, /(#)]) which may
be attributed to X. The coefficients of the powers of X in the function F (X)
may be integral numbers or functions of x. The linear congruence
in which A and B denote two modular functions, is, in particular, always
resoluble when A is prune to f(x), mod p, and admits, hi that case, of only
one solution.
We shall now suppose that the function f(x) in the compound modulus
\.P>f( x )] i 8 irreducible for the modulus p, a supposition which involves the
consequence that, if a product of two factors be congruous to zero for the
modulus [p,f(x)~\, one, at least, of those factors is separately congruous to
zero for the same modulus. We thus obtain the principle (cf. Art. 11) that
no congruence can have more solutions, for an irreducible compound modulus,
than it has dimensions. For, if X=%, mod \_p, f(x)\ satisfy the congruence
F m (X) = 0,mod[p,f(x)],
we find F n (X) = F m (X)-F m ($=(X- F^X), mod ]>,/(*)],
F m _ l (X) denoting a new function of order m 1; whence it follows that if
the principle be true for a congruence of m 1 dimensions, it is also true for
one of m dimensions ; i. e. it is true universally.
70. Extension of Fermat's Theorem. Let 6 denote any one of the p m 1
residues of the modulus [p, f(x)~\ which are prime to f(x)\ ', it may be shown,
by a proof exactly similar to Dirichlet's proof of Fermat's Theorem, that
e m -i=l,mod[p,f(x)] ......... (A)
This result, which is evidently an extension of Fermat's theorem, involves
several important consequences.
It implies, in the first place, the existence of a theory of residues of powers
of modular functions, with respect to a compound modulus, precisely similar
to the theory of the residues of the powers of integral numbers with regard
to a common prime modulus. A single example (taken from M. Dedekind's
memoir) will suffice to show the exact correspondence of the two theories.
The modular function is or is not a quadratic residue of f(x), for the
modulus p, according as it is or is not possible to satisfy the quadratic con-
gruence X- = 6, mod [p, /(#)]. In the former case 6 satisfies the congruence
0i<p-o = i j mo d [ p> /(a;)] ; i n the latter, 0i<"-> = - 1, mod [p, f(x)]. And,
further, if ^ and 6 a be two primary irreducible modular functions of the orders
Art. 70.] REPORT ON THE THEORY OF NUMBERS. 153
TO and n respectively, and if we use the symbols [1 and f 1 to denote
L 6% J L QI J
the positive or negative units which satisfy the congruences
OP''-* = (), mod (p,ej, and a i<*--i> = (), mo d (p, 0,),
respectively, these two symbols are connected by the law of reciprocity
But the equation (A) admits also of an immediate application to the theory
of ordinary congruences with a simple prune modulus.
In that equation let us assign to 6 the particular value x ; we conclude
that the function x pm ~ l \ is divisible for the modulus p by f(x), i.e. by
every irreducible modular function of order m. Further, if d be a divisor
of TO, x prr '~ ' 1 is algebraically divisible by x pd - 1 - 1 ; whence it appears that
x pm ~ l -l is divisible, for the modulus p, by every function of which the order
is a divisor of TO. But it is easily shown that x fm ~ l 1 is not divisible (mod p)
by any other modular function, and that it cannot contain any multiple modular
factors. Hence we have the indeterminate congruence
x m -i-l = nf(x), modp, ........ (B)
in which f(x) denotes any primary and irreducible function, the order of
which is a divisor of m, and the sign of multiplication II extends to every
value of f(x). This theorem, again, is a generalisation of Lagrange's inde-
terminate congruence (Art. 10). We may infer from it that, when m is>l,
the number of primary functions of order m, which are irreducible for the
modulus p, is
_ wi nt w
\p m - ZpT + Zp^T - Zj>i23 + . . . 1 ,
TO- L J
q lt q 2 , ... denoting the different prime divisors of TO. As this expression is
always different from zero, it follows that there exist functions of any given
order, which are irreducible for the modulus p.
A congruence F (x) = 0, mod p, may be considered resolved when we have
expressed its left-hand member as a product of irreducible modular factors.
The linear factors (if any) then give the real solutions ; the factors of higher
orders may be supposed to represent imaginary solutions. We have already
observed that even when all the modular factors of F(x) are linear, we possess
no general and direct method by which they can be assigned ; it is hardly
-
154 REPORT ON THE THEORY OF NUMBERS. [Art. 71.
necessary to add that the problem of the direct determination of modular
factors of higher orders than the first, presents even greater difficulties.
Nevertheless the congruence (B) enables us to advance one step toward the
decomposition of F(x) into its irreducible factors; for, by means of it, we
can separate those divisors of F(x) which are of the same order, not, indeed,
from one another, but from all its other divisors. We may first of all suppose
that F(x) is cleared of its multiple factors, which may be done, as in algebra,
by investigating the greatest common divisor of F(x) and F'(x) for the
modulus p. The greatest common divisor (mod p) of F(x) and x p ~ l 1 will
then give us the product of all the linear modular factors of F(x); let F(x)
be divided (mod p) by that product, and let the quotient be F l (x) ; the
greatest common divisor (mod p) of F 1 (x) and x pt - 1 -l will give us the
product of the irreducible quadratic factors of F(x); and by continuing this
process, we shall obtain the partial resolution of F(x) to which we have
referred.
71. Imaginary Solutions of a Congiiience. We have said that the non-
linear modular factors of F(x) = 0, mod p, may be considered to represent
imaginary solutions. These imaginary solutions can be actually exhibited,
if we allow ourselves to assign to x certain complex values. The following
proposition, which shows in what manner this may be effected, is due to
Galois :
' If /(a?) represent an irreducible modular function of order m, the con-
eraeDOe ^(0)=0, mod [!,/(*)],
is completely resoluble when F(x) is an irreducible modular function of
order m, or of any order the index of which is a divisor of m.'
To establish this theorem, write 6 for x in equation (B) ; we find
0p--i_l = IT.F(0), mod_p,
the sign of multiplication II extending to every irreducible modular function
having m or a divisor of m for the index of its order. But the congruence
admits of as many roots as it has dimensions ; therefore also every divisor
of &P"- 1 1 (and, in particular, the function F(ff) considered as a congruence
for the same compound modulus) admits of as many roots as it has dimensions.
{Add that no two irreducible congruences whose indices divide m can have
a root in common. }
Let the order of the congruence F(Q)=0, mod [p,f(xj], be S, and let
Art. 72.] REPORT ON THE THEORY OF NUMBERS. 155
any one of its roots be represented by r ; it may be shown that all its roots
are represented by the terms of the series r, r p , r 1 *, ...,r pS ~ l . For, if
F(r) =0, mod [>,/()],
we have also F(r> ) = [F(ry]* = 0, mod [p,f(xj],
and similarly F(r f '}= [F(r)Y = 0, mod [p,f(x)~\ ; and so on ;
so that r, r p , r pt , ..., r 1 * 5 " 1 are all roots of
It remains to show that these S functions are all incongruous, mod \_p,f(x)\.
If possible let r pk + tf = r k , mod [ p, f(x}~\ ,
k and k' being less than S ; we have, raising each side of this congruence to
the power p s ~", r p + = r i>* } m0( J [p,f(xj],
i.e. r p>c = r, or r p1t ~ l =l, mod [j>, /(*)],
observing that r p =r, mod [p,f(x)~\,
because r p ~ 1 l is divisible by F(r) for the modulus p. We conclude, there-
fore, that r is a root, mod \_p,f(xj], of some irreducible modular divisor of the
function 6 f ~ l l, i.e. of some irreducible function of an order lower than S,
because k is less than S ; r is therefore a root, mod \_p, f (x)], of two different
irreducible modular functions, which is impossible *.
If, therefore, we suppose x to represent, not an indeterminate quantity,
but a root of the equation f(x) = 0, we may enunciate Galois' theorem as
follows :
'Every irreducible congruence of order m is completely resoluble in complex
numbers composed with roots of any congruence which is irreducible for the
modulus p, and which has m or a multiple of m for the index of its order.
' And all its roots may be expressed as the powers of any one of them.'
72. Congruences having Powers of Primes for their Modules. It remains
for us to advert to the theory of congruences with composite modules a subject
to which (if we except the case of binomial congruences) it would seem that
the attention of arithmeticians has not been much directed. We shall suppose,
first, that the modulus is a power of a prime number.
The theorem of Lagrange (Art. 11), and the more general proposition of
* {Let to = the common divisor of k and 8, since r p * = 1, r p l = 1, we can prove that
r pl " = 1, mod [p,/(a;)], but fl^"" 1 1 = a product of irreducible functions whose indices divide to;
therefore r is a root of two different irreducible functions whose indices divide m, and this is
impossible.)
X 2
156 REPORT ON THE THEORY OF NUMBERS. [Art. 72.
Art. 69, in which it is (as we have seen) included, cannot be extended to
congruences having powers of primes for their modules.
Let the proposed congruence be F(x) = Q, mod p m ; and let us suppose
(what is here a restriction in the generality of the problem) that the coeffi-
cient of the highest power of x in F(x) is prime to p, or, which comes to the
same thing, that it is unity. Let F (x) = P x Q x R . . . mod p, where P,Q,R,...
are powers of different irreducible modular functions. It may then be shown
that F (x) = P" x Q' x I?..., mod p m , where P', Q', R,... are functions of the
same order as P, Q, R, ..., respectively congruous to them for the modulus^,
and deducible from them by the solution of linear congruences only. We
have thus the theorem that F(x), considered with respect to the modulus p m ,
can always be resolved in one way and in one way only, into a product of
modular functions, each of which is relatively prime (for the modulus p) to
all the rest, and is congruous (for the same modulus p) to a power of an irre-
ducible function. We may therefore replace the congruence F (x) = 0, mod p nt ,
by the congruences _P = 0, mod_p m , (7=0, mod p m , R'=0, mod^>, But no
general investigation appears to have been given of the peculiarities that
may be presented by a congruence of the form P'=0, mod p m , in the case in
which P is a power of an irreducible function (mod p), and not itself such
a function a supposition which implies that the discriminant of F(x) is
divisible by p. If, however, P be itself an irreducible function, the congruence
P' = 0, mod p m , gives us one and only one solution of the given congruence
if P be linear ; or, if P be not linear, it may be considered as representing
as many imaginary solutions as it has dimensions. In particular, if we consider
the case in which all the divisors P, Q, R, ... are linear, we obtain the
theorem :
' Every congruence which, considered with respect to the modulus p, has as
many incongruous solutions as it has dimensions, is also completely resoluble for
the modulus p m ; and has as many roots as it has dimensions, and no more.'
If x=a lt mod p, be a solution of the congruence F (x) = 0, mod p, and
if that congruence have no other root congruous to a 1} the corresponding
solution x = a m , mod p m , of the congruence F(x) = 0, mod p m , may be obtained
by the solution of linear congruences only a proposition which is included
in a preceding and more general observation. The process is as follows :
If, in the equation
Art. 73.]
REPORT ON THE THEORY OF NUMBERS.
157
we determine k by the congruence
-F(a 1 ) + kF' (a,) = 0, mod p,
(which is always possible, because the hypothesis that (x a t ) 2 is not a divisor
of F(x), mod p, implies that F' (a^) is not divisible by p*), and then put
a 2 =a 1 + kp, mod p 2 , we have F(a 2 ) = 0, mod p 2 . Similarly, from the expansion
F (a, + kp*) = F (a 2 ) + kp* F' (a,) + ...,
a value of k may be deduced which satisfies the congruence F(a 2 + kp 2 ) = Q, or
F(a^) = 0, mod p 3 ; and so on continually until we arrive at a congruence of the
form F (a m ) = 0, mod p m . But when F (x) is divisible (for the modulus p) by
( a) 2 or a higher power of x a, the congruence F(x) = 0, mod p m , is either
irresoluble, or has a plurality of roots incongruous for the modulus p m but all
congruous to a for the modulus p. Thus the congruence
(a; a) 2 + kp (x 6) = 0, mod p 2 ,
is irresoluble, unless a =b, mod p ; whereas if that condition be satisfied, it
admits of p incongruous solutions, comprised in the formula
x = a + np, modp 2 , M = 0, 1, 2, 3, ...,p-l.
73. Binomial Congruences having a Power of a Prime for their Modulus.
If M be any number, and \^ (M) represent the number of terms in a system
of residues prime to M, it will follow (from a principle to which we have
already frequently referred : see Arts. 10, 26, 53, 70) that every residue of that
system satisfies the congruence x^ u) =\, mod M, a proposition which is well
known as Euler's generalisation of Fermat's theorem f. In particular, when
M=p m , we have x' p ~ l)pm " 1 = 1, mod p m . This congruence has, consequently,
precisely as many roots as it has dimensions a property which is also pos-
sessed by every congruence of the form a^=l, modp m , d denoting a divisor
of (p l)p m ~ l . This has been established by Gauss in the 3rd section of
the Disquisitiones Arithmeticse, by a particular and somewhat tedious method J.
The simpler and more general demonstration which he intended to give in
the 8th section j, was perhaps in principle identical with the following; we
exclude the case p = 2, to which indeed the theorem itself is inapplicable :
* If F(x) = (x aj) <f> (x), mod p, where </> (a^ is not divisible by p, we have
f" (x) = <f> (x) + (x - a,) 0' (x), mod p, or F'(ai) = $ (aj, mod p.
t Euler, Comment. Arith. vol. i. p. 284.
$ Disquisitiones Arithmetic*, Arts. 84-88. See also Poinsot, Reflexions sur la Theorie des
Nombres, cap. iv. Art. 6.
Disquisitiones Arithmetic*, Art. 84.
158 REPORT ON THE THEORY OF NUMBERS. [Art. 74.
Let d = Sf> n , $ representing a divisor of p - 1, and n being < m - 1 ; and let
us form the indeterminate congruence
of - 1 = (x- c^) (x- a 2 ) ... (x- aj), mod p m ~ n ,
which is always possible, because of 1 = 0, mod p, has S incongruous roots.
It is readily seen that, if A and B represent two numbers prime to p, and if
A = B, mod p r , A*=B* t mod p r +'; and conversely, if A^ = B^, mod p r +',
A = B, mod p r .* By applying this principle it may be shown that
rfp" _ i = ( x p n _ a/ 1 ) (a"" - a/") . . . (x* n -. a/ 1 ), mod jj.
For if we divide a 8 *"-! by a*" -a/*, the remainder is of"*-!.- But, because
a, a =l, mod _p m -", a^Hl, mod p" ; i.e. x^'-a/ 1 divides x 8 *"-! for the
modulus p m . Similarly x? p " - 1 is divisible (mod p m ) by a; 1 "" - a/ 1 , etc. ; and
since all these divisors are relatively prime for the modulus p, a?*"-! is
divisible (mod p m ) by their product; i.e.,
y n _ i = ( x i> n _ /) ( x p n _ a/ 1 ) . . . (x* - a/"), mod p n .
We have thus effected the resolution of xP pn 1 into factors relatively prime,
each of which is congruous (mod p) to a power of an irreducible function ; since
evidently (x p " - a*")= (x - a) 1 "*, mod p. To investigate the solutions of
we have therefore only to consider separately the S congruences included in
the formula x pn = a fn , mod p m . But each of these congruences (by virtue of
the principle already referred to) admits precisely p* solutions, viz. the p*
numbers (incongruous, mod p m ) which are congruous to a, mod p m ~ n . The
whole number of solutions of oc?*" -1 = 0, mod p m , is therefore equal to the
index Sp n of the congruence. It further appears that the complete solution
of the binomial congruence cc 8 *" 1 = 0, may be obtained by a direct method,
when the complete solution of the simpler congruence X s 1 = 0, mod p, has
been found. For we may first (by the method given in the last article) deduce
the complete solution of a? 1 = 0, modp m ~ n , from that of 0^ 1 = 0, mod p;
and then the roots of of pn 1 = 0, mod p m , can be written down at once.
74. Primitive Roots of the Powers of a Prime. All the elementary pro-
* If A = B, mod p", but not, mod p r+1 , we have A B + kp r , where k is prime to p. Hence
A* = (B + kpY = B f ' + kB*- 1 p > + r +p* t - r ,
K denoting a coefficient divisible by p ; or A& = B**, modip' +r , but not, mo&p**'* 1 ; because kB pl ~ l is
prime to p. This result implies the principle enunciated in the text.
Art. 76.] REPORT ON THE THEORY OF NUMBERS. 159
perties of the residues of powers, considered with regard to a modulus which
is a power of a prime number, may be deduced from the theorem just proved.
In particular, the demonstration of the existence and number of primitive
roots (Art. 12) is applicable here also ; so that we have the theorem :
'There are p m ~ 2 (p l)^f(p 1) residues prime to p m , the successive
powers of any one of which represent all residues prime to p m .' These
residues are of course the primitive roots of p m .
If 7 be a primitive root of p, of the p numbers included in the formula
7 + Tcp (mod p-), p 1 precisely will be primitive roots of p 2 . For 7 + kp
is a primitive root of p 2 unless (y + kp) p ~ l = 'L, mod^ 2 ; and the congruence
x v ~ l = \, mod p 2 , has always one, and only one, root congruous to 7 for the
modulus p. But every primitive root of p 2 is a primitive root of p 3 , and of
every higher power of p, as may be shown by an application of the prin-
ciple proved in a note to the last article, or, again, by observing that every
primitive root of m + 1 is necessarily congruous, for the modulus p m , to some
primitive root of p m , and that there are p times as many primitive roots
o f pm + 1 gg O f pa ^g ee Jacobi's Canon Arithmeticus, Introduction, p. xxxiii ;
also a problem proposed by Abel in Crelle's Journal, vol. iii. p. 12, with
Jacobi's answer, ibid. p. 211.)
75. Case when the Modulus is a Power of 2. The powers of the even
prime 2 are excepted from the demonstrations of the two last articles in
fact, if m^3, 2 m has no primitive roots. Gauss, however, has shown (Disq.
Arith., Arts. 90, 91) that the successive powers of any number of the form
Sn + 3 represent, for the modulus 2 m , all numbers of either of the forms 8n + 3
or 8n + l; similarly all numbers of the forms 8n + 5 and 8n+l are repre-
sented by successive powers of any number of the form 8n + 5. If, there-
fore, we denote by 7 any number of either of the two forms 8n + 3 or 8n + 5,
we may represent all uneven numbers less than 2"* by the formula ( I) a 7^,
in which a is to receive the values and 1, and /3 the values 1, 2, 3, ..., 2 m ~' 2 .
A double system of indices may thus be used to replace the simple system
supplied by a primitive root when such roots exist.
Tables of indices for the powers of 2, and of uneven primes inferior to
1000, have been appended by Jacobi to his Canon Arithmeticus.
70. Composite Modules. No general theory has been given of the repre-
sentation of rational and integral functions of an indeterminate quantity as
products of modular functions with regard to a composite modulus divisible
by more than one prime. And it is possible that no advantage would be
160 REPORT ON THE THEORY OF NUMBERS. [Art. 77.
gained by considering the theory of congruences with composite modules from
this general point of view. A few isolated theorems relating to particular
cases have, however, been given by Cauchy (Comptes Rendus, vol. xxv. p. 36,
1847). Of these the following may serve as a specimen :
'If the congruence F(x) = Q, mod M, admit as many roots as it has
dimensions, and if, besides, the differences of these roots be all relatively
prime to M, we have the indeterminate congruence
F(x) = k (x - rj (x - r 2 ) (x-r 3 )...(x- r n ), mod M,
k denoting the coefficient of the highest power of x in F(x).'
But if, instead of considering the modular decomposition of the function
F(x), we confine ourselves to the determination of the real solutions of the
congruence F(x) = 0, mod M, it is always sufficient to consider the congruences
F(x) = 0, mod A, F(x) = 0, mod B, F(x) = 0, mod C, etc., . . . (G)
where A x .Bx C... =M, and A, B, C, ... denote powers of different primes.
For if x= a, mod A, x=b, mod B, x = c, mod C, denote any solutions of the
first, second, third, ... of those congruences respectively, it is evident that,
if X be a number satisfying the congruences
X=a, mod A, X=b, mod B, X=c, mod C
(and such a number can always be assigned), we shall have F(X) = for
each of the modules A, B, C, ... separately, and therefore for the modulus M ;
and further, if the congruences (G) admit respectively a, /3, y, ... incongruous
solutions, the congruence F(x) = 0, mod M, will admit ax /3 x 7 x ... in all;
for we can combine any solution of F(x)=0, mod A, with any solution of
F(x)=0, mod B, and so on*.
77. Binomial Congruences with Composite Modules. The investigation of
the real solutions of binomial congruences depends (in the manner just stated)
on the investigation of the real solutions of similar congruences the modules
of which are the powers of primes. With regard to the relations by which
these real solutions are connected with one another, little of importance has
* 'Infra [i.e. in the 8th section] congruentias quascumque secundum modulum e pluribus priinis
compositum, ad congruentias quarum modulus est primus aut primi potestas reducere, fusius docebi-
mus' (Disq. Arith., Art. 92). It is difficult to see why Gauss should have employed the word 'fusius'
if his investigations extended no further than the elementary observations referred to in the text.
Nevertheless it is remarkable that Gauss in the 3rd section of the Disq. Arith. sometimes speaks of
demonstrations as obscure, which are of extreme simplicity when compared with one in the 4th and
several in the 5th section (see in particular Arts. 53, 55, 56).
Art. 78.]
EEPOET ON THE THEORY OF NUMBERS.
161
been added to the few observations on this subject in the Disquisitiones
Arithmetics (Art. 92). If the modulus M be = p a q 1 ^..., where p,q,r,... repre-
sent different primes, the congruence x^^^l, mod M, possesses no primitive
roots ; for if n be the least common multiple of
n will be less than, and a divisor of, -^(M). But evidently, if x be any
residue prime to M, the congruence x n 1 = will be satisfied separately for
the modules p", q b , r, ..., and therefore for the modulus M; i.e., no residue
exists, the first ^ (M) powers of which are incongruous, mod M. If, however,
M= 2p a this conclusion does not hold, since the least 1 common multiple of
^(2) and ^(p m ) is ^ (2p m ) itself; and we find accordingly that every uneven
primitive root of p m is a primitive root of 2 p m . When, as is sometimes the
case, it is convenient to employ indices to designate the residues prime to a
given composite modulus, we must employ (as in the case of a power of 2) a
system of multiple indices. To take the most general case, let M=2 e p a q b r c ... ;
let u be any number of either of the forms 8n + 3 or 8n + 5, and P, Q, R, ...
primitive roots of p a , <f, r*, ... respectively. Then, if n be any given number
prime to M, it will always be possible to find a set of integral numbers
e n , <a n , a n , /3 n , -y n , ... satisfying the conditions
( - 1)*" t*"- = n, mod 2" ; 0^f ri <2, O^ <2"- 2 ,
P a " = n, modp a ; 0^ a n <p a ~ 1 (p- 1),
Q^ = n,modq 1 '; </3 n < g-i ( q- I),
R" 1 ' = n, mod r ; 0^y n < r c ~ l (r-1);
and these numbers form a system of indices by which the residue of n for
each of the modules 2 e , p a , q h , r, ... (and consequently for the modulus M)
is completely determined. (See Dirichlet's memoir on the Arithmetical Pro-
gression, sect. 7, in the Berlin Memoirs for 1837.)
78. Primitive Roots of the Powers of Complex Primes. Dirichlet has
shown * that, in the theory of complex numbers of the form a + bi, the powers
of primes of the second species (see Art. 25) have primitive roots ; in fact, if
a + bi be such a prime, and N(a + bi) = a 2 + b 2 =p, every primitive root of
p m is a primitive root of (a + bi) m . On the other hand, if q be a real prime
of the form 4n + 3, q m has no primitive roots in the complex theory. For in
general, if M be any complex modulus, and M=a a Wc"t..., a, b, c, ... being
* See sect. 2 of the memoir, TJntersuchungen tiber die Theorie der complexen Zahlen, in the
Berlin Memoirs for 1841.
162
REPORT ON THE THEORY OF NUMBERS. [Art. 78.
different complex primes, and if A=N(a), B = N(b), C=N(c), etc., the
number of terms in a system of residues prime to M. \ is
A"-^(A-l)B^(B-l)C-'- l (C-l)...;
and if we denote this number by ^(M), every residue prime to M will
satisfy the congruence x* w = l, mod M,
which here corresponds to Euler's extension of Fermat's Theorem. If M=q m ,
this congruence becomes ^z^-D (-!) = i m0( j qm .
but it is easily shown that every residue prime to q m satisfies the congruence
o; m ~ 1( 2 - 1) = l, modg m ;
i.e., q m has no primitive roots, because the exponent q m ~ l (q 2 l') is a divisor
of, and less than, q 2{m - 1) (q* 1). Nevertheless two numbers, y and y, can
always be assigned, of which one appertains to the exponent q m ~ l (q 2 1)
and the other to the exponent g 1 "" 1 , and which are such that no power of
either of them can become congruous to a power of the other, mod q m , without
becoming congruous to unity ; from which it will appear that every residue
prime to q may be represented by the formula <f y' v , if we give to x all values
from to (<j' 2 -l)g' m ~ 1 -l inclusive, and to y all values from to q m - l -l
inclusive.
The corresponding investigations for other complex numbers besides those
of the form a + bi have not been given.
We here conclude our account of the Theory of Congruences. The further
continuation of this Keport will be occupied with the Theories of Quadratic
and other Homogeneous Forms.
[The Additions to Arts. 16, 20, 22, 24, 25, 36, 37, and 38 to Part I of the Report (1859),
inserted in their proper places, see footnote p. 57, were published at the end of this Part
of the Report.]
VII.
REPORT ON THE THEORY OF NUMBERS.
PAET III.
[Keport of the British Association for 1861, pp. 292-340.]
(B) Theory of Homogeneous Forms.
79. fROBLEM of the Representation of Numbers. A rational and integral
homogeneous function (a quantic according to the nomenclature introduced by
Mr. Cayley), of which the coefficients are integral numbers, is, in the Theory
of Numbers, termed a form (Disq. Arith., Art. 266). The form is linear, quad-
ratic, cubic, biquadratic or quartic, quintic, &c., according to its order in respect
of the indeterminates it contains ; and binary, ternary, quaternary, &c., according
to the number of its indeterminates. Thus x 2 + y 2 is a binary quadratic form,
x 3 + y 3 + z 3 3xyz a ternary cubic form. A form is considered to be given, when
its coefficients are given numbers ; and a number is said to be represented by
a given form, when integral values are assigned to the indeterminates of the
form, such that the form acquires the value of the number. If the values of
the indeterminates are relatively prime, the representation is said to be primi-
tive ; if they admit any common divisor beside unity, it is a derived repre-
sentation. Thus 13 and 8 can be represented by x* + y*; for 3 2 + 2 2 = 13,
2 2 + 2 2 = 8; and the first of these representations is primitive, the second is
derived. The first general problem, then, that presents itself in this part of
the Theory of Numbers, is the following, ' To find whether a given number
is or is not capable of representation by a given form, and, if it is, to find
all its representations by that form.' The number of different representations
of a given number by a given form may be either finite or infinite ; in the
former case the complete solution of the problem of representation consists in
Y 2
1 64 REPORT ON THE THEORY OF NUMBERS. [Art. 80.
the actual exhibition of the different sets of values that can be given to the
indeterminates of the form : in the latter case it consists in assigning general
formulae, in which all those values are comprised. It is in either case sufficient
to consider primitive representations only ; for if the given form / be of order
m, and the given number N be divisible by the m th powers dj n , d 2 n , ..., the
derived representations of N by f coincide with the primitive representations
. N N ,
of -75 , -r= , . . . by the same torm.
1 2
80. Problems of the Transformation and Equivalence of Forms. A form
/(a/!, x' z , ...,x' n ) is said to be contained in another form / (x lt x 2 , ...,), when
/ arises from/! by a linear transformation of the type
3/1 = ^i,\X i i a~i t2 X 2 ...... "T ^*i,n*k n>
X 2 = a 2> iX i + a 2i2 X 2 T T (^2,n X n>
in which the coefficients a iti are integral numbers and the determinant is dif-
ferent from zero *. This transformation we may, for brevity, describe as the
transformation |a|. When |a| is a unit-transformation, i.e. when the deter-
minant of |a| is a positive or negative unit, the inverse transformation of |a|,
which will be a transformation of the same type as ] a | , will have all its co-
efficients integral numbers; so that in this case/, which contains/, is also
contained in it. When each of two forms is thus contained in the other, they
are said to be equivalent. If/ contain/, and/ contain/,/ will contain/;
for if / be changed into / by the transformation [ a \ , and / into / by the
transformation |6|, it is clear that/ will be changed into/ by a transformation
\T\, of which the constituents are defined by the equation
The transformation | T \ is said to be compounded of the transformations | a |
and | b | , and this composition is expressed by the symbolic equation
in which it is to be observed that the order of the symbols | a \ and | b ] is not,
in general, convertible. When, in particular, / is equivalent to /, and / to
* Gauss says that f 3 is contained in /,, even when the determinant of transformation is zero
(Disq. Arith., Art. 215). But we shall find it more convenient to retain the restriction specified
in the text.
Art. 80.]
REPORT ON THE THEORY OF NUMBERS.
165
Jl, fi is equivalent to f 3 ; i.e. forms which are equivalent to the same form are
equivalent to one another. All the forms, therefore, which are equivalent to
one and the same form, may be considered as forming a class. All the in-
variants of any two equivalent forms have the same values ; but it is not
true, conversely, that two forms which have the same invariants are necessarily
equivalent. Nevertheless it may be conjectured that all forms of the same sort
(i.e. of the same degree, and the same number of indeterminates), the invariants
of which have the same values, distribute themselves into a finite number of
classes ; and this conjectural proposition is certainly true for binary forms of
all orders, and for quadratic forms of any number of indeterminates. It is
readily seen that if a number be capable of representation by one of two equi-
valent forms, it is also capable of representation by the other ; and that the
number of representations is either finite for both, or infinite for both, and, if
finite, is the same for each. The general problem, therefore, of the representa-
tion of numbers (which we have already enunciated) suggests naturally the
following, which we may term that of the equivalence of forms : ' Given two
forms (of the same sort), of which the invariants have equal values, to find
whether they are, or are not, equivalent, and if they are, to assign all the
transformations of either of them into the other.' The number of transforma-
tions may be either finite or infinite ; if finite, the transformations themselves,
if infinite, general formulae containing them, are required for the complete
solution of the problem.
When /! is not equivalent to, but contains f 2 , the invariants of f 2 are
derived from those of f^ by multiplication with certain powers of the modulus
(i.e. of the determinant) of the transformation by which f t is changed into f 2 ;
viz. if / be an invariant of f lt and if i and ra be the orders of /, and o
or
f 2 , the corresponding invariant of f 2 is a"/, a denoting the modulus of trans-
formation, and the number - - being always integral. This observation enables
7i
us to enunciate with precision a problem in which the preceding is included :
' Given two forms, of which the invariants have values consistent with the
supposition that one of them contains the other, to find whether this suppo-
sition is true or not, and, if it is, to find all the transformations of the one
form into the other.' But, in every case, the solution of the problem in this
more general form may be made to depend on the solution of the problem of
equivalence. For every transformation of order n, and modulus a, arises, in
166
REPORT ON THE THEORY OF NUMBERS.
[Art. 81.
one way and in one only, from the composition of two transformations a \ and
||, of which the latter is a unit-transformation, and the former one of the
finite number of transformations included in the formula
0, M S ,
o, o,
1 ,n
M
::.i<
(C)
0, 0, 0, , M n
in which ^ y.n 2 x ... x n n = a, and 0<k itj <ft i (Phil. Trans., vol. cli. p. 312).
To determine, therefore, whether the form / x can be transformed into f z by a
transformation of modulus a, we apply to f^ all the transformations (C) in suc-
cession, obtaining a series of transformed forms (p 1} < 2 > If none of the forms
</> are equivalent to f 2 , f^ cannot contain f 2 ; but if one or more of the forms
<p be equivalent to f 2 , f^ will contain f 2 , and all its transformations into f z may
be obtained as soon as the transformations of the forms < into f 2 have been
determined. This is the method proposed by Gauss for binary quadratic forms
(Disq. Arith., Arts. 213,214); it is evidently of universal application; but the
following modification of it possesses a certain advantage. Instead of represent-
ing | T\ by the formula ] T\ = \a\ x \v\, we may employ the formula | T\ = \v\ x ]aj,
in which | v \ is a unit- transformation as before, and | a \ is one of the transforma-
tions included in the formula (C), where, however, the inequality ^ k itj < M,-
is to be replaced by <&, ,J<MJ ; the transformations thus defined we shall call
the transformations (C'). If we now apply to f z the inverse of each trans-
formation included in (C'), we shall obtain a series of forms fa, fa, fa, ... of
which the coefficients will not necessarily be integral numbers, because the
coefficients of the inverse transformations are not necessarily integral. If all
the forms <f) ly fa,... be fractional, or if none of those which are integral be
equivalent to f 1} f v cannot contain f 2 ; but if some of them be integral, and
equivalent to^, it is plain thatyj contains^, and that all the transformations
of fi into^ roay be obtained by means of the transformations of f into those
forms < which are equivalent to it. The advantage above referred to consists
in the circumstance that the rejection of the fractional forms <p diminishes the
number of the problems of equivalence which must be solved to obtain the com-
plete solution of the question proposed (compare Disq. Arith., Art. 284, and note.)
81. Automorphic Transformations. The unit-transformations by which a
2 3
form passes into itself are the automorphics of the form ; thus '' ' is an
Art. 81.] REPORT ON THE THEORY OF NUMBERS. 167
autornorphic of x* 3y 2 . When every invariant of a form is zero, the form
may pass into itself by transformations of which the modulus is different from
unity; for example, x- 4:xy + 4y 2 , a binary quadratic form of which the dis-
criminant is zero, passes into itself by the transformation
3 2
of which
1.
the modulus is 4. In like manner it is to be observed that when two 'forms
of the same sort have all their invariants equal to zero, it may happen that
each of them passes into the other by transformations of which the modulus
is not a unit. But in this Report we shall have no occasion to consider these
exceptional cases, whether of equivalence or of automorphism, and we shall
therefore employ these terms with reference to unit-transformations exclusively.
If 1 7 1 ! | and | T z \ be automorphics of a form /, | Tj | x | T 2 1 and T 2 x 1 7\ ] are also
automorphics of f; so that, in particular, every power of an automorphic is
also an automorphic. (The positive powers of a transformation are, of course,
the transformations which arise from compounding it continually with itself;
its negative powers are the positive powers of its inverse. See Mr. Cayley's
Memoir on the Theory of Matrices, Phil. Trans., vol. cxlviii. p. 17.) Hence,
if a form have a single automorphic, of which no two powers are identical,
it will have an infinite number of automorphics. The importance of auto-
morphic transformations in the solution of the problems of equivalence and
transformation will be apparent from the following considerations. If f- and
f 2 be two equivalent forms, h \ a given transformation of f^ into f 2 , a x | and
|a 2 | the general formulae representing all the automorphics of/ and/ respect-
ively, all the transformations of/ into / will be represented by either of the
formulae e^ | x | h \ or | h \ x | a 2 1. And again, if / contain /, and if we represent
by &j|, |A 2 ],... certain particular transformations of/ into /, obtained by
compounding each transformation (C), which gives a form ^> equivalent to/,
with some one transformation of (f) into f 2 , then all the transformations of /
into/ will be comprised in a finite number of formulae of the type
|a 2 | still denoting indefinitely any automorphic of/. Or, if we employ the
second method of the preceding article, the same transformations will be
represented by
where a, [ is any automorphic of/!, and \h^\, h 2 \, \h 3 '\, ...... are certain
particular transformations of / into f 2 , obtained in a manner sufficiently in-
dicated by the method itself. It appears, therefore, that when we know all
168 REPORT ON THE THEORY OF NUMBERS. [Art. 82.
the automorphics either of /j or f 2 , we can deduce all the transformations
of /i into / 2 , from one of those transformations when/ is equivalent to / 2 ,
and from a certain finite number of them when / t contains, but is not equi-
valent to, / 2 . We may add, that when one transformation of two equivalent
forms, and the automorphics of either of them are known, those of the other
are known also, for we evidently have the equation
|*]-|Aj-'x|4lx|A'|.
82. Problem of the Representation of Forms. We give the enunciation of
one other general problem, which may be said to occupy a middle place
between the problems of the representation of numbers, and of the equi-
valence of forms. By using a defective substitution of the type
X 1 = Q>\ t \X !+ Ctj >2 33 2+ T <*!, 2-^ n r
X 2 = 0/2.1'*' 1 ' ^2,2^ 2~r ' 2,n 2'E n r>
" ~~ ^n,!'" 1 ' < *n,2'* ; 2~i~
a (orm f l (x l , x 2> ...,x n ) may be changed into another f 2 (x\, x' 2 , ...,x' n _ r ) of
the same order but containing fewer indeterminates. The form f 2 is said to
be represented by f^ ; and the representation is proper or improper according
as the determinants of the system do not, or do, admit of any common divisor
besides unity. Our third general problem therefore is, ' Given two forms of
the same order, of which the first contains more indeterminates than the
second, to find whether the second can be represented (properly or improperly)
by the first, and, if it can, to assign all the representations of which it is
susceptible.' If the second form contain only one indeterminate (i.e. if it be
an expression of the form Ax n ), the problem reduces itself to that of the
representation of the number A by the form f v . If, again, f z contains as* many
indeterminates as f l} the problem becomes that of the transformation of yi
into f 2 . We may add that the problem of improper representation may be
made to depend on that of proper representation, by methods analogous to
those by which the problem of transformation depends on that of equivalence.
(See Disq. Arith., Art. 284, where Gauss treats of the improper representation
of binary by ternary quadratic forms.)
83. It is hardly necessary to state that what has been done towards ob-
taining a complete solution of these problems is but very little compared with
what remains to be done. Our knowledge of the algebra of homogeneous forms
(notwithstanding the accessions which it has received in recent times) is far
Art. 84.]
REPORT ON THE THEORY OF NUMBERS.
169
too incomplete to enable us even to attempt a solution of them co-extensive
with their general expression. And even if our algebra were so far advanced
as to supply us with that knowledge of the invariants and other concomitants
of homogeneous forms which is an essential preliminary to an investigation of
their arithmetical properties, it is probable that this arithmetical investigation
itself would present equal difficulties. The science, therefore, has as yet had
to confine itself to the study of particular sorts of forms ; and of these (ex-
cepting linear forms, and forms containing only one indeterminate) the only
sort of which our knowledge can be said to have any approach to completeness
are the binary quadratic forms, the first in order of simplicity, as they doubtless
are in importance. Of all other sorts of forms our knowledge, to say the least,
is fragmentary.
We shall arrange the researches of which we have now to speak in the
following order, according to the subjects to which they refer :
1. Binary Quadratic Forms.
2. Binary Cubic Forms.
3. Other Binary Forms.
4. Ternary Quadratic Forms.
5. Other Quadratic Forms.
6. Forms of order n decomposable into n linear factors.
The theory of linear forms (i.e. of linear indeterminate equations) we shall
refer to hereafter. That of forms containing only one indeterminate will not
require any further notice.
(1) Binary Quadratic Forms.
84. Instead of confining our attention exclusively to the most recent re-
searches in the Theory of Quadratic Forms, we propose, in the following articles,
to give a brief but systematic resume of the theory itself, as it appears in the
Disq. Arith., introducing, in their proper places, notices, as full as our limits
will admit, of the results obtained by later mathematicians. We adopt this
method, partly to render the later researches themselves more easily intel-
ligible, by showing their connexion with the whole theory; but partly also
in the hope of facilitating to some persons the study of the Fifth Section of
the Disq. Arith., which, probably owing to the obscurity of certain parts of
it, is even now too much neglected by mathematicians. This section is com-
posed, as Lejeune Dirichlet has observed (Crelle, vol. xix. p. 325), of two very
distinct parts. The results contained in the former of the two (Arts. 153-222)
z
1 70 REPORT ON THE THEORY OF NUMBERS. [Art. 85.
are for the most part those which had been already obtained by Euler, La-
grange, and Legendre; but they are completed in many respects; they are
derived, in part at least, from different principles, and are expressed in a
terminology which has been adopted by most subsequent writers. The second
part (Arts. 223-307) is occupied, after some preliminary disquisitions (Arts.
223-233), with the ulterior researches of Gauss himself. We proceed then to
give a summary of the definitions and theorems contained in the first of these
two portions.
85. Elementary Definitions. The quadratic form ax 2 + 2bxy + cy 2 is sym-
bolised by the formula (a, b, c) (x, y) 2 , or, when it is not necessary to specify
the indeterminates, by the simpler formula (a, b, c). The second coefficient is
always supposed to be even ; and an expression of the form px 2 + qxy + ry 2
(in which q is uneven) is not considered by Gauss as itself a quadratic form,
but as the half of the quadratic form (2p, q, 2r). The discriminant b 2 ac of
the form (a, 6, c) is called by Gauss the determinant of the form ; an expression
which at the present time it would be neither possible nor desirable to alter.
When two forms are equivalent, they are said to be properly equivalent if the
modulus of transformation is + 1, but improperly equivalent if it is 1. Only
those forms which are properly equivalent to one another are considered to
belong to the same class ; two forms which are only improperly equivalent are
said to belong to opposite classes. This distinction between proper and im-
proper equivalence is due to Gauss, and is of very great importance. In what
follows, unless the contrary is expressly specified, we shall use the terms equi-
valence and automorphism to denote proper equivalence and proper automor-
phism. It is readily seen that the greatest common divisors of a, 26, c, and
of a, b, c are the same for (a, b, c) and for every form equivalent to (a, b, c) ;
if each of those greatest common divisors is unity, (a, b, c) is a properly
primitive form, and the class of forms equivalent to (a, b, c) a properly primitive
class ; if the first greatest common divisor be 2, and the second 1, the form,
and the class of forms equivalent to it, are termed improperly primitive. Every
form which is not itself primitive, is a numerical multiple of some primitive
form of a less determinant, and is therefore called a derived form. Thus
x 2 + 3y 2 is a properly primitive form of det. -3, but 2x 2 + 2xy + 2y 2 is an impro-
perly primitive form of the same determinant; while 2x 2 + 6y 2 , 4x 2 + 4:xy + 4:y t
are derived forms of det. 12.
In all questions relating to the representation of numbers, or the equi-
valence of forms, it is sufficient to consider primitive forms, as the solution of
Art. 85.] REPORT ON THE THEORY OF NUMBERS. 171
these problems for derived forms is immediately deducible from their solution
for primitive forms ; but in certain investigations connected with the trans-
formation of forms the consideration of derived forms is indispensable. (The
problem of Art. 82- coincides with that of the representation of numbers, in the
case of binary forms of any order.)
The nature of the quadratic form (a, 6, c) depends very mainly on the
value of its determinant, which we shall symbolise by D. (1) If D = 0, the
form (a, b, c) reduces itself to an expression of the type m (px + qy) 2 , p and q
denoting two numbers relatively prime, and m being the greatest common
divisor of a, b, c. The arithmetical theory of such expressions, which are not
binary forms at all, since they are adequately represented by a formula such
as m X 1 , is so simple, and at the same time diverges so much from that of true
binary quadratic forms, that we shall not advert to it again in this Report,
and in all that follows the determinant is supposed to be different from zero.
(2) When D is a perfect positive square, the form (a, b, c) reduces itself to
an expression of the type m (p 1 x + q 1 y) (p 2 x + q 2 y), i.e. it becomes a product
of two linear forms. Owing to this circumstance the theory of forms of a square
determinant is so much simpler than that of other quadratic forms, that we
shall not enter into any details with regard to them, though it is not necessary
to exclude them (as is the case with forms of determinant zero) from those
investigations which relate simultaneously to the two remaining kinds of quad-
ratic forms ; viz. (3) those of a negative determinant, and (4) those of a positive
and not square determinant. An essential difference between these two kinds
of forms is, that whereas both positive and negative numbers can be repre-
sented by any form of positive and not square determinant, forms of a negative
determinant can represent either positive numbers only, or negative numbers
only. For if the roots of a + 2 b 6 + c 6 2 = be real, it is clear that ax 2 + 2bxy + cy*
will have values of different signs, when the ratio y : x falls between the two
roots and when it falls outside them ; but if the roots be imaginary, the form
will always obtain values having the same sign (viz. that of a or c), whatever
the ratio y : x may be. If (a, b, c) be a positive form (i. e. a form representing
positive numbers only) of a negative determinant D = A, ( a, b, c) is a
negative form of the same determinant, and can represent negative numbers
only. We see, therefore, that there are as many pos tive as negative classes
for any negative determinant ; and as everything that can be said about positive
forms or classes may be transferred at once, mutatis mutandis, to negative forms
and classes, we shall in what follows exclude the latter from consideration, and,
Z 2
172 REPORT ON THE THEORY OF NUMBERS. [Art. 86.
when we are speaking of forms of a negative determinant, confine ourselves
to the positive forms.
Since x*-Dy 2 , or (1, 0, - D), is a form of determinant D, we see that one
class at least of properly primitive forms exists for every determinant ; and the
class containing the form x z Dy* is called the principal class. Improperly
primitive forms only exist for those determinants which satisfy the condition
D = l, mod 4 ; since, if (a, 6, c) be improperly primitive, we have 6 = 1, mod 2,
a = c = 0, mod 2. But for every determinant satisfying this condition, one class
at least of improperly primitive forms exists; for (2,1, -- - ) is an im
properly primitive form of determinant D, and the class containing it may be
called the principal class of improperly primitive forms.
86. Reduction of the Problem of Representation to that of Equivalence.
The problem of the representation of numbers depends, first, on the solution
of a quadratic congruence, and, secondly, on the solution of a problem of equi-
valence. This dependence is established by the two following theorems :
(i.) ' When the number M admits of a primitive representation by (a, 6, c),
the quadratic congruence x 2 D = 0, mod M, is resoluble.'
For if am 2 + 2 bmn + en 2 = M be a primitive representation of M, let n, v
be two numbers satisfying the equation mv n/* = 1 ; we then find
(am 2 + 2bmn + cn 2 ) (a/x 2 + 26 / ui/ + ci' 2 ) = (am/* + 6 [mv + nfji] + cnv) 2 D;
or Q 2 = Z), mod M ; if i2 = am/u + 6 \mv + np"] + cnv.
We have already referred to this result in Art. 68.
The representation am? + 2 bmn + en 2 of the number M by the form (a, 6, c),
is said by Gauss to appertain to the value of the congruential radical +JD,
mod M. To understand this definition with precision, it is to be observed that
if in the expression of Q we replace n and v by any two other numbers satis-
fying the equation mv /* = !, the new value of Q will be of the form Q + kM ;
and conversely, values for /* and v can always be found which shall give to
amn + b [mv + nrii] + cnv any assigned value of the form Q + kM. Two different
representations of M appertaining to the same value of ,^/D, mod M, are said
to belong to the same set.
(ii.) ' If M admit of a primitive representation by the form (a, 6, c) apper-
taining to the value Q of ^/D, mod M, the two forms (a, b, c) and
are equivalent ; and conversely, if these two forms are equivalent, M admits
Art. 86.]
REPORT ON THE THEORY OF NUMBERS.
173
of a primitive representation by (a, b, c) appertaining to the value Q of
mod M:
To establish the first part of this theorem, we observe that the assertion
that M admits of a primitive representation by the form (a, b, c) appertaining
to the value & of ^/Z), mod M, implies the existence of four numbers TO, n, n, v,
satisfying the equations
mv nn = l,
If, therefore, we apply to (a, b, c) the transformation
TO,
n, v
, the resulting form
will have M and Q for its first and second coefficients respectively; its third
Q 2 D
coefficient will therefore be ^F > because its determinant must be D ; i. e. the
T?
two forms (a, b, c) and (M, Q, ^, ) are equivalent. And, conversely, the
equivalence of the two forms (a, b, c) and
<* * V)
fYYl II
implies the existence of a transformation of (a, b, c) into
n, v
i.e. it implies the existence of four numbers m, n, n, v, satisfying the equations
(k) ; or, finally, of a primitive representation of M by (a, b, c) appertaining to
the value Q of *JD, mod M.
If (A, B, C] be a form equivalent to a form (a, b, c) by which
M= am? + 2 bmn + en 2
is represented, and if ' be a transformation of (A, B, C) into (a, b, c), it
is clear that
(A, B, <J) (am + fin, ym + Sri) 2 = (a, b, c) (TO, rif = M.
Two such representations of M by equivalent forms are called corresponding
representations ; and we may enunciate the theorem, ' Corresponding repre-
sentations of the same number M by equivalent forms appertain to the same
value of the expression ^/D, mod M ; ' the truth of this is evident from the
174 REPORT ON THE THEORY OF NUMBERS. [Art 87.
nature of the function Amu + B\mv + nfi] + Cnv, which is a covariant (in respect
of m, n and n, v) to Ax* + 2Bxy+Cy*.
To obtain, therefore, all the primitive representations of a given number
by a given form (a, b, c), we investigate all the values of the expression ^/D,
mod M. If Qj, Q 2 , . . . be those values, we next compare each of the forms
(* * V)
with (a, b, c). If none of them be equivalent to (a, b, c), M does not admit of
primitive representation by (a, 6, c) ; but if one or more of them, as
be equivalent to (a, 6, c), let
formations of (a, b, c) into
y,l
be the formula exhibiting all the trans-
then all the primitive representations of M by (a, b, c), which appertain to the
value Qj of ^/D, mod M, are contained in the formula
(a,b,c)(a, y y = M.
87. Determination of the number of Sets of Representations. It appears
from what has preceded, that if S denote a system of representative forms of
determinant D (i.e. a system of forms containing one form, and only one, for
every class of forms of determinant D), the number of different sets of pri-
mitive representations of M by the forms of S is equal to the number of
different solutions of the congruence x 2 = D, mod M. If, in particular, M be
uneven and prime to D, it is clear that M can only be represented by properly
primitive forms ; and in this case the number of solutions of the congruence
sc 2 = D, mod M, i.e. the number of sets of primitive representations of M by
the properly primitive forms contained in S, is expressed by either of the two
formulae _ / Z\\ /Z>\
n (! + (-)), or Z( T ),
in which p and S denote respectively the prime divisors of M, and those divisors
of M which are divisible by no square ; while ( ) and (-r) are the quadratic
symbols of Lagrange and Jacobi (see Arts. 16, 17, 68, 76). If p. denote the
number of different primes dividing M, the common value of the two expressions
n(i+<f )) -a z,
Art. 88.] REPORT ON THE THEORY OF NUMBERS. 175
is 2* 1 or zero, according as the condition ( J = 1 is satisfied by every prime
divisor of M, or is not satisfied by one or more of them. When D is = 1, mod 4,
S will certainly contain improperly primitive forms ; and the unevenly even
number 2M (where M is still supposed prime to D) will admit of primitive
representation only by the improperly primitive forms contained in S (for if Q
Q2 T)
denote any root of the congruence x 2 =D, mod 2M, Q will be uneven, ^vj--
even, and the form (M, Q, ) will be improperly primitive). And the
number of sets of primitive representations of 2 If by these improperly primitive
forms will be the same as the number of sets of primitive representations of
M by the properly primitive forms in S.
The problem of obtaining the derived representations of M by (a, b, c)
depends on that of finding the primitive representations of a given number
by a given form (see Art. 79). Two derived representations of M are said to
belong to the same set, when the greatest common divisor of the indetermi-
nates, which we will symbolise by ta, is the same for each, and when the two
, M
primitive representations of , from which they are derived, appertain to
/ M
the same value of +JD, mod . Adopting this definition, we may enunciate
the theorem, ' If M be an uneven number prime to D, the whole number of
sets of representations of M (and if D = 1, mod 4, of 2Jf) by a system of re-
presentative forms of determinant D is 2 (-,-) ; d denoting any divisor of D.'
We may add that, as before, M will be represented only by properly primitive
forms ; and, when D = 1, mod 4, 2 M only by improperly primitive forms *.
88. Reduction of the Problem of Transformation to that of Equivalence.
It has been shown in Art. 80, that the general problem, ' Given two forms of
unequal determinants, to decide whether one of them contains the other, and
if so, to find all the transformations of the containing into the contained form,'
* The theorems of .this Article will not be found in the Disq. Arith. If, in their expression,
we transform the symbols (-~-)> (-r) by the law of reciprocity, we obtain results which coincide
with those given by Lejeune Dirichlet in his memoir, ' Kechercb.es sur 1'application etc.,' sect. 7
(Crelle, vol. xxi. pp. 1-6).
176
REPORT ON THE THEORY OF NUMBERS.
[Art. 89.
can be reduced to the simpler problem of the equivalence of forms. For the
sake of clearness we shall here point out how the first of the two general
methods of that article is to be applied to quadratic forms. If of two forms
/ and F the former contain the latter, the determinant of F is a multiple of
that of /by a square number, viz. by the square of the modulus of transforma-
tion. Let the determinant of / be D, and that of F, De z ; also let m and M
be any two conjugate divisors of e, so that mi*. = e. Then every transformation
of which the modulus is e may be expressed in one way, and one only, by the
in which k denotes one of the numbers 0, 1, 2, 3, ...,
is any unit-transformation whatever. If, therefore, we apply
,k
formula
m, k
0,M
X
,
r,
|8
a
m 1, and
a, ft
is i
im
7 ,S
*/
to the form f all the transformations included in the formula (of which
0, /x
the number is equal to the sum of the divisors of e), we shall obtain a series
of forms <f> lt <^ 2) of determinant De z . If none of these forms be equivalent
to F, F is certainly not contained in /; but if one or more of them, for example,
m, k
<|>, arising from the transformation
, is equivalent to F, let
repre
one of the transformations included in the formula
If we take
sent indefinitely any transformation of <f> into F; then/ passes into F by any
m, k
O,M!
in succession for < every form in the series <f> lt <j) 2 , ... which is equivalent to F,
it is readily seen that the transformations of/ into F, which are thus obtained,
are all different, and that they include all possible transformations of /into F.
We have supposed the number e to be positive, i.e. we have supposed that
/ contains F properly. To decide whether / contains F improperly, we have
only to examine whether any of the forms <j) 1 , (j) 2 , .., be improperly equivalent
to F ', and if any one of them be so, to combine the transformation of / into
it, with its (improper) transformations into F.
89. Problem of Equivalence. It remains to speak of the problem of equi-
valence. Of the three parts of which this problem consists, viz. (1) to decide
whether two given forms are equivalent or not, (2) if they are, to obtain a single
transformation of one form into the other, and (3) from a single transformation
to deduce all the transformations, the last only admits of being treated by a
method equally applicable to forms of a positive and negative determinant.
We shall therefore consider it first. The solution which Gauss has given of it
Art. 89.] REPORT ON THE THEORY OF NUMBERS. 177
(Disq. Arith., Art. 162) depends on principles which are concealed (as is fre-
quently the case in the Disquisitiones Arithmeticse) by the synthetical form in
which he has expressed it. We shall not therefore repeat the details of his
solution, but shall endeavour to point out the basis on which it rests.
Let /=(, b, c) (x, y) 2 be transformed into F=(A, B, (J) (x, y) 2 by two
different, but similar, transformations,
o , A
and
i ft
; i.e. by two trans-
7o> <J
formations of which the determinants are equal in sign as well as in magnitude
to the same positive or negative number e. Let also, for brevity,
so that f(X , F ) =f(X 1} FJ) = F(x, y) ; we have then the algebraical theorem
' The homogeneous functions F (x, y) and X Y l X-^ F differ only by a
numerical factor, not containing x or y.'
The truth of this theorem is independent of the supposition that the
coefficients of the given forms and given transformations are integral numbers.
Its demonstration is implicitly contained in the formulae given by Gauss ; or
it may be verified more indirectly by the consideration, that if w be a root of
the equation a + 2ba> + cca 2 = Q, we have, simultaneously,
12 denoting in each case the same root of the equation + + = , an
assertion which would not be true, if the equal determinants
a i ^i ft 7i were of opposite signs. Hence the equation
Q
coincides with the equation
A+ZBQ+CQ'-O;
i.e. JToFj JTjFo is identical (if we neglect a factor not containing x or y) with
F(x,y).
Comparing this conclusion with the identity
[F(x,y)]*=f(X ,Y )
we obtain a second result of the same kind
'The function a X X, + b ( X F, + X, F ) + c F Y, differs from F (x, y) only
by a numerical factor not containing x or y.'
Let m be the greatest common divisor of A, 2 B, and (7; U and T the
A a
178
REPORT ON THE THEORY OF NUMBERS.
[Art. 89.
greatest common divisors of the coefficients of x 2 , xy, and y 2 in X Y 1 X 1 Y
and aX Xi + b(X Y l + X l Y ) + cY Y 1 respectively; m being a positive integer,
but the signs of U and T being fixed by the equations
F(x,y) _ X.Y.-X.Y,
~ ~~
T
which are implied by the two algebraical theorems that have preceded ; the
numbers T, U, and m will satisfy the equation 7"-Z)Z7' = m J , which is obtained
by combining the equations (A) and (&), and will serve to express the relation
which subsists between the transformations and l . Solving the
7o, S 7 1( <J,
equations
lib
for Xj. and Y lt we find mX, = (T- bU)X - cU Y ,
mY,= all X + (T+bU)Y ;
or, finally, equating the coefficients of x and y,
1
= x
m
Ta - U (ba + C 7o ), m - U (6ft + C S )
1
= x
m
(C)
T-bU, -cU
aU, T+bU
If we suppose the complete solution of the indeterminate equation
T 2 DU" 1 fff to be known, the formula (C) supplies us with a complete solu-
tion of the problem, ' Given one transformation of / into F, to deduce all
the similar transformations of / into F! For if we suppose in that formula
that T and U denote indefinitely any two numbers satisfying the indeterminate
equation, it will appear (1) that every transformation of/ into F is contained
in (C) ; (2) that every transformation contained in (C) is a transformation of
/into F ; (3) that no two transformations contained in (C), and corresponding
to different values of T and U, are identical. Only it is to be observed that
the transformations (C) are not, in general, all integral. They are so, however,
when e, the modulus of transformation, is a unit, a supposition which we have
not yet introduced ; i.e. when the forms / and F are either properly or im-
properly equivalent ; because - , , and - - are then evidently integral ;
. , , T+bU , T-bU
whence it may be inferred that - and - - are so too.
Art. 90.]
REPORT ON THE THEORY OF NUMBERS.
179
90. Expression for the Automorphics of a Quadratic Form. To find the
automorphics of any quadratic form it is sufficient to consider the case of a
primitive form. Putting then f = F, and taking for
formation
1,0
0,1
To,
the identical trans-
, we obtain from the formula (C) the following general expres-
sion for the automorphics ofy,
j_ T-bU, -cU
~~ m X aU, T+bU
where m = l, or 2, according as f is properly or improperly primitive. The
nature of this expression for the automorphics depends on the value of D.
If D be positive and not square, let us represent the least positive numbers
satisfying the equation T 72 Z)7 2 = m 2 by 2\ and U^ ; we then have, by a known
theorem, the following formula for all the solutions in which T is positive,
m m
k denoting any positive or negative integral number.
ft
From this we can infer that if
be the automorphic in the formula
(D), arising from the values T ly U t of T and U, all the other proper auto-
morphics are powers of
i, ft
, and are included in the formula
i,ft
representing one or other of the identical transformations
1,0
0,1
and
-1,
0, -1
If D be a negative number, the only solutions of the equation
identical transformations
(except in two cases presently to be noticed) are T= m, 7 = 0. Hence
the only proper automorphics of a form of negative determinant are the two
and . The two excepted cases are
(1) D=-l, m = l; (2) D=-3, m = 2.
In the former case we have for T and U the four values +1,0, and 0, + 1 ;
whence the proper automorphics of a form of det. 1 are the four transforma-
Aa 2
180 REPORT ON THE THEORY OF NUMBERS. [Art. 91.
-6, -C *
tions supplied by the formula
a, b
If D = 3, m = 2, the solutions of
T = 4 are six in all, viz. +2, 0; +1,1; and + 1, -1; whence six
automorphics, comprised in the formula
k
exist for an improperly primitive form of det. 3. We may add that in each
of these two cases, in addition to the proper automorphics we have found, there
exist an equal number of improper automorphics.
From the formula (C), compared with the theory of representation contained
in Art. 86, it follows that if (a, b, c) (a, <y) 2 = M be any representation of M by
(a, 6, c), all the representations of the same set are included in the formula
Ty +
L m m
For forms of a positive and not square determinant the number of represen-
tations in each set is therefore infinite. For forms of a negative determinant
the number of representations in each set is, in general, two ; and if [a, 7] be
one of them, the other is [ a, 7]. But if the determinant be 1, or if the
form be derived from a form of det. 1, the number of representations in
each set is four ; and if the form be an improperly primitive form of det.
3, or be derived from such a form, the number of representations in each
set is six.
91. Expression for the Automorphics Method of Lejeune Dirichlet. We
have inferred the expression (D) of the automorphics of f, from the formula
(C) of which it is a particular case. But it is plain, from the general theory
of Art. 81, that, when f and F are equivalent, we can conversely infer the
formula (C) from (D). This method has been preferred by Lejeune Dirichlet,
who obtains the automorphics of a primitive form/=(a, b, c), of which the
determinant is not a positive square, by the following process (Crelle, vol. xxiv.
p. 324). If
be any rational automorphic of/, we have evidently,
7,S
a(ax 2 + 2 bxy + cy*) = [ax + (b+ JD}y] [ax + (b
= [(aa+[b +
[(aa+[b-
an equation which, for brevity, we may write
Art. 91.]
REPORT ON THE THEORY OF NUMBERS.
181
and which implies one or other of the two following systems :
h) n n P P Pi Si- Pz _ &
v / fiw x i^j ft f\ > T> ~ ~ri i
If (1) be the system which is satisfied by
P2
, let
a, ft
7 ,S
A m- Jf 2 m
T and U denoting rational numbers, and m still representing the greatest
common divisor of a, 26, c. These assumptions are legitimate, because ~
and - contain no irrationality but */D, and are conjugate with regard to
\/Z). Substituting in the equations
Pi $1 n
^ = & = _L ( T - UVD),
for PI, p 2 ', q l} q 2 ; P lt P 2 ; Q lt Q 2 , the expressions which these letters represent,
and equating the rational and irrational parts, we find
a, ft
J L<
m*
T-bU, -cU
aU,T+bU '
In this expression T and U satisfy the equation T 2 - DU 1 = m\ because
Pi p 2 = P 1 P 2 . From this we infer that acS 7 = 1 ; further, if we now introduce
the condition that a, /3, y, $ are to be integral and not merely rational numbers,
it will Mow, because 7, S -a, -ft are integral, that U, U, U are
m mm
also integral ; i.e. that U itself, and consequently T, is integral ; so that the
formula at which we have arrived coincides exactly with the formula (D). The
system (2), treated in a similar manner, leads to the conclusion a$ fty= 1 ;
whence it follows that that system can be satisfied by no proper automorphic
This method, as Dirichlet observes, has the advantage of putting in a clear
light the difference between proper and improper automorphism. A proper
automorphic changes each of the two factors, into which the form may be de-
composed, into a multiple of itself by a complex unit of the form
f
182 REPORT ON THE THEORY OF NUMBERS. [Art. 92.
whereas improper automorphics, which only exist for particular kinds of forms,
change each factor into a multiple of the other. A similar distinction subsists
between proper and improper equivalence ; the radical VD entering with the
same sign, or with opposite signs, into the factors which are transformed into
one another, according as the transformation is proper or improper.
92. Problem of Equivalence Forms of a Negative Determinant. To com-
plete the solution of the problem of equivalence, we consider, first, forms of a
negative, and then those of a positive and not square determinant.
A form (a, b, c) of a negative determinant D = A, which satisfies the
conditions enunciated in the following Table, is called a reduced form. The
symbols [26] etc. are used to denote the absolute values of the quantities
enclosed within the brackets.
General Conditions. Special Conditions.
1. [2&]S [a].
2.
1. If a = c, 6 = 0.
2. If [26] = [a], 6 = 0.
3. [a]<[c].
The essential character of a reduced form is sufficiently expressed by the two
symmetrical conditions [2 6]^ [a], and [26]^[c]. The third general condition
(which combined with the first implies the second), and the special conditions,
are, it may be said, artificial restrictions, intended to enable us to enunciate
with precision the theorem that ' every class contains one, and only one, re-
duced form.'
To show that one reduced form always exists in any given class, we select
from the given class all those forms in which the coefficient of x 2 is the least ;
and again, from those forms we select that one form, (a, 6, c), or those two
forms, (a, 6, c) and (a, b, c), in which the coefficient of y 2 is the least. The
single form (a, b, c), or the two forms (a, b, c), (a, 6, c), thus obtained, will,
it is easy to see, satisfy the general conditions ; and since, if a = c, or again
if [2 b] = [a], the opposite forms (a, b, c) and (a, 6, c), each of which satisfies
the general conditions, are equivalent, and therefore both belong to the given
class, it is clear that a form always exists satisfying the special conditions
proper to these cases. That only one reduced form exists in each class may
be proved by employing a principle due to Legendre (The"orie des Nombres,
voL i. p. 77) :
'If/=(a, b, c) be a form satisfying the general conditions for a reduced
form, /(I, 0), or a is the least number (other than zero) which can be rep re-
Art. 92.]
REPORT ON THE THEORY OF NUMBERS.
183
sented by/; and/(0, 1) or c is the least number which can be represented
by/ with any value of the second indeterminate different from zero.'
For, if we wish to find the least numbers that can be represented by /,
it will be sufficient to attribute positive values to x and y in the formula
/= ax 2 2 bxy + cy 2 ,
in which we suppose b positive as well as a and c. But
f(x-l,y)=f(x, y)-2b(x-y)-(a-2b)x-a(x-l),
f(x,y-l)=f(x,y)-2b(y-x)-(c-2b)y-c(y-\);
from which equations it appears that if in the formula / (x, y) we diminish by
a unit that indeterminate which is not less than the other, we diminish, or
at least we do not increase, the value of / (x, y) ; a conclusion which leads
immediately to the principle enunciated by Legendre.
From this principle it follows that a form satisfying the general conditions
of reduction is the form, or one of the two opposite forms, to which we are
led by the process of selection above described. If, therefore, there be two
reduced forms in the same class, they must be two opposite forms (a, b, c)
and (a, b, c). But it is easily proved that two such opposite forms, each
satisfying the general conditions of reduction, cannot be equivalent, unless
either [2 b] = a, or a = c ; in which cases only one of the two forms satisfies the
special conditions. In every case, therefore, there exists one, and only one,
reduced form in each class.
To obtain the reduced form equivalent to a given form, we form a series
of contiguous forms, beginning with the given form and ending with the
reduced form (Disq. Arith., Art. 171). Two forms of the same determinant,
(a, b, c) and (a, b', c'), are said to be contiguous when
c = a', and b + b' = 0, mod a.
Two contiguous forms are always equivalent; for if b+'b f = na, the former
passes into the latter by the transformation '
Let, then, ( , 6 , a x ) be the given form of det. A, which is supposed not
to satisfy the general conditions for a reduced form. Let 6 + 6 1 = Miai, &i
denoting the minimum residue of &, mod a 1; so that pfej^ajj and let a 2
represent the integral number -
The form (c^, 6 1} 2 ) will be contiguous,
i
and therefore equivalent, to (a , &, j). Let a third form, (a 2 , 6 2 , a 3 ), be
similarly derived from (a l5 b lt a 2 ), and let the series of contiguous forms
(a , 6 , a,), (oj, &j, a 2 ), (a 2 , 6 2 , a 3 ), ... be continued until we arrive at a form
184
REPORT ON THE THEORY OF NUMBERS.
[Art. 92.
(a n , &, + ,), in which a n + ,^a n . We shall certainly arrive at such a form, or
we should have a series of numbers a,, 02, a,,, ... all represented by the form
(oo, 6 , Oj), and yet continually decreasing for ever ; whereas a form of negative
determinant can acquire only a finite number of values inferior to any given
limit. The form (a n , b n , a n+1 ), in which a n <a n + l , satisfies the general con-
ditions for a reduced form. For by the law of the series of forms [2 &]<;
and since a n <a n + 1 , we have also
Again, the process can always be terminated in such a manner as to give a
form satisfying the special conditions for a reduced form. If a n a n + 1 , and b n is
negative, instead of stopping at the form (, b n , a n ), we continue the process
one step further and obtain the reduced form (a n , b n ,a n ). If 26 n = a n ,
instead of the form (a n , &, a n + i), we take the form (a n , b n , a n + 1 ), which is
contiguous to (a n _i, b n _ l} a n ), for the concluding form of the series.
The transformation | T n by which (a , b , Oj) passes into the equivalent
reduced form (a n , b n , a n + 1 ), is
o, -i
1, v-i
where
0, -1
1, M2
M,- =
0, -1
1, M,
W
a.
or if we represent the successive convergents to the continued fraction
P P P
* o * i * I
-7*r- >' > so that
r ' o"'
/I V2
we may express | T n \ by the formula
p P
-*-!> -* n
The theory of the reduction of quadratic forms was first given by Lagrange.
(See his ' Kecherches d'Arithme'tique ' in the Nouveaux Me"moires de 1'Acade'mie
de Berlin for 1773 ; see also his additions to Euler's Algebra, Art. 32 ; a memoir
of Euler's, ' De insigni promotione scientiae numerorum,' Opusc. Anal., vol. ii.
p. 273, or Comment. Arith., vol. ii. p. 140 ; Legendre, The"orie des Nombree,
premiere partie, sect, viii ; Disq. Arith., Arts. 171-173 ; M. Hermite in Crelle's
Journal, vol. xli. p. 193.) The method is applicable to forms of a positive,
Art. 93.]
REPORT ON THE THEORY OF NUMBERS.
185
as well as to those of a negative determinant ; but when the determinant is
positive, the reduced forms are not, in general, all non-equivalent. When
the determinant is negative, it is as applicable to forms, of which the coeffi-
cients are any real quantities whatever, as to those of which the coefficients
are integral numbers. We shall revert hereafter to the consequences which
M. Hermite has deduced from this important observation.
We have now a complete solution of the problem of equivalence for forms
of a negative determinant. To decide whether two forms /j and / a of the
same negative determinant are equivalent or not, we have only to investigate
the reduced forms fa and fa equivalent to f^ and / 2 : according as fa and fa
are or are not identical, f^ and / 2 are or are not equivalent ; and if they are
equivalent, all the transformations of /j into / 2 are obtained, by compounding
the reducing transformation of f lt first, with the automorphics of fa, and then
with the inverse of the reducing transformation of / 2 .
93. Problem of Equivalence for Forms of a Positive and not Square De-
terminant. The solution of the problem of equivalence for forms of a positive
and not square determinant occupies a considerable space in the Disq. Arith.
(Arts. 183-196). But, as Lejeune Dirichlet has observed, in a memoir which
he has devoted to this problem (' Vereinfachung der Theorie der binaren
quadratischen Formen,' in the Memoirs of the Academy of Berlin for 1854,
or in Liouville, New Series, vol. ii. p. 353), the demonstrations relating to it
may be greatly abbreviated by employing certain known results of the theory
of continued fractions. The following method does not differ materially from
that proposed by Lejeune Dirichlet ; nor indeed is it, in principle, very distinct
from that of Gauss, the connexion of which with the theory of continued
fractions he has suppressed.
We shall suppose that the forms which we consider are primitive a sup-
position which involves no loss of generality; and we shall understand, in what
follows, by a 'quadratic equation,' an equation of the form a a + 2b + a 1 2 = 0,
in which 6 2 a a a 1 is positive, and a , &, <% are integral numbers without any
common divisor. Such a quadratic equation we shall symbolise by the formula
[#o b , oj, and we shall regard the two quadratic equations [a , b , Oj],
[ o> ~b , a i] as different. If ^/D denote the positive square root of
6 2 ctpOj, it is convenient to call
and -
the first and second roots of [ , 6 , aj respectively; so that if we change
Bb
186 REPORT ON THE THEORY OF NUMBERS. [Art. 93.
the sign of the equation throughout, we change at the same time the deno-
mination of the roots. Whenever, therefore, a root of a quadratic equation,
and the denomination of the root, are given, the quadratic equation itself is
given. It is readily seen that if two forms ( , fy i)> (A, -#o. A) be pro-
perly or improperly equivalent, so that
transforms (a , 6 , a t ) into
(A , B , AI), the corresponding roots of the quadratics
i.e. those which are connected by the relation to = - k , are of the same, or
a + pit
of opposite, denominations, according as the equivalence is proper or improper.
Let the first root of the equation [a , 6 , aj be developed in a continued
fraction, of which all the integral quotients are positive except the first,
which has the same sign as the root. In this process we obtain a perfectly
determinate series of transformed equations, each having a complete quotient
of the development for its first or second root, according as it occupies an
uneven or an even place in the series, counting from the proposed equation
inclusive. The complete quotients eventually form a period of an even number
of terms ; there exists therefore a corresponding period of transformed quadratic
equations, which will be of the type
[ a o> A i], [n & 02], [ 2 > &, s]> > [ a 2fc-i> &!t-i. o]-
Every equation of the period has one of its roots positive and greater than
unity, the other negative and less in absolute magnitude than unity; and if
we suppose (as we shall do) that we begin the period with an equation
occupying an uneven place in the series of transformed equations, the positive
root of any equation of the period will be its first or second root, according
as it occupies an uneven or an even place in the period.
To apply what has preceded to our present problem, we require the fol-
lowing lemma (see sect. 2 of Dirichlet's memoir, or M. Serret in Liouville,
vol. xv. p. 153).
'If (a and & be two irrational quantities connected by the relation
_L J?O
u> = , where a, /8, 7, S are integral and aS py= 1, the developments
a + pSJ
of and Q in a continued fraction will ultimately coincide, and the same
quotient will occupy an even or an uneven place in both developments alike,
if a5 l3y= +1, but an even place in the one, and an uneven place in the
other, if aS-Py= - 1.
Art. 93.]
REPORT ON THE THEORY OF NUMBERS.
187
A quadratic form (a , |S , a x ) of positive determinant is said to be reduced*
when the roots of [a , /3 , 04] are of opposite signs ; the absolute value of the
first root being greater, that of the second less than unity. A series of reduced
forms equivalent to any proposed form (a , 6 , c^) can always be found. For,
if the first root of [a , 6 , a-j] be developed in a continued fraction, and if its
period of equations (beginning with an equation occupying an uneven place
in the series of transformed equations) be represented as before by
the forms (a , $,, a^, (a x , -ft, a 2 ), ..., (a^^, - (3 2k _ 1} a )
will be all reduced and all equivalent to (a , b , a^). These forms, so deduced
from the development of the first root of the equation [a , &, aj, we shall
term the period of forms equivalent to (a , b 0> ctj), or, more briefly, the period
of (a fl , 6 , e^). It will be seen that each form of the period is contiguous
to that which precedes it, and that the first is contiguous to the last.
We can now obtain a complete solution of our problem. If (a , b , a^)
and (A , B , A^) are equivalent, the first roots of [a , b , aj and [A , B , A^\
will be corresponding roots, and the developments of these two roots will
ultimately coincide, giving one and the same period of complete quotients.
And, p : nce the same complete quotient will occur in an even or in an uneven
place alike in each development, it will be a root of the same denomination
in the quadratic equation determining it in each development. The period
of equations will therefore be precisely the same for each development ; and
the same equation may be taken as the first equation of each period. . Con-
sequently the periods of (a , &, a^, (A , B , A^) are identical. Two forms
therefore are or are not equivalent, according as their periods are or are not
identical. To obtain the transformations of (a , b , aj into (A , B , AJ, when
these two forms are equivalent, let the complete quotients in the development
of the first root of [a , b , aj be w 1; w 2 , > an d let the convergent immediately
preceding u> n + l be Similarly, let Q n + 1 and -~ be a complete quotient and a
Pn *
convergent in the development of the first root of [A , B , A^\. Then, if
"V = Q M (where n=M, mod 2), all the transformations of (a , 6 , c^) into (A , B , A^
are comprised in the formula
P P
x|r|x
* These reduced forms are not to be confounded with the reduced forms of the last article.
B b 2
REPORT ON THE THEORY OF NUMBERS. [Art. 93.
denoting any automorphic of the form corresponding to the equation of
which a>n + i or Q M +i IS a ro t-
It should be observed that a reduced form is always a form of its own
period. To prove this, we remark that reduced forms are of two kinds;
they are either such as (a , ft, aj), where the first root of [a , ft, aj is positive,
or such as (a lf -ft, a 2 ), where the first root of [a l5 -ft, a 2 ] is negative. Now
a reduced form such as (a , ft, c^) is evidently a form of its own period, for
the equation [a , ft, aj is itself an equation of the period in the development
of its first root. And a reduced form such as (a 1} -ft, a 2 ) is also a form of
its own period.* For if we develope the second root of [a 1( ft, a 2 ], we obtain
a period of equations of which [o 1} ft, a 2 ] is itself one. Let [a 2 , ft, a 3 ] be the
equation immediately following [a,, ft, a 2 ] in this period; then [a^ ft, a 2 ] is
an equation occupying an even place in the period of equations arising from
the development of the first root of [a 2 , ft, a 3 ], and consequently (a x , -ft, a 2 )
is a form in the period of (a 2 , ft, a 3 ) ; i.e. it is a form in its own period, because
it is equivalent to (a 2 , ft, a s ).
It follows from this that no reduced form can be equivalent to a given
form, unless it occur in the period of that form.
The inequalities satisfied by the roots of any equation of a period give
rise to certain inequalities which are satisfied by its coefficients. These in-
equalities (which are not all independent) are,
(i) [aj < 2 v'Z> ; [ft] < JD ; [,] < 2 VZ> ;
(ii) x/D - [ft] < [a ] < VD + [ft] ;
(iii) VD - [ft] < [aj < VD + [ft].
The same inequalities are, of course, satisfied by the coefficients of a reduced
form ; its middle coefficient is, moreover, positive. And, conversely, every
form whose middle coefficient is positive and whose coefficients satisfy these
inequalities is a reduced form.
* {Or thus [May, 1876]: Since (04, /3,, a 2 ) is properly equivalent to (a , /3 , a,), the first
root of [aj, /3j, aj gives the same period as the first root of [a , /3 , a,], the same equations occu-
pying even or uneven places in both periods alike. Hence the period of (a^ /3,, a 2 ) is the same
as the period of (a , /3 , a,).
[July, 1876.] By developing the first root of [a,, /3,, a 2 ] we do indeed obtain the period of
(Oj, /3,, a a ) : but not immediately. "We ought, therefore, here to prove the theorem that a reduced
form has always one antecedent and one consequent reduced form contiguous to it. }
Art. 94.]
EEPORT ON THE THEORY OF NUMBERS.
189
94. Improper Equivalence Ambiguous Forms and Classes. If it be required
to find whether two forms (a, b, c) and (a', b', c'} of the same positive or negative
determinant are or are not improperly equivalent, it will suffice to change
one of them, as (a, b, c), into its opposite (a, - b, c), and then to solve the
problem of proper equivalence for (a, - b, c) and (a', b', c). If it be found that
these two forms are properly equivalent, let | T represent any transformation
of the first into the second ; then the improper transformations of (a, b, c)
into (a', b', c') will be represented by the formula '
It may happen that two forms are both properly and improperly equivalent
to one another ; when this is the case, each of the two forms, and every form
of the class to which they belong, is improperly equivalent to itself, i.e. admits
of improper automorphics. A class consisting of such forms is said to be am-
biguous (classis anceps classe ambigue). An ambiguous form is a form (a, b, c)
in which 2b is divisible by a; if 26 = /ua, the ambiguous form is transformed
into itself by the improper automorphic ; and if T\ be the general
expression of its proper automorphics, all its improper automorphics are included
1,
by the formula
x | T\. Every ambiguous form belongs to an ambiguous
/*
0, -1
class, and, as we shall presently see, every ambiguous class contains ambiguous
forms.
To complete the theory of equivalence, we shall here briefly indicate the
solution of the problem, ' To decide whether a given form is improperly equi-
valent to itself or not, and if it is, to find its improper automorphics.'
When the determinant is negative, it follows from the principle that two
reduced forms cannot be equivalent, that no reduced form, the opposite of which
is different from it and is also a reduced form, can be improperly equivalent
to itself. Hence the only reduced forms which have improper automorphics
are those in which 6 = 0, or 2 b = a, or a c. In the two former cases the
0,1
reduced form is ambiguous, in the latter it has the improper automorphic
1,0
and is moreover contiguous and therefore equivalent to the ambiguous form
(2a 2b, a b, a). These considerations supply a sufficient criterion for de-
ciding whether a form of negative determinant is {improperly} equivalent to
itself or not. If it is, its improper automorphics are given by the formula
1 ; |T| denoting the reducing transformation of the given form,
190 REPORT ON THE THEORY OF NUMBERS. [Art. 94.
and | r I any improper automorphic of the reduced form. For forms of a
positive determinant,* we observe that if
(0 Po, l) (1> -ft, 2), , (2fe-l, -&1-1, o)
be the period of (a, b, c), the period of (a, b, c) is
( a O> fti-H 2&-i) ( a 2i-l> ftfc-2> a 2fc-2)> , ( a l, A a o)-
For (a, &, c) is equivalent to (a , ^ Z k-\> a 2t-i), because (a, 6, c) is equivalent
to (a 2 t-i> ftzk-K a o) ; an d, by a known theorem, the period of equations in the
development of the second root of (a, b, c) is
[ a O> Plk-H a 2fc-l]> [ a 2fc-I, &fc-2> 2fc-2]> > [ a l A a o].
the equation [a , /3 2 jt_i, a 2j-i] occupying an even place in the development;
this period is therefore the period of equations in the development of the
first root of [a , -/8 2 j;_i, 2fc-i]; *- e - tne period
( a O> Afc-U a 2fc-l)> ( a 2t-l> P2fc-2) a 2fc-2)> > ( a l> A)> a fl)
is the period of (a , /3 2 fc-i, a 2 fc-i), or, which is the same thing, of (a, b, c).
If we now suppose that (a, b, c) is improperly equivalent to itself, it will be
properly equivalent to (a, b, c) ; and these two forms will have the same period,
which we shall represent by (p , q , pj, (p lt q lt p 2 \ &c. If (_p x , q x , p^ + 1 ) be
any form of this period, the associate of (p^, q^, p\+\), i. e. the form (p\ + i, q^, p*),
will also be a form of the period, and the indices of these two forms in the
period will differ by an uneven number, because the signs of the numbers
P\> .PA. + I. are alternate. From this we can infer that there will be two
different forms in the period, each of which will be immediately preceded by
its own associate ; so that the type of the period will be
(Po, q ,pi), (Pi, qi,p 2 ), , (Pk-i, <lk-i,Pk),
(Pk, qk-i,pk-i}, (Pk-i, qk-2,pk-*), , (PI, qo,p<>},
where for simplicity we have supposed that (p , q , pj is one of the two
forms which is preceded by its associate; the other is (p k , q k _ 1} Pk-i)- These
two forms are ambiguous, for it follows from the contiguity of each form
to that which precedes it, that 2q = 0, mod^> ; 2(fr._ 1 = 0, mod^.. We arrive
therefore at the conclusion that the period of every ambiguous class contains
two ambiguous forms ; either of which enables us, as in the case of forms
of a negative determinant, to obtain all the improper automorphics of any
form of the class.
' {Here, again, it is not necessary to recur to the definition of the period of reduced forms equi-
valent to a given form; the associated period is a period of reduced forms equivalent to (a, b, c)}.
Art. 95.]
REPORT ON THE THEORY OF NUMBERS.
191
Gauss has shown (Disq. Arith., Art. 164), by an analysis which it is not
necessary to explain here, that if f contain F both properly and improperly,
an ambiguous form contained in f, and containing F, can always be assigned.
This theorem comprehends the result which we have incidentally obtained in
this article, that every ambiguous class contains ambiguous forms. (See also
a note by Dirichlet, in Liouville, New Series, vol. ii. p. 273.)
95. The important theorem, that for every positive or negative determi-
nant the number of classes is finite, is a consequence of the theory of reduc-
tion. To establish its truth, it is sufficient to employ the reduction of Lagrange
(Art. 92), which is applicable to forms of a positive ; determinant having inte-
gral coefficients no less than to forms of a negative determinant, and which
shows that in every class of forms of determinant D there exists one form at
least the coefficients of which satisfy the inequalities [26]s[a], [2fc]^[c].
These inequalities give, if D be negative, ac< f-D, [?>] = \/ - D ; and if D
be positive, [ac]< D, [5]^ \/\D. The number of forms whose coefficients satisfy
these inequalities is evidently limited ; therefore, a fortiori, the number of non-
equivalent classes is finite.
To construct a system of representative forms of det. D, we have only to
write down all the forms of det. D whose coefficients satisfy the preceding
inequalities, to which we may add []^[c]. If the determinant be negative,
it only remains to reject the forms which do not satisfy the special conditions ;
if it be positive, we must examine whether any of the forms which we have
written down are equivalent ; and, if so, retaining only one form out of each
group of equivalent forms, we shall have the representative system required.
A few particular cases of the theory merit attention from their simplicity.
If D= 1, there is but one class of forms, represented by x 2 + y 2 ; and by
the theorems of Arts. 87 and 90, the number of representations of any uneven
(or unevenly even) number by the form x 2 + y 2 is the quadruple of the excess of
the number of its divisors of the form 4 + 1, above the number of its divisors of
the form 4n + 3. (See Jacobi in Crelle's Journal, vol. xii. p. 169 ; Dirichlet, ibid.
vol. xxi. p. 3. In counting the solutions of the equation x 2 + y 2 = 2p, Jacobi con-
siders two solutions, such as x-f + y^ = 2p and x/ + y 2 2 = 2p, to be identical, when
a-, 2 = x 2 ", yj 2 = y 2 2 ; the number of solutions is thus a fourth part of the number
of representations.) In particular every prime of the form 4w + l (and the
double of every such prime) is capable of decomposition in one way, and one
only, into two squares relatively prime ; and, conversely, every uneven number
capable of such decomposition in one way only is a prime of the form 4w + 1.
192 REPORT ON THE THEORY OF NUMBERS. [Art. 96.
If D= -2, x 2 + '2y 2 represents the only class of forms; and every uneven
number can be represented by x* + 2y*, in twice as many ways as it has divisors
of either of the forms 8 n + 1, or 8n + 3, in excess of divisors of the forms Sn + 5,
or 8 + 7. (Dirichlet, loc. cit.) In particular every prime of either of the forms
8n + l or Sn + 3 is decomposable in one way, and in one only, into a square and
the double of a square.
Again, for each of the determinants 3 and - 7, there is but one properly
and one improperly primitive class, which may be represented by the forms
(1, 0, 3) and (2, 1, 2) ; (1, 0, 7) and (2, 1, 4). Uneven numbers are therefore
represented by x* + 3y 2 , in twice as many ways as they have divisors of the
form 3n + l, in excess of divisors of the form 3n-l ; and by x* + 7y 2 in twice
as many ways as they have divisors of the forms 7n + l, 2, 4, in excess of
divisors of the forms 7n + 3, 5, 6. Similarly, x 2 + ly 2 represents the only primi-
tive class of det. 4.
For each of the eleven positive determinants of the first century 2, 5, 13,
17, 29, 41, 53, 61, 73, 89, 97, there is but one properly primitive class ; there
is also for each of the ten uneven determinants one improperly primitive class.
Representing any one of these eleven numbers by D, by [T, U~\ the least
solution of T*-DU* = 1, and by If, an uneven positive number prime to D,
we may enunciate the theorem,
' The equation x 2 Dy z = M is capable of as many solutions in positive
numbers x and y, satisfying the conditions x<T*/M, y<U\/M, as M has
divisors of which D is a quadratic residue in excess of divisors of which D is
a quadratic non-residue.'
Thus the number of solutions of the equation x 2 2y* = M, where M is an
uneven number, and < x ^ 3 -JM, < y < 2 VM, is the excess of the divisors
of M of the forms 8n I above its divisors of the forms 8n + 3.
The conditions <x^ T\/M, <y < U VM, which are satisfied by one re-
presentation, and only one, in each set, are obtained by considerations to which
we shall hereafter refer (Art. 100).
96. The Pellian Equation. The two indeterminate equations,
T 2 -DU* = l and r-W = 4,
are, as we have seen, of primary importance in the theory of quadratic forms
of a positive and not square determinant. When the complete solution of these
equations is known, we can deduce, from a single representation of a number
by a form, every representation of the same set ; and, from a single trans-
formation of either of two equivalent forms into the other, every similar trans-
Art. 96.] REPORT ON v THE THEORY OF NUMBERS. 193
formation. The same equations also present themselves in the solution in
integral numbers of the general equation of the second degree containing two
indeterminates, and enable us in the principal case in which it admits an infinite
number of solutions to deduce them all from a certain finite number. This
fundamental importance of the equation T* DU* = 1 was first recognised by
Euler, who has left several memoirs relating to it (see Comment. Arith., vol. i.
pp. 4, 316 ; vol. ii. p. 35 ; also Euler's Algebra, vol. ii. cap. vii.) ; but the equa-
tion itself had already given rise to a discussion which forms a well-known
passage in the scientific history of the seventeenth century. Its solution was
proposed by Fermat (see the Commercium Epistolicum of Wallis. Ep. 8) as a
challenge to the English mathematicians, and especially to Wallis. The problem
was at first misunderstood by Lord Brouncker and Wallis, who each gave a
method for its solution in fractional numbers ; not attending to the restriction
to integral numbers implied, though not expressed, in Format's enunciation,
without which the problem is of a very elementary character. Ultimately,
however, they obtained a complete solution by a method, which Wallis describes
in the Comm. Epist., Epp. 17 (postscript) and 19, and in his Algebra, capp.
xcviii. and xcix., attributing it to Lord Brouncker, though he seems himself
to have had some share in its invention. This method is the same as that
which is given by Euler in his Algebra, and in the first of the memoirs above
cited, and which is attributed by him to Pell *. It differs, in form at least,
from that now employed, and was evidently suggested by the artifices of sub-
stitution employed in Diophantine problems. It is most easily explained by
an example. If T* 13/" 2 = 1 be the equation proposed, the process would
stand thus :
(1) 3U<T<4:U; let T=BU + v l ; - 4 T + 6 7^ + w, 2 = 1,
(2) v 1 <U<2v 1 ~, let U= v l -\-v 2 ', 3v^ 2v 1 v 2 4i? 2 2 =1,
(3) v 2 <v l <2v 2 ; let v l = v 2 + v 3 ; 3v 2 2 +4rVv 3 +3v 3 2 =l,
\ / ^ r 1 i ' O 7 & 6 O ' O
(4) V 3 <v 2 <2v 3 ; let v 2 = v 3 -\-v t ',. 4v 3 2 2v 3 v 4 3v 4 2 = 1,
(5) v t < v 3 < 2i> 4 ; let v 3 = v 4 + 1> 5 ; v 4 2 + 6v 4 v 5 + 4 v s 2 = 1,
(6) 6v 5 < v 4 < 7v 5 ; let v^ = 6v 5 +v 6 ', 4v 5 2 Gv 5 v e v 6 2 =1,
* There does not seem to be any ground for attributing either the problem or its solution to Pell ;
and it is possible that Euler may have been misled by a confused recollection of the contents of "Wallis's
Algebra, in which an account is given of the method employed by Pell in solving Diophantine problems.
Nevertheless the equation T^DU" 1 = 1 is often called the Pellian equation after him, probably upon
Euler's authority.
C C
104 REPORT ON THE THEORY OF NUMBERS. [Art 96.
(7) r 6 < r, < 2r 6 ; let r 4 = r, + 1' 7 ; - 3 6 * + 2 r c r 7 + 4t' 7 2 = 1,
(8) v 7 < Vg < 2t- 7 ; let t' = v 7 + v 8 ; 3v, 2 - 4 r 7 1> 8 - 3i- 8 2 = 1,
(9) v, < v 7 < 2v 8 ; let v 7 = v s
(10) v 9 <v s <2v g ; let v 8 = v 9
In the last equation we may put , = !, r 10 = 0; whence 7*= 649, 7=180.
It will be seen that the success of the method depends on its leading at last
to an equation in which the coefficient of one of the indeterminates is + 1 .
Wallis does not prove that such an equation will always occur ; and the de-
monstration which he has given of the resolubility of the equation T* DU* = \
is inconclusive. (See his Algebra, cap. xcix ; the reader will find the paralogism
which vitiates 1 his reasoning in the proof of the lemma, upon wliich it depends ;
see also Lagrange's criticism in the 8th paragraph of the Additions to Euler's
Algebra ; and Gauss, Disq. Arith., Art. 202, note.) It is evident that the
method of solution employed by Wallis really consists in the successive deter-
T
mination of the integral quotients in the development of jj in a continued
T
fraction ; in addition to this, Euler observed that jj- is itself necessarily a con-
vergent to the value of \ f D ; so that to obtain the numbers T and U it suffices
to develope </D in a continued fraction. It is singular, however, that it never
seems to have occurred to him that, to complete the theory of the problem,
it was necessary to demonstrate that the equation is always resoluble, and that
all its solutions are given by the development of *JD. His memoir (Comment.
Arith., vol. i. p. 31 ) contains all the elements necessary to the demonstration,
but here, as in some other instances, Euler is satisfied with an induction
which does not amount to a rigorous proof. The first admissible proof of
the resolubility of the equation was given by Lagrange in the Melanges de
la Socie'te' de Turin, vol. iv. p. 41. He there shows that in the development
of VD, we shall obtain an infinite number of solutions of some equation of
the form T* DU* = A, and that, by multiplying together a sufficient number
of these equations, we can deduce solutions of the equation T^ DU^ V.
But the simpler demonstration of its solubility, which is now to be found
in most books on algebra, and which depends on the completion of the theory
(left unfinished by Euler) of the development of a quadratic surd in a con-
tinued fraction, was first given by Lagrange in the Hist, de 1' Academic de
Berlin for 1767 and 1768, vol. xxiii. p. 272, vol. xxiv. p. 236; and, in a
simpler form, in the Additions to Euler's Algebra, Art. 37. Lastly, Gauss,
Art. 96.]
REPORT ON THE THEORY OF NUMBERS.
195
who in the Disq. Arith. avoids the use of continued fractions, has shown
that if we form, by the method which he indicates, the period of any quadratic
form of det. D, we may infer the complete solution of the equation
from the automorphics of any reduced form, according as the form is properly
or improperly primitive. (Disq. Arith., Arts. 198 202.)
To express conveniently the principal theorems relating to these equations,
we employ the following notation *. The numerator of the continued fraction
11 1
Ql I ': 7~
q 2 + q 3 + ... q n
is called the cumulant of the numbers ft, q 2 , ...,q n> and is represented by the
symbol (q lt q 2 , q 3 ,...,q n ) ; the denominator is evidently the cumulant (q 2 , q 3 ,...,q n ).
Accents are sometimes employed to indicate that the first or last quotient
of a cumulant is to be omitted ; thus
'(ft, ft, ft, > ft) = (ft, ft, -. ?), (ft, ft, ft, > ?)' = (ft, ft, ft, > ft.-i),
'(ft 22, ><?)' = (#2) <?3, ,9'n-l)-
A periodic cumulant is represented by the notation (g l , q 2 , ...,q n ) x , the suffix
indicating the number of times which the period is repeated, and a point
being placed over the first and last quotients of the period. In what follows
m represents 1 or 2, according as we are considering the equation
r-Dtr=i, or =4.
(i.) If /KU /u 2 , ...,/u 2fc be the period of integral quotients in the development
of either root of a quadratic equation of determinant D, which we suppose
properly or improperly primitive according as m = l, or m = 2, the positive
numbers T x and U x which satisfy the equation T 2 DU 1 = ii^ are all contained
in the formulae j 1
m
Jk
o
-2 A,
where
* This notation is due to Euler (see Nov. Comm. Pet. vol. ix. p. 53, and the memoir already
cited, 'De usu novi algorithm! in Problemate Pelliano solvendo.' Comment Arith., vol. i. p. 316).
The convenient term ' cumulant ' has been introduced by Professor Sylvester (Phil. Trans., vol. cxliii.
p. 474), who has also suggested the use of accents to indicate the omission of initial or final quotients.
C C 2
196 REPORT ON THE THEORY OP NUMBERS. [Art. 96.
and a + 2/8 + aj0 2 = is the quadratic equation determining the quotient /u,,
in which we suppose for simplicity that 04 is positive.
If, in particular, we consider the quadratic equation 2 D = 0, or rather
2 -D-2a0 + 2 = 0, where a 2 .<Z)<(a+l) 2 , we have m=l, ^ = 2a, and we
find, by the symmetry of the period in this case,
+ A x ) = (a, /x 2 , n 3 , ...,M 2 i, 2 a, n 2 , ...,^k) x -i,
which are Euler's formulae for the solution of the equation T 1 DU* = 1.
(ii.) We have already observed (Art. 90) that when T-i and Z7, are known,
T x and U x are denned by the equation
= r^+U^D-e*
L m \'
m
JEither from this equation, or from the cumulantive formulae for T x , U x , we
infer that T x and U x satisfy the equation of finite differences,
m
so that the two series, of which T x and U x are the general terms, are each a
2 T
recurring series, the scale of relation being 1, -, 1.
It is convenient to observe that T_ X =T X \ but U_ x = U x .
(iii.) If we denote by -^ the imaginary arc
we have evidently
T x
m mi m mi
The analogy implied by these formulae enables us to transform many trigono-
metrical identities into formulae containing T x and U x . For example, from
the formulae cos (0 + 0) = C os <j> cos 6 + sin $ sin 0,
sin (( + 0) = sin (/> cos 6 + sin 6 cos $,
we have, putting ^> = x\^, = //, where a; and y are any positive or negative
integers, i
T XV =[T X T V
[T X U V T V U X }.
Art. 96.] REPORT ON THE THEORY OF NUMBERS.
(iv.) It is also found that
197
m
>" j * I
in mm
m
(v.) If 5 be any integral number whatever, we can always find a solution
[T x , Z7 X ] satisfying the congruences T^ = T = m, mod q, and U X =U = 0, mod <?.
If [7\, C/^] be the least solution satisfying these congruences, X will be less
than 2q, and the residues (mod q) of the terms of the two series T x and U x
will each form a period of X terms, so that we shall always have
If Z7 X ' be the first number of its series which is divisible by q, we shall have
either X' = X, or 2X' = X. In either case, the only numbers U which are divisible
by q, are those whose indices are divisible by X'; and the formula T mK ',
comprises all the solutions of the equation T 2 Dq 2 U 2 = m 2 . Thus, in solving
the equation T 2 DU 2 = m 2 , we can always substitute for D its quotient when
divided by its greatest square divisor. (See Lagrange, Additions to Euler's
Algebra, Art. 78. Gauss, Disq. Arith., Art. 201, Obs. 3 and 4.)
We may add, that if q be a prime (an uneven prime when m = 2), and if
q" and q* be the highest powers of q dividing U K and n respectively, (f+v- will
be the highest power of q dividing U nK . (Dirichlet, in Liouville's Journal, New
Series, vol. i. p. 76.)
(vi.) The methods of Lagrange and Gauss are applicable to the equation
J n DU 2 = 4, only when D = l, mod 4; because they suppose the existence
of an improperly primitive form of det. D. In all other cases the equation
F-DU* = 1 may be divided by 4, and reduced to the form r-DU' = l:
viz. if Z) = 0, mod 4, T is even; and if D = 2, or =3, mod 4, T and U are
both even. A similar reduction takes place if D = l, mod 8; the equation
T* DU* = 4: admitting in that case only even solutions. But if D = 5, mod 8,
T* DU* = 4 may or may not have uneven solutions ; and no criterion is known
for distinguishing ft priori these two cases. If T 72 DU* = 4: admit of uneven
solutions, its least solution [7\, U^\ will be uneven ; its even solutions will be
comprised in the formula [T 3n , U 3n ~\, and consequently [ijTU, 2^a] will represent
the solutions of T - DU* = 1.
(vii.) The equations T*-DU*= -4, T-DU*=-1 are not resoluble for
all values of D, but only for those values for which 1 is capable of represen-
IDS
REPORT ON THE THEORY OF NUMBERS.
[Art. 96.
tat ion by the principal form of det. D. Whenever the period of integral quo-
tients in the development of VD consists of an uneven number of terms, these
equations will be resoluble, and conversely. This will always happen when
D is a prime number of the form 4n + l, and may happen in many other cases,
but never can happen when D is divisible by any prime of the form 4n + 3.
If T'-DIF^ -1 be resoluble and [7\, Z7J be its least solution, the formula
[T 2n+l , 1/2 n + J contains all its solutions, and [T 2n , U 2n ] all the solutions of
T*-DU* = \. If, in addition to the supposition that T'-DU* - 1 is reso-
luble, we suppose that T* DU* = 4: admits of uneven solutions, T^. DU 2 4
will also admit of uneven solutions ; and if [ T lt U^\ be its least solution,
[^2n-M> /2 + lJ> L-*2n> ^2J> L2-*6n+3) 2^6n+3j> |_2 *6n> 2 ^J
will represent all the solutions of I 72 DU* = 4, =4, = 1, and =1, respect-
ively. It is evident that these considerations will frequently serve to abbreviate
the process of finding the least solution of T 2 DU^ = 1. (See a memoir of
Euler's in the Comment. Arith., vol. ii. p. 35.)
(viii.) The 'Canon Pellianus' of Degen (Havnise, 1817) contains a Table,
giving for every not square value of D less than 1000, the least solution of
the equation T 2 DU 2 = I, together with the development of \'D in a continued
fraction. Its arrangement will be seen in the following specimens :
357 18, 1, 8, (2)
1, 33, 4, 17
180
3401
97 9, 1, 5, 1, 1, (1, 1)
1, 16, 3, 11, 8, (9, 9)
6377352,
62809633.
The numbers hi the third and fourth rows are the least values of U and T
in the equation T* DU* = I. The first row of numbers is the period of integral
quotients in the development of */D : it is continued only as far as the middle
quotient, or the two middle quotients, after which the same quotients recur in
an inverse order. Thus,
180= (1,8,2,8,1);
3401 = (18, 1, 8, 2, 8, 1);
6377352 = (1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1) ;
62809633 = (9, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1).
Art. 96.]
REPORT ON THE THEORY OF NUMBERS.
199
The numbers in the second row are the denominators of the complete quo-
tients ; i.e. taken alternately positively and negatively, they are the extreme
coefficients in the equations of the period. Thus the period of equations for
v/357is [-33, -18,1], [1,18, -33], [-33, -15, 4], [4, 17, -17], [-17, -17, 4],
[4, 15, - 33]. The first half of the period of equations for V97 is [ - 16, - 9, 1],
[1, + 9, - 16], [ - 16, - 7, 3], [3, 8, - 11], [ - 11, - 3, 8], [8, 5, - 9], [ - 9, - 4, 9],
[9, + 5, - 8], [- 8, - 3, 11], [11, 8, - 3], [- 3, - 7, 16] ; the second half being com-
posed of the same equations in the same order but with their signs changed.
The middle coefficients of the equations are not given in the Table ; but if
be two consecutive equations, of which the former determines the integral
quotient M X , they may be successively calculated by the formula
x
Lagrange has proved that if x 2 D y- = H, and H be < VD, - is always a
*s
convergent to VD ; so that a number less than VD is or is not capable of repre-
sentation by the principal form of det. D, according as it is or is not included
among the numbers of the second row.
The second Table of the ' Canon' contains the least solution of the equation
T DV 1 1 for those values of D less than 1000 for which that equation
is resoluble.
Mr. Cayley (Crelle, vol. liii. p. 369) has calculated the least solution of
the equation T : Z)/ 2 = 4, or T 2 DV = 4, for every number D of the form
8n + 5 less than 1000, for which those equations are resoluble in uneven num-
bers. This Table, as well as Degen's second Table, is implicitly contained in
the first Table of the ' Canon,' as appears from the theorem of Lagrange just
cited.
(ix.) The theory of the equations T 2 DU* = 1 and = 4 is connected in a
remarkable manner with that of the division of the circle*. Let X = 2 / u + l
represent an uneven number divisible by k unequal primes, but having no
square divisor ; let also the numbers less than A and prime to it be repre-
sented by a or b, according as they satisfy the equation (r) = l> o
* See Dirichlet, ' Sur la maniere de resoudre 1'equation t*pu?= 1 au moyen des fonctions
circulaires,' Crelle, vol. xvii. p. 286. Also Jacobi B note on the division of the circle, Crelle,
vol. xxx. p. 173.
200 KEPORT ON THE THEORY OF NUMBERS. [Art. 97.
and let X= be the equation of the primitive X-th roots of unity. The form of
this equation (see Art. 59) implies that
In :-tt 26w
Ze~ + Ze *=(-!)*;
we have also the relation
2 of* 26tir
Ze^-Ze * = t>VA,
which is easily deducible from the formulae of Gauss (see Arts. 20 and 104
of this Report, or Dirichlet, Crelle, vol. xxi. pp. 141, 142). From these values
lain 2ft'ir 2aiir 2tfir
of Ze x and Ze * we infer that 2 II (x e x ) and 2H(x e A ) are two
quantities of the form Y+i^ZvK, and Yi^Z-SX, Fand Z denoting integral
functions of x with integral coefficients; i.e. that 4X = F 2 ( l^XZ 2 . From
this equation, which is a generalisation of that obtained by Gauss for the case
when X is a prime (Disq. Arith., Art. 357), we can deduce a solution of the
Zniw
equation 7 I2 -XF 2 = 4. In the formula 2D(x-e" T ~)= Y+i?Z>S\, let us first
write { for x, and then i for i, and let us denote by X it Y ( , Z t , X_ it Y_ it Z_ f
the values which X, Y, and Z acquire when i and i are written for x. We
thus find, denoting the number of numbers less than X and prime to it by X',
2 at* 2at'jr
or, writing
T for i[F,- Y_ i + \Z,Z_:\, U for \\i^Z, Y_ i + i-^Z_ { F,.],
and observing that X { X_ { = 1,
where it is easily seen that T and U are integral numbers. When n is even,
we may obtain a solution of the equation more simply by writing +1 or 1
for x. (See the notes of Jacobi and Dirichlet already referred to.)
It is to be observed, however, that the solution obtained by these methods
is not in general the least solution. Its ordinal place in the series of solutions
depends (as we shall hereafter see) on the number of classes of forms of det. D.
97. Solution of the General Indeterminate Equation of the second degree.
The solution of the indeterminate equation
ax 2 + 2 bxy + cy 2 + 2 dx + '2ey +/=
depends on the problem of the representation of a given number by a quadratic
Art. 97.]
REPORT ON THE THEORY OF NUMBERS.
201
form. We confine ourselves to the case which presents the greatest complexity,
that in which b 2 ac = D is a positive and not square number. The methods
of solution contained in Euler's Memoirs relating to it (see Comment. Arith.,
vol. i. pp. 4, 297, 549, 570, vol. ii. p. 263 ; and the Algebra, vol. ii. cap. vi.) are
incomplete in several respects : first, because Euler always assumes that a
single solution is known, and only proposes to deduce all the solutions from
it ; secondly, because it is not possible, from a given solution, to deduce any
other solutions than those which belong to the same set with the given
solution, whereas the equation may admit of solutions belonging to different
sets ; and lastly, because he gives no method for distinguishing between the
integral and fractional values contained in the formulae by which x and y
are expressed. The first complete solution of the problem was given by
Lagrange in his Memoir ' Sur la solution des Problemes Inde'termine's du
second degre' (Hist, de 1' Academic de Berlin for 1767, vol. xxiii. pp. 165-311).
But the following method of solution, which is different in some respects
and much simpler, will be found in a subsequent memoir, ' Nouvelle methode
pour re"soudre les problemes indetermines en nombres entiers ' (Hist, de 1' Aca-
demic de Berlin for 1767, vol. xxiv. p. 181) ; and in the Additions to Euler's
Algebra (paragraph 7). If we multiply by aD and write p for ax + by + d,
q for (b 2 ac) (y + bd ae), Mfor (bd ae) 2 (b 2 ac) (d 2 of), the given equation
becomes q 2 Dp 2 = M. Confining ourselves to the primitive representations
of M by q 2 Dp 2 (the derived representations, corresponding to the different
square divisors of M, are to be treated separately by the same method),
we see that, since p and M are prime, q is of the form Mr + Qp, where
r and Q are two new indeterminates of which the latter may be supposed
Q 2 D
On substituting this value for q, it will appear that N= =r^-
is necessarily integral, i.e. that Q is one of the roots of the congruence
fi 2 D = 0, mod M ; and the equation will assume the form
in which every admissible value of Q is to be employed in succession. The
development of either root of the equation N+2&6 + M6 2 = will give all
the values of p and r which satisfy the equation
Np* + 2Qpr + Mr 2 = 1,
because 1 is the minimum value which the form (A r , Q, M ) can assume. (See
the Additions, paragraph 2, and especially Arts. 33-35.) Or again, if we
apply the transformation of Art. 92 to the form (N, Q, M), we obtain an
Dd
202 REPORT ON THE THEORY OF NUMBERS. [Art. 98.
equation of the type Px'* + 2Qx / y > + Ry* = l, in which Q*-PR = D, and
P<JD; whence, if x" = Px'+Qy, we finally deduce x" 2 -Dy' 2 = P, all the
solutions of which (see Art. 96, viii.) are necessarily given by the development
of \/D in a continued fraction. Applying either of these methods (the latter
is not given in the Memoir, but only in the Additions to Euler's Algebra)
to every equation of the form Np* + 2lp* + Mr 2 = 1 which can be deduced
from the equation < Dp 2 = M, or from the equations of similar form obtained
by replacing M by the quotient which it leaves when divided by any one
of its square divisors, we obtain a finite number of formulae of the type
[T, U~\ denoting any solution of the equation 7 12 - DZ7 2 = 1. These formulae
are fractional ; but by attending to the principle of Art. 96, v., we can ascertain
for each pair of formulae whether they contain any integral values or not,
and if they do contain any, we can substitute for the single pair of fractional
formulae a finite number of pairs not containing any fraction.
The form in which the solution of this problem has been exhibited by
Gauss is remarkable for its elegance. Let
a, b, d
b, c, e = A,
d, e,f
and, representing by S the greatest common divisor of 6 2 ac, cd be, ae bd, let
J) , A , cd be ae bd
then, putting D'x = X+p, D'y=Y+q, we find aX* + 2bXY+cY* = D'X.
If [X ni Y n ~\ denote indefinitely any representation of D'A' by (a, 6, c), we have
only to separate (by Lagrange's method) those values of X n , Y n which satisfy
the congruences X n +p = 0, Y n + q = 0, mod D', from those which do not, and
we shall obtain a finite number of formulae, exhibiting the complete solution
required.
98. Distribution of Classes into Orders and Genera. The classes of forms
of any given positive or negative determinant D are divided by Gauss into
Orders, and the classes belonging to each order into Genera. Two classes,
represented by the forms (a, b, c), (a', b', c), belong to the same order,
when the greatest common divisors of a, b, c and a, 26, c are respectively
equal to those of a', b', c', and of a', 26', c'. Thus the properly primitive
Art. 98.]
EEPOET ON THE THEORY OF NUMBERS.
203
classes form an order by themselves ; and the improperly primitive classes form
another order. To obtain the subdivision of orders into genera, it is only
necessary to consider the primitive classes ; because we can deduce the sub-
division of a derived order of classes from the subdivision of the primitive order
from which it is derived. The subdivision into genera of the order of properly
primitive classes depends on the principles contained in the following equa-
tions, in which q is an uneven prime dividing D, m and m' uneven numbers
prime to q, and capable of representation by the same properly primitive form
of determinant D.
(ii.) If Z> = 3, mod 4, (-
(iii.) If D = 2, mod 8, (-
(iv.) If D = 6, mod 8, (-
(v.) If D = 4, modS, (_!)*- = (_i)J(*-i).
(vi.) IfD=0, modS,
(1 I 5 ' W ^ 1 J f I i 5 *"* 1) Q T~l f~I ( 1 1 5 \Wl ~ ' 1) | "I^T?''"1/
, ( II IV I I ^^ JL I I ^^ .LI .
The interpretation of these symbolic formulae is very simple. Thus, the
formula (i.) expresses that
' The numbers prime to any prime divisor q of D which can be represented
by f, the same properly primitive form of det. D, are either all quadratic residues
of q, or else all quadratic non-residues.'
Again, the formula (iv.) expresses that
'If D be of the form 8n + Q, the uneven numbers that can be represented
by /are either all included in one of the two forms 8?i + l, 8 + 3, or else in one
of the two forms 8n 1, 8n 3.'
All the formulae are deducible by the most elementary considerations from
the three equations
m = ax 2 + 2 bxy + cy 2 , m = ax' 2 + 2 bx'y' + cy' 2 ,
(ax 2 + 2bxy + cy 2 )(ax' 2 + 2bx'y' + cy* 2 ) = { (axx' + b[xy' + x'y] + cyy') 2 - D(xy - x'y) } 2
Thus we find immediately
/mm'
or =
And again, if D = 6, mod 8, the last equation shows that axxf + b [xy + x'y] + cyy'
D d 2
204
REPORT ON THE THEORY OF NUMBERS.
[Art. 98.
is uneven ; and consequently
mm ' = l-Q(xy-a?yY, mod 8, i.e. mm' = + 3, or = + 1, mod 8,
according as xy x'y is uneven or even ; whence m and mf are either both
of the forms 8n + l, 8. + 3, or else both of the forms 8w-l, 8-3.
The form / is said to have the particular character
or
according as the numbers (prime to q) which are represented by it satisfy the
equation
or
and we are to understand in the same way the expressions that f has the
particular character ( l)*(/ 2 -i)= +1, O r = 1, &c.
Every particular character of a form belongs equally to all forms of the same
class, and is therefore termed a particular character of the class. The complex
of the particular characters of a form or class constitutes its complete or generic
character ; and those classes which have the same complete character are con-
sidered to belong to the same genus : so that the complete character of a form
is possessed not only by every form of the same class, but by every form of any
class belonging to the same genus.
To enable the reader to form with facility the complete character of any
given properly primitive class, we add the following Table, taken from Dirichlet
(Crelle, vol. xix. p. 338), in which S 2 denotes the greatest square dividing D ;
P or 2P is the quotient e2 , according as that quotient is uneven or even;
o
p, p', ... are the prime divisors of P ; and r, r the uneven primes dividing S,
but not P.
I. D = PS\ P = l, mod 4.
(a) S = l, mod 2.
'/'
r
JL. JL
(0) S = 2, mod 4.
//
-^ f\
Vr/' \y)'""
(?) -8=0, mod 4.
(la)' W )''
Art. 98.]
(a) S = l, mod 2.
(7) S=0, mod 4.
(a) S = l, mod 2.
' = 0, mod 2.
(a) S = l, mod 2.
(-D"-
O) S = 0, mod 2.
REPORT ON THE THEORY OF NUMBERS.
II. D = PS*, P = 3, mod 4.
/'
205
. () ().-.
()8) S = 2, mod 4.
r-a fZ
III. D = 2PS\ P = l, mod 4.
IV. D = 2PS\ P = 3, mod 4.
// / /
\ /A /A /A
"l> T It '"} ~ l> 7" /)
/ vjj / v r / v r /
It appears from this Table, that if n be the number of uneven primes
which divide D, the total number of generic characters that can be formed
by combining the particular characters in every possible way is 2^ when
D = l or 5, mod 8 ; 2 A<+2 when .0 = 0, mod 8 ; and 2''" 1 " 1 in every other case.
But it follows from the law of quadratic reciprocity, that one-half of these
complete characters are impossible ; i. e. that no quadratic form characterised
by them can exist. To see this, we observe that if m be a positive and uneven
number prime to D, and capable of primitive representation by /, the congruence
Q 2 - D = 0, mod m, is resoluble; and consequently ( )=+!. Therefore also
(-) = !, or () = !, according as D is of the form PS 2 or 2PS\ In the
Vm/ V m '
first case we have
206 REPORT ON THE THEORY OF NUMBERS. [Art. 98.
or
in the other case () (} = 1 ; i.e.
Vm/ Vm/
, or ( ) (y..=(-l)i<-iHP-i)+i(ro-i)
A comparison of these equations with the preceding Table will show
that the product of the particular characters which stand before the line
of division in the Table is equal to +1 in the case of any really existing
genus ; i. e. that precisely one-half of the whole number of complete generic
characters are impossible. We shall hereafter see that the remaining half
of the generic characters correspond to actually existing genera, and that
each genus contains an equal number of classes. That genus, every particular
character of which is a positive unit, is called the principal genus ; it evi-
dently contains the principal class, and is therefore, in every case, an actually
existing genus.
Since the extreme coefficients of a form are numbers represented by it,
and since, further, if the form be properly primitive, one or other of them is
prime to 2 and to any prime divisor of the determinant, we see that the
generic character of a form can always be ascertained by considering the
values of its first and last coefficients. Thus the complete character of the form
(11, 2, 15), of which the det. is - 161 = - 7 x 23 (case II. (a) in the Table), is
(y)" 1 - <) -- 1 ' (-
that of (5, 2, 33), of the same determinant, is
Two forms, which have different generic characters, cannot be equivalent ;
nor can a number be represented by a form if its character is incompatible
with the generic character of the form. It is therefore convenient, in any
problem of equivalence or representation, to begin by comparing the generic
characters of the given forms with one another, or with the characters of the
given numbers.
The uneven numbers prime to the determinant, which are represented by
forms of the same genus, are contained in one or other of a certain number
of linear forms. If R denote the product of the primes r, r', ... already defined,
and if 6 be any term of a system of residues prime to 2 k PR, where & is =1,
Art. 98.] REPORT ON THE THEORY OF NUMBERS. 207
when D = 1 or 5, mod 8, is = 3 when D = 2, 6, or 0, and is = 2 in every other case,
the numbers contained in the formula WPR + Q can be represented only by forms
belonging to that genus the character of which coincides with the character of
the number 6. It is clear that one half of the linear forms, included in the
formula 2 k PJi + 6, do not satisfy the condition of possibility indicated in the
Table, and are therefore incompatible with any quadratic form of determinant D ;
while the remaining half of those linear forms will be equally distributed
among the actually existing genera ; so that there will be either
n{to-l).K*-l)} or 2 n{l( IJ -l).l(r-l)}
linear forms proper to each genus. But although no number contained in
any one of the first-named linear forms can be represented by a form of
determinant D, yet it is not to be inferred that every number m contained
in the other half of the linear forms is capable of such representation ; for
from the linear form of m, we can indeed infer the equation ( ) = 1 ; but,
^m /
if m be not a prime, or at least the product of a prime by a square, we cannot
from this equation infer the resolubility of the congruence O 2 =J), mod m,
or of any congruence of the form & = D, mod ->- We may add that if we
assume the theorem that every arithmetic progression, the terms of which are
prime to their common difference, contains prime numbers, the consideration
of the case in which m is a prime establishes the actual existence of every
genus the character of which satisfies the condition of possibility. (Crelle,
vol. xviii. p. 269.)
If m be an uneven number not divisible by q, a prime divisor of D,
and if the double of m can be represented by an improperly primitive form
/
f of det. D, we attribute to f the particular character ()=+!, or = 1,
according as (--}=+!, or = 1 ; and to form the complete character of f,
we may use the Table
D = PS\ P = l, mod 4, S = I, mod 2.
A /
A /
* All the results of this article are given in the Disq. Arith., Arts. 223-232 ; but as Gauss does
not employ the symbol of reciprocity, we have preferred to follow the notation of Dirichlet. It is also
to be noticed that Gauss does not use the law of quadratic reciprocity to demonstrate the impossibility
208 REPORT ON THE THEORY OF NUMBERS. [Art. 99.
99. In the preceding Articles we have briefly recapitulated the definitions
and principles which constitute the elements of the theory of quadratic forms.
We have hitherto followed closely the 5th section of the Disq. Arith. (Arts.
153-222 and 223-233) ; but before we proceed to an examination of the re-
mainder of that section, it will be convenient to place before the reader an
account of the method employed by Lejeune Dirichlet in his great memoir,
' Recherches sur diverses applications de 1'analyse infinite'simale a la thdorie
des nombres,' for the determination of the number of quadratic forms of a
given positive or negative determinant.
It appears from the Additamenta to Art. 306, X. of the Disq. Arith., that
Gauss, at the time of the publication of that work, had already succeeded in
effecting this determination ; and the method by which he effected it will at
length appear in the second volume of the complete edition of his works, the
publication of which is now promised by the Society of Gottingen. Never-
theless the originality of Dirichlet in this celebrated investigation is unques-
tionable, as there is nothing whatever in the Disq. Arith. to suggest either the
form of the result, or the method by which it is obtained *.
of one-half of the generic characters ; for, as we shall hereafter see, this impossibility is proved in the
Disq. Arith. (Art. 261) independently of the law of reciprocity, and is then employed to establish that
law. (Gauss's second demonstration, see Disq. Arith., Art. 262.) There is also an unimportant differ-
ence between Dirichlet and Gauss with respect to the definition of the generic character of an
improperly primitive form; for Gauss obtains the generic character (see Art. 232) by considering the
numbers represented by the form, and not the halves of those numbers. But he also observes
(Arts. 227, and 256, VI.) that each improperly primitive class is connected in a particular manner (to
which we shall again refer) with one or with three properly primitive classes ; and that this consider-
ation may be employed to divide the improperly primitive classes into genera. And it will be
found that the complete character which Dirichlet's definition attributes to an improperly primitive
form is, in fact, the complete character of the properly primitive class or classes with which it is
connected.
* The following is a list of the papers of Lejeune Dirichlet which relate to the theory of
quadratic forms :
1 . Sur 1'usage des sdries infinies dans la th6orie des nombres. Crelle, vol. xviii. p. 259.
2. Recherches sur diverses applications de 1'analyse infinitesimale a la theorie des nombres.
Crelle, vol. xix. p. 324, and xxi. pp. 1, 134.
3. Auszug aus einer der Akademie der Wissenschaften zu Berlin am 5 Marz 1840 vorgelesenen
Abhandlung. (Crelle, vol. xxi. p. 98, or the Monatsberichte for 1840, p. 49.)
This paper is an abstract of an unpublished memoir containing the demonstration of the theorem
that every properly primitive form represents an infinite number of primes.
4. Untersuchungen iiber die Theorie der complexen Zahlen. (Crelle, vol. xxii. p. 375, or in the
Monatsberichte for 1841, p. 190 ) An abstract of the following memoir.
Art. 99.] REPORT ON THE THEORY OF NUMBERS. 209
We propose, in what follows, to give as full an analysis as our limits will
permit of the contents of the memoir. Its first section contains certain prin-
ciples relative to the theory of series.
(i.) 'If &! ^ 2 < & 3 <; & 4 ... be a series of continually increasing positive
/n
quantities ; and if the ratio -r- continually tend to a finite limit a (that is to say,
*n
if, 3 denoting a given positive quantity, however small, we can always assign a
finite value of n = N, such that for all values of n surpassing N the inequalities
. n
a d < -j- <a
k n
n oo -t
are satisfied), the limit of the expression p 2 - -- , when the positive quantity
p is diminished without limit, is a.' *
= oo n = N n = o>
For ^
N denoting a finite number ; and by virtue of the inequalities written above
Observing that lim p 2 is intermediate between
r dx f dx
lim p I rr^ and lim
5. Recherches sur les formes quadratiques a coefficients et a indetermin6s complexes. Crelle,
vol. sxiv. p. 291.
6. Sur un th^oreme relatif aux series. (Liouville, New Series, vol. i. p. 80, or Crelle, vol. liii.
p. 130.)
7. Sur une propriete des formes quadratiques a determinant positif. (Monatsberichte for July 16,
1855, or Liouville, New Series, vol. i. p. 76, or Crelle, vol. liii. p. 127.)
8. Vereinfachung der Theorie der binaren quadratischen Formen von positiver Determinante.
(Memoirs of the Berlin Academy for 1854, p. 99, or, with additions by the author, in Liouville, New
Series, vol. ii. p. 353.)
9. Demonstration nouvelle d'une proposition relative a la throne des formes quadratiques.
Liouville, New Series, vol. ii. p. 273.
10. De formarum binarium secundi gradus compositione. Crelle, vol. xlvii. p. 155.
The three last papers contain important simplifications of theories which appear in a very compli-
cated form in the Disq. Arith. To two of them we have already referred (Arts. 93, 94).
* This theorem is a generalisation of that in the memoir (Crelle, vol. xix. p. 326). It is given by
Dirichlet in No. 6. of the preceding list.
E 6
210 REPORT ON THE THEORY OF NUMBERS. [AH. 99.
and is consequently unity, we infer from the last inequalities that
l+p
and therefore also lim p 2 , 1 + p
n = 1 H
which is identical with it, because
n-l
differs from a by a quantity comminuent with S ; i.e.
since by hypothesis S is a quantity as small as we please.
(ii.) A convergent infinite series may be convergent in two very different
ways. It may be convergent, and always have the same sum irrespective of the
arrangement of its terms ; or it may be convergent for certain arrangements of
its terms, giving the same or different sums for these different arrangements,
and divergent for other arrangements. We suppose, however, that we con-
sider only such different arrangements of the terms of a series as are compa-
tible with the condition that any term which occupies a finitesimal place in
any one arrangement should occupy a finitesimal place in every other arrange-
ment *. Thus the series
i I i n >
31 + ,, ^
is convergent, and has the same sum in whatever order we sum its terms ;
but of the two series
* This condition is necessary, because without it the sum of no series whatever would be inde-
pendent of the arrangement of its terms, if by the sum of a series we understand the limit to which
we approximate by the continual addition of its terms in the order in which they are given.
For example, the series cited in the text,
is convergent, and its sum is irrespective of the arrangement of its terms, provided that arrangement
satisfy the condition enunciated in the text. But if we were to arrange the terms of the series in an .
order regulated (say) by the number of primes dividing their denominator!,, the limit to which we
should continually approach by adding together the terms taken in their new order would be 2 -y^
1 P
in which p denotes any prime, and not 2 -^- , in which n denotes any integer.
Art. 100.]
REPOET ON THE THEORY OF NUMBERS.
211
J_ JL l JL l
~"~ r ~r r ~~ r *T* r "*" r ~r . . . ,
JL l JL JL 1
f 3? " 2?~ h 5* " 7?" 4* H
only the first is convergent ; while the two series
l.l+l.i+l. 1 ,
2 + 3 4 + 5 6"^
11111
f 3~2 + 5 + 7~6 + -
are both convergent, but have two very different sums *.
These observations will show the importance of the following proposition t :-
' If c n be a periodic function of n, satisfying the equations
C n+ fc = C n ,
g
the series 2 - in which the terms are taken in their natural order, is con-
.-i n'
vergent for all values of s superior to zero, and its sum is a continuous func-
tion of s.'
For if we add together the k consecutive terms
1 2 fc
we obtain a fraction of which the denominator is of the order ks in respect
of m, while the numerator is only of the order (k 1) s 1, because the coefficient
of m (k ~ 1)> is zero. We may therefore replace the given series by a series of
the form 2
in which <b(m) is a function of the order 1+s in respect
of m. This series is always convergent for positive values of s ; its convergence
is irrespective of the arrangement of its terms, and its sum is a continuous
function of s, because <(m) is a continuous function of s. The given series is
therefore also convergent, and its sum is a continuous function of s.
100. The second section of the memoir refers to the symbols of reciprocity
of Jacobi and Legendre (Arts. 15, 16, and 17 of this Report).
* These illustrations are taken from the Memoir on the Arithmetical Progression in the Berlin
Memoirs for 1837, pp. 48 and 49.
t The demonstration in the text is a little simpler than that given by Dirichlet, who uses the
function F to express the sum of the series.
E 6 2
212 REPORT ON THE THEORY OF NUMBERS. [Art. 100.
The third and fourth sections contain the principal theorems relating to the
generic characters of quadratic forms, and to the representation of numbers.
There is only one of these theorems to which we need direct our attention here,
as the others have already come before us in the preceding articles.
Let (a, 6, c) be a primitive form of the positive determinant D ; let
also (a, 6, c) (x a , y ) M a positive number represented by (a, b, c); m the
greatest common divisor of a, 26, c; [T, Z7] the least positive solution of
T t -Dir = m*-, so that if
x n = [T n x - U n (bx + cy )] , y n = [T n y + U n (ax + by,,)],
7/C* til'
the two formulae [X H , T/,,] and [ x n , y n ~] will together express every repre-
sentation of M, which belongs to the same set as [a; , y ]. Similarly, let
\x n , y' n ], [-x' n , y'n] denote a complete set of representations of the positive
number M' by (a, b, c).
If we trace the hyperbola represented by the equation ax 2 + 2 bxy + cy 2 = 1
referred to rectangular axes, the diameters included in the formula y=~x, in
X k
which & is to receive all values from oo to + cc, will form a pencil of
lines, which all meet the curve, and which, commencing with the asymptote
y = -- j: r x, continually recede from it, and approximate to the asymptote
""
11 = , _ , x. The sectorial area contained between any two consecutive lines
v/D-6
of this pencil and either branch of the hyperbola is constant and equal to
^ YJ; log - ; as may be ascertained by employing polar coordinates.
rn f
Since the same observations apply to the pencil y = ^-?x, we infer that the
x n
lines of these two pencils lie alternately, unless the two pencils coincide.
Let us now suppose that in the form (a, 1), c), a is positive and c negative ;
so that the axis of x does, and the axis of y does not cut the curve. On
this supposition the values of and of ^ continually increase from
x n x n
to 7j)_-L as n increases from - oo to +00. The alternate position of the lines
of the two pencils gives, in this case, the theorem,
' The inequalities 7 / ,,
yjc < y_ < y k
x k x n = x' k
Art. 100.] REPORT ON THE THEORY OF NUMBERS. 213
in which k represents any given number, are satisfied for one value of n, and
one only.' If, taking a for M' and [1, 0] for [z' , y' ~\, we put k = 0, we obtain
the conclusion,
' Each set of representations of the positive number M by the form (a, b, c),
in which a is positive and c negative, contains one and only one representation
which satisfies the inequalities
It is in this form that the theorem appears in Dirichlet's memoir. We
may add that any values of x and y which satisfy these inequalities will
give a positive value to (a, b, c) ; for such a pair of values will correspond
to a point situated in the internal angle between the asymptotes of the
hyperbola.
The fifth section contains the demonstration of the theorem, that if A
denote the absolute value of D, and \^ (2 A) be the number of numbers less than
2 A and prime to it, a properly primitive form of determinant D will acquire
a value prime to 2D, if its indeterminates x and y satisfy any one of a certain
set of 2A\J/(2A) congruential conditions included among the 4 A 2 conditions
represented by the formulae
x = a, mod 2 A ; y = /3, mod 2 A,
in which both a and /3 represent any term of a complete system of residues,
mod 2A; but will acquire a value not prime to 2Z>, if x and y satisfy any of
the other congruential conditions.
If the form be improperly primitive, the number of congruential conditions
that will render its value unevenly even and prime to A will be A \J<- (A), or
3 A \J<- (A), according as Z> = 1, or =5, mod 8.
These theorems are easily demonstrated by considering separately the
prime divisors of A. For example, if the form (a, b, c) be improperly primitive,
and p be a prime divisor of D, since either a or c is prime to p, let a be
prime to p ; then (ax + by} 2 Dy- will be prime to p, when ax + by is so ;
i.e. it will be prime to p, for p(p 1) combinations of the residues (mod p)
of x and y; or, if p n be the highest power of p dividing D, for p zn ~ l (p1)
combinations of the residues of x and y, mod p n . Again, the 4 combinations
of residues for the modulus 2 will give -| (a, b, c) the values
0, ia 2 , ic 2 , ^a + b + ^c,
of which it is easily seen that one or three will be uneven, according as ac = 0,
214 BEPOET ON THE THEORY OF NUMBERS. [Art. 101.
or 4, mod 8; i.e. according as D = l, or 5, mod 8. The combination of these
results will give Dirichlet's theorem.
101. Series expressing the number of Primitive Classes. The sixth section
of the memoir contains the demonstration of the formulae which express in
the form of an infinite series the number of classes of properly and improperly
primitive quadratic forms of a given determinant. We shall abbreviate the
demonstration of these formulae by using the theorem of Art. 87.
Let h be the number of properly primitive classes of determinant D ; we
shall first suppose D to be negative, and = A ; let also
(<*!, &!, Cj), (a 2 , & a , c 2 ), ..., (a h , b h , c h )
be a system of forms representing the properly primitive classes of that deter-
minant ; and let us consider the sum S =
,+ + 2,
the sign of summation 2 t extending to all values of x and y from oo to
+ oo, which give the form (a k , b k , c k ), a value prune to A. By the theorem
of Art. 87, any uneven number n prime to A is capable of 2 2 (~r) repre-
sentations by the properly primitive forms of determinant D (for there are
2 (-T) sets of representations, and each set contains two *.) We have there-
fore the equation S=22[2(~)i] (a)
(the inner sign of summation referring to every divisor d of n ; and the outer
sign extending to every positive value of n prime to 2 A). If we write n for d,
and nn for n, so that n and n each represent any positive number prune to
2 A, this equation assumes the simpler form
j (6)
the sign 2 indicating two independent summations with respect to n and n' ;
or, if we perform the two summations separately, and omit the accent,
S=2S-~Z( )-^- ( C )
/yi8 \ lYi / fn \ '
* If A = 1, each set contains four representations. To obtain a correct result in this case,
we must therefore double the right-hand members of the equations (a), (6), (c), and (A).
Art. 101 ]
REPORT ON THE THEORY OF NUMBERS.
215
To deduce an expression for h from this equation, we write 1 4- p for s, and
multiplying, each side by p, we suppose p to be positive and to diminish with-
out limit. In order to find the limit of pS on this supposition, we consider
separately the partial sums, such as pZ (atf + z'bx +c 2 V + ' f wnich ** is
composed. * ^ '
If
be the n-th term of the series 2
, in which we
(ax 2 + 2bxy + cy' 2 ) 1 + P
suppose that the terms are so arranged that no term surpasses any that precedes
n Tr\W2A)
it, it can be shown that lim ^ = - v . For if 2A + , 2A)? + ^ represent
k n 2 A -/A
generally any one of the 2 A \^ (2 A) systems of values that can be attributed
to x and y consistently with the condition that (a, b, c] assumes a value prime
to 2 A, the number of terms up to k n inclusive (i.e. the number n) is evidently
equal to the number of points having coordinates of any one of the forms
[2A+ , 2Aj + J7 ] that lie within the ellipse ax 2 + 2bxy + cy* = k n , together
with one, or all, or some of the similar points lying on the contour of the ellipse,
1
according as
is the first or the last, or neither the first nor the last of
c
the terms equal to it in the series. The area of the ellipse is ^ ; whence,
if n be very great, the number of the points we have defined is approximately
n
- , the error being ot the same order as %/# : i.e.
lim H
Q ^
Hence by Dirichlet's first Lemma (Art. 99),
Again, by the same Lemma, the expression
has
2A
for its limit,
when p diminishes without limit. And, lastly, the limit of the series 2 ( )
J \ n ' n 1
is the series 2 ( 1 - , in which the terms are taken ha their natural order.
v / n
To establish this, we observe that the symbol ( j is a periodic function of n,
^ ftf *
and that the sum of the terms of which one of its periods is composed is zero.
Using the notation of Art. 98, and attributing the value +1 or 1 to the
symbol S according as P = l or =3, mod 4, and to the symbol e according as
216 REPORT ON THE THEORY OF NUMBERS. [Art. 101.
D = PS* or =2 PS*, we have, by Jacobi's law of reciprocity,
Hence ( ) = (~)> ^ n = n', mod 2 1 ' PR*; or ( ) is a periodic function of n.
Again, if a and 6 denote the general terms of a system of residues prime to
2 k and p respectively, we find
f n \ 'a ' a*-l) / &> \
u - 6 I ~^FT I ^ O^ ~^ X 11 . I IXllt?* *"" J I,
\p / \ p /
where in the left-hand member the summation extends to every value of n prime
to 2 k PQ and less than it, while in the right-hand member the signs of sum-
mation refer to a and b, and the signs of multiplication to p and r respectively.
This equation is easily verified ; for if
n = a, mod 2 k , =b, mod p, = b', modp', ...,
we have , n ^ /b\ /
(p)~ \p)(p.
so that each member of the equation consists of the same units. But one at
least of the factors of which the right-hand member is composed is zero ;
unless we have simultaneously 5 = 1, e = l, P l, a supposition which is inad-
missible, because it implies that D is a perfect square. We infer therefore
that 2 ( ) = 0, i. e. that the sum of the terms of a period of the symbol
\ 77- '
( j is equal to zero. If, then, we suppose the terms of the series
2 ( ) rr to be taken in their natural order, it will follow from Dirichlet's
V n J n 1 + f
second Lemma (Art. 99) that its sum represents a finite and continuous
function of p for all values of p superior to 1 ; i. e. the limit of the series
2 i ) . for p = is the series 2( ) -, in which the terms are taken
V/w 1 + p VTO/W
hi their natural order. We thus obtain the equation
')- (A)
* The index k is not the same as in Art. 98 ; it is 1 when 5 = 1, = 1 ; 2 when 8 = - 1, e = 1,
and 3 when e = 1.
Art. 101.]
REPORT ON THE THEORY OF NUMBERS.
217
Secondly, let the determinant D be positive ; and let us retain the same
notation as in the former case. If in the series S =
1 1 1
(in which it is convenient to suppose that the forms (a k , b k , c k ), representing
the properly primitive classes of determinant Z>, have their first coefficients
positive, and their last coefficients negative) we suppose the sign of double
summation 2 fc to extend only to those integral values of x and y which
render the value of the form (a k , b k , c t ) prime to 2D, and which further
satisfy the inequalities
x > 0, y>0, y ^
we obtain, by a comparison of Arts. 87 and 100, the equation
n' \ n
in which n denotes any positive number prune to 2D, and which corresponds
to equation (c).
1
be the n-ih term of the series
1
If
n is equal to the number of points having coordinates of any one of the forms
which lie in the interior of the sectorial area, bounded by the positive axis of x,
the arc of the hyperbola ax 2 + 2bxy + cy* = k n , and the straight line
aU
nt . _ />
y ~T-bU '
together with one, all, or some of the similar points on the contour of the sector.
The area of the sector is
whence, reasoning as before, we find
t)
h-
log [T+ UVD} 2 ( TO ) n '
for the number of properly primitive forms of a positive determinant D. The
corresponding formulae for improperly primitive forms are obtained by a pre-
cisely equivalent process. The results are, if D = A,
Ff
218 REPORT ON THE THEORY OF NUMBERS. [Art. 102.
[2 -(- 1)^-0] h'= ?^AZ1* ...... (C)
and if D = + A,
, .... (D)
[T f , IT] denoting the least solution of the equation T-LIT = 4.
102. Proof that each Genus contains the same number of Classes. The
sixth section of the memoir also contains a demonstration of the proposition
to which we have already referred (Art. 98), that all the possible genera actually
exist, and contain an equal number of classes. This demonstration is not
deduced from the expression for the number of properly primitive forms, but
depends on an equation between two infinite series similar to the equation (a)
of the last article. Let x denote any one of the particular characters proper
to the determinant, and let < be any term in the product II (1 + x), with the
exception of the first term, which is unity, and also of that particular com-
bination of the values of x, the value of which, by the condition of possibility,
is also a positive unit. If X be the number of particular characters, 2 X 2 will
be the number of expressions symbolised by <f>. Let H and H' be the numbers
of classes satisfying the conditions <p = 1 and <p = 1 respectively. It can be
shown, as follows, that H=H'. Confining ourselves, for perspicuity, to the
case of forms of a negative determinant, we have, by the principle of Art. 87,
, ,
where in the right-hand member ( ) is +1 or 1, according as the number
> Td '
n satisfies the condition < = 1 or ( = 1; and similarly, in the left-hand member
<j) k = 1 or = +1, according as the generic character of the form (a k , b k , c k )
satisfies the condition < = 1 or c = 1. In this equation the signs of sum-
mation have the same signification as in the similar equation (a) of the last
article ; and, as in that equation, the right-hand member may be expressed
in the simpler form ^ -^ _# ^ -^
\nJ if \tt,/v7i/w"
* If A =3, we must triple the right-hand member of this equation ; as each set of representations
of a number by a form of determinant 3 contains six representations, instead of two.
Art. 103.]
REPORT ON THE THEORY OP NUMBERS.
219
If we now write 1 4- p for s, and, multiplying by p, allow p to converge to
2 Ax/A
Tk
Ine series
- converges to a finite limit ; for ( ) and (-*-} are each of
n 1 + f \n/ v/
zero, the limit of the left-hand number is (HH')
V /D\ /<b\ 1 . ..,.., n /D*
2 w v
them expressions of the form S^ (n ~ 1} e^ (nt ~ l) (-^=r\ S and e denoting positive
v y/
or negative units, and Q an uneven number composed of unequal primes
dividing D ; their product is therefore another expression of the same form,
in which 8, e, and Q are not simultaneously equal to 1, because we have
expressly excluded that combination of the particular characters which causes
( ) to coincide with ( ) It can therefore be shown, by reasoning as in
the last article, that the second Lemma of Art. 99 is applicable to the series,
and that it converges to the finite limit 2 ( J ( ) ~" Similarly, it may be
shown that 2 () ^r converges to a finite limit. The limit of the right-
hand member of the equation (d) is consequently zero on account of the
evanescent factor p ; from which it follows that H=H'. Let G l} G 2 , ... be
the different possible genera; h 1} h 2 ,... the number of classes they severally
contain ; -r) the value of < for the genus G.
The equation H H' = comprises 2* 2 equations of the type
corresponding to the 2 X 2 different expressions symbolised by (j). If we
multiply each of these equations by the coefficient of h k in it, and add the
products to the equation
2^ + 2A 2 + 2/* 3 + ... = 2h,
we arrive at the conclusion 2*h k = 2h. For the coefficient of h r in the result-
ing equation is the product II 1 + (^r) (TT) ! an( i this product is 2 X , if G>
and G k are identical, but is zero in every other case, as one at least of the
factors will be zero.
103. The seventh section (Crelle, vol. xxi. p. 1) commences with the proof
of the theorem that the number of sets of representations of any number M
F { 2
220 REPORT ON THE THEORY OF NUMBERS. [Art. 103.
prime to 2 D by quadratic forms of determinant D, is equal to the excess of
the number of those divisors d of M which satisfy the equation
above the number of those divisors which satisfy the equation
-D .*- (A). .1,
the symbols S and e having the same signification as in Art. 101. Of this
theorem, which coincides with that of Art. 87, since
two demonstrations are given, one purely arithmetical, the other derived from
the equation (&) of Art. 101, the proof of which in Dirichlet's memoir does
not involve the theorem of Art. 87, but is deduced from the arithmetical
principles on which that theorem itself depends. We have already referred
(Art. 95) to some of the particular results which can be deduced from the
general theorem.
It is evident from the mode of formation of the equation (fe), or of the
corresponding equation for a positive determinant, that it may be generalised
by taking instead of the power (ax* + 2bxy + cy)~, any function of
ax 2 + 2 bxy + cy z
which renders the two members of the equation convergent ; {. e. we may
write, in the case of a negative determinant,
Dirichlet illustrates this observation by giving to </> the exponential form <f,
which satisfies the condition of convergence, if the analytical modulus of q
be inferior to unity. Each double sum, such as Zg* B ' ! + 2l *' + i' 2 in the left-
hand member of the equation
n
can then be replaced by 2aA\^(2A) (or sometimes by fewer) products of the
= 00
q a x 2 q a
V= 00
in which each simple series such as
2'
v= oo
Art. 103.]
REPORT ON THE THEORY OF NUMBERS.
221
can be expressed by means of the elliptic function ; the right-hand member
can also be expressed by means of elliptic series. If, for example, D = 3,
we have the equation
t?=00
2
2
= 00
t>=a+CO
2
1>= CO
64+1
It does not appear that this remarkable transformation, which is only
very briefly noticed by Dirichlet, has been further examined. (See a note
by Mr. Cayley in the Cambridge and Dublin Mathematical Journal, vol. ix.
p. 163.)
In the eighth section Dirichlet assigns the relation between the numbers
of properly and improperly primitive classes. When the determinant is ne-
gative we find, by a comparison of the formulae (A) and (C), h = h', or
h = 3h', according as D = \, or =5, mod 8 ; observing only that if D 3
we have, exceptionally, h = h'. When the determinant is positive, we infer
from the formulae (B) and (D),
log (T+
log(T+UVD)
according as D = l or =5, mod 8. Comparing these expressions with the
observations in Art. 96 (vi.), we find, if D = 1, mod 8, h = h' ; and if D = 5, mod 8,
h = h', or h = 3h', according as the least solution of the equation 7 72 -Dt/ 2 =4
is uneven or even.
Dirichlet also deduces from the formulae (A) and (B) the relation which
subsists between the numbers of properly primitive classes for any two deter-
minants which are to one another as two square numbers. It is sufficient
to consider two determinants such as D and DS\ of which the former is not
divisible by any square. If h and H be the numbers of classes for these two
determinants, we have evidently, when the determinants are negative,
H
h
(*)'
V n ' n
222 REPORT ON THE THEORY OF NUMBERS. [Art. 104.
the two series in the numerator and denominator not being identical, because
in the one n is any number prime to IDS*, in the other any number prime
to 2D. But, by a principle due to Euler,
V n
~
/IHi'
V 71 / p
p representing any prime, except those dividing 2DS* or 2D. Hence
if s denote any prime dividing S but not dividing D.' For a positive deter-
minant we find D i o ^ +
'-
[7", Z7 7 ] denoting the least solution of the equation T* DS*U 3 = 1 ; i.e. the
least solution [T k , Z7J of 7 -DT=1, which satisfies the condition U k =0,
mod S ; so that we may write
In a subsequent note (No. 7 in the list) Dirichlet infers from this expres-
sion that, given any positive determinant D, we can always deduce from it
an infinite number of determinants of the form DS* having each the same
number of classes. For if we attribute to S a series of values of the form
II . s, all composed of the same prime numbers s, and having continually
increasing numbers for the indices of those primes, it appears from a remark
S
to which we have already referred (see Art. 96, (v.)), that the quotient y will
eventually be constant ; i. e. there will exist an infinite series of determinants,
all composed of the same primes, and all having the same number of properly
primitive classes. As it is possible to find determinants contained in a series
of this kind, and having only one class in each genus, it appears that the
number of the positive determinants, which have only one class in each genus,
is infinite. This result, which was anticipated by Gauss (Disq. Arith., Art. 304),
is remarkable, because it is probable, from the result of a very extensive
induction, that there are but 65 negative determinants, of which the greatest
is 1848, having the same property.
104. Summation of the series expressing the number of Properly Primitive
Classes. It appears from the last article that, to obtain expressions in a finite
Art. 104.]
REPORT ON THE THEORY OF NUMBERS.
223
form for the number of classes, we may confine our attention to the order
of properly primitive forms, and may suppose that the determinant is not
divisible by any square. To sum the series 2 ( ) - upon this supposition,
Dirichlet employs the formulae given by Gauss in his memoir, ' Summatio
Serierum quarundam singularium,' to which we have already referred in this
Report (Art. 20). The ninth section is occupied with the demonstration of
these formulae ; in the tenth they are applied to the summation of the series
2 ( ) - Two different methods are given by Dirichlet, by either of which
N yj, / y/,
this summation can be effected.
(i.) If Tc be the index of periodicity of ( Y so that
= (2), and 2f^) = 0,
n
v/
Hi
the summation indicated by the symbol 2 extending to all values of n prime
to 2Z> from 1 to k, we have, writing V for 2 ( ) - ,
^ ft ' ID/
V- -
.o
dx
/\
where f(x) = 2 ( J x n , so that /(I) = 0. Integrating by the ordinary method
of decomposition into partial fractions, we find
-kV= 2
x e
To simplify this complicated expression, it is requisite to transform the
symbol ( j by the law of reciprocity, and to consider separately the eight
cases which arise from every possible combination of the hypotheses, (a) D
positive or negative, (,8) D even or uneven, (7) D, or ^D, =1, mod 4, or =3,
mod 4. As an example of the process, we shall take the two cases in which
D = 3, mod 4, so that
224 REPORT ON THE THEORY OF NUMBERS. [Art. 104.
2mw<
A still denoting the absolute value of D. The value of f (e 4A ) is assigned
by the formulae of Gauss ; it is
2;*u + A)* (_
or zero, according as m is, or is not, prime to 4 A *. We thus find
* If p be any prime divisor of A, an uneven number admitting of no square divisor, and if, for
brevity, P = ; we have, by Gauss's formula,
* 1 7 ZkmPwi T
"
according as m is or is not prime to p. If we multiply together the equations of this type, cor-
responding to every prime divisor of A, and observe
(1) that = S . kP* represents a system of residues prime to A,
(2) that (4-) = (-) (-) -. = (^) (^)- ,
VA' Vft'.Vft' \Pi' V 2V
(3) that II ( ){*<P-D 2 = (- 1)SP1-1)(P2-D x 2*(p-iy = ii[2{p-i)P tlCA-i?,
we find S A -.fMA-w'A, or = 0, ........ (2)
according as m is or is not prime to A. We have already met with this equation in Art. 96, (if.).
If in the equations (1) we write 4P for P, and join to them the equation
2(-l)J(*- 1 )ei*'AT<=2;(-l)i('"- 1 )- 1 -i<A- I >, (m uneven), or =0 (m even),
in which k is either term of a system of residues prime to 4, we obtain after multiplication the
. Smni
equation which is employed in the text. And similarly may the function f(e * J be evaluated,
whatever be the form of D.
The formulae (A) and (A') of Art. 20 are only particular cases of the general result obtained
by Gauss in the ' Summatio Serierum &c.' The general formula, including (A), is
* = 2~ 1 r*' = ( )i Vn,
t = o V '
h denoting any number prime to n. When n is even, the formula (A') of Art. 20 is similarly include d
in the following,
'=0, or =()i-4<*-<0(l+;)v^, {= (~
according as n is unevenly or evenly even.
When n is uneven and not divisible by any square, the two sums
*~ij~V and S( V
* = o v 7
are identical, as appears from a comparison of (2) with the generalisation of (A), and has been already
observed in the case when n is a prime (Art. 21).
{[Aug. 8, 1877.] The generalised formulae (A) and (A') here given coincide with the formulae
Art. 104.]
REPORT ON THE THEORY OF NUMBERS.
225
-l)*<- (f
the summation extending to all values of m prime to 4 A and less than it. In
this expression the sum 2(- l)* (m ~ 1} (-*- ) is zero, because the terms correspond
to m and 2 A + m destroy one another ; so that
Distinguishing the two cases D = A, and D = A, and observing that the
imaginary parts vanish identically, as they ought to do, because V is real,
we have, finally, if D = A,
- 4Z>F=2 </DZ(- !)*(- Q) log sin (^J)
) log sin
m/
and if D = - A,
m
2s/A
/ D \
( )m.
Vm/
(ii.) The series 2 ( j - can also be summed by substituting for ( j its
trigonometrical value deducible from the formulae of Gauss. We will take
as an example the case in which D = A = 3, mod 4. Writing n for m, and
m for n, in the equation
X
e
we find, observing that ^(1 + A) is uneven,
2mnirt
p 4A
7- 2 ( ) sin
A/A Vm/
2VA V m -
4A /'
of M. Lebesgue (Liouville (I), vol. xii. p. 509). If n = P = ap, h = Q = bq, a and 6 being powers
of 2, p and ^ uneven, we have
this is wrong when P is even, and p = 1, mod 4. "We have, however, in every case,
226 REPORT ON THE THEORY OF NUMBERS. [Art. 104.
the summation extending to every value of ra prime to 4 A and less than it.
Substituting this expression for (--) in V, we have
V i\f *
Since the expression which we have substituted for ( Jis zero, when n is
not prime to 4 A, we may attribute to n, in the series
2 - sin \ - - ,
n V 4A /
either all uneven values, or all integral values. The sum of the series
sin x sinSa; sin5o;
1 ^ K ""
is, by a known theorem, ^TT or %TT, according as Q<x<ir, or ir<x<2Tr.
Hence attributing to n only uneven values, and denoting by mf and m" the
values of m inferior and superior to 2 A,
F=
8 /A
7T
= 4/A
because ( >) = ( - .)
\m / v^A + m/
If we attribute to n all integral values, the equation
j, , sin x sin2o; sin 3 a:
~T~ ~2~ ~3~
which subsists for all positive values of x less than 2 IT, will give the value
already obtained for V by the former method, viz.,
2
The mode of application of this method may be still further varied ; for,
f 'Yl \
instead of substituting for ( I)?*"- 1 ' (-r-J, we may leave the factor ( 1)2 U
(fV\ v
-j- \ by means of the equation
Art. 104.]
REPORT ON THE THEORY OF NUMBERS.
227
which, as well as the substitution which we have employed, is deducible from
the formulae of Gauss *. We should thus obtain a third expression for V,
different in form from both of those which we have already found.
The forms which the expression of h can assume are very numerous ; we
select the following as examples, D still denoting a determinant not divisible
by any square.
I. If D = l, mod 4.
For a positive determinant, D = A,
2 /D\,
2 log ton
For a negative determinant, D = A,
&-
the summations extending to every uneven value of m prime to A and less
than A.
II. If D be not = 1, mod 4.
For a positive determinant,
1
h=-
log(T+UVD)
For a negative determinant,
/mir\
\D)'
the summations with respect to m and m' extending to all values prime to
2 A, and inferior to 4 A and 2 A respectively.
Dirichlet observes that when the determinant is positive, the coefficient of
a lo g arithm of the form l g( T i+ U hVD); (T h , U lt ) being
los(T+lT7S)
one of those solutions of the equation T 2 DU 1 = 1 which are deducible from
the theory of the division of the circle. Thus h is in fact determined as the
index of the place occupied in the series of solutions of T 1 DU* = 1, by an
assigned trigonometrical solution. (See a note by M. Arndt in Crelle, vol. Ivi.
p. 100.)
* See equation (2) of the preceding note.
Gg2
228 REPORT ON THE THEORY OF NUMBERS. [Art. 104.
In the particular case in which the determinant is a prime of the form
4 n + 3 taken negatively, an expression for the number of classes had already
been given by Jacobi (Crelle, vol. ix. p. 189). It would seem, from his note
on the division of the circle (Crelle, vol. xxx. p. 166), that the unpublished
method, by which his result was obtained, formed a part of that theory.
VIII.
REPORT ON THE THEORY OF NUMBERS.
PAET IV.
[Report of the British Association for 1862, pp. 503-526.]
105. GENERAL Theorems relating to Composition. The theory of the
composition of quadratic forms occupies an important place in the second part
of the 5th section of the ' Disquisitiones Arithmeticse,' and is the foundation
of nearly all the investigations which follow it in that section. In accordance
with the plan which we have followed in this portion of our Report, we shall
now briefly resume the theory as it appears in the ' Disquisitiones Arithmetics, '
directing our special attention to the additions which it has received from
subsequent mathematicians. We premise a few general remarks on the Problem
of composition.
If F l (x 1 , Xj, ...,x n ) be a form of order m, containing n indeterminates,
which, by a bipartite linear transformation of the type
= 1, 2, 3, ...,n,
= 1, 2, 3,. ..,,
y = l, 2, 3, ...,,,
is changed into the product of two forms F 2 (y 1} y 2 , ...,y n ) and F 3 (z 1> z 2 ,
of the same order, and containing the same number of indeterminates,
said to be transformable into the product of F 2 and F s ; and, in particular, if
the determinants of the matrix i n
\ u a, 0, y .
which is of the type n x n 2 , be relatively prime, F l is said to be compounded of
F 2 and F 3 . Adopting this definition, we may enunciate the theorem ' If F :
,z n )
s
230 REPORT ON THE THEORY OF NUMBERS. [Art. 105.
be transformable into F 2 xF 3 , and if F lt G 2 , G 3 be contained in G lt F 2 , F 3
respectively, G l is transformable into G 2 x G a ; and, in particular, if F 1 be
compounded of F a and F 3 , and tbe forms F ly G 2 , G 3 be equivalent to the forms
G lt F 2 , F a respectively, G l is compounded of G 2 and G 3 .'
It is only in certain cases that the multiplication of two forms gives rise to
a third form, transformable into their product. Supposing that F 2 and F 3 are
irreducible forms, i.e. that neither of them is resoluble into rational factors,
let /!, 7 2 , /g, be any corresponding invariants of F 1} F 2 , F 3 , and let us represent
by B and C the determinants
dx n
o = l, 2, 3, ...,n
= 1, 2, 3,...,n
a = l,2, 3,. ..,11,
7 = 1, 2,3,..., n.
,)
, f
The transformation of F l into F 2 x F 3 then gives rise to the relations
mi
/! X B^ = I 2 X ^
Ml
/ lX C = I 3 xF 2 { ,
? denoting the order of the invariants /,, / 2 , 7 3 . If one of the two numbers
/, and / 3 be different from zero, we infer that m is a divisor of n. For if
- be the fraction reduced to its lowest terms, the equations
v n
1^ x B^ = I 2 " x F 3 vi ,
7/x C^^IJxFj*
imply that F 2 and F 3 (cleared of the greatest numerical divisors of all their
terms) are perfect powers of the order /x ; i.e., M = l, or m divides n, since F 2
and ^ 3 are by hypothesis irreducible. We thus obtain the theorem (which
however applies only to irreducible forms having at least one invariant dif-
ferent from zero) ' No form can be transformed into the product of two forms
of the same sort, unless the number of its indeterminates is a multiple of its
order.' For example, there is no theory of composition for any binary forms,
except quadratic forms, nor for any quadratic forms of an uneven number of
indeterminates.
Again, when m is a divisor of n, let n = km, and let b, c, d 2) d 3 represent
the greatest numerical divisors of B, C, F 2 , F 3 respectively ; v,e find
mi mi
c
Art. 106.]
REPORT ON THE THEORY OP NUMBERS.
231
The first two of these equations show that the invariants of the three forms
FI, F 2 , F 3 are so related to one another, that we may imagine them to have
been all derived by transformation from one and the same form (see Art. 80) ;
the last two (which, it is to be observed, present an ambiguity of sign when
- is even) show that the forms B and F 3 k , C and F 2 k , are respectively identical,
mw
if we omit a numerical factor.
Lastly, let & l} <J> 2 , <I> 3 be any corresponding covariants of F l} F 2 , F 3 .
relation of covariance gives rise to the equations
The
x B
mpq
" =<J>
*i (i, *2, -, O x C " =<J> 3 (!, 2.,, ..., 2,,) x Ff (y lt ..., y n \
where p and q are the orders of the covariants in the coefficients and in the
indeterminates respectively. Combining with these equations the values of'
1 1
B and C already given, we see that <J> 2 x F s m and <J> 3 x F 2 m are identical, ex-
cepting a numerical factor ; i.e. that 4> 2 and $ 3 are either identically zero, or
else numerical multiples of powers of F 2 and F 3 . If therefore two forms can
be combined by multiplication so as to produce a third form transformable into
their product, their covariants are all either identically zero or else are powers
of the forms themselves. There is, consequently, no general theory of com-
position for any forms other than quadratic forms, because all other sorts of
forms have covariants which cannot be supposed equal to zero, or to a multiple
of a power of the form itself, without particularizing the nature of the form.
And even as regards quadratic forms, we may infer that composition is possible
only in cases of continually increasing particularity, as the number of indeter-
minates increases.
106. Composition of Quadratic Forms. Preliminary Lemmas. The follow-
ing lemma is given by Gauss as a preliminary to the theory of the composition
of binary quadratic forms (Disq. Arith., Art. 234) :
(i.) ' If the two matrices
and
be connected by the equation
A
A l A 2 ..
B
B 1 B 2 ..
a
1 * *
b
&1 &2
i
A
B
7 a
-l 6
REPORT ON THE THEORY OF NUMBERS. [Art. 107.
in which the sign of equality refers to corresponding determinants in the two
matrices ; and if the determinants of
unity ; the equation
admit of no common divisor beside
A
n
in which the sign of equality refers to corresponding constituents in the two
matrices, is always satisfied by a matrix | k \ of the type 2x2, of which the
determinant is k, and the constituents integral numbers.' *
The subsequent analysis of Gauss can be much abbreviated if to this lemma
we add three others.
In their enunciations we represent by X, Y, x, y, four functions, homo-
geneous and linear in respect of each of the n binary sets, , m, 2 ? 2 > > ? i n ;
and
n
the matrices composed of the coefficients of X, Y and x, y,
respectively ; by (P, Q, R), (P f , Q', R) quadratic forms of which the coefficients
are any quantities whatever ; and by k an integral number.
(ii.) ' If X, Y, x, y, satisfy the n equations included in the formula
dX dY dX dY
i
~
dy dx dy\
A
B
and
a
b
satisfy the equation
A
-k
a
j
B
A/
b
the matrices
(iii.) ' The greatest numerical common divisor of the n resultants
dXdY dXdY
is equal to the greatest common divisor of the determinants of
(iv.) ' If the n resultants of X and Y be not all identically equal to zero,
the equation px 2 + 2QXY+RY' = P'X* + 2Q'XY+R'y'
implies the equations P = P I , Q=Q f , R R-'
107. Gauss's Six Conclusions. Taking F, /, /' to represent the forms
(A, B, C) (X, F ) 2 , (a, b, c) (x, y}\ (a', &', c') (*', yj, of which the determinants are
* For a generalisation of this theorem, see a paper by M. Bazin, in Liouville, vol. xix. p. 209 ;
or Phil. Trans., vol. cli. p. 295.
Art. 107.]
REPORT ON THE THEORY OF NUMBERS.
233
D, d, d' ; let also M, m, m' be the greatest common divisors of A, 2B,C, of
a, 2b, c, and of a, 26', c ' ; fflL, m, m' the greatest common divisors of A, B, C,
of a, b, c, and of a, b', c', respectively. Supposing that F is transformed into
f x f by the substitution
X =p xx' + p!xy' + p 2 x'y + p 3 yy', Y=q xx' + q t xy' + q 2 x'y + q 3 yy',
let us represent the two resultants
dX dY
dx dy dy dx
and
dX dY
^_
dx' dy' dy' dx'
by A and A' ; the six determinants of the matrix
Po, Pi, P*>
(taken in their
natural order) by P, Q, R, S, T, U ; the greatest common divisor of these six
numbers by k, and the greatest numerical divisors of A and A' by S and S'
respectively, so that (Lemma 3) k is the greatest common divisor of S and S'.
From the invariant property of the determinants of F,f and/' we infer
jf j/
Hence the quotients yr and are squares. (Gauss's 1st conclusion.) Also D
divides d'm? and dm' 2 . (Gauss's 2nd conclusion.) But k is the greatest com-
mon divisor of S and & ; therefore Dk 2 is the greatest common divisor of d'm 2
J Jf
and dm 2 . (Gauss's 4th conclusion.) Let -=- = n 2 , -^ = n' 2 , and let the signs
of n and n' be so taken that A' = n'f, A = nf" ; these two equations are equi-
valent to the six following :
P^ = R-S
a
U
T-
c'
(0)
16 c a 2b'
(Gauss's 3rd conclusion.)
Multiplying together the two resultants A and A', we obtain an identity,
which we shall write at full :
1 + (PO ?3 ~p* qo+pz qi ~PI q^) xy + (p% q 3 p 3 <fo) y 2 ]
-Pa q) x' 2 + (po q 3 -p 3 q +^ q 2 -p 2 qj x'y' + (p v q 3 -p 3 qj y' 2 ~\
= (%i 1z - 2o ^3) ( Po xx + p l xy + p 2 x'y + p s yy') 2
,y) .... (I)
-Pi
x
x (q xx + q l xy' + q 2 x'y + q 3 yy')
xx + q l xy'+ q 2 x'y + q 3 yy') 2 .
Hh
234 REPORT ON THE THEORY OF NUMBERS. [Art. 108.
Comparing this identity with the equation AA' = nn'ff = nn'F, we find by
Lemma 4
The 5th and 6th conclusions relate to the order of the form compounded
of two given forms. The equation
shows that M divides mm'. But also mm divides M k\ For operating on the
... d 2 d 2 rf 2
equation just written with -j z , , , , -y-^ successively, we find
dXdX dXdY dX
vf ^^ n ^_
-j -- 7 + B(-j -- j h -j -- -j )+C-j -- -j \=0, mod mm , . (j)
dx dy V dx dy dy dx / dx dy J
, yJ} , n
+ 2 B -j -- j h C -j - = 0, mod mm .
-y-r- -j -- j -j
dy 2 dy dy dy*
Whence A A 2 , IB A 2 , (7A 2 , and consequently MS 2 , are congruous to zero, mod mm.
Similarly MS' 2 = Q, mod mm'; i.e. mm divides Mk 2 . If then k = 1, i.e. if F be
compounded offandf, M=mm'. (Gauss's 5th conclusion.)
Again, if in the congruences (j) we take m'm as modulus instead of mm',
we may omit the factor 2 in the second congruence, and may infer that A A 2 ,
BA 2 , (7A 2 are all divisible by m'm, i.e. that mm' divides ffllk 2 , or Jtt, when F
is compounded of/ and f. It is also readily seen that JflJl divides mm' and
mm'; whence observing that m = m or \m, m' = m' or ^m', ^H M, or \M,
according as/,/', and F are derived from properly or improperly primitive forms,
we conclude that if f and f be both derived from properly primitive forms, the
form compounded of them is also derived from a properly primitive form; but
if either f or /' be derived from an improperly pinmitive form, the form coin-
pounded of them is derived from a similar form. (Gauss's 6th conclusion.)
In the transformation of F into /x/', the form / is said to be taken
directly or inversely, according as the fraction n is positive or negative. And
similarly for /' and n.
108. Solution of the Problem of Composition. It appears from the identity
(I) that if A, B, C, p ,pi, p 2 ,p 3 , go, gt, g 2 > ga, be integral numbers satisfying the
nine equations (0), the form (A, B, C) (X, Y) 2 will be transformed into the pro-
duct of the two forms (a, b, c) (x, y) 2 and (', b', c') (x', y') 2 by the substitution
X =p xx' +P! xy +p 2 yx' +p 3 yy, Y= g xx + g! xy + g 2 yx + g 3 yy.
Art. 108.]
REPORT ON THE THEORY OF NUMBERS.
235
In order, therefore, to find a form, F, compounded directly or inversely of two
given forms of which the determinants are to one another as two squares,
we have to find eleven integral and two fractional numbers, satisfying the
equations (Q) and (Q'), in which a, b, c, a, b', c, and the signs of n and ri,
are alone given ; the numbers p , p 1 ,p. 2 ,p 3> q ,qi,q 2 ,q 3 , being further subject to
the condition that the determinants of the matrix
Po, Pi, P2,
9o ) qi, q 2 ,
are to admit
of no common divisor. To determine n and n, we observe that the six deter-
minants satisfy the identical relation PUQT+RS = 0; from which we infer,
first, that P,Q,R-S,R + S,T,U must be relatively prime, if P, Q, R, S, T, U
are to be so ; and secondly, substituting for the determinants their values
given by the first six of the equations (Q), that dn' 2 = d'n 2 . Denoting by S'
and the greatest common divisors of P, R S, U and of Q, R + S, T, so that
S and S' are relatively prime, we have evidently n + , n = + 7 ; the
positive or negative signs being taken according as / ' and f enter the com-
position directly or inversely ; and the absolute values of S and S' being deter-
mined by the equation S 2 d'm 2 = S' 2 dm 2 . The fractions n and n being thus
ascertained, the values of P, Q, R, S, T, U are known from the equations (0) :
these values are all integral : for P, Q, R S, R + S, T, U, this is evident from
the equations (0), and may be proved for R and S by means of the identity
PUQT+RS=0. We have next to assign such values to the constituents
of the matrix ' " *' "* , that its determinants may acquire the known
values of P, Q, R, S, T, U. To do so, it is sufficient * to obtain a fundamental
set of solutions of the indeterminate system,
x 1 U-x 2 T+x 3 S=Q,
-x U + x 2 R-x s Q = Q,
T T V Ji 4- T P
*/jj ' tfj-^ -il> ~ t^jj -i V,
~" XQ O "1" X] \g """ it/2 = ,
which is equivalent to only two independent equations. From the skew sym-
* For a solution of the general problem, ' To find all the matrices of a given type, of which the
determinants have given values,' see a paper by M. Bazin, in Liouville, vol. xvi. p. 145; or Phil.
Trans., vol. cli. p. 302. For the definition of a fundamental set of solutions of an indeterminate
system, see ibid, p. 297. It may be observed that the analysis of Gauss, which is exhibited in the
text, is applicable to any matrix of the type n x ( + 2).
Hh2
236 REPORT ON THE THEORY OF NUMBERS. [Art. 108-
metrical form of the matrix of this system, it appears that if O g , 0,, 2 , 3 be any
multipliers whatever, any four numbers (x , x^, a^, 0%) proportional to
-e p +e 2 s+e 3 T,
-e.Q-e.s +e s u,
-0 R-e 1 T-e 2 U
will satisfy the system (S), and in addition the equation
6 x + 0i x 1 + 2 x 2 + 3 x 3 = 0.
Assigning, then, to ,6 1 ,6 2 , 6 3 any arbitrary values whatever, let q ,qi,q 2 , q 3
be four numbers relatively prime, and proportional to the four numbers (2) ;
let also -T q + Tr 1 q 1 + ir 2 q 2 + Tr s q 3 =l; and employing ir , v lt Tr 2 , Tr 3 in the place
of , 0,, 2 , 6 3 , let us represent by p , p 1} p 2 , p 3 the solution of (S) thus obtained.
We have thus two solutions of (S), satisfying respectively the relations
>Topo + '*'iPi + ' 3r 2p2 + '*ap3 = Q, and Tr q + Tr 1 q 1 + -ir 2 q 2 + Tr 3 q 3 = l,
which prove that the two solutions form a fundamental set, i.e. that the
determinants
PO,P1,P2,P 3
= [P,Q,R,
It only remains to show that the values of A, B, C, which are now sup-
plied by the equations ()'), are integral. Operating on the identity (I) with
d 2 d 2 d 2 d 2 d 2 d 2
-5-r, and also with -j-r a , -r-rr-,, -j~^, we
similar to that which we have employed to establish the 5th conclusion, that
2 Ann, 2Bnn, 2Cnn, which are certainly integral numbers, are divisible by
7?4- Sf 7? Sf
2<5<T if - r and , are both even, and by <W if either of these numbers
o o
is uneven. In the former case A, B, C are evidently integral ; in the latter,
either - - or 7 is uneven, i.e. either m or m' is. even, and the quotients of
2A 2B 2(7
2 Ann, 2Bnn, 2Cnri divided by W are - -,, ,, , ; whence, again,
, .- ' J mm mm' mm
A, B, C are uitegral.*
* Gauss shows that A, B, C are integral by substituting the values of />,..., q , . . . in
and observing that the results, after division by nn, are integral. The values of J> ,... are always
obtained free from any common divisor by the process in the text; but Gauss has to determine
four new multipliers, , 0,, 0,, 3 , to obtain from the formulae (2) the exact values of ?,..., and
not equimultiples of those values. M. Schlafli (Crelle, vol. Ivii. p. 170) has shown that Gauss's
demonstration is connected with a remarkable symbolical formula.
Art. 109.]
REPORT ON THE THEORY OF NUMBERS.
237
109. Composition of severed Forms. It will now be convenient to extend
the definition of composition to the case in which more than two forms are
compounded. If a quadratic form, F, be changed by a substitution, linear
in respect of n binary sets, into the product of n quadratic forms, f^f^, ...,f n ,
so that =
F(X,Y) = n (a
we shall say that F is transformable into f^ xf 2 x ... xf n ; and if the deter-
minants of the matrix of the transformation are relatively prime, we shall
say that F is compounded of fi,f 2 , ...,/. We shall retain, with an obvious
extension, the notation of Art. 107. The invariant property of the deter-
minant of F supplies the n equations
from which we infer (1) that D, d 1} d 2 , ... are to one another as square numbers,
(2) that Dk 2 is the greatest common divisor of the n numbers ^
Ac-
cording as the equation A,.yj = Ayr- H/ is satisfied by a positive or negative
value of the radical, we shall say that f { is taken directly or inversely.
Adopting this definition, we can enunciate the theorem
'If .Fbe compounded of/j,/^ ...,/ rt , and F' be transformable into
Ji x J 2 x x />
the forms being similarly taken in each case, F' contains F.' For we infer from
(2) that D'k' 2 = D, whence A'^'A,-, or by the Lemmas 2 and 1 of Art. 107,
X' = aX + pY, Y' = yX+SY,
a, /3, 7, $ denoting integral numbers which satisfy the equation. a fiy = lc.
We thus obtain the equation
F'(aX+pY, yX+SY) = F(X,Y),
'
whence, by Lemma 4, F' is transformed into F by
.
If F be compounded of/!,/ 2 > ./> an d a single transformation of F into
yj xf 2 x ... xf n be given, we may, by the same principles, find all the transforma-
tions of F into the product off l} f 2 , ..-,f n , taken as in the given transformation.
For '} F(X ,Y ) = Uf represent the given transformation, and F(X, Y) =
be any other transformation, we find
238 REPORT ON THE THEORY OF NUMBERS. [Art. 109.
and consequently F(aX
Q
or
is, by Lemma 4, a proper automorphic of F. The formula
is an automorphic of F, will therefore represent all the trans-
iii which
7, i
formations required.
If F be transformable into/j xf 2 x ... xf n , and $ contain F, while /j,/ 2 ,...,/,
contain <pi, <f> 2 , ...,$, *$ will be transformable into ^ x ^> 2 x ... x <p n . This
follows from a preceding general observation (Art. 105) ; but we must add
here that if T, r { denote any positive or negative units, according as the
transformations of < into F, and f { into <p ( are proper or improper, while v ;
denotes a positive or negative unit according as f f is taken directly or inversely,
<p { will be taken directly or inversely according as T 7 x T,- x v { is positive or
negative. This is apparent if we observe that the sign of the quantity
/ A
-' * is altered by an improper transformation of X, Y, or x { , y it but is not
/
altered by a transformation of any of the other sets.
The theorem that ' forms compounded of equivalent forms, similarly taken,
are themselves equivalent' is included in the preceding. We may, therefore,
speak of the class compounded of any number of given classes.
It is an important and not a self-evident proposition, that if F be com-
pounded of <t>,/ 3 ,f t , '--ifn, an d <p be compounded of fi,f 2 , F is compounded of
fi>fz> >& Let < = a 2 + 2 fify + y^ s let /u. be the greatest common divisor of
a, 2/3, 7, and V the determinant of ; let also X, Y transform F into
Writing in X and Y for and i\ the bipartite expressions linear in x l y l , X 2 y 2 ,
by which (f> is transformed into f t xf 2 , we obtain a transformation of F into
yi x_/2 x ... xf n . If k be the greatest common divisor of the determinants of the
matrix of this transformation, Dk 2 is the greatest common divisor of the n
numbers II m 2 . But this common divisor is the same as the greatest common
m?
divisor of V x II m,- 2 , and the n 2 numbers
d if i**-* 2
m i 8 = 3
because V is the greatest common divisor of d l m z 2 and d z m^ (4th conclusion),
Art. 110.] REPORT ON THE THEORY OF NUMBERS. 239
and because fj. = m 1 m 2 (5th conclusion) ; i. e., Dk 2 = D, or k 2 = 1, and F is com-
pounded of fi,f z , ,f n . Also, if i> 2, ft is similarly taken in both compositions,
for A< A; ff
~f ? ~p~ and -, 2 s
T v r v v r rn v r v v /
./I < ^./2 * * * j n r / 3 " " * .In
are identical ; and if 4 = 1, or 2 r
cfce^ cf?/,- cfo/j cfce f \c cZ^ c>; d' ^dx { dy { dy { dx^
whence, if Q and , be positive or negative units, according as <p andy) are taken
directly or inversely in the composition of F and < respectively, f { will be taken
directly or inversely in the composition of F, according as Q x to,- is positive or
negative.
By this theorem, the problem of finding a form compounded of any number
of given forms is reduced to the problem of finding a form compounded of two
given forms. For if fi,f z , >./ be the given forms, we may compound the
first with the second, the resulting form with the third, and so on until we
have gone through all the forms, when the form finally obtained will be com-
pounded of the given forms, as will immediately appear from successive appli-
cations of the preceding theorem. We also see that we may compound the
forms in any order that we please, or we may divide them into sets in any way
we please, and compounding first the forms of each set, afterwards compound
the resulting forms. If any of the given forms are to be taken inversely, we
may substitute for them their opposites (Art. 92) taken directly. We may thus,
without any loss of generality, and with some gain in point of simplicity, avoid
the consideration of inverse composition altogether ; and, for the future, when
we speak of the form compounded of given forms, or the class compounded of
given classes, we shall understand the form or class compounded directly of the
given forms or classes.
110. The solution of the problem of composition given in Art. 108 may be
put into a form better suited to actual computation.
The system (S) is evidently satisfied by [0, P, Q, R], and also by
[P, 0, S, T~\ ; and these solutions are independent, because the determi-
nants of their matrix cannot all be zero unless P = 0, a supposition which may
be rejected as it implies that a = 0, i. e. that d is a square. From this set of
independent solutions a set of fundamental solutions is deduced, as follows.
Let fj. be the greatest common divisor of P, Q, R; and let k be determined by
the congruences Q R P
k-' S=0, k T=0, mod ,
240 REPORT ON THE THEORY OF NUMBERS. [Art. 110.
O Jf
which are simultaneously possible, because and -- have no common divisor
with the modulus, while the determinant
/t*
is divisible by it. The solutions
kQ-pS kR-nT^ r P Q R
are then a fundamental set, and may be taken for [p a ,pi, p a ,ps], [<?<>, ?!> 9 f 2>
respectively. We thus find
M \Ju\.v f\ Tl / Tl C1 f\ 7 *t
or A = --; 2Bnn = R + S-2k .
fl" fJL 2 ft
P Q 7?
Multiplying this equation by , , : - in succession, and attending to the
congruences satisfied by k, we obtain the congruences
P D db' Q v a'b R . bb' + Dnn'
B = , B = , -B = - ,inodA;
"P /D 7?
which determine B for the modulus A, because , , are relatively prime.
fl. ft. fJL
These determinations [viz. of A, and of B, mod A] are sufficient for our purpose ;
because, if R = B + \A, the forms
(A, B, ^) and (A, B, """
are equivalent. To obtain, therefore, the form compounded of two given forms
(a, b, c), (a, b', c'), we first take the greatest common divisor of d' ra 2 and d m' 2
for D (giving to D the sign of d or c?') ; we then determine n and n by the
equations W = \/T)> n ' = < ^~Ti> an ^> representing by n the greatest common
divisor of an, a'n, bn' + b'n, we obtain A, B, C, from the system
A _ aa '
V* '
an' r>_ab'
~" _[j ^"^ j
a'n a'b
- B = , ! mod
M M
bn' + b'n D _ bb' + Dnn
Art. 111.]
REPORT ON THE THEORY OF NUMBERS.
241
These formulae, which are applicable to every case of composition, and are
therefore more general than the analogous formulae given by Gauss (Disq.
Arith., Art. 243), are due to M. Arndt*, who has also given an independent
investigation of them, though our limits have compelled us here to deduce
them from Gauss's general solution of the problem of composition. That
(A, B, (7) is transformed into (a, b, c) x (a, b', c) by the substitution
1 , V-Bn'
-X=xx-\
b-Bn , W + Dnri - B (bn + b'n)
- --
a
a
aa
yy
nY = arixy' + a'nx'y + (b'n + bn) yy',
may be inferred from the values of p , ..., q u , ...; or may be verified directly
by observing that
M [A X + (B + VD] Y] = [ax + (b + n v/D) y] x [ax + (b' + nVD] y']-
111. Composition of Forms Method of Dirichlet. Lejeune Dirichlet, in
an academic dissertation (' De formarum binariarum secundi gradus compositione,'
Crelle, vol. xlvii. p. 155), has deduced the theory of the composition of forms
from that of the representation of numbers. The principles of this method
are applicable to any case of composition ; but Dirichlet has restricted his
investigation to properly primitive forms of the same determinant D. Let
(a, 6, c), (a', b', c) be two such forms ; let M and M' be two numbers prime
to 2 D, and capable of the primitive representations
M = am 2 + 2 bmn + en 2 , M' = am'* + 2 b'm'n + en'*,
by the forms (a, b, c) and (a', b', c} respectively ; also let these representations
appertain to the values w and u> of \/D, so that
w 2 = Z>, mod M, <a' 2 =D, mod M',
* Crelle's Journal, vol. Ivi. p. 64. In the new edition of the Disq. Arith. (Gottingen, 1863),
a MS. note of Gauss is printed at p. 263, containing the congruences by which B is determined in the
case of the direct composition of two forms of the same determinant.
The account of the theory of composition in the preceding Articles (106-109) differs from that in
the Disq. Arith. (Arts. 234-243) chiefly in the use which is here made of the invariant property of
the determinant. A different mode of treatment of Gauss's analysis is adopted by M. Bazin, in
Liouville, vol. xvi. p. 161.
In Arts. 108 and 110 we have endeavoured to supply the analysis of a problem which Gauss,
as is not unusual with him, has treated in a purely synthetical manner (Disq. Arith., Arts. 236
and 242, 243); and it is for this reason that we have introduced the consideration of fundamental
sets of solutions of indeterminate systems, which are not explicitly mentioned in the Disq. Arith. It
is perhaps singular that Gauss does not employ the identity P(7QT+RS= 0; it was first given
by M. Poullet Delisle, in a note on Art. 235 in his Translation of the Disq. Arith.
I i
242 REPORT ON THE THEORY OF NUMBERS. [Art. 111.
and so that the forms (a, b, c), (a', b', c') are respectively equivalent to the forms
If the values u> and ' are concordant, i.e. if it is possible to find a number Q
satisfying the three congruences
Q 2 = D, mod MM', = , mod Jf, Q = a,', mod M',
(in which case the solution Q of the congruence Q 2 = .D, mod MM', may be
said to comprehend the solutions o> and w' of the congruences co 2 = Z), mod Jf,
and w' 2 = D, mod Jf ',) the form
(MM; a,
will be a properly primitive form of determinant D, and will belong to one
and the same class (which may be termed the class compounded of the classes
containing (a, b, c) and (a', b', c')) whatever two numbers (subject to the con-
ditions prescribed) are taken for M and M '. To prove this, a few preliminary
remarks are necessary. (1) If the solutions and u> are concordant, there is
but one solution Q (incongruous mod MM') comprehending them. (2) The
necessary and sufficient condition for the concordance of u> and ta is w = w',
for every prime modulus dividing both M and M '. (3) If , u>, u>' satisfy the
congruence x 2 = D for the modules MM', M, and M' respectively; and if, besides,
Q = to, Q = w', for every prime divisor of M and M' respectively, o> and ' are
concordant, and Q is the solution comprehending them. (4) The value of \/D,
to which any given primitive representation (such as M= am? + 2 bmn + en 2 )
appertains, may be defined by congruences, without employing the numbers
n and v which satisfy the equation mv n/j. = 1 (see Art. 86) ; in fact, we find
am + (6 + o)w = 0, mod M, (b u>)m + cn = Q, mod M ;
whence also = b, mod d, w = + b, mod d', if d and d' are common divisors
of M and m and of M and n.
We may suppose that the given forms (a, b, c) and (a, b', c') are so prepared*
* It is readily proved that a properly primitive form can represent numbers prime to any given
number ; thus a form can always be found equivalent to a given properly primitive form, and having
its first coefficient prime to a given number. This transformation will be frequently employed in the
sequel : in the present instance, we have only to substitute for the given forms any two forms
respectively equivalent to them and having their first coefficients relatively prime.
Art. 111.]
REPORT ON THE THEORY OF NUMBERS.
243
that the representations of a and a' by them appertain to concordant values
of v/-t> ; i.e. that we can find a number B satisfying the congruences
-B 2 = D, mod aa', B = b, mod a, B = V, mod a'.
ni _ r\
Let - -, = C; the forms (a, b, c), (a, b', c) are then equivalent to (a, B, a'C\
ctct
(a, B, aC) respectively; and if
X = xx - Cyy', Y = axy' + a'x'y + 2 Byy',
we find by actual multiplication
From this equation (which is included as a particular case in the formulae
of M. Arndt) it appears that MM' is capable of representation by (aa', B, (7) ;
it can also be shown (1) that this representation is primitive ; (2) that it
appertains to a value of \/D, mod MM', comprehending the values to and to',
to which the representations of M and M' by (a, b, c) and (a', b', c') respectively
appertain. (1) If x, y, x', y', and X, Y are the values of the indeterminates
in the representations of M, M', and MM' by (a, B, a'C), (a, B, aC), and
(aa, B, (7), the hypothesis that X and Y admit of a common prime divisor p
is expressed by the simultaneous congruences
xx' Cyy = 0, axy' + a'x'y + 2 Byy' = 0, mod p.
These congruences are linear in respect of the relatively prime numbers x' and y' ;
their coexistence implies, therefore, that p divides their determinant M ;
similarly it may be shown that p divides M ' ; so that to = to', mod p, because
and to' are concordant. The congruences satisfied by to and to' now give the
relations ax+(B + u)y = Q, ax + (B + w)y'= 0, mod p ;
whence, eliminating x and x' from the congruence Y=0, and observing that 2 to
is prime to M, and therefore to p, we find yy'= 0, mod p. If y is divisible by p,
we infer, from the congruence X = 0, that x' is also divisible by p; but the*
congruences satisfied by to and to' give in this case the contradictory results
to = + B, to = B ; i.e. y is not divisible by p, and similarly it may be shown
that y' is not divisible by p. The congruence yy'=0, mod p, is therefore im-
possible ; or the representation of MM' by (aa', B, C) is primitive. (2) Let Q'
be the value of v/Z), to which this representation appertains ; and let p be any
divisor of M ; then Q' satisfies the congruences
aa'X+(B + Q')Y = 0, (B-Q')X+CY=0, mod p;
and it will be found, on substituting the values of X and Y, that these con-
i i 2
244 REPORT ON THE THEORY OF NUMBERS. [Art. 112.
gruences are also satisfied by ; whence it follows, since either X or Y is prime
to p, that Q' = w, mod p. Similarly, if p be a prime divisor of M', Q'=u, mod p ;
or Q' is a solution of the congruence Q"=D, mod MM', comprehending the
solutions to and <o. Hence Q'= Q, mod MM', and the form
(JOT. Q,
is equivalent to (aa, B, C], because either of them is equivalent to
The equivalence of all the forms included in the expression
(MM; , )
is therefore demonstrated.
It will be seen that Dirichlet's method may be applied to the composition
of any number of forms, and that the theorems of Art. 109 present themselves
as immediate consequences of his definition of composition.
112. Composition of Classes of the same Determinant. We shall now con-
sider more particularly the composition of classes of the same determinant D.
We represent these classes by the letters /, <, ..., and we use the signs of
equality and of multiplication to denote equivalence and composition respec-
tively*. The following theorems are then immediately deducible from the
six conclusions of Art. 107, and from the formulae of Art. 110.
(i.) ' If /be a properly primitive class, /x $ is of the same order as <.'
(ii.) 'A class is unchanged by composition with the principal class.' In
consequence of this property, it is sometimes convenient to represent the
principal class by 1.
(iii.) ' The composition of two opposite t properly primitive classes produces
the principal class.'
If, then, f denote any properly primitive class, we may denote its opposite
by/" 1 , and we may write/x/" 1 = 1.
* Gauss uses the sign of addition instead of that of multiplication ; thus /x < is /+ < in the
Disq. Arith., and /" is nf. The change appears to have been introduced by his French translator, and
to have been acquiesced in by subsequent writers.
t Two classes which are improperly equivalent are called opposite, because they contain opposite
forms (see Art. 92).
Art. 112.]
REPORT ON THE THEORY OP NUMBERS.
245
(iv.) 'If f be a given properly primitive class, and $ any given class, the
equation Fxf=& is always satisfied by one class, F, and by one only; viz. by
the class F=3>xf~\'
(v.) ' If <&!, <J> 2 , ... be all different classes, and/ be a properly primitive class,
fy. ^j./x < 2 , ... are all different classes.'
(vi.) 'A properly primitive ambiguous class produces by its duplication the
principal class ; ' for an ambiguous class is its own opposite. Conversely, if
2 = 1, i.e.jfff) be a class which, by its duplication, produces the principal class,
<p is a properly primitive ambiguous class; for we find <p 2 X(f>~ 1 = <~ 1 , whence
<f) = <f>-\ or < and its opposite are properly equivalent.
(vii.) ' The class compounded of the opposites of two or more forms is the
opposite of the class compounded of those forms.' It follows from this, or from
(vi.), that a class compounded of ambiguous classes is itself ambiguous.
(viii.) Let <, ^, ..., ^ a _ 1 represent all the classes of det. D, and of a given
order 12; and let 1, fi, f%, , f n -\ represent the properly primitive classes of
the same determinant ; it may then be shown that w is a divisor of n, and that,
n
given two classes of the order Q, there always exist - properly primitive classes,
which, compounded with one of them, produce the other. Assuming, for a
moment, that a form < exists, such that the equations included in the
formula 3> xf=<fr lt can all be satisfied, we see that each of these equations
is satisfied by the same number of properly primitive classes f; for if the
equation $> xf=3> be satisfied by k primitive classes, 1, tp lt < 2 , ..., <jt_i, the
equation ^ O xf=^^, which is, by hypothesis, satisfied by a single class, f^, is
also satisfied by the kl classes f^xfa, ..., f^ x fc _ 1; but by no other class.
Since, then, the classes </> x/, of which the number is n, represent every class
of the order Qk times, we have evidently n = ku>. It is also readily seen that
every equation of the type <I> I ,x/=4> M admits of k solutions; and thus it only
remains to justify the assumption on which the preceding proof depends. If
the order Q be derived by the multiplier m from a properly primitive class of
determinant A = , we may take for < the class represented by the form
lit/
(m, 0, Am) ; if Q be derived from an improperly primitive class, we take for
* the class represented by the form (2m, m, \m (A- 1)). Representing $ M in
the first case by the form (ma, mb, me], and in the second by the form
(2ma, mb, 2 me), and supposing (as we may do) that a in each case is prime
246 EEPORT ON THE THEORY OF NUMBERS. [Art. 113.
to 2D, we see that the forms (a, mb, m 2 c) and (a, bm, 4cm 2 ) are properly
primitive ; and we find by the formulae of composition (Art. 110),
(m, 0, - Am) x (a, bm, cm 2 ) = ( ma, mb, me),
(2m, w, ^m(A - 1)) x (a, bm, 4cm 2 ) = (2m, mb, 2mc) ;
i.e. the equation 4> x/=4> /1 can be satisfied for every value of M.
113. Comparison of the numbers of Classes of different Orders. To deter-
71
mine the quotient - of the last Article, Gauss investigates the properly primitive
classes of det. D, which, compounded with the classes
(m, 0, Am) and (2m, m, ^m(A-l)),
reproduce those classes themselves. He thus employs the theory of composition
to compare the number of properly primitive classes of a given determinant
with the number of classes contained in any other order of the same deter-
minant ; or, which comes to the same thing, to compare the numbers of classes,
of any given orders, of two determinants which are to one another as square
numbers (Disq. Arith., Arts. 253-256). We have already seen (Art. 103) that
the infinitesimal analysis of Dirichlet supplies a complete solution of this pro-
blem ; whereas, in the case of a positive determinant, the result in its simplest
form was not obtained by Gauss. It has, however, been recently shown by
M. Lipschitz (Crelle, vol. liii. p. 238) that the formulae of Dirichlet may be
deduced, in a very elementary manner, from the theory of transformation.
We propose in this place to give an account of this investigation, and to
point out its relation to the method pursued by Gauss. We begin with the
theorem
' Every properly primitive class of determinant De 2 is contained in one, and
only one, properly primitive class of determinant D.'
Let (A, B, C) be a properly primitive form of det. De 2 , in which A is prime
to e ; let B' be determined by the congruence eB' = B, mod A, and C" by the
B' 2 D
equation C' = j ; then the forms (A, B, C) and (A, Re, C'e 2 ) are equi-
valent ; but (A, B'e, C'e 2 ) is contained in (A, B', C"), therefore also (A, B, C)
is contained in (A, B', C'), that is, in a properly primitive form of determinant
D. Again, if (a, b, c), (a', b', c') are two forms of det. D, each containing
(A, B, C) these two forms are equivalent. For applying to (A, B, C) the
system of transformations of modulus e, included in the formula
m, k
0, A*
(Art. 88),
Art. 113.]
REPOBT ON THE THEOEY OF NUMBERS.
247
we readily find that, of the resulting forms, one, and only one, will have its
coefficients divisible by e 2 * ; therefore the class represented by (A, B, C) con-
tains one, and only one, class of det. De*, and of the type (e 2 p, e 2 q, e 2 r). But,
applying to (A, B, C) the transformations inverse to those by which (a, b, c) and
(a', b', c) are changed into (A, B, C), then (A, B, C) is changed thereby into
(e 2 a, e z b, e 2 c) and (e 2 a, e 2 b', e z c); these two forms are therefore equivalent;
i.e. (a, b, c) and (a', b', c') are equivalent.
We have next to ascertain how many different properly primitive classes
of determinant De 2 are contained in the class represented by (a, b, c}, a properly
primitive form of det. Z>, in which a may be supposed prime to e. Applying to
(a, b, c) a complete system of transformations of modulus e, we inquire, in the
first place, how many of the resulting forms are properly primitive. For this
purpose we observe that if e = e 1 xe 2 x e 3 x ..., (e lt e 2} ... representing factors of
which no two have any common divisor), a complete system of transformations
for the modulus e is obtained by compounding, in any definite order, the systems
of transformations for the modules e 1} e 2 , ...; i.e. if ] e 1 , \e 2 \, ... be symbols
representing complete systems of transformations for the modules e lt e 2 , ..., every
transformation of modulus e is equivalent by post-multiplication f to one, and
only one, of the transformations | e a | x | e 2 1 x | a, | x It will, therefore, be
sufficient to determine the number of properly primitive forms obtained by
applying to a properly primitive form a complete system of transformations
for a modulus which is the power of a prime. Let p be an uneven prime, and
-y, k
let (a, b, c) be changed into (A, B, C) by
o,
a formula which will repre-
sent a complete system of transformations for the modulus p a , if y receive every
value from to a inclusive, and if k be the general term of a complete system
If
m, k
transform (A,B, C) into (P,Q,fi), we have
P = Am\ Q = m(Ak + Bp), R =
Observing that A is prime to e, we infer from the congruence P = 0, mod e 2 , that m = e, p = 1 ; the
congruence Q = 0, mod e 2 , then becomes Ak + B = 0, mod e, giving one, and only one, value of
k, mod e ; and this value satisfies the remaining congruence R = 0, mod e 2 , since AR = (A k + B) 1 De 1 .
t If | A | and | B \ are two transformations connected by the symbolic equation
\B\ = \A\ x\V\,
in which | V \ is a unit transformation, | A \ and B \ are said to be equivalent by post-multiplication,
or to belong to the same set. A complete system of transformations for any modulus contains one
transformation belonging to each set.
248
REPORT ON THE THEORY OF NUMBERS.
[Art. 113.
of residues, mod p a ~~i ; we find
A = ap* ( "-i\ B = (ak + bpi) p a -", C= dk* + 2 bkpi + cp^ ;
whence, if y = a, (A, B, C) is properly primitive; and if so, or not, for every
other value of 7, according as C is not, or is, divisible by p. If 7 = 0, we have
(7=0, for p a ~ 1 \ 1 + ( ) values of k, incongrouous mod p a ; if y have any
value intermediate between and a, we have C=0, for p a -i~ l values of k,
incongruous mod p a ~t. Hence the number of properly primitive forms is
and similarly if p = 2 it will be found that the number of properly primitive
forms is 2. Hence the number N of properly primitive forms, arising from
the application of a complete system of transformations of modulus e to the
form (a, b, c), is e II 1 ( ) , p denoting any uneven prime dividing e.
It remains to determine the number of non-equivalent classes in which these
N forms are contained. For brevity, we consider the case of a positive deter-
minant. Let [T x , f/J represent any solution of the equation T ! DU' = l, and
let <r be the index of the least solution of that equation which is also a solution
of T 2 e*DU* = l, i.e. let <r be the index of the first number in the series
Z7,, Z7 2 , ... which is divisible by e ; also let (A, B, C} represent any one of the
N properly primitive forms into which (a, 6, c) is transformed. The trans-
formations of modulus e by which (a, b, c) is changed into (A, B, C) belong to
a- different sets, the transformations of the same set being equivalent by post-
multiplication, but those of different sets not being so equivalent. For if
be a transformation of (a, b, c) into (A, B, C), any other transformation is repre-
sented (Art. 89) by the formula
T t -bU x ,
aU x ,
-cU
and these two transformations will or will not belong to the same set, according
, satisfying the equation
as a unit transformation
a, ft
T x -bU x ,
-cU x
Art. 113.]
EEPOBT ON THE THEORY OF NUMBERS.
249
does or does not exist,
we find
e x
, -
-7,
Premultiplying each side of this equation by
A, ft eT x -BU x , -CU X
v lf > = AU,,eT, + BU x '
whence, observing that A, B, C are relatively prime, we see that A, n, v, p are
or are not integral according as U x is, or is not, divisible by e ; a conclusion
which implies that the transformations of (a, b, c) into (A, B, C} are contained
in cr different sets. It thus appears that, of the N transformations which
applied to (a, b, c) give properly primitive forms, there are o- which give forms
equivalent to (A, B, C) ; i.e. the number of properly primitive classes of det. De 2 ,
contained in (a, b, c), a properly primitive class of det. D, is
p \p
a result which is in accordance with the formula of Dirichlet (Art. 103). If D
he negative, we have only to put a- = 1, as is sufficiently apparent from the
preceding proof; if, however, D = 1, a- = 2.
The properly primitive classes of det. De 2 , into which a given properly
primitive class (a, b, c) of det. D is transformable, are always such that, com-
pounded with the class (e, 0, -De), they produce the class (ea, eb, ec). For
let (a, 6, c) be transformable into (A, B, C) of det. De 2 , and let us take a form
of the type (A, Re, C'e 2 ), equivalent to (A, B, C) ; then (a, b, c) and (A, B', C")
are equivalent. But
(e, 0, -De) x (A, B'e, C'e*) = (eA, eB', eC'),
therefore also (e, 0, - De) x (A, B, C) = (ea, eb, ec).
And, conversely, the classes which, compounded with (e, 0, -De), produce
(ea, eb, ec) are precisely the classes into which (a, b, c) is transformable. Thus
the properly primitive classes of det. De 2 , which, compounded with (e, 0, De),
reproduce that class itself, are no other than the properly primitive classes of
det. De 2 into which (1, 0, D) is transformable. And it is by this substitution
of a problem of transformation for a problem of composition that M. Lipschitz
has simplified and completed the analysis of Gauss.
A method similar in principle is applicable to the comparison of the num-
bers of properly and improperly primitive classes. We can first show that if
D = l, mod 4, the double of every properly primitive class of det. D arises by
a transformation of modulus 2 from one, and only one, improperly primitive
class of the same determinant; viz. if (a, b, c) is a given properly primitive
Kk
250
REPORT ON THE THEORY OF NUMBERS.
[Art 113.
form*, in which a and b are uneven, (2 a, b, c) is improperly primitive, and is
1,0
changed into (2a, 2b, 2c) by
Q
and, again, if (2p, <?, 2r), (2p', g>', 2r / ) are
two improperly primitive forms, each of which is transformable into (2 a, 2&, 2c)
these two forms are equivalent, because (a, 6, c) is transformable into (Ip, 2q, 4r)
and also into (4p', 2j', 4r'), while it can be shown that (a, 6, c) is transform-
able into the double of only one improperly primitive class. Also, applying the
system of transformations,
1,0
0,2
2,0
0,1
2,1
0,1
, to the improperly primitive
form (2p, q, 2r), we obtain, if D=l, mod 8, the double of only one properly
primitive form : in this case therefore the numbers of properly and improperly
primitive classes are equal. If D = 5, mod 8, we obtain the doubles of three
properly primitive forms ; and we have to decide to how many different classes
B,/8
these three forms belong. It appears from Art. 89, that if
be a trans-
* *
formation of (2p, q, 2r) into the double of a properly primitive form (a, 6, c), all
the transformations are included in the formula
l(T x -qU x ),
-rU x
a, ft
7 ,S
\T X , Uy\ denoting any solution of the equation T 1 D U" 2 = 4. Taking the case
of a positive determinant, and employing the same reasoning as before, we infer
that if U a be the first of the numbers U l , U 2 , ... which is even, these trans-
formations are contained in a- different sets. But a- is either 1 or 3 according
as t/i is even or uneven (see Art. 96, (vi.)) ; the three forms will therefore re-
present three classes or one, according as Z7j is even or uneven ; and the number
of properly primitive classes, in these two cases respectively, will be three times
the number of improperly primitive classes, or equal to it. If D be negative,
the three forms will belong to different classes ; and there will be three times
as many properly as improperly primitive classes. From this statement, how-
ever, we must except the determinant 3, which has one properly and one
improperly primitive class.
It will be found that the properly primitive class or classes, into the double
* { If (a, b, c) is properly primitive, a and c uneven, b even, _ ' transforms (a, b, c) into
H~ > *
2 x an improperly primitive form ; i.e. when there are improperly primitive forma at all, which
implies a + c = 0, mod 4. Either sign may be taken.}
Art. 114.]
REPORT ON THE THEORY OF NUMBERS.
251
of which a given improperly primitive class can be transformed, and which in
turn can be transformed into the double of the given class, are also the class
or classes which, compounded with the class (2, 0, ^(D 1)), produce the given
class. Thus every improperly primitive class is connected either with one or
three properly primitive classes (see Art. 98, note, and Art. 118).
114. Composition of Genera. Let /and/' be two properly primitive classes
of det. D, m and m two numbers prime to one another and to 2 D, and repre-
sented by/ and/' respectively; then mm' is represented by/x/'. Hence the
generic character of /x/' is obtained by multiplying together the values oi the
particular characters of / and /'. For those generic characters which are
expressed by quadratic symbols this is evident, since
(mm'\ / m \ f m \
p ) \p / \p ) '
and it is equally true for the supplementary characters, since it will be found
that
The genus T, in which /x/' is contained, is said to be compounded of the
genera 7 and 7', in which / and /' are contained ; and this composition is
expressed by the symbolic equation F = 7 x 7'. It will be seen that the
composition of any genus with itself gives the principal genus.
The same considerations may be extended to improperly primitive classes.
Thus, if/ and /' be respectively properly and improperly primitive, m and m
uneven numbers prime to one another and to D, represented by / and \f ',
the genus of the improperly primitive class, /x/', may be inferred from the
number mm', i.e. it is obtained by the composition of the generic characters
of/ and f. Or, again, if / and /' be both improperly primitive, so that the
class compounded of them is the double of an improperly primitive class, the
generic character of this improperly primitive class is obtained by compound-
ing those of the two given classes.
It follows, from these principles, that the number of classes in any two
genera of the same order is the same. For if 4> 1( <J> 2 , ...,<& be all the
classes of any genus of properly or improperly primitive forms, F^ a class
belonging to any other genus of the same order, and (j) a properly primitive
class satisfying the equation <I> 2 x < = .Fj , the classes ^ x <f>, ...,< x (f> are all
different, and all belong to the genus (F); consequently (F) has at least as
many classes as (<), and vice versd () has at least as many as (F), i.e. they
both contain the same number of classes.
K k 2
252 REPORT ON THE THEORY OF NUMBERS. [Art. 115.
115. Determination of the Number of Ambiguous Classes, and Demon-
stration of the Law of Quadratic Reciprocity. The number of actually existing
genera of properly primitive forms cannot exceed the number of properly
primitive ambiguous classes. For let n be the number of classes in each
genus, k the number of actually existing genera, so that kn is the number
of properly primitive classes; let also 1, A lt A z , ..., A h _ 1 be the properly
primitive ambiguous classes. Every class produces, by its duplication, a class
of the principal genus ; and if K be a class of the principal genus produced
by the duplication of X, K is also produced by the duplication of
XxA lt XxA z , ..., Jx^ 7l _i,
but by the duplication of no other class. If, therefore, there be n' classes
in the principal genus which can be produced by duplication, the whole
number of properly primitive classes is hxn, i.e. hn =hn. But n ^ n,
therefore k < h.
It may be inferred from Art. 112, (vii.), that all genera which contain any
ambiguous classes contain an equal number of them. We shall immediately
see that the number of ambiguous classes is equal to the number of genera,
and is consequently a power of 2. The number of ambiguous classes in any
genus is, therefore, either zero or a power of 2 ; and if any genus contain 2"
ambiguous classes, such classes will exist only in ^ genera.
Gauss determines the number h of properly primitive ambiguous classes
by very elementary reasoning. He first finds the number of properly primitive
ambiguous forms of one or other of the two types (A, 0, C) and (25, B, C),
and then assigns the number of non-equivalent classes in which these forms
are contained. Let D be divisible by /x different primes ; and let us except
the case D= 1. Resolving D in every possible manner into two positive or
negative factors, having no common divisor but unity, we find 2^ + 1 properly
primitive forms of the type (A, 0, (7) ; but we shall diminish this number by
one-half by rejecting one of the two equivalent forms (A, 0, C) and (C, 0, A),
viz. that in which [A\ >\C~\. There are no properly primitive forms of the
type (2B, B, C) unless D = 3, mod 4, or Z) = 0, mod 8; for one or other of
these congruences is implied by the equation D = B(B 2C), because C is
uneven. Resolving D into any two factors relatively prime, if D=3, mod 4,
and having 2 for their greatest common divisor, if D = 0, mod 8, we take one
of them for B, the other for B 2C; and we obtain, in either case, 2>* + 1 properly
primitive forms of the type (2J5, B, C). If BB' = -D, it is easily seen that
Art. 115.] REPORT ON THE THEORY OP NUMBERS. 253
the forms (25, 5, (7) and (25', B', <?')* are equivalent. We may thus diminish
by one-half the number of forms of the type (25, B, C), rejecting those in
which [5]>x/[-D]. We conclude, therefore, that if we now denote by n the
number of uneven primes dividing D, we have in all 2' i+2 ambiguous forms
when D = 0, mod 8, 2* 1 when D = 1, or =5, mod 8, and 2^ + 1 in every other case.
These ambiguous forms we shall call Q, and we observe that their number
is equal to the whole number of assignable generic characters (Art. 98).
To find the number of non-equivalent classes in which these forms are
contained, we consider separately the case of a positive and of a negative
determinant. For a negative determinant, we diminish by one-half the number
of the forms by rejecting the negative forms. The remaining forms, if of the
type (A, 0, (7), are evidently reduced, because A < C ; if of the type (25, 5, (7),
they are also reduced, unless 2 5 > C, an inequality which implies that
(C, (7-5, C), to which (25, 5, (7) is equivalent, is reduced (Art. 92). The
number of [positive] ambiguous classes is, therefore, one-half the number of
the ambiguous forms 0.
For a positive determinant, we deduce from the forms Q an equal number
of reduced ambiguous forms. Thus (A, 0, (7) is equivalent to (A, kA, C') ;
and because [A] < \'D, this form is reduced, if kA be positive and be the
greatest multiple of [A] not surpassing \/D. Similarly (25, (2 + 1)5, C')
is equivalent to (25, 5, (7), and is reduced if (2& + l)5 be positive, and be
the greatest uneven multiple of [5] not surpassing \/D. There are, therefore,
as many reduced ambiguous forms as there are forms in 12 ; and there are no
more, because it is readily seen that every reduced ambiguous form is included
in one or other of the two series of forms (A, kA, C') and (25, (2& + l)5, C')
which we have obtained. But every ambiguous class contains two reduced
ambiguous forms (Art. 94) ; we infer, therefore, that for positive as well as
for negative determinants the number of ambiguous classes is one-half the
number of the forms Q, i.e. one-half of the number of assignable generic
characters.
Combining this result with the theorem at the commencement of this
* When the first two coefficients of a form are given, the third is given also ; thus C" is here
&* D
used for . Similar abbreviations will be employed occasionally in the sequel. The symbols
[A], &c. are used, as in Art. 92, to denote the absolute values of the quantities enclosed within the
brackets.
254 REPORT ON THE THEORY OF NUMBERS. [Art. 116.
article, we obtain a proof of the impossibility of at least one-half of the
assignable generic characters. As this proof is independent of the law of
quadratic reciprocity, we may employ the result to demonstrate that law.
[Gauss's second demonstration, Disq. Arith., Art. 262.] Let p and q be two
primes, and first let one of them, as p, be of the form 4n + l. If ( j = 1,
we infer that ( )= 1; for if ( )=+!, we should have a> 2 =p, mod q,
/ ft) 2 D\
and consequently there would exist a form (q, to, - -*-) of det. p, of which
f
the character would be ( ) = 1, *'& there would be 2 genera of forms of
determinant p. Similarly, if ( )=+!, we have a> 2 =q, mod p; and
(p, ">, (- ) is a form of det. +q. If +q be of the form 4n + l, there
will be but one genus of forms, i.e. the principal genus; whence ()= + !
These two conclusions are sufficient to establish the theorem of reciprocity
when one of the two primes is of the form 4n + l. If both p and q be of
the form 4w + 3, there are four assignable characters for the determinant >y.
Of these
are possible, as is shown by the existence of the forms
(1,0, -pq), (-1,0,^);
the other two are therefore impossible. Hence in the form (p, 0, q) we must
havener or
which expresses the theorem of reciprocity for this case. The supplementary
theorems relating to 2 and 1 can be similarly proved.
116. Equality of the Number of Genera and of Ambiguous Classes.
In the preceding article it has only been shown that k cannot exceed h.
But, as we have already seen (Art 102) that the number of actually existing
genera is one-half the whole number of assignable generic characters, we
know that k = h. To prove this, by the principles of the composition of
forms, it is sufficient to show that n = n, i.e. that the problem 'to find a
class which by its duplication shall produce a given class of the principal
genus' is always resoluble. This problem Gauss actually solves (Disq. Arith.,
Arts. 286, 287) ; he shows, first, that any proposed binary form, belonging to
the principal genus of its own determinant, can be represented by the ternary
Art. 116.] REPORT ON THE THEORY OP NUMBERS. 255
quadratic form X* 2 YZ ; and, secondly, that from this representation we can
always deduce a binary form, which shall produce by its duplication the
proposed form. This solution implies a previous investigation of the theory
of ternary quadratic forms, and cannot be properly introduced here.
A more elementary method, however, has been given by M. Arndt (Crelle,
Ivi. p. 72). Let D = AS 2 , S 2 representing any square dividing D; M. Arndt
observes that the ratio of the number of actually existing genera to the
whole number of assignable generic characters is the same for each of the
two determinants D and A. To prove this we make use of the following
subsidiary proposition :
' If f= (a, 6, c) be a properly primitive form of any det. D, and if 8 M and
be two numbers relatively prime, the necessary and sufficient condition for
the resolubility of the congruence
ax 2 + 2by?y + cy 2 = 6, mod SM (A)
is that the supplementary characters of f (if any), and the particular characters
of/ (if any) which relate to uneven primes dividing both M and D, should
coincide with the corresponding characters of 9.'
We may add (though this is not necessary for our present purpose), that if
0j and 2 he two values of 6 for each of which the congruence (A) is resoluble,
it is resoluble for each an equal number of times.
On reference to the Table in Art. 98, it will be seen that the particular
characters proper to the determinant A are included among the particular
characters proper to D. Let then (F) and (F, I") represent any two com-
plete generic characters for the determinants A and D, the particular cha-
racters common to the two complete characters having the same values attri-
buted to them in each. It may then be shown that the genus (F, F') is or
is not an existent genus, according as (F) is or is not existent: For (1) if
(F, I") be actually existent, let 6 be a number prime to 2D and capable of
primitive representation by some class of that genus ; the congruence &> 2 = D,
mod 6 is therefore resoluble; i.e. the congruence w 2 = A, mod 6, is resoluble,
so that 6 can be represented by a class of properly primitive forms of det. A,
or the genus (F) is actually existent. And (2) if (F) be an existing genus,
let / be a form included in (F), and 6 a number prime to 2D and satisfying
the generic character (F, I"). It appears from the subsidiary proposition
that some number 9 of the linear form SmD + d is capable of representation
by/; if S be the greatest common divisor of the indeterminates in the repre-
sentation of 9 by/, the congruence w 2 = A, and consequently the congruence
256 REPORT ON THE THEORY OF NUMBERS. [Art. 117.
to 2 =D, is resoluble for the modulus -^ ; i.e. -rr, the character of which coincides
with the character of Q, and therefore with that of the genus (F, F'), is capa-
ble of representation by a form of det. D, or (F, I") is an actually existing
genus.
If, then, K be the number of particular characters contained in (F, F') and
not in (F), the numbers of actually existing genera and assignable generic
characters for the det. D are each 2" times the corresponding numbers for the
det. A.
It appears from this result that it will be sufficient for our present purpose
to consider determinants not divisible by any square. If (a, 6, c) be a form
of the principal genus of such a determinant (we suppose that a is prime to D),
the equation ax 2 + 2bxy + cy* = <o 2 is resoluble with values of to prime to D ; for
if a = a'S 2 , S 2 representing the greatest square divisor of a, the equation
is certainly resoluble in relatively prime integers, by virtue of a celebrated
theorem of Legendre * ; and the values of which satisfy it are prune to D ;
whence, if -&;
a? = M- -, 2/ = M>7, <O = MJ>
Cv
H denoting a multiplier, which renders the values of x, y and to integral and
relatively prime, the equation ax 2 + 2bxy + cy 2 = w 2 will be satisfied, and the
values of to will be prime to D. The form (a, b, c) is therefore equivalent to
a form of the type (to 2 , X, v) ; and this form is produced by the duplication
of (w, X, wo) if to be uneven, and of (2 w, \ + <a, /) if to be even.
117. Arrangement of the Classes of the Principal Genus. If C be a
class of the principal genus, the classes C, C 2 , C 3 , ... will all belong to that
genus. And it will be found, by reasoning similar to that employed in
Euler's second proof of Format's theorem (see Art. 10 of this Report), (1)
that the classes 1, C, C*, ... are all different until we arrive at a class C*,
equivalent to the principal class ; (2) that /u is either equal to, or a divisor of,
the number n of classes in the principal genus ; (3) that if C r = l, r is a mul-
tiple of ft. The ft. classes C, C 2 , C 3 , ..., O" 1 , 1, are called the period f of the
class C ; C is said to appertain to the exponent M ; and the determinant is
* Th6orie des Nombres, ed. 3, vol. i. p. 41 ; Disq. Arith., Art. 294.
t These periods of non-equivalent classes are not to be confounded with the periods of equivalent
reduced forms of Art. 93.
Art. 117.]
REPORT ON THE THEORY OP NTTMBERS.
257
regular or irregular according as classes do or do not exist which appertain
to the exponent n. With the former case we may compare the theory of the
residues of powers for a prime modulus ; with the latter the same theory for
a modulus composed of different prunes (see Art. 77).
(i.) When the determinant is regular, we may take any class appertaining
to the exponent n as a basis, and may represent all the classes of the principal
genus (to which we at present confine ourselves) as its powers. It will then
appear (1) that if d be a divisor of n, the number qf classes appertaining to
the exponent d is \f/- (d) ; so that, for example, the number of classes that
may be taken for a base is ^ (n) : (2) that if ef= n, the equation X e = 1 will
be satisfied by e classes of the principal genus ; and if these classes be repre-
sented by A l} A 2 , ...,A e , each of the equations X* = A will be satisfied by
f different classes of the same genus : (3) that the only classes of the prin-
cipal genus which satisfy the equation X k = l are those which satisfy the
equation X d = 1, where d is the greatest common divisor of k and n.
It will be seen in particular that the equation X 2 = 1 admits of only one,
or only two solutions, according as n is uneven or even ; i. e. the principal
genus of a regular determinant cannot contain more than two ambiguous
classes.
To obtain a class appertaining to the exponent n, Gauss employs the same
method which serves to find a primitive root of a prime number (Art. 13 ;
Disq. Arith., Arts. 73, 74), and which reposes on the observation, that if A
and B be two classes appertaining to the exponents a and /3, neither of which
divides the other, and if M, the least common multiple of a and ft, be re-
solved into two factors p and q, relatively prune and such that p divides a
a
and q divides /3, the class A" x B q will appertain to the exponent M.
(ii.) When the determinant is irregular, the classes of the principal genus
cannot be represented by the simple formula C\ and we must employ an
expression of the form C^'i x C/z x C/s.... To obtain an expression thus
representing all the classes of the principal genus, we take for Ci a class
appertaining to the greatest exponent X to which any class can appertain ;
and in general for C^ we take a class appertaining to the greatest ex-
ponent 8p to which any class can appertain when its period contains no
class, except the principal class, capable of representation by the formula
(7 2 '2X ... x(7 M _!'"-'. The number -=0 2 x0 3 x... is called by Gauss the
MI
Ll
258 REPORT ON THE THEORY OF NUMBERS. [Art. 118.
n n
exponent of irregularity ; and similarly we might term 5-5-, 5 p-jr , &c., the
PI "2 "l #2 "3
second, third, &c., exponents of irregularity. From the mode in which the
formula C^'i x CJ* x ... is obtained, it can be inferred that Q l is divisible by 3 ,
Q* by #3. and so on ; whence it appears that a determinant cannot be irregular
unless n be divisible by a square ; nor can it have r indices of irregularity
unless n be divisible by a power of order r + 1. Moreover, whenever the
principal genus contains but one ambiguous class, the determinant is either
regular or has an uneven exponent of irregularity; if, on the contrary, the
principal genus contain more than two ambiguous classes, the determinant is
certainly irregular, and the index of irregularity even ; if it contain 2* ambi-
guous classes, the irregularity is at least of order K, and the K exponents of
irregularity are all even.
A few further observations are added by Gauss. Irregularity is of much
less frequent occurrence for positive than for negative determinants ; nor
had Gauss found any instance of a positive determinant having an uneven
index of irregularity (though it can hardly be doubted that such determinants
exist). The negative determinants included in the formulae,
-D = 21Qk + 27, =1000& + 75, =1000&+675,
except - 27 and - 75, are irregular, and have an index of irregularity divisible
by 3. In the first thousand there are five negative determinants (576, 580,
820, 884, 900) which have 2 for their exponent of irregularity, and eight
(243, 307, 339, 459, 675, 755, 891, 974) which have 3 for that exponent;
the numbers of determinants having these exponents of irregularity are 13
and 15 for the second thousand, 31 and 32 for the tenth. Up to 10,000
there are, possibly, no determinants having any other exponents of irregularity;
but it would seem that beyond that limit the exponent of irregularity may
have any value.
118. Arrangement of the other Genera, In the preceding article we have
attended to the classes of the principal genus only; to obtain a natural
arrangement of all the properly primitive classes, we observe that, if the
number of genera be 2?, the terms of the product
in which I\- represents any genus not already included in the product of
the i1 factors preceding 1 + F,-, will represent all the genera. If, then,
A lt A 2 , ...yAp represent any classes of the genera I\, F 2 , ..., F M respectively,
Art. 118.]
REPORT ON THE THEORY OF NUMBERS.
259
and | C \ be the formula representing all the classes of the principal genus, the
expression K = C
supplies a type for a simple arrangement of all the classes of the given
determinant. When every genus contains an ambiguous class, it is natural
to take for A 1} A 2 , ..., A^, the ambiguous classes contained in the genera
r is F 2 , ..., F^ respectively. When the principal genus contains two ambiguous
classes (and when, consequently, one-half of the genera contain no such classes),
let (?! be the class taken as base (or, if the determinant be irregular, as first
of the bases) in the arrangement of the classes of the principal genus, and let
Qj 2 = Cj ; it may then be shown that Q 1 will belong to a genus containing no
ambiguous class, and that the formula
\K\= C|x(l + Q 1 )(l+^)...(l+4,),
in which A 2 , ..., A^, are ambiguous classes, represents all the classes*. In
general, if the principal genus contain 2* ambiguous classes (a supposition which
implies that the determinant is irregular, having K even exponents of irregu-
larity, and that there are only 2'*-" genera containing ambiguous classes) let
2
it will be found that all the classes are represented by the formula
\K\-\C x (1+00(1 + 00- (1 + 0.) (l + 4, + i)-(l+^U
in which A K + 1 , ....A^ are ambiguous classes, and Q 1} O a , ...,O K classes belonging
to genera containing no ambiguous class f.
A similar arrangement of the improperly primitive classes (when such
classes exist) is easily obtained. Let 2 denote the principal class of improperly
primitive forms, i.e. the class containing the form
(2, 1, -l(Z)-l));
we have seen (Art. 113) that the number of properly primitive classes which,
* Gauss employs a class H, producing C l by its duplication, both when one and when two
ambiguous classes are contained in the principal genus. The number of classes requisite for the
construction of the complete system of classes is therefore ^ in either case, since (7, may be replaced
by <V-
t The principles employed by Gauss for the arrangement of the classes of a regular determinant
are extended in the text to irregular determinants. If the determinant have K' uneven exponents
of irregularity, the number of classes requisite for the construction of the complete system of classes
is fx+x'.
Lla
260 REPORT ON THE THEORY OF NUMBERS. [Art. 119.
compounded with 2, produce 2, is either one or three. When there is only
one such class, the number of improperly primitive classes is equal to that
of properly primitive classes ; and if | K \ be the general formula representing
the properly primitive classes, the improperly primitive classes will be repre-
sented by 2x|Jf|. When there are three properly primitive classes, which,
compounded with 2, produce 2, the principal class will be one of them, and
if be another of them, <f> 2 will be the third ; also < and <p* will belong to
the principal genus, and will appertain to the exponent 3. When the deter-
minant is regular, instead of the complete period of classes of the principal
genus, 1, C, C 2 , ..., C n ~ l , we take the same series as far as the class (7* n ex-
clusively; when the determinant is irregular, we can always choose the bases
C 1} C z , ... in such a manner that the period of one of them shall contain (p
and <f>*, and this period we similarly reduce to its third part by stopping just
before we come to (f> or < 2 . Employing these truncated periods, instead of the
complete ones, in the general expression for the properly primitive classes,
we obtain an expression, which we shall call \K'\, representing a third part of
the properly primitive classes, and such that 2 x K' \ represents all the im-
properly primitive classes.
119. Tabulation of Quadratic Forms. In Crelle's Journal, vol. Ix. p. 357,
Mr. Cayley has tabulated the classes of properly and improperly primitive forms
for every positive and negative determinant (except positive squares) up to
100. The classes are represented by the simplest forms contained in them * ;
the generic character of each class, and, for positive determinants, the period
of reduced forms (Art. 93) contained in it, are also given. The arrangement
of the genera and classes is in accordance with the construction of Gauss, ex-
plained in the preceding articles ; and the position of each class in the arrange-
ment is indicated by placing opposite to it, in a separate column, the term
to which it corresponds in the symbolic formula (such as | K \ or 2 x | K \ ) which
forms the type of the arrangement. To the two Tables of positive and negative
determinants Mr. Cayley has added a third, containing the thirteen irregular
negative determinants of the first thousand.
* The simplest form contained in a class is that form which has the least first coefficient of all
forms contained in the class, and the least second coefficient of all forms contained in the class and
having the least first coefficient. If a choice presents itself between two numbers differing only in
sign, the positive number is preferred. In the case of an ambiguous class of a positive determinant,
the simplest ambiguous form contained in the class is taken as its representative.
Art. 119.]
REPORT ON THE THEORY OP NUMBERS.
261
In a letter addressed to Schumacher, and dated May 17, 1841, Gauss
expresses a decided opinion of the uselessness of an extended tabulation of
quadratic forms. ' If, without having seen M. Clausen's Table, I have formed
a right conjecture as to its object, I shall not be able to express an opinion
in favour of its being printed. If it is a canon of the classification of binary
forms for some thousand determinants, that is to say, if it is a Table of the
reduced forms contained in every class, I should not attach any importance
to its publication. You will see, on reference to the Disq. Arith., p. 521 (note),
that in the year 1800 I had made this computation for more than four thousand
determinants ' [viz., for the first three and tenth thousands, for many hundreds
here and there, and for many single determinants besides, chosen for special
reasons] ; ' I have since extended it to many others ; but I have never thought
it was of any use to preserve these developments, and I have only kept the final
result for each determinant. For example, for the determinant 11,921, I have
not preserved the whole system, which would certainly fill several pages*, but
only the statement that there are 8 genera, each containing 21 classes. Thus,
all that I have kept is the simple statement viii. 21, which in my own papers
is expressed even more briefly. I think it quite superfluous to preserve the
system itself, and much more so to print it, because (1) any one, after a little
practice, can easily, without much expenditure of time, compute for himself a
Table of any particular determinant, if he should happen to want it, especially
when he has a means of verification in such a statement as viii. 21 ; (2)
because the work has a certain charm of its own, so that it is a real pleasure
to spend a quarter of an hour in doing it for one's self; and the more so,
because (3) it is very seldom that there is any occasion to do it My
own abbreviated Table of the number of genera and classes I have never
published, principally because it does not proceed uninterruptedly.' f Probably
the third of Gauss's three reasons will commend itself most to mathematicians
who do not possess his extraordinary powers of computation. An abbreviated
Table of the kind he describes, extending from -10,000 to +10,000, would
occupy only a very limited space, and might be computed from Dirichlet's
formulae for the number of classes (see Art. 104), without constructing systems
of representative forms. But it would, perhaps, be desirable (nor would it
' Mr. Cayley's Table of the first hundred negative determinants occupies about four pages of
Crelle's Journal; the determinant 11,921 would occupy about one page.
t Briefwechsel zwischen C. F. Gauss und H. C. Schumacher, vol. iv. p. 30.
262 REPORT ON THE THEORY OF NUMBERS.
increase the bulk of the Table to any enormous extent) to give for each deter-
minant not only the number of genera, and of classes in each genus, but also
the elements necessary for the construction, by composition only, of a complete
system of all the classes. For this purpose it would not be necessary to specify
(by means of representative forms) more than 5 or 6 classes, in the case of any
determinant within the limits mentioned.
IX.
REPORT ON THE THEORY OF NUMBERS.
PART V.
[Report of the British Association for 1363, pp. 768-786.]
120. (jrEOMETRICAL Representation of Forms of a Negative Determinant.
Before quitting the subject of binary quadratic forms, we have still to men-
tion several investigations of great interest, relating chiefly to forms of a
negative determinant. We shall first refer to the geometrical considerations
which Gauss has employed to illustrate the nature of these forms *.
Let an infinite plane area be divided by two systems of parallel lines into
similar and equal parallelograms. The vertices of these parallelograms we
shall call nodes ; and we observe that every system of nodes possesses the
characteristic property, that if it be displaced without rotation in its own
plane, so as to bring any one node into a position originally occupied by any
other node, then every node will also occupy a position originally occupied by
another node ; and the system in its second position will entirely coincide with
the system in its original position. From this property we infer that the
system of nodes admits of an infinite number of parallelisms besides the given
parallelism; i.e. that it may be regarded, in an infinite number of different
ways, as dividing the plane area into similar and equal parallelograms. For
let and (7 be any two nodes such that no node lies on 00' between and
(7 ; let P be one of those nodes which lie the nearest to the line 0(7 produced
indefinitely both ways, and let PP' be drawn parallel and equal to 00'; then
* See Gauss's review of Seeber's ' Untersuchungen uber die Eigenschaften der ternaren quad-
ratischen Formen,' in the Gottingen 'Gelehrte Anzeigen' for 1831, No. 108, or in Crelle's Journal,
vol. xx. p. 312 ; also Lejeune Dirichlet, Crelle, vol. xl. p. 209.
264 REPORT ON THE THEORY OF NUMBERS. [Art. 120.
P' is a node, and OO'P'P is a parallelogram of which the vertices are nodes,
and which has no other node either on its contour or in its interior ; such a
parallelogram we shall call an elementary parallelogram. It is then evident
from the characteristic property of the system, that every elementary paral-
lelogram supplies us with a parallelism of the system ; also we can obtain
an infinite number of dissimilar elementary parallelograms ; for if Ox and Oy
are the two lines of the given parallelism which intersect in 0, and if m and n
are any two integers relatively prime, the intersection of the mth parallel
from Ox with the nth parallel from Oy will give a node (7 such that no
node can lie on 00' between and (7 ; and, again, instead of P in the pre-
ceding construction, we may take any node lying on either of the two lines
of the system which are the nearest to 0(7. The areas, however, of all
elementary parallelograms are equal. To prove this, we observe that if AOB
is an elementary triangle (i.e. a triangle of which the vertices are nodes, but
which has no other node either on its contour or inside it), the parallelogram
OAO'B, obtained by drawing parallels to any two of its sides OA and OB
through the opposite vertices B and A, is an elementary parallelogram. For
if AO and BO are produced to A' and B', so that O bisects A A' and BB',
A' and B' are nodes, and the triangle A' OB' is elementary; because if there
were a node x (other than its vertices) in A'OB', we could immediately con-
struct a node x (other than its vertices) in AOB. But A'OB' can be made to
coincide with BO' A by a displacement without rotation ; therefore BOA is
elementary as well as AOB ; or the parallelogram AOBO' is elementary. Hence,
if two elementary triangles have a common base, they are certainly equal. For
if through the vertex of either triangle we draw a parallel to the base, an
elementary parallelogram will be contained between that parallel and the base ;
that is, the altitude of either triangle will be the distance of the base from the
parallel nearest to the base; or the triangles will be equal. Again, let AOB,
aOb, be any two elementary triangles, which we may suppose to have a common
vertex ; if BOa is an elementary triangle, they are each of them equal to it
and to one another ; if not, let x be that node contained in BOa which lies
the nearest to OB, then BOx is elementary, and has the side BO in common
with AOB ; by proceeding in this manner we shall form a series of elementary
triangles, of which the first is AOB, and the last a OB, each triangle having
a side in common with that preceding it, whence AOB = aOb; i.e. any two
elementary parallelograms are equal.
We shall next show that it is always possible to find a reduced paral-
Art. 120.] REPORT ON THE THEORY OP NUMBERS. 265
lelogram, i.e. an elementary parallelogram, the sides of which are not greater
than its diagonals. Let he any node ; A a node as near to O as any other ;
S a node on one of the parallels nearest to OA, and as near to as any node
on either of those parallels; complete the elementary parallelogram OAO'B;
it will have the property required. Produce OB to 0", making BO'=0'B;
then AB = 00' ; hut hy hypothesis OA < OB, and OB ^ 00', OB < OO" ; i.e. the
sides of OAO'B are not greater than its diagonals.
Again, if OAO'B is a reduced parallelogram in which OA < OB, it can be
proved that no node lies nearer to than A, and that no node, out of the
line OA, lies nearer to than B ; for, first, no node on the line 0"BO' lies
nearer to O than B, because by hypothesis OB S 00', OB ^ 00", and because
the extremity of the perpendicular drawn from to 0"0' falls between the
points of bisection of the segments 0"B and BO', or on one of those points :
secondly, no node on any parallel beyond O"BO' can lie as near to O as B,
for the limits of the angle AOB are evidently 60 and 120; whence the
perpendicular distance of OA from the parallel nearest to it but one is ^ 05^/3 ;
i. e. the distance of any node on that parallel from is > OB.
If then we join any node 0, first to a node A, which lies as near to as
any other node, and, secondly, to a node B, which lies as near to as any
node out of the line OA, the joining lines are adjacent sides of a reduced
parallelogram ; for, by what precedes, B must lie on one or other of the
parallels nearest to OA.
In general, a system of nodes has but one reduced parallelism, because in
general there is a pair of opposite nodes A A', each of which is nearer to than
any other node whatever, and a second pair of opposite nodes BB', not lying
in the line AOA', each of which is nearer to than any node not lying in
that line. Even if A and B are equidistant from 0, provided only that their
common distance from is less than the distance of any other node from 0, the
system has but one reduced parallelism. But there are two special cases in
which a nodal system admits of more than one reduced parallelism.
1. If there is one pair of opposite nodes A A' nearer to than any other
node, and two pairs BB', W, equidistant from 0, not lying in the line AOA',
and nearer to O than any other node not in that line, the system admits of
two reduced parallelisms, having one set of parallels hi common, and having
their common set of parallels equally inclined to the other two sets.
2. If there are three pairs of points at the minimum distance from 0,
the system of nodes forms a system of equilateral triangles ; and, suppressing
M m
266 REPORT ON THE THEORY OF NUMBERS. [Art. 120.
in turn each one of the three systems of parallel lines by which these triangles
are formed, we obtain the three reduced parallelisms of which the system
admits.
That, in these two cases, the reduced parallelisms are such as we have
described, and that, except in these two cases, there is but one reduced
parallelism, may be inferred from the existence of a reduced parallelogram
in every system, and from the properties which have been shown to belong
to it.
To apply these results to the theory of quadratic forms, let a
be a form of the negative determinant A ; let cos <a= _ , and with a pair
x /ac
of axes inclined to one another at an angle w, let us construct all the points
whose coordinates are integral multiples of \/a and \/c respectively ; thus
forming a nodal system. The expression ox 2 + 2 bxy + cy* will then represent
the square of the distance between any two nodes, the differences of whose co-
ordinates are x \/a and y */b : and the area of an elementary parallelogram will
be A/A. If the transformation x = aX+[3Y, y = yX + SY, where a^-/3y= +1,
change ax 2 + 2 bxy + cy 2 into AX* + 2BXY+CY* ; and if, in the same plane
as before, we construct a nodal system corresponding to the latter form the
directions of rotation from the axis of X to the axis of Y, and from the axis
of x to that of y, being the same it will be found that the two systems may
be made to coincide. For if we consider the point in the first system whose
coordinates are x*/a, y /c as corresponding to the point in the second system
whose coordinates are X \/A, YVC, the distance between any two points of
the first system is equal to that between the corresponding points of the second
system ; therefore the two systems are identical, and are either similarly situ-
ated, i.e. are capable of being made to coincide by moving either of them
about in their common plane, or else are symmetrically situated, i.e. are
capable of being made to coincide after the plane of one of them has been
turned over and applied again to the plane of the other. On comparing any
two corresponding triangles in the two systems, for example the triangle
obtained by giving to X and Y the values (0, 0), (1, 0), (0, 1), with the
triangle obtained by giving to x and y the values (0, 0), (a, 7), (/3, S), it will
be seen that the two systems are similarly or symmetrically situated, according
as aS-/3y = +1, or = -1.
It thus appears that a class of quadratic forms of a negative determinant
may be considered to represent a nodal system, and that each form of the class
Art. 120.] REPORT ON THE THEORY OF NUMBERS. 267
corresponds to a parallelism of the system. Conversely, to each parallelism of
the system a form of the class corresponds. For let Ox, Oy be lines of any
parallelism of the system, and OX, OY lines of any other parallelism, the
directions of rotation from Ox to Oy and from OX to OF being the same; let
also A/a, \/c be the lengths of the sides of an elementary parallelogram in the
first system, and --=- the cosine of the angle between them ; and let \/A,
s v
.- : have the same signification with regard to the second system ; then, if
(x\/a, yVc), (X VA, YVC)
are the coordinates of the same node P, we must have two equations of the form
x = aX+pY, y = yX+SY,
in which x and y are integral if -X" and Y are so, and vice versd ; hence a, /3, y, S
are integral, and a<$ /3-y= +1; the sign of the unit being determined by the
supposition we have made as to the situation of the axes with respect to one
another. Also
OP 2 = ax 2 + 2bxy + cy 2 = AX 2 + 2BXY+ CY 2 ;
or the two given parallelisms are represented by two properly equivalent forms.
The theorem that in every nodal system a reduced parallelism exists, has
for its arithmetical expression, ' In every class a form exists in which [26] ^ [a],
[26] ^ c.' We thus obtain an independent proof of the theory of reduction
of Art. 92 ; the geometrical signification of the special conditions in the definition
of a reduced form is as follows : If a = c> [26], the corresponding nodal system
has only one reduced parallelism ; but either of the two directions in this
reduced parallelism may be taken for the axis of x, consistently with the
condition that the rotation from Ox to Oy should have a given direction;
the condition 26^0 implies that if the angle between the axes is not right,
that direction is to be assumed for the axis of x which renders the angle
between Ox and Oy acute. Similarly, if a<c, but a = [2 6], the system has
two reduced parallelisms, and the condition 26^0 distinguishes one of them
from the other. If a = [26] = c, the system has three reduced parallelisms,
which are identical and similarly placed ; the condition 2 6 > does not dis-
tinguish between these, but only between the two modes in which any one
of them can be taken.
The number of automorphics of a class may be ascertained by causing the
nodal system which represents it to revolve in its own plane round one of its
nodes and examining the number of positions in which it coincides with its
M m 2
268 REPORT ON THE THEORY OF NUMBERS. [Art. 121.
original position. After a revolution of 180 it will always do so ; but in
order that it should do so in any other position, the first and second sides
of its reduced parallelogram must be equal, and must include an angle of 90
or 60, i.e. the system must be one of squares or of equilateral triangles. Hence
we infer (Art. 90) that there are in general but two automorphics for a form of
a negative determinant, but that for the classes containing the forms x 2 + y 2 and
2x 2 + 2xy + 2y 2 (or multiples of those forms) there are four and six respectively.
Similarly we may investigate the conditions for the ambiguity of a class.
In order that a class should be ambiguous, the nodal system representing it
must be symmetrically equivalent to itself. If therefore there is but one
reduced parallelogram, that parallelogram must be symmetrically equivalent
to itself, i.e. it must be either a rectangle or a rhombus. When there are two
reduced parallelograms, we have seen that they are symmetrically equivalent
to one another; and when there are three, they are each of them rhombs.
We thus obtain the conclusion that if (a, b, c) is the reduced form of an am-
biguous class, either 6 = 0, or a = c, or a = 26 (Art. 94).
121. Application of Formulae relating to the Division of the Circle to the
Theory of Quadratic Forms. We have already referred to the trigonometrical
solutions of the equation T 2 Z>Z7 2 = 1 (Art. 96, (ix).) and to the connexion
existing between them, and the number of classes of quadratic forms of de-
terminant D (Art. 104).
If p is a prime of the form 3n + I or 4n + l, the coefficients of the cubic,
or biquadratic, equation of the periods depend on the values of the indetermi-
nates in the equation &p = x 2 + 3y 2 , or p = x 2 + y* (Art. 43). Thus in these two
cases, if, for any given value of p, we calculate the equation of the periods,
we obtain, by a direct though tedious process, the values of the indeterminates
in certain simple quadratic decompositions of 4^ or p. But the theory of the
division of the circle supplies a method equally direct and of more general
application for the investigation of such decompositions in certain cases. The
principles of this method were discovered by Gauss, who deduced from them
the first of the three following theorems :
'If
x = * j ,mod; x = 1, mod 4;
* Hn.lln
H2n.U2n
y= -\ -- n -n , mod p.
Hn.lln
(Gauss, Theor. Ees. Biq. Comm. prima, Art. 23.)
Art. 121.]
REPORT ON THE THEORY OF NUMBERS.
269
X ~~
Un.Tin'
y=0, mod 3.'
(Jacobi, Crelle, vol. ii. p. 69 ; Stern, ib. vol. vii. p. 104, vol. ix. p. 97, vol. xviii.
p. 375 * ; Clausen, ib. vol. viii. p. 140.)
'If p = 8n + l=x 2 + 2y 2 ,
H5n
x =k Tf - rF7~ > mod P ' Sl, mod 4.
a 1171.114%
(Jacobi, Crelle, vol. ***. p. 168 ; Stern, ib. vol. xxxii. p. 89.)
In all these formulae the absolute value of x is evidently <-%p', so that x
is determined without ambiguity as the minimum residue for the modulus p
of the binomial coefficient. And the combination of the two congruences
satisfied by x gives rise in each case to a remarkable property of the coefficient :
thus, from the two congruences satisfied by a; in the first theorem, we infer
that 'if p is a prime of the form 4w + l, the minimum residue of ^^ - =
for the modulus p is of the form 4ra + l.'
To show, by an example, how these formulae are obtained, we shall consider
the last of them in particular. Resuming the notation of Art. 30, let & be a
primitive root of the equation x*' 1 1 = ; and let
x p
x representing a root of the equation -
SO ^
congruence x p ~ 1 = 1, mod p. Then
= 2
8 =
^- = 0, and 7 a primitive root of the
^ X
is an integral function of a> only (Art. 30, m.) ; let
(eo) = a + ba> + C) 2 + duo 3 .
The function
fore
or
so that
F e . n
) ;
= 0, & = d, and
is not changed, if for 8~ n we write 0- 3n ; there-
a 2 + 2 b 2 (Art. 30, iv.).
* {This reference relates to p = 8ni 1 =
270 REPORT ON THE THEORY OF NUMBERS. [Art. 121.
' modp (Art. 30, v.) 5 -
whence _ _ Il5n
~
To show that a = 1, mod 4, we observe that by the definition of the func-
tion ^IT, \p- (w) = 2w~ 1 'i~ 4| '2, y l and y 2 representing any two numbers of the
series 1, 2, ...,p 2, which satisfy the congruence y"i + y v i = l, mod^). Hence
a = Z( I)"***!, where n\ is one of the numbers 1, 2, ..., 2n-l, and i/ 1} 7/ 2
satisfy the congruence 7* '1 + 7 "2 = !, mod p. Let a- be any one of the numbers
1, 2, ...,-!, and let ^4, .B be the values of 7*2 corresponding to the values
n <r, n + a- of ?! ; then
^1 x5 = (l -7 4( "- <r) ) x (1 -7*( + *>) = _ y -*< + ) x (1 - 7 <+*>), mod^) ;
therefore AxB ia a quadratic residue of p, and the values of y 2 corresponding
to the values n tr, n + or of ^ are either both even or else both uneven ;
also, if >]i = n, 7 "a = 2, mod|>, and y z is even, because (-J = l. Let k be the
number of values of n\, included in the series 1, 2, ...,n 1, for which 3/2
is uneven ; then
a = 2(-l)'2- | -'? 1 = 2(ri-l)-4A; + (-l)"; i.e. a=-l, mod 4.
We might also determine a; in the equation p = x 2 + 2y 2 by the congruence
or by the congruence ( ^^^^ "2n mo
^riw.iin
These determinations, which have been given by M. Stern, may either be
obtained directly by considering the functions
F(e- n )F(e- 3n )
or may be deduced from the formula of Jacobi. The formulae for the deter-
mination of x in the first two theorems also admit of various modifications. It
will be observed that, in the first, y is determined by a congruence as well as x.
This determination is obtained by a comparison of the two congruence
1+^ = 0, mod p, l + (II2w) 2 = 0, mod^,
Art. 121.]
REPORT ON THE THEORY OF NUMBERS.
271
(the latter arising from Sir J. Wilson's theorem) ; with regard to it Gauss
observes, 'quum insuper noverimus quo signo affecta prodeat radix quadrati
imparis, eo scilicet ut semper fiat formae 4m + 1, attentione perdignum est,
quod simile criterium generale respectu radicis quadrati paris hactenus inveniri
non potuerit. Quale si quis inveniat et nobiscum communicet magnam de nobis
gratiam feret.'
These congruential determinations possess great interest, not only because
direct methods of solution present themselves very rarely hi the theory of
numbers, but also on account of the singular connexion which they establish
between certain binomial coefficients and certain quadratic decompositions of
primes. Nor is it less remarkable that the properties of the resolvent function
of Lagrange form the intermediate links in this connexion ; although it is
proper to observe that Gauss has exhibited his demonstration of the theorem
relating to the equation p = x 2 + y z in a form in which its connexion with the
theory of the division of the circle is disguised.
Results of a more general character have been obtained by Jacobi and
Cauchy. Cauchy has treated of the subject with great fulness of detail in his
Memoir on the Theory of Numbers, in the 17th volume of the Memoirs of the
Academy of Sciences (pp. 249-768) ; while Jacobi has barely indicated his
method in his note on the division of the circle (Crelle, vol. xxx. p. 166);
nevertheless, as in some respects it seems preferable to that employed by
Cauchy, we shall endeavour to adhere to it in what follows.
Retaining the other notations which we have employed in this article, let
) F(0-*)
'
or
when there is no occasion to consider 6 explicitly; we observe that
^(m,n) = ^(n,m); ^(0, ri) = ^(m, 0) = -^(0, 0)= -1 ;
also vf' (m, n') = -$r (m, n), if m'=m, mod p 1, n' = n, mod p 1 ;
^(ra, n) = ( l) m + *p = (-!)" + 1 p, if m + n = Q, modjp- 1,
but in and n are not =0, modp-1. Let m 1} m/, ...,m a) be any set of cr + 1
numbers, each of which satisfies the conditions < m^ <p 1 ; let
m 1 + m 1 '+ ... +m 1 (<r) = % (p-
where < j <p 1 ; and put
Writing, for brevity,
^', ..., modp-1,
272 REPORT ON THE THEORY OF NUMBERS. [Art. 121.
and determining MI> Mi', //', ... so as to satisfy the conditions Q<n l <p 1, we
In this expression if MI (O + W + 1)> f~l> we write for ^(MI (<> , m 1 (l '' M) ) its equi-
valent p-r^(p lp l ( '\ p l-m 1 (<+1) ); and if /x 1 (0 + m 1 (f ' +1) =_p 1, we write
for \|/ (MI (O , m 1 ( ' + 1) ) its equivalent ( l) 1+m i H * ^>. It is evident that the con-
dition /i (0 + m ( ' + 1) >j9 1 will be satisfied n^ tunes precisely; so that x(#) assumes
<, (0)
the form p n i V/, i(0) and ^(0) denoting products of factors of the form
^ (h, h'), in each of which h + h' < p 1. It will now be found that
, mod p.
For (1), if j u i w + m 1 (l-+1) <_p-l, we have
) -v^= - ^ 1>H . mod
- 1 - 1 ' ^'
(2), if MI M + WI, W+I) > 1, we have -1-7 5 pr 5 p- ; r
^(p-1- Pi, p-l- mj*+ l \ 7)
n(p-i-/B 1 )) . n^-i-wH^^), n^'-")
n (2 P - 2 -M!- h w+1) ) E n^w . nv+D' r
since, by Sir J. Wilson's theorem,
C - IV
n(^-l-y)=- v I / , modj?, if j<p-l;
(3), if j 1 W + m 1 (<+1 >=^-l, we have
because n / u i
by Sir J.Wilson's theorem, while II yu^'^ 1 ^ 1, since /* 1 ( ' + 1) = ; whence, mul-
tiplying and writing Sj for /* I (ff) , we obtain the congruence written above. Let
r represent any term of a system of residues prime to p 1 ; let the numbers
m r , m/, ..., m r (ff) be determined by the congruences m r (0 =m 1 (0 r, mod(j> 1),
combined with the condition ^ m r (f} <p l; and let
where again < s r < p 1 : we have for every value of r an equation of the form
^ (0^
y \@ r ) = 7)**** X ^ - ,
and a congruence of the form
cb fA TI S
/n^. /^ m d ^-
nm r nm/nm r "...
Art. 121.] REPORT ON THE THEORY OF NUMBERS. 273
Let ^(6) =
k + 1 denoting the number of terms in a system of residues prime to p 1 ; let
n r be the least of the numbers n lt ...,n k + l , and j the exponent of the highest
power of p dividing A , A lt ..., A k : then shall j = n v . For, first, if j>n v ,
from the equation ^f v (0) = $ (&), in which the coefficients of the powers
of 9 are integral numbers, we infer the congruence
but -^ = 0, mod p ; therefore, < (7) = 0, mod p,
which is impossible. Secondly, if j < n v , writing A- for A { -j- p', and observing
that ty r (7) is prime to p, for every value of r, we find
A ' + AI Y + A,' 7 2r + . . . + A V = 0, mod p\-i,
for every value of r : but the determinant of this system is prime to p, therefore
A '=0, Ai=0, ..., A k '=Q, mod p n v ~', which is contrary to the hypothesis that
j <n v , and thatp" is the highest power of^> dividing A , A 1} ..., A k .
The application of these results leads to the following general theorems ;
in the enunciations of which p is an uneven prime, and A a number not divisible
by any square.
'If A = 4m + 3, p = A + l, and if we represent by a and b numbers less
than A and prime to A, respectively satisfying the equations
" 1 * wehave
55-So
4
'Ifjp = 4Aw+l, A being of any other linear form than 4m + 3, and if we
represent by a and 6 numbers less than 4 A and prime to 4 A, respectively
satisfying the equations f- -) = + 1, ( r~ 1 = 1> we have
A =
(B),
__
2 n a [II an]
In these formulae the signs of summation extend to every value of a and
N n
274 REPORT ON THE THEORY OP NUMBERS. [Art. 121.
respectively; and in the expression II a [II an] the exterior sign of multiplica-
tion II extends to every value of a, while the interior sign is the factorial
symbol, so that Tlan = l . 2. 3 ... an. The number 3 is excluded from the first
formula ; the numbers 1 and 2 from the second.
It will suffice to show how the first of these two theorems is to be de-
monstrated. For this purpose we consider the product HF(6~ an ); taking an,
a'n, a"n, ... for m^, m/, ... we find \(&)= JlF(9- an ); because (as may easily
be proved) Za = 0, mod A, whence Zaw = 0, mod p 1. We shall now show
that x(0) is of the form A^G an + S^O ln . Actually multiplying the expressions
F(9~ an ), F(Q-* n ), ..., the coefficient, in the product, of any term such as a?'9 mn
is equal to the number N of the solutions of the simultaneous congruences
7 w + 7"' + 7 v "+... = ^, modp, ay + a'y f + a"y"+ .'..= -m, mod A.
(7* \
j = + 1, N will
not be changed, if we write rm, ra, ra', ... (or rather the least positive residues
of those numbers, mod A) for m, a, a, .... Hence, in x (#) all powers of 6 whose
exponents are of the form an have the same coefficient A', and all powers of 9
whose exponents are of the form bn have the same coefficient B'. Again, con-
sider a power of Q of which the exponent is of the form aSn; S representing a
given divisor of A (other than 1 or A), and a representing any number less than
A A
-5- , and prime to -y- ; all such powers of 6 will have the same coefficient. For
we can always find a number r prime to A, satisfying the equation (-T-) = 1,
and yet congruous, for the modulus -- , to any given number prime to -5- ;
whence it follows that the number N will remain the same for all values of
m, included in the formula a S. But a sum of the form 2 a a5n is equal to +1
or 1, according as the number of primes dividing -y is even or uneven, be-
A
cause it is the sum of the primitive roots of the equation x s = 1. Thus, the
function x($) assumes the form A"2 6 an + B' ^2 6 ln + C', whence, attending to the
equation Z0 a " + Z0 z " 1 = ( 1)\ in which X is the number of primes dividing A,
we find, as has been said,
If we write Q~ l for 6 in this equation, it becomes
snce
Art. 121.]
REPORT ON THE THEORY OF NUMBEBS.
275
Multiplying the two equations together, and observing that
F(Q- an ) F(6 an ) = ( iy n p=p, because n is even, we obtain
x -
or, since (20 - Ze^ = -A*,
4_p^<*> = (A + BY + A (A - BY,
\J/- o (A) representing the number of numbers less than A and prime to A.
We have next to determine the highest power of p dividing A+B and A B,
or, which is the same thing, A and B. By the principles indicated above,
we have
=p
so
*-l(Q
Writing in these equations 7 for 0, and observing that the determinant
as well as the four numbers
is prime to p, we infer that the exponent of the highest power of p dividing
^ci ^C 1 7,
A and B is the less of the two numbers r- , Of these the former is
the less J ; if therefore we write x and y for
"A", and (A
* See Art. 96, (ix.) of this Report, or the note on Art. 104.
t Since 20 fln + 20 frn = ( 1) A , we have Syn + Sy 6 " = (-1) A ,
andsince (20 aB Sfl 6 *) 2 = A, we have (2y a>1 Sy*") 2 ^ A, mod p.
Thus the two factors of the determinant are each of them prime to p.
The principle that any rational equation containing only powers of 6 and integral numbers
may be changed into a congruence for the modulus p, if y be written in it for 0, has already been
employed in this Article. Its truth is evident, if we observe that the irreducible equation satisfied
by 6, if considered as a congruence for the modulus p, is satisfied by y. This principle is of more
general application than a similar one which has been already employed in Art. 5 1 of this Report ;
but its proof supposes the irreducibility of the equation of the primitive roots, which is not necessary
to the proof of the principle of Art. 51.
^1 V
J 26 2 a is certainly positive, because -r is equal to the number of improperly primitive
classes of the negative determinant A. Or (as it is desirable to avoid making use of this result
N n 2
276 REPORT ON THE THEORY OF NUMBERS. [Art. 121.
respectively, our equation becomes
Also, since ^ (?) __/_-, \ *
*,(?)- ' n a [Han]'
we have _^ _?f
Sa Sa
here) 26 2a is positive because (26 2a) is the sum of the series 2 (-T-) -, the summation
extending to every value of n prime to A, and the terms being taken in their natural order. This
series is positive, because the series 2 (-T-) -^r p , of which it is the limit, when p is diminished without
limit, is certainly positive, being the reciprocal of the product
in which the sign of multiplication extends to every prime q not dividing A, and in which every
factor is positive. The series 2 ( ^ - is one of those summed by Dirichlet in the memoir ' Re-
! ^A'
cherches sur di verses applications <fec.' (Crelle, vol. xxi. p. 141 et seq.): for the case in which
A is a prime, he had already summed it in the memoir on the Arithmetical Progression (Memoirs
of the Academy of Berlin for 1837, p. 55). Cauchy (Memoires de 1'Academie des Sciences, vol. xvii.
p. 673 et seq.) inverts Dirichlet's process, and transforms sums of the form 2/(a) 2/(6) into
infinite series. The transformation is effected by substituting for / (a;), in the expression
the equivalent infinite series
2 m =
whence, observing that
f m^ .. , *^~ l fV\
A ' and "
we obtain i V A (/()-/(&)) = ( J) sin
a formula from which Dirichlet's result is immediately deducible, by putting /(x) = x, and performing
the integrations. It is a remarkable fact that the inequality 26>2a has never been proved by
elementary considerations, or without the use of infinite series (see the Memoir on the Arithmetical
Progression, p. 57). If A is a prime, 26 2a is certainly not zero, for 26 + 2o is uneven (because
A is of the form 4n + 3); but even this remark cannot be extended to the case in which A is
composite.
Art. 121.] REPORT ON THE THEORY OF NUMBERS.
whence by addition
277
26
If A is a prime, x also satisfies the congruence ^x = 1, mod A ; for the
sum of the coefficients in any function -v^ (m, n) is = 1, mod p 1, and
therefore mod A ; whence the sum of coefficients in x ($)> which is a product
of an even number of such functions, is =1, mod A; because the reduction of
x(#) to the form A^6 an + ~Z6 bn is effected only by means of the equations
A -1
' n -l =
whereof the former does not alter the sum of the coefficients at all, and the
latter alters them only by a multiple of A. Consequently
(A-1) (A+B) = 1, mod A,
or, since _p A =1, mod A, and ( l) x = 1, ^x = l, mod A.
It will be observed that if A is of the form 8m + 7, whether A is a prime
or not, x and y are necessarily even in the equation (A) ; whence, dividing
by 4, we may put the equation in the form
p A =x 2 + &y 2
, mod p.
Let A = 7,
8 n a [IIan]'
> = 7 n + 1 ; the values of a are 1, 2, 4 ; of b, 3, 5, 6 : hence
> ; also x = '.
(Jacobi, Crelle, vol. ii. p. 69.)
Whenever the exponent of p is 1, the formulae (A) and (B) completely de-
termine the value of x ; when the exponent of p is 2, we can only be sure
that the absolute value of x is less than p, so that x is not completely deter-
mined, but is either the least positive or the least negative residue of \he
binomial coefficient ; though in this case if A be a prime of the form 4 n + 3,
the ambiguity may be removed by the congruence \ x = 1 , mod A. But when
the exponent of p is > 2, x is never completely determined by the congruence
for the modulus p.
It is very remarkable that the exponent of p in the formula (A) is pre-
cisely the number of improperly primitive classes of determinant A, and in
278 KEPOET ON THE THEORY OF NUMBERS. [Art. 121.
the formula (B) is precisely the number of properly primitive classes of deter-
minant A*.
* See Art. 104 of this Report. When A is of the form in + 3, the two expressions given by
Dirichlet for the number of properly primitive classes of determinant A are
where A and B represent the numbers of residues inferior to A, and satisfying the conditions
()= + ! and ()= ! respectively. Hence - ^ is the number of improperly primitive
classes ; because that number is equal to or is one-third of the number of properly primitive classes,
according as A = 7, or = 3, mod 8 (see Art. 103 or 113).
There is no difficulty in showing that Dirichlet's two expressions are identical. If ( ^ = 1
\A'
the congruence 2 6'= b, mod A, is always resoluble; and if 6 receive in succession all positive values
less than A which satisfy the condition ( -\ = 1, 6' will obtain the same values in a different order.
,6 26' _ 6 v> 26'-6
Hence 2 = 2 --- 2 = 2 - --
A A A A
But if 6'< JA, 26'-6 = 0; if6'>$A, 26'- 6= A, i.e. 2-^ = 4,
for there are A values of 6 greater than JA. Similarly 2 = 5, so that - -- =.AB. In
precisely the same manner it may be shown, if f-r-) = 1, by considering the congruences
26'+ 6 = 0, mod A, 2a' + a = 0, mod A,
that 3^ = 2^ = 2^ + 5 and ^ = 2 5fi = 2B + A;
26 2a
whence 3 - - - = AB,
A
Also the expression given in Art. 104 coincides with AB. For that expression may be written
in the form
a' and 6' representing numbers less than J A.
IT [F(0- an )~\ 2
If we consider, as Cauchy has done, the product L a ^ , the exponent of p in the formula
' n r F (d-"")!*
(A), will be AB. That product is evidently equal to nF(6-<">), or to -:| pr^.Jr , according
,2^ \
as (j^j = + lor= 1 ; a result which is in accordance with the equation
In the formula (B), the exponent of p, obtained by the consideration of the same product, is
A' ^_ 2x (S6-Sa).
4A '
A' and B' denoting respectively the numbers of residues of the classes a and 6 respectively, which
are inferior to 2 A.
Art. 121.]
REPORT ON THE THEORY OF NUMBERS.
279
Before Dirichlet's discovery of the formulae expressing the number of
classes of quadratic forms of a given determinant, Jacobi, having succeeded
in determining the exponent of p in the formula (A), for the case in which
A is a prime number, was led with singular sagacity to conjecture that
^C* I, ^^
-r - must represent the number of improperly primitive classes of deter-
minant A*. If A is the number of classes in the principal genus of impro-
perly primitive forms of determinant A, it follows from the theory of com-
position of quadratic forms that 2p ft can always be represented primitively
by the principal form in that genus, i.e. by the form (2, 1, J(A + 1)), and that
the exponent of the lowest power of p which is capable of such representa-
tion is either h or a submultiple of h. Again, the equation
26-Sa
if we write in it 2 X+ Y for x, and Y for y, becomes
2p~^~ = (2, 1, i (A + 1)) (X, YY,
the values of X and Y being integral. Assuming, then, that there exist
primes of the linear form wA + 1, the doubles of which are capable of repre-
sentation by a class appertaining to the exponent h (an assumption which
implies that A is not an irregular determinant, at least in respect of its
improperly primitive classes), we see that in the case in which A is a prime
of the form 4w + 3, and in which therefore there is but one genus of impro-
"y TI ^/r
perly primitive forms, - -r - must be equal either to the number of im-
properly primitive classes, or to a multiple of that number ; and as Jacobi
^i 7, ^*
found, upon a sufficient induction, that h was always equal to - r -- , he did
not scruple to enunciate the theorem as true. We know, however, from an
account which Dirichlet has given of a communication made to him by Jacobi,
that Jacobi never obtained a demonstration of the theorem ; and, indeed, it
* Crelle, vol. is. p. 189. Jacobi counts the classes of the prime determinant A on the prin-
ciple of Legendre, not distinguishing opposite classes from one another. If n is the numher of
improperly primitive classes so counted, we have & = 2w 1, because there is but one improperly
primitive ambiguous class. When A is of the form 8m +7, Jacobi enunciates the theorem with
reference to the number of properly primitive classes, which in this cdse is equal to the number of
improperly primitive classes.
280 REPORT ON THE THEORY OF NUMBERS. [Art. 122.
would seem probable, as has been observed by Dirichlet, that its demonstration
requires other principles (Crelle, vol. Hi. p. 206).
It is hardly necessary to add that when there is more than one genus of
forms of determinant A, i.e. in every case except when A is a prime of the
form 4/i + 3, the exponent of p in the formulae (A) and (B) is always a
multiple of the least exponent for which those formulae can be satisfied.
122. Extension of the preceding Theorem by Eisenstein. In the theory of
which an account has been given in the last article, the prime number p is
throughout supposed of the linear form %A + 1 or 4wA + l ; thus in the equa-
tions p = x* + 7y 2 , p = x 2 + 8y 2 , we have supposed p to be of the forms 7n + l
and 8 n + \ respectively. But we know that some power of every prime of
which A is a quadratic residue is capable of representation by the form
05 2 + Ay 2 ; and, in particular, that primes of the form Sn + 3 are capable of
representation by x 2 + 2y z , and primes of either of the forms 7n + 2 or 7w + 4
by cc 2 + 7y 2 . M. Stern found by induction that the value of x in the equation
p
satisfies the congruences
> * = (-!)", mod 4*;
and Eisenstein succeeded in demonstrating this theorem, as well as the two
following f :
'If p = 7n + 2 = x 2 + ly\
TL3n
'If p =
U3n+l
X = *
These demonstrations are obtained by expressing the prime number p as
the product of two complex factors, composed of 8th or 7th roots of unity.
But the decomposition of p is no longer supplied by the formula of Art. 30 ;
nor are the complex factors included in the definition of the functions ^, which
have been considered in Art. 30 and in the last Article.
If p = 8n + 3 is a real prime, p is also a prime in the theory of complex
* Crelle, vol. xxxii. p. 89. "We enunciate the latter part of the theorem in the form in which
it has been given by Eisenstein 1 .
t Crelle, vol. xxxvii. p. 97.
Art. 122.]
REPORT ON THE THEORY OF NUMBERS.
281
numbers of the form a + bi; let 7 be a primitive root of p in that theory, and
let 7" = 1 + iz, mod p, z representing one of the real integers, 0, 1, ,..,p l.
Also let \J" () = Sai", w denoting a primitive 8th root of unity, and the sum-
mation extending to every value of y. Eisenstein establishes the equations
whence ^ (o>) is of the form .4 + B (1 + i) &>, and
1> = >K) >Jr (-') = 4* + 2.B*.
To find the residue of A, mod p, let
and write successively 7" and y 6e for <*> in the function \[<- (eo). We find
^f (7*) = 27"" = 2(1 + ^'z) e = 2(1+ iz) 3 " + J (1 {)", mod p,
because in general (a + bi) p = (a bi), mod p. In this expression no power
of z has an exponent divisible by p 1 ; but
\
--.]
2 2^ = 0, modp,
unless 6 is different from zero, and is a multiple of p 1 ; therefore
4" (7*) = 0, mod p.
un, because 5e = 7n + 2 + (5n + l)p,
vp- (7 5e ) = 2 (1 +zi) 7n + 2 (1 zi) 5B + 1 , mod p ;
in this expression the coefficient C of z p ~ l is
H7n+2.H5n+l
2 r
where
II^ . 117/1 + 2 -M . II/.II5W + 1 -/*' '
'=j) 1, and the summation extends from yw = 3 + l to
Writing 3n + 1 + v for /x, 5n + l v for /u", and observing that
n,.II//=( 1) 1+/1 ', modp,
we find
<7=*
-=o
_
-
,
l'
observing that 2 4n + 1 = 1, mod p, and transforming each of the three factorials
by Sir J. Wilson's theorem. Hence, finally,
o o
282 REPORT ON THE THEORY OF NUMBERS. [Art. 122.
in accordance with the enunciation of M. Stern. The congruence
A =(-!)", mod 4,
is inferred by Eisenstein from the values of
but we may omit these determinations here.
Ifp = 7 + 2, or 7 + 4, Eisenstein considers the complex numbers formed
with the roots of the equation >y 3 21 9 7 = 0. If w is an imaginary seventh
root of unity, and t] k = 3(u) k + <o~ k ) + l, the roots of this equation are n\, 1%, is',
and every complex number formed with them is of the type a + bm + c^, a, b, c
denoting real integral numbers. Let 7 be a primitive root of p in this complex
theory ( p is a prune of the theory, because the congruence
f - 21 ,-7=0, modp,
is irresoluble : see Art. 44 of this Report), and let
7 = l + z 1 i7 1 + z 2 ; 2 , modp,
Z-L and z 2 each representing any term of the series 0, 1,2, . . ., p 1. The function
v|^ (to) = ~Zu>" (the summation extending to all the p 2 values of y) is shown by
Eisenstein to satisfy the equations
*() = * K> = * ("*)> ^ ( w3 ) = 4- O 5 ) = * (")> *() x *( ' =1> ;
whence ^ (<o) is of the form a + b (to + eo 2 + <o 4 ) + c (w 3 + w 6 + w 6 ), and p = A* + 7ff;
if A = a ^(b + c), = ^(b c). The equation p* = a + 3b + 3c, considered as a
congruence, mod 7, becomes A =p 2 , mod 7 ; i. e. A= 4, or = 2, mod 7, according
as p is of the form 7n + 2 or 7 + 4. To obtain the congruence, modp, which
is satisfied by A, we consider the congruence
2 A = ^ (> e ) + 4- ( 7 3e ), mod p ;
in which e = ^(p 3 -l) = a + j8p + yp z ,
a, /3, y representhig positive integers less than p, of which the sum will be
found to be p 1. Now \^ (7*) = ~Zy ev
P-IP-I
= 2 2 (1 -j- Z x i?! + 2 2 >? 2 ) a X (1 + Zj i? p + 2 2 i? 2l) ) X (1 + 2j I7 p s + 2 2 i? 2p 2)7 mo d p ;
because in general
[/Oh)?=/(a*iodp.
Hence -^ (7*) = 0, mod p, because a + /3 + y=p 1, and because
P-IP-I
2 2 zfiZj**=0, modp,
o o
unless 0j and ^ 2 are both different from zero, and both divisible by p-1.
Art. 123.]
REPORT ON THE THEORY OF NUMBERS.
283
Again, if 3 e = a' + ftp + yp 2 , we find a +p + y = ( ^ _ 1 ) ; and omitting terms in
which the sum of the indices of 24 and z 2 is inferior to 2 (p 1),
=2 2
+ z 2 >? 2
4 z 2 ,,!)/, mod
Substituting for z l tj p i + z 2 n^ its value z 1 ^ - z 2 ^ z x >/ p z 2 *j 2p , we obtain
^ ( 7 ") = Ha', n^. n r '. P S P 2 ( Zl ^ + z 2 ,^1-1 ( Zl ,, + z 2 , 8p )i-i, mod p ;
because every such sum as
P
in which 6 is one of the numbers 1, 2, 3, ..., p 1, taken positively or negatively,
is certainly = 0, mod p, as may be seen by substituting (z l n p + z 2 7 2p ) for
(zj >7! + z 2 172)*. Lastly, the coefficient C of (z 1 z 2 ) p ~ 1 in the expression
P-IP-I
is evidently
"~
^ representing the coeificient of a;'* in the expansion of (1 + x) p ~ l . Hence
C = (ii 1 2P ~ ^Vp) 1 " 1 = 1, mod p,
because feq^ %^*21; so that finally
A = i n a '. IT/?, ny . ; (z, z,)"- 1 = | n'. n/r. n 7 ', mod p -,
an expression which, on substituting for a', /8', 7' their values in the two
cases p = 7n + 2, p = 7n + 4:, will be found to coincide with the formulae given
by Eisenstein.
There can be no doubt that the principles of this method are capable of
many other applications ; but nothing has as yet been added to these researches
of Eisenstein.
123. Applications of Continued Fractions to the Theory of Quadratic Forms.
Representations of a number by quadratic forms are in certain cases deducible
from the development of its square root in a continued fraction. If A is any
number not a square, -^-j^ - the (n + l)th complete quotient in the develop-
ment of -/A, and the convergent fraction immediately preceding that complete
002
284 REPORT ON THE THEORY OF NUMBERS. [Art. 123.
quotient, so that pl-Aql = (-!)" D n , then the form (ql, -p n , A), of which the
determinant is ( l) n D n , is either properly or improperly primitive, and belongs
in either case to the principal genus of its order. If we investigate the trans-
formation by which this form is reduced to the simplest form in its class, we
shall obtain, by an operation exempt from all tentative processes, a repre-
sentation of A by that simplest form. The following proposition, however,
supplies a method by which, when q n is uneven, and (ql, p n , A) belongs
to the principal class of properly primitive forms, or when q n is even, and
(^2> ~Pn> 2 A) belongs to the principal class of improperly primitive forms,
we can frequently infer from the development of \/A itself the solution of
the equations
' If (a, b, c), (a, V, c) are two primitive forms of the determinants D
and D', whose joint invariant ac 2bb' + ca is zero, and if ra and m are the
greatest common divisors of a, 26, c; a, 2b', c ; then w 2 Z)' and m' 2 D are
capable of primitive representation by the duplicates of (a, b, c) and (a', b', c')
respectively.'
Thus if (a, b', c') is properly primitive and ambiguous, D can be repre-
sented primitively by (1,0, IX); if (a',b',c) is improperly primitive and
ambiguous, 2Z) can be represented by (2, 1, -^(1 D')). For (a, b, c) and
(a, b', c) let us take (1, 0, -A) and (q n , -p n , q n A), whose joint invariant is
zero, and of which the first is properly primitive ; while the second is properly
or improperly primitive according as q n is uneven or even, and has for its
duplicate in the former case (ql, p n , A), in the latter 2 x (^ql, p n , 2 A) :
so that it is ambiguous in both cases alike. Further, let us represent by
( e > 4> e s-i) the form into which (q n , p n , q n A) is transformed by
we infer, from the property of the invariants, the equations
(-l) n + 1 A, = e, *._!-.$?, e t _ 1 D s -2S e J,-e,D e _ 1 = 0.
Let us first suppose that n is uneven, so that ( l) n Z) n is a negative deter-
minant which we shall call A ; since
q n (q n 3? - %P n xy + q n Af) = (q n x -p n yf + Ay 2 ,
it is evident that when q n a? 2p n xy + q n Ay 2 attains its minimum value,
X 75
- is a convergent to --; not, we may add, the last convergent, if the last
y 2 w
integral quotient in the development of is unity. If therefore (q n , p n , q n A)
Art. 123.]
REPORT ON THE THEORY OF NUMBERS.
285
is properly primitive and of the principal class, we shall have, for some value
of s, e, = 1 ; whence
A-i=-2^J 8 +e s _ 1 D 8 , and A=J* + D.D._ 1 = (J.-i l D.r + && l .
If (2n> -p, q n A) is improperly primitive, and of the principal class of its
order, we shall have for some value of s, e 8 = 2, D,_!= -<S 8 e7 3 + ^e g _ 1 .D (!)
/f\
We may therefore enunciate the theorem : ' If - ? is an inferior convergent
to \/A, and t = qlA-pl; when (q n , p n , q n A) is of the principal class of
forms of determinant A, A is of the form X 2 + A F 2 , and Y is the denomi-
nator of a complete quotient in the development of VA ; when (q n , p n , q n A)
is of the principal class of improperly primitive forms of determinant A,
A is of the form 2X* + 2XY+^(& + l)Y 2 , and Y is the denominator of a com-
plete quotient in the development of */A.'
When (q n , p n , q n A) is ambiguous and properly primitive, but of some
other class than the principal class, we must distinguish between two cases,
that in which the reduced form equivalent to (q n , p n , q n A) is itself an am-
biguous form, and that in which it is of the type (a, 6, a), In the former
case we shall arrive at a form (e g , S,, e g _j), in which e g , being the least number
which can be represented by (q n , p n , q n A), is a divisor of 2 s , and consequently
of D, and 2 A ; and we shall find
In the latter case we shall, in the series of forms (e g , - S s , e,^), arrive at a
sequence of one or other of the three types :
(1) (2[a-6],-[a-6], a ), (a, (a - b), 2 [a - &]) ;
(2) ( a ,-[a-6], 2[a-&]), (a, b, a );
(3) ( a , - 6 , a ), (a -b , a ) ;
i.e. we shall arrive at a form in which e g is the least number but one, which can
be represented by (q n ,p n , q n A), and is a divisor of D, and 2 A ; we shall then
(1) A = (J, 2
(2)
(3)
286 REPORT ON THE THEORY OF NUMBERS. [Art. 123.
Similar results may be enunciated for the case in which (q n , p n , q n A) is
improperly primitive and ambiguous, but not of the principal class.
In applying the preceding formulae to particular cases, the following
theorem of Goepel's is very useful. Since
e > = ?n$ ~ 2p n p. q. + Aq n ql - 4 = q n p e p,-i -p (p. q.-y. +^,_i q.) + A q n q, ?,_
we find, if ft, is the integral quotient immediately succeeding , that
q>
4 +i = 4 /** Hence S lf S 2 , ... form a continually decreasing series. But
$i=Pn is positive, and S n = Aq n _ 1 is negative; there exists, therefore, a pair
of consecutive terms 4 and 4 + i> of which the former is positive, or zero, and
the second negative ; Goepel shows that 4 S e+l < A. For we find
whence q.e,^ + q a _ l S, > f^ t (
or multiplying by e,, Aq, > -^
that is, A> S t S l+l , because ^ g+1 is negative.
Thus if A = 1, we have necessarily
<$ e = 0, e. = e._ 1 = l, A-i =
If A = 2, we have either
(1) 4 = 0, ^.,- 2, c f = l, D 8 -i
or (2) 3, = 0, e 8 _ 1= 1, e, = 2, Z>,
or (3) 4 = 1, 4 + i= -1, M, = 2, e,
If A = 3, we have either
(i) 4 = o, e 8 _ 1= 3, e ,=i, A-i
or (2) 4 = 0, %_!- 1, e. = 3, D. = 3D t _ 1}
or (3) 4 = 1, 4 + i =-2, M, = 3, e, = l, e 8 _ 1 =
or (4) 4 = 2, 4+i= -1, M, = 3, e, = l,
^ = (J, +1 -Z),
or (5) 4 = 1, 4+i= -1, M, = 2, e, = l, ._! = . + 1 = 4,
A=(J,-D t )* + 3D*;
or (6) 4 = 1, 4 + i= -li M, = l, f, = f,-i = f, + i = 2;
Art. 123.]
REPOBT ON THE THEORY OF NUMBERS.
287
the last case occurring always and only when q n is even. If A = 7, and if we
suppose q n even, so that (q n , -p n , q n A) is improperly primitive, we shall cer-
tainly arrive at a form (e s , - S,, e,_j), in which S, = + 1, and either e, = 2, e,^ = 4,
or vice versd e,_ 1 = 2, e, = 4 ; so that there are four cases
(1,2) J, = 2D ! _ 1 -D S , A=
(3,4) J, = 2D,-D s _ l , A =
Let us next suppose that n is even, so that ( 1)" D n = A is a positive
determinant. Then it is evident S 1} 8 2 , ... are all positive, for
q s .= - (qP> -p q,) (q n p,-i -p n q,-i) + &q, q,-i,
of which both parts are positive. Again, the numbers e 1} e 2 , ... form a conti-
nually decreasing series; for q n e, = (q n p,p n q,y Aq*; of which the positive
part continually decreases, and the negative increases in absolute magnitude.
But e 1 = q n , and e n = Ag n ; there exists, therefore, a term e,_ 1 which is positive,
while the following term e, is negative; whence $? = A + e,e,_ 1 < A. Thus if
A = 2, we shall have
If A = 3, we shall have either
If a is the integral number immediately inferior to VA, the period of
integral quotients in the development of \/A is of the type
and it is sometimes possible to assign a priori the value of D k , the denominator
of the complete quotient corresponding to b ; for that denominator is always
a divisor of 2 A, and is besides <2^A. Thus if A is a prime, D k = l or 2;
if \A is a prime, D k = l, 2, or 4. Hence if A or \A is a prime of the form
4n + l, ( - I)* D k = - 1 ; for the equations x 2 -Ay 2 = 2, =+4 are impos-
sible on the supposition that x and y are relatively prune, and the equation
x 2 Ay* = 1 is inadmissible, because 6 is not the last quotient of a period.
Similarly if A or \A is a prime of the form 4m + 3, ( l) k D k = 2 or 2,
according as the prime is of the form 8m + 7 or 8m + 3; if ^A is a prime of
the form 4m + 3, (- \} k D k = +3 or -3, according as the prime is of the form
12m + 11 or 12m + 7 ; and, in general, if X and - : are each of them a prime of
288 REPORT ON THE THEORY OF NUMBERS. [Art. 123.
the form 4w + 3, and if 2\<</A, ( l)*Z) t = X, or X, according as X is or is
A
not a quadratic residue of - We thus obtain a direct method for the repre-
x
sentation of primes of the forms 4m + 1, 8m + 3, 8m + 7, or the doubles of
such primes, by the forms x 2 + y 2 , x 2 + 2y*, x 2 2y 2 : when X is a prime of the
form 12m + 7, the developments of ^/JX and 2^/^A will give representations
of 3X by the forms x* xy + y 2 , x 2 + 3y 2 : when X is a prime of one of the forms
28m +11, 28m + 15, 28m + 23, the development of ^/I-X will give a repre-
sentation of 7 X by the form x 2 xy + 2y 2 , &c.
The theorem relating to primes of the form 4+l is very celebrated; it
was established independently by Gauss and Legendre, and it no doubt sug-
gested the researches of Goepel in his doctoral dissertation 'De quibusdam
aequationibus indeterminatis secundi gradus' (Crelle, vol. xlv. pp. 1-13).
Goepel confined his investigation to the case D n = 2, though his method,
which in the main is that here described, is of a much more general character.
The theorems relating to the case A = 3 were first given by M. Stern, who
employs Goepel's method with very little modification (Crelle, vol. liii. pp. 87-98).
A paper by M. Hermite, which appeared in Crelle's Journal (vol. xlv. p. 191)
prior to the republication there of Goepel's dissertation, contains a method
(see pp. 211-213) which is very similar to that of Goepel, but which does not
connect itself so readily with the common theory of continued fractions. In
these researches of M. Hermite the invariant ac 2bb' + a'c appears explicitly;
which is not the case in Goepel's paper.
REPORT ON THE THEORY OF NUMBERS.
PART VI.
[Report of the British Association for 1865, pp. 322-375.]
124. APPLICATION of the Theory of Elliptic Functions to Quadratic
Forms. The Theta Functions of Jacobi. It will be for the convenience of the
reader to give in this place a brief statement of a few principles and results
which belong to the theory of elliptic functions, and to which we shall have
occasion to refer in the following articles.
The Theta functions of Jacobi are denned by the equation
= 2 _
if e ira = q, by the equation
e^ v (x, ) = 2 (-i)">
m= oo
In these equations, n and v are given integral numbers ; is an imaginary
constant, having for the coefficient of i in its imaginary part a quantity dif-
ferent from zero and positive ; so that the analytical modulus of q is inferior
to unity, and the series defining the Theta functions is convergent for all
values of x real or imaginary ; lastly, a is a constant at present undetermined,
but to which we shall hereafter assign a particular value depending on that
of to. When it is not necessary to specify the value of (a, we shall write
OP. " ( x )> instead of 6^ v (x, to). The following equations are immediate con-
sequences of the definition of the Theta functions :
pp
290
REPORT ON THE THEORY OF NUMBERS.
[Art. 124.
(1)
=(-!)"
(*)
(5)
(6)
Thus there are only four different Theta functions, ^(x), 0tl (x), 6 1>0 (x), #1,1 ()
(equations 1 and 2) ; of these, the first three are even functions, the last an
uneven function (equation 3) ; they are all periodic, having a or 2 a for their
period, according as p. is even or uneven (equation 4) ; the quotient M , V (x) 4- M < X (a;)
is doubly periodic, having a<a or 2 aw for its second period, according as v v'
is even or uneven (equation 5) ; finally, any one of the four can be expressed
as the product of any other by an exponential factor (equation 6).
The identical equations
+ q (v + v-^ + qtfy*
,\ / -t *\
- - (7)
qv) (1 +q 3 v"
(v 3 + v ~ 3 )
-... x
(8)
in which v is any quantity whatever, and q any quantity of which the ana-
lytical modulus is inferior to unity, express an important property of the
Theta functions. Elementary demonstrations of the first have been given by
Jacobi and Cauchy* ; the second is immediately deducible from it, by writing
* Jacobi, Fuudamenta Nova, pp. 176-183 ; Crelle's Journal, vol. xxxvi. p. 75 ; Cauchy, Comptes
Rendus, vol. xvii. pp. 523, 567. See also the note (by M. Hennite), ' Sur la Theorie des Fonctions
Elliptiques' in. the 6th edition (Paris, 1862) of Lacroix, Traite Eleinentaire du Calcul Differential,
vol. ii. p. 397.
2''irr t'irjr
{In (7), v = e~~ ; in (8), v = i~~ '. Hence it would be better in (7) to write v* for v : then to
obtain (8) by putting q^v for v, and multiplying by q$v. This transformation of (7) into (8) is
immediately suggested by a comparison of
Zmiirz
Ziirx
irx
and
Art. 124.]
REPORT ON THE THEORY OP NUMBERS.
291
qv z for v, and multiplying by q*v. We infer from these identities the four
formulae * :
=
' ) _
c s
Q
3
Q / 7rX \] + V/
=3 ( ) = 2 (-
N (* ' ) _,
Ct
COS
a
(10)
a
cos
a
; . (11)
;. . . .(12)
by which the Theta functions are expressed as convergent products of an in-
finite number of factors.
Other important consequences are deducible from the equation
- 3 X Q a _
~ a:; x
x Qo-
4 y (is)
X t'(r-
X ^-
* {This ^-notation is that employed by Jacobi in his Lectures; see Enneper, Sect. 15, p. 78.
OO
It gives 3, (as) =22 cos 2mx. (See also p. 95.) The notation further gives
1 *
Perhaps it might be best to use it with double suffixes. To these notations we must now add that of
Glaisher's tables ; = 0, &, = ~ H, a = ^ H u 3 = yV !}
P p 2
292
REPORT ON THE THEORY OF NUMBERS.
[Art. 124.
which contains four independent arguments, x lt x 2 , x 3 ,x t , and in which
2s = X 1 + X a + X 3 + X t , 2<r = Mi + M2 + M3 + M4, 2 </=/! + 1/ 2 + v 3 + c 4 ;
the numbers /*,, ju 2 , MS, M and v lt v 2 ,v a , v t being subject to the restriction that
their sums are respectively even, so that a- and </ are integral *. Let \//c, \/c'
be two quantities defined by the equations
,o (o)
attributing hi (13) to the elements
X lt X 2 , X 3 , X t
Mi, M, Ms, M
the systems of values
CO
"l, "2, *3> "4
0, 0, 0,
0, 0, 0,
0, 0, 0,
(ii)
x, x, 0,
1, 1, 0,
1, 1, 0,
(iii)
x, x, 0,
0, 0, 0,
1, 1, 0,
* This very symmetrical formula is, it would seem, nearly the same as that employed by Jacobi
in his Lectures on Elliptic Functions at the University of Konigsberg (see his letter to M. Hermite in
Crelle's Journal, vol. xxxii. p. 177). It may be proved by actually multiplying the four Theta series,
ir
and transforming the indices of 1, e <lr ", and e" in the general term of the product by means of the
elementary formulae
22 = a
where 2s = a
{Put (i)
and add ; we find
(ii) C ft = ft = ft = M,= l,
( " = " = "s = "4 = o ;
n
= "n
*
This is given by Rosenhain as Jacobi's Fundamental Formula (M6moires dep Savants Etrangers,
vol. xi. p. 61.)}
[The formula (13) forms the subject of Professor Smith's paper 'On a formula for the multipli-
cation of four Theta Functions, No. XVI.]
Art. 124.] REPORT ON THE THEORY OF NUMBERS. 293
we obtain successively
/C 2 + / 2 = l, (15)
Again, attributing to the same elements the values
x + y, x-y, 0,
, 1 , 1,
1 , 1 , 0,
we find 0i, i (as - y) <V i ( + y) 0j o (0) 0| o (0)
Dividing by y, and diminishing y without limit, we obtain
as(Hlf) = e" l( o ) ?' l( o ) ^ x o(a) ' < 18 >
Similarly, we might form the differential coefficient of any other quotient of
two Theta functions ; of these we require only the two following :
d / 01.0 (as) \ _ 0o,o(OKi(0) 0i.i (a) 00,0 (s) , 1S v
dx \e 0>1 (x)) ~ 1>0 (o) e 0>1 (o) ei tl (x) (it
We shall now attribute to a, which has hitherto been left indeterminate,
the value 2K, K being a constant, the square root of which is determined by
the equation
A/ =MO) = 2l 2 m2 = n ro (l-2 2m )(l+2 2m - 1 ) 2 ; . . . (19)
7T
+00
T^"
we shall also write K' for - . Attending to the values of VK' and -/*, we
i
find from (10) and (11),
-2 2m - 1 ) 2 J (20)
7T
2 . . (21)
Multiplying together the infinite products (19), (20), (21), and reducing by
an identity of Euler's,
294 REPORT ON THE THEORY OF NUMBERS. [Art. 124.
we obtain also
. 1 _ -f -1
= i0; il (0). (22)
These equations (19-22) are of great importance in the arithmetical appli-
cations of the theory.
The constant a having the particular value 2K, the functions
0o,o (*), 0tl (x), e li0 (x), jO^x)
(f
are denoted by Jacobi by the symbols Q! (x), (x), H l (x), H (x) ; we shall find
it convenient occasionally to employ this notation.
The elliptic functions (properly so called), sin am x, cos am x, A am x, are
defined by the equations
1 H(z) A'H^z) ,,Qi(x)
sin am x = r- ~. ; ( ; cos am x = - ^-^- ; A am x = /* .-> / \- ( 23 )
V*c &(x) -A "
These functions are all doubly periodic, having for their periods 4K, 2{K';
{2 (K + iK.')} ; 2K, 4iK respectively; introducing them into the equations
(16-18), we obtain
cos 2 am x + sin 2 am x = 1, ) / 9 .v
A 2 am x + K 2 sin 2 am x = 1, J
d . sin am x
- = cos am x A am x,
dx
d . cos am x
dx
d . A am x
= sin am x A am x,
= K 2 sinamxcosamx.
(25)
dx
From these formulae it appears that if y sin am x, x is one of the values
of the integral / . 1= = . All the values of that integral are re-
presented by the formula a: + 4mK + 2m'i'K', in which m and m' represent any
integral numbers whatever. Since sinamK = l, K is one of the values of the
x*l (y
integral / 7- y _ ; and it can be proved that K' is one of the
/o V(i-y)(iy*)
/I ^2
y . When the real part of vanishes
i-i-
(in which case g r , K, K', *, / are real and positive, and K, K less than unity),
K and K' are the ordinary values of those definite integrals ; i.e. the values
Art. 125.] KEPORT ON THE THEORY OF NUMBERS. 295
obtained by causing y to pass from the inferior to the superior limit, through
a series of real values.
The well-known formulae of Addition and Subtraction which express the
elliptic functions of the sum or difference of two arguments in terms of the
elliptic functions of the arguments themselves, are easily deduced from (13).
But as we shall not require these formulae in the following articles, we may
omit them here.
125. The Modulus and its Complement. The Theory of Transformation.
In the arithmetical application of the theory, the functions K and K, which are
respectively termed the modulus of the elliptic functions, and the complement
of the modulus are of primary importance. They are respectively fourth powers
of the quantities
oo -- 2m 2m1
(26)
which are themselves perfectly determinate functions of o>, if we understand the
positive square root of 2 by ./2, and e* iir<0 by gi Of these functions, which we
shall designate by (f) () and ^ (), the following equivalent expressions have
been given by Jacobi (Crelle's Journal, vol. xxxvii. pp. 75-77) :
u
= V 2 ?* -
u =
./g j7i
~^ ?
2 (- 1 ) B> g 2n " +ro
v/2 M (0,2)
-
-72 -^07^)" :
, 1-" -(" f -n)
V(27)
= -, iT ei.o (Q,
y (28)
(1 - 34 ) (1 - g*-) 2 ( - 1)" g" 2 0>1 (0, 2) '
^ n (i-g 4ro - ii ) 2 (i-g 4 '") = z(-i)-g 2 "' = eo.i(
-
296 REPORT ON THE THEORY OF NUMBERS. [Art. 125.
These expressions of u and u' may be verified by a comparison of their
general factors with the general factors in the formulae (26) : for some of them,
this comparison requires the Eulerian identity already cited (E). Limits of II
and 2 are 1, + oo , and oo , +00; the transformation of the products into sums
is effected by means of (7).
If w = a + bi, and if the positive quantity b increases without limit, a re-
maining finite, we infer, from (26), that
,. v
lim \f/- (a + oi) = + 1, lim - = cos ^a-n- + 1 sm \av.
V &
We shall presently see that </> (w) = ^ ( ) ; hence if = j- and b increase
without limit,
lim < (-^) = lim -f (bi)= +1, lim ^ (f ) = lim < (ib) = 0, lim
^u'
^ ( )
The principal properties of < (to) and -^ (&>) are deducible from the Theory of
the Transformation of Elliptic Functions. The general problem considered in
that theory is ' Given w = , T. Q > where a, b, c, d are positive or negative in-
tegral numbers, to express the Theta functions containing Q by means of the
Theta functions containing w.' The determinant ad be must be different from
zero and positive, because the coefficients of i in the imaginary parts of w and
must both be different from zero and positive ; if ad be = n, the transformation
is said to be of order n. Let A, A', \, X', v, v be the same functions of Q that
Kv' ' ' f r ** " , C + cZQ
, K. , K, K , u, u are ot ta ; since i2 = i-r- u> = i-^-. the equation to = -
A K a+bQ
implies the existence of two others of the form
( 29 )
in which M is a coefficient termed the multiplier ; when A has been found, M is
determined by the equation
l- = A (a + 6 Q )= ^ (c + rfQ ); ...... (30)
it also satisfies the relation
-
~n (!-**) dX ..........
* Fuudameuta Nova, p. 75.
Art. 125.]
REPORT ON THE THEORY OF NUMBERS.
297
If n= 1, the theory of the transformations of the first order has been com-
prised by M. Hermite in the single formula *,
in which
- (x " >) ..... (82)
or
,7 .-i'>-
VO o
= ( )i~* a , if a is uneven,
\\ff
= -r i-* a x i-ifo-OG-D, if 6 is uneven f ;
* Liouville, New Series, vol. iii. p. 26 ; and, with less detail, in the Comptes Eendus, vol. xlvi.
p. 171.
t These determinations of the value of J coincide with those given by M. Hermite in Liouville's
Journal, vol. iii. p. 29 ; where, however, it would seem that the formulae relating to the two cases of
' a pair ' and ' a impair ' ought to be transposed.
{Observe that, if we denote *-<" 0^,, by 0^, (32) acquires on the right-hand side the factor
i"" 1 -**; and
Observe also that -/ i(a + bQ) = i~4 */a + bl, the real parts in both radicals being positive.
It is convenient to divide by f ~* ; so that
J 1 =itJ (} i-4(a-i), a uneven,
or, snce
(-J-} i-W"- 1 ), 6 uneven,
' T *. 11) =
putting A = 2 K,
as in the text.}
K
= C x
x m , n (as,
298 REPORT ON THE THEORY OF NUMBERS. [Art. 125.
the radical N /-i(a + 6S2) represents that square root of -i(a + bQ), of which
the real part is positive ; lastly, A is determined by the equation
*''**' ..... (33)
which is a particular case of the formula (32) ; and M by the equation
1 . - eVj(0.) (34)
M- ** 0* , (o,a,)
The formula supposes that 6 is different from zero and positive ; if 6 = 0, we may
suppose a = d = l, so that = c + Q, and the formula of transformation is
,*), ...... (35)
whe 1 -^'ft'*)
M- 0*0,0 (<>,*>)
The equations of the annexed Table, which, for any transformation of the
first order, express the relation subsisting between the given and the trans-
formed modulus, are also due to M. Hermite, and are of great importance in
the theory of the functions <f> (to) and -^ (<o) *. They may be obtained by apply-
ing the formula of transformation (32) to the expressions of ^> (&>) given by
Jacobi (27). There are six cases, answering to the six solutions, of which the
congruence ad bc = l, mod 2 is susceptible. We add, in each case, the value
of the multiplier f.
* 'Sur la resolution de liquation du cinquieme degreV Comptes Rendus, vol. xlvi.
p. 508; or in a separate reprint (including other memoirs from vols. xlvi. and xlviii.) with
the title ' Sur la theorie des Equations modulaires, et la resolution de liquation du cinquieme
degre,' p. 4.
t [The column giving the transformations of ^ (to) was added in manuscript by Professor
Smith. He mentions that the values in this column were taken from Koenigsberger, Clehsch's
Annalen, vol. iii. p. 10, and verified by
*(>-*(-;)
The subject-matter of 124 and 125 is considered in much greater detail by Professor Smith
in his 'Memoir on the Theta and Omega Functions,' on which he was engaged at the time
of his death.]
Art. 125.]
REPORT ON THE THEORY OF NUMBERS.
TABLE A.
299
c+dfl
ad be =1, o- = es <lr .
II.
I =
d=
*{)=
(f) T
*()-
III.
IV.
V.
(_!)*(-!)/
VI.
(_1 )*(-!) '
It would be easy to write these equations so as to express (f> (Q) in terms
of (p (o>), thus completing the solution of the Problem of Transformation of the
first order; hut it is more convenient to retain them in their actual form.
Similar formulae exist expressing \f<- (w), , , ( , in terms of (j) (Q) and -^ (Q) *.
The propositions implied in the equations of the Table may also be enun-
ciated conversely. Thus to case I. corresponds the theorem ' If to and are
imaginaries in which the coefficient of i is positive, and if < 2v (<o) = (f> 2v (0),
four integral numbers a, b, c, d can be found satisfying the relations
-bc = l; a = d = l, mod 2 ; 6 = 0, mod 2; c = 0, mod 2*-".'
a + 6S2'
* M. Hermite has also shown that the function
which is a cube root of <(<o) x ^('o), possesses a similar property; viz. if w = , ad bc=l,
X (to) can be expressed in terms of x (&), $ (ty, and
dulaires, p. 15.)
( Sur ^ theorie des equations Mo-
300 KEPOBT ON THE THEORY OF NUMBERS. [Art. 125.
If <f>(w) = (j> (Q), four integral numbers a, b, c, d can be found satisfying the
relations = j-~ ; ad be = 1; 6 = 0, mod 2 ; and either a = d=l, mod 8 ;
c = 0, mod 16, or a = d=3, mod 8, c = 8, mod 16.'
These converse propositions may be demonstrated by means of the differ-
ential equations satisfied by the elliptic functions ; by a similar process we
obtain the following equally important theorem :
' If A is any quantity, real or imaginary, other than zero or positive unity,
there exist values of o>, having the coefficient of i in their imaginary parts
different from zero and positive, which satisfy the equation < 8 (&>) = A.'
When n is an uneven integer other than 1, the formula of tranformation is
. (,), (36)
in which m and n are determined as before, and T is a homogeneous function
of order ^(n 1) of the squares of two of the functions O^^x, ). We need not
occupy ourselves here with the determination of A and T, but shall confine
ourselves to the consideration of the modulus and multiplier alone. Represent-
ing by <& (n) the sum of the divisors of n, every binary matrix of order n is
included in the formula
a, b
c, d
= | A x | e |, in which | e | is an unit matrix, and | A |
one of the <& (n) matrices
7,
, y and y being conjugate divisors of n, and k
representing any term of a complete system of residues, mod y. It is thus
sufficient to consider a system of 4> (n) transformations of order n, since all others
arise from compounding transformations of the first order with the transforma-
tions of that system. If we take, in particular, the system of transformations,
-16& + *
-, corresponding to the matrices
(since n, and there-
7
fore 7' is uneven, we may take a system of residues, mod y, of which every term
is divisible by 16), we have for the determination of the transformed modulus,
the fundamental theorem *,
* M. Hermite, Sur la theorie des equations Modulaires, p. 36 ; M. Joubert, Comptes Rendus,
vol. 1. p. 774 ; or, in a separate reprint with the title ' Sur la Theorie des Fonctions Elliptiques,
et son application a la Theorie des Nombres,' p. 21. The demonstration o." this theorem for the
case in which n is a prime, is contained in Sohnke's important memoir ' ^Equationes modulares pro
transformatione functionum ellipticarum,' Crelle, vol. xvi. p. 97. From this particular case, the truth
of the theorem for any value of n is inferred without difficulty.
Art. 125.] REPORT ON THE THEORY OF NUMBERS.
301
on
' The quantities (-) <f>(Q) = (-) $ ( y<<>+ , -) are the roots of an equati
of order $ (n), in which the first coefficient is unity, and the other coefficients
are rational and integral functions of < (o>) having integral coefficients.'
This equation is termed the modular equation of the transformation of the
nth order; designating c (o>) by u, and (_)0('2^_ \ by Vf we s h a ll re-
present it by f(n, u, v) = 0, or more simply by f(u, v) = 0. The function f(u, v)
is characterized by the following, among many other properties,
.... (37)
If in the equation f(u, v) = we put u = ^ (<a), the roots are represented by
If we put u = e3 i *'<t> (w), where s is any integral number, the roots are re-
presented by 2 ^ ni ^ xy
If u = e* il " ! , there are 7' roots represented by (\ e* iirni } y denoting any divisor
/ \ / i
of n. If we put u = e* i ?r-j-4, where s is any uneven number, the roots are
T ( ft ')
represented by ,y+16K
-- \
y^
The equations whose roots are respectively the squares, fourth powers, and
eighth powers of the roots off(u, v) = 0, contain only the squares, fourth powers,
and eighth powers of u ; we shall represent these modular equations by
(w 2 , v 2 ) = 0, / 4 (u*, v*) = 0, or f t ( K , \) = 0, and / 8 (u*, v 8 ) = 0, or / 8 ( K 2 , \ 2 ) = 0.
* It is easily seen that v = </> (<o) is one of the roots of / (7) < ( -- / - ) ^ = 0: this
establishes the first of the equations (37). The other properties given in the text are deducible from
the equations M = <^>((o), v = (-)</>(- -, ), by applying to a) different transformations of the
first order, and employing the formulae of the Table A.
302 REPORT ON THE THEORY OF NUMBERS. [Art. 125
The last equation (by what has preceded) remains unchanged if we write (1)
K for X, and vice versd, (2) 1 * 2 for 2 , 1 X 2 for X 2 , (3) - for K, - for X.
K X
If n admits of a square divisor S 2 , f(n, u, v) is divisible by f(~p, u, (T) v) I
c 'f ' n /2\ /ya> + l6k\ . _ ,/n \
for if yy = , p{-J$^- 7 ) is a root of f(jp, u, wJ-0, and
.
* a root o
It is sometimes convenient to suppose that the modular equation has been
freed by divison from the factors corresponding to the quotients of n divided
by its square divisors ; its degree will then be
if PI, pz, ... represent the primes whose squares divide n, or nil (l + -\ if p
represent any prime dividing n. The roots of this reduced modular equation
are expressed by the same formula as before ; only that y, y', and k are now
subject to the condition that they must not have any common divisor.
With regard to transformations of an even order, we shall only have
occasion to consider the case in which n is a power of 2. K n = 2, we have the
modular equations, 2 u z 1 u*
l - . v'* = - , (38)
4 ' *'
1+u
of which, if u = <p (&>), the roots are given by the equations
If we represent the modular equation of order 2^, when cleared of fractions, by
f(2/*, u, v lt ) = 0, the modular equation of order 2' t+1 , or f(2f + 1 , u, v M+ i) = 0, is
obtained by eliminating i> M from the two equations
f (2*, u, v^ = 0, and
We may thus successively calculate the modular equations of the orders 4, 8,
16, ...; and, attending to the expression, by means of the transcendent <, of
the roots of the equation v 4 = - we may establish the following proper-
1 + u
ties : the function /(2/ 1 , u, V M ) is of the order 2*-- in v s , or of the order 2^ in u 2 ;
the coefficient of u 2lt+l is v 2>t + l , and the equation is not altered by writing
Art. 126.]
1
EEPORT ON THE THEOBY OF NUMBEKS.
303
for u* and multiplying by u 21 ** 1 ; if u = (f>(u>~), the values of v are given by
the eq uation
( 39 )
in which k represents any term of a complete system of residues, mod 2' J ~ 2 , and
correspond to the transformations defined by the formula
to =
a, b
c, d
1,0
-Sk, 2*
1,0
2h, 1
where h is any term of a system of residues, mod 8 ; if v = < (0), the values
of u are given by the equation
where h is any term of a system of residues, mod 2**.
For the determination of the multiplier in a transformation of an uneven
order n, we have the theorem,
' If M is the multiplier corresponding to the transformation u> = - ,
1 7
the < (n) quantities z = ( 1) z ( T ~ 1} -^ satisfy an equation of order <& (n), in
which the coefficient of the highest power of z is unity, and the coefficients of
the other powers of z are rational and integral functions with integral coefficients
of K 2 ; the absolute term, in particular, being + n ' *.
126. The Complex Multiplication of the Argument. The problem of the
multiplication of the argument is ' Given an integral number n, to express
the Theta functions of nx and by means of the Theta functions of x and .'
The solution of this problem may be made to depend on that of the addition
of arguments ; for to add n equal arguments is to multiply the argument by n.
The problem is also included in that of transformation ; for if we consider
n,
the transformation of order n 2 , of which the matrix is
0, n
, we have
When to is not the root of a quadratic equation having integral coefficients,
* Jacobi in Crelle's Journal, vol. iii. p. 308. M. Joubert (Comptes Rendus, vol. xlvii. p. 341)
has calculated the equations of the multiplier for the orders 3, 5, 7, 11. See also M. Brioschi in
Tortolini's Annals, vol. i. (New Series) p. 175, M. Hennite, Equations Modulaires, pp. 12 and 31.
No complete demonstration of the theorem appears to have been given.
304
REPORT ON THE THEORY OF NUMBERS.
[Art. 126.
the transformations, of any square order ri*, and of the type
n,
0, n
are the only
transformations which do not alter the value of u>. For if <o =
, and
io = Q, we have &" + ( d) c = 0. But, hy hypothesis, w is not the root
of any quadratic equation having integral coefficients ; neither is &> rational ;
therefore the three numbers b, a d, and c are all zero, and the matrix
a, 6
c, d
is of the type
n,
0, n
But if w is the root of a quadratic equation hav-
ing integral coefficients, an infinite number of transformations, other than those
included in the formula
n,
0, n
, can be assigned, which do not alter the value
of o>. Let A + 2Ba>+ Cto 2 = be the equation satisfied by ; and let A C B 2 = A ;
then A is different from zero and positive ; also A and C are of the same sign,
and may be supposed to be positive, so that = ~ ; lastly, let 6 = 1,
(j
or = 2, according as (A,B, C) is properly or improperly primitive. Let n be any
number such that 6 2 n admits of representation by (1, 0, A) ; and let <r, r be the
values of the indeterminates in any such representation ; then the transformation
6 ' 6 ^
of order n will not alter the value of &>, because &> =
have for the reciprocal of its multiplier
-rA + (<r-TB)(a
.] , and will
ff+TB+rCw
The transformations derived from different values of n, or from different repre-
sentations of the same value, are all different ; and every transformation of
order n which does not alter the value of &>, is derived from some representation
of 6 2 n by (1, 0, A) ; so that the transformations and representations correspond
to one another one by one, and are equal in number. It will be observed that
the multiplier corresponding to any of these transformations is a complex factor
(composed with N / A) of the number expressing the order of the transforma-
tion ; so that the transformation is equivalent to a complex multiplication of the
argument. And the Theta functions containing w do, or do not, admit of com-
plex multiplication, according as is or is not a quadratic surd.
Art. 126.]
REPORT ON THE THEORY OF NUMBERS.
305
If we consider the values of contained in Theta functions admitting of
multiplication with i V A, we see that these values are infinite in number ; each
form of determinant A supplying one. But the values of < 8 (w), corresponding
to these values, are finite in number, being six times as many as the classes of
forms of det. A ; provided that in the enumeration of the classes a class of
det. 1, or a class derived from a class of det. 1, is counted as ^ instead of 1 ;
and an improperly primitive class of det. 3, or a class derived from such a
class, is counted as -j instead of 1. For it appears from the Table (A) that the
ues of <j>* (eo) corresponding to two equivalent forms, are equal or not, accord-
ing as the transformation, by which one form passes into the other, is or is not
the type
0,1
mod 2. We have therefore only to ascertain how many
bclasses each class contains, a subclass consisting of forms equivalent by
1,0
formations of the type
0,1
A simple discussion shows that the number
of subclasses is six (corresponding to the six types of binary matrices for the
modulus 2) ; except in the two cases just referred to, when the number of
subclasses is reduced to 3 and 2 respectively, owing to the existence in those
1,0
two cases of automorphics which are not of the type
0,1
mod 2. Thus the
whole number of values of < 8 (<o) is 66r(A), 6r(A) representing the number of
classes of det. A, counted in the manner stated above *. It will be seen that
the six values of </> 8 (w) corresponding to the forms of the same class are of the
1 1 K 2 1 /C 2
type /c 2 , , , 1 K", - , - (being in fact related to one another as
K 2 ' 1-K 2 ' K 2 ' K 2
the six anharmonic ratios of four points). The three values corresponding to the
forms of det. 1 are 1, 2, ^ ; and the two values corresponding to the im-
properly primitive forms of det. 3 are the imaginary cube roots of 1.
It is an important theorem (to which we shall again refer) that the 6 G (A)
values of </> 8 (o>) satisfy an equation of that order, of which the coefficients are
integral numbers (but the first coefficient not, in general, unity).
The whole number of values of < (<o), corresponding to the forms of deter-
minant - A, is 48 G (A). For if a be the value of <f> () corresponding to any
form of a given subclass, and n be any eighth root of unity, >? will be a value
of $ (&>) corresponding to another form of the same subclass.
* G (A) is the sum of the densities of the classes of det. A ; the density of a class, according to
the definition of Eisenstein, being the reciprocal of the number of its automorphics.
30(5 REPORT ON THE THEORY OF NUMBERS. [Art. 127.
127. Jacobi's Formulae for the number of decompositions of a number into
squares. The first applications of elliptic formulae to the theory of numbers
were made by Jacobi. The developments, in series proceeding by powers of (/.
of the squares, fourth, sixth, and eighth powers of the functions
v^-
V
2*K =
TT
which are found in the ' Fundamenta Nova ' (sections 40-42, and 65, 66), are
the analytical expression of arithmetical propositions relating to the composition
of numbers by the addition of two, four, six, and eight squares. In these
developments n represents any number from 1 to GO, v any uneven number from
1 to oo ; d is any divisor of n, S any uneven divisor of n or any divisor of v ;
d' and <T are the divisors conjugate to d and S ; and the summations indicated by
2 n , 2 y , 2 d , and 2 extend to every value of n, v, d, and S respectively.
-TT "'' l-q v '' ' n l + 2 2n
Art. 127.] REPORT ON THE THEORY OF NUMBERS. 307
,. 16K*_ / " 3 ~"
(8)
Of these formulae, the first two are the analytical expression of the prin-
cipal theorems relating to the composition of numbers by the addition of two
squares (see Art. 95 of this Report) ; the others may be paraphrased as
follows *.
(3) 'The number of representations of any number N as a sum of four
squares is eight times the sum of its divisors if N is uneven, twenty-four
times the sum of its uneven divisors if N is even.'
(4) ' The number of compositions of the quadruple of any uneven number
N by the addition of four uneven squares is equal to the sum of the divisors
of N:
(5) ' The number of representations of any number N as a sum of six
squares is 42 ( l)j (8 ~ 1) (4^ /2 ^ 2 ), S denoting any uneven divisor of N, S' its
conjugate divisor. In particular if JV1, mod 4, the number of represen-
tations is 122(-l)i- 1 ); if N= -I, mod 4, it is -202 (- 1)^ 8 - 1 ^ 2 .'
(6) ' The number of compositions of the double of any uneven number N
by the addition of six uneven squares is
if N = l, mod 4, this number is zero ; if N= 1, mod 4, it is
-|2(-lp-i><$ 2 .'
(7) ' The number of representations of any uneven number as a sum of eight
squares is sixteen times the sum of the cubes of its divisors ; for an even number
it is sixteen times the excess of the cubes of the even divisors above the cubes of
the uneven divisors.'
(8) ' If N is any number whatever, the number of compositions of 8 N by
the addition of eight uneven squares is equal to the sum of the cubes of those
divisors of N whose conjugates are uneven.'
* The expansions of (1) x (2), (1) x (4), (3) x (2), (3) x (4), are also given in sections 40 and 41 of
the ' Fundamenta ' ; and may be similarly interpreted.
R r 2
308 REPORT ON THE THEORY OF NUMBERS. [Art. 127.
In counting the number of compositions by addition of squares, two com-
positions are to be considered as different if, and only if, the same places in
each are not occupied by the same squares ; but in counting the number of
representations we have to attend also to the signs of the roots of the squares.
Thus each composition by the addition of four squares, none of which is zero,
is equivalent to sixteen representations. Only one or two of the preceding
theorems are enunciated in the published writings of Jacobi : see Crelle's
Journal, voL iii. p. 191 ; vol. xii. p. 167. Some of the others have been
given by Eisenstein (Crelle, vol. xxxv. p. 135), who had also obtained purely
arithmetical demonstrations of them from the theory of quadratic forms con-
taining several indeterminates. ' In my investigations,' he says ' these theorems
are proved by purely arithmetical considerations, and appear as special cases
of more general theorems ; at the same time we see why these developments
close with the eighth power ; since, in fact, eight is the greatest number of
indeterminates for which only one class of forms, represented by a sum of
squares, appertains to the determinant 1.'
In the second of the notes to which we have just referred (Crelle, vol. xii.
p. 167), Jacobi has given an arithmetical demonstration of the theorem (4).
It consists in a kind of translation of the analytical proof into an arith-
metical one ; and is of great interest and importance, as the first example of a
new method, and as having suggested important researches to MM. Liouville
and Kronecker (see Liouville's Journal, New Series, vol. vii. p. 48 ; M. Kro-
necker, ' Monatsberichte,' May 26, 1862, p. 307).
The doubly periodic functions of argument - - obtained by dividing any
VT
Theta function by any other, or the product of any two of them, by the pro-
duct of the other two, admit of development in series proceeding by sines or
cosines of multiples of the argument x. These developments, which, unlike
the developments of the Theta functions themselves, are not convergent for all
values of x, real or imaginary, will be found for the most part in section 39
of the ' Fundamenta Nova ' ; and the complete system has been given by M.
Hermite (Comptes Rendus, July 7, 1862). One, which we require in this place,
will serve as an example of the rest,
2Kx _, q% v sin vx \
sin am
It is from these developments that the expansions (1) ... (8) of the powers
Art. 127.]
EEPOBT ON THE THEOUY OF NUMBERS.
309
VO~I? /O 1?
- and /\/ - - are deduced. Thus, writing \ v for x in (A), we find,
since sin am K = 1,
-K
_ /-
~
l-q v '
which is the formula (2). We shall now show how the equation (4) can be
obtained .by squaring this formula. For this purpose we represent by a and /3
any two unequal positive uneven numbers congruous to one another for the
modulus 4, and by a' and /3' any two positive uneven numbers not congruous
to one another for the modulus 4. We then have
"
g"
v
= P+Q-R, for brevity.
Here
again in Q, if we double each term we may suppose /3>a; let |8 = a
observing that a may be any positive uneven number, and n any positive
number whatever, we find
9T
Lastly, in R let a' + /3' = 4n ; so that
4-l
2,,
Consequently
=
" i
= P+Q-R
nq n
"- 4 "
I-?
which is the formula (4).
310 REPORT ON THE THEORY OF NUMBERS. [Art. 127.
Thus by a purely analytical process we deduce from an equation which
exhibits the number of compositions of the double of an uneven number by the
addition of two uneven squares, an equation exhibiting the number of com-
positions of the quadruple of an uneven number by the addition of four uneven
squares. This analysis Jacobi has expressed arithmetically as follows. Re-
presenting by N an uneven number, by [4JV] the number of compositions of 4N
by the addition of four uneven squares, we resolve 4_A^ in every possible way
into two unevenly even numbers ^N l and 2^, and each of these in every
possible way into two uneven squares ; we thus obtain the equation
[4JV> 2 [2^ = (2x+ 1) 2 + (2y + 1) 2 ] x \2N Z = (2x + l)* + (2y + 1) 2 ],
in which the summation extends to every pair of uneven numbers A^ and JV 2
which satisfy the equation 2N=N 1 + N 2 , and the square brackets represent
the number of solutions in positive integers of the equations included in them.
Observing that [2N 1 = (2x + l) 2 + (2y + l) 2 ] is the excess of the number of
divisors of N 1 which are of the form 4&+1, above the number of its divisors
which are of the form 4& 1, retaining the signification of a, /3, a', /3', and
denoting by a and b any positive uneven numbers, we may transform the
expression of [4^V] into the following,
in which the square brackets still retain the same signification. Supposing,
as before, ft > a, and /3 = a + 4 n, we have
or, putting a = v + 4 kn, v being less than 4 n,
y being uneven and v < 2n. Again, if in [2N=aa' + bfr r ] we write 4n for
a + /3', and suppose a > b (the supposition a = b is inadmissible as it would
render N even), we have
as before. Hence
[2N = aa + bp]-[2N = aa' + bp] = 0, and [4 N] = [2N = (a
i.e. [47V] is the sum of the divisors of N. In this arithmetical process we
determine the coefficient of q N in P, Q, R, instead of determining those functions
2 n o 2 n
themselves ; and as the difference Q R= 2 n - - is au even function in
1 q tn
the analytical process, so the difference [2 N= a a + bfi] [2N = a a' + 6/3'] vanishes
in the arithmetical one.
Art. 128.]
REPORT ON THE THEORY OF NUMBERS.
311
Lejeune Dirichlet, in a letter addressed to M. Liouville (Liouville's Journal,
few Series, vol. i. p. 210), has put Jacobi's demonstration into a form in which
it is more easily followed, but is a little further removed from the analysis.
He shows that to every solution of the equation aa + b(3 = 2N, in which a>/3,
there corresponds a solution of the equation a a + b'f3' ' = 2N, in which a>($',
id vice versd, the two solutions being connected by the relation
V
x + 1, x + 2
x , x + 1
b,
s'here x \s the integral number immediately inferior to
thing, to
Hence, as before,
or, which is the
= a'a+b'F], and
[4^] is equal to the sum of the divisors of N.
128. Theorems of Jacobi on Simultaneous Quadratic Forms. In an elabo-
rate memoir ' On Series whose Exponents are of two Quadratic forms ' *,
acobi has established a great number of elliptic formulae, which are the ana-
ical expression of theorems relating to the representation of numbers by
irtain quadratic forms. A comparison of the two criteria of Gauss for the
biquadratic character of 2 with respect to a prime p of the linear form 8k + 1,
leads to a result which will serve as an example of these theorems. By the
first criterion, 2 is or is not a biquadratic residue of a prime p of the form
8&+1 according as a is even or uneven in the equation p = (4a + l) 2 + 8b 2 ;
by the second, 2 is or is not a biquadratic residue of p according as /? is even
or uneven in the equation j> = (4a + l) 2 + 16/3 2 f. We infer therefore that a
* Crelle's Journal, vol. xxxvii. pp. 61 and 221 ; or Mathematisehe Werke, vol. ii. p. 67.
t Theoria Kesiduorum Biquadraticorum, Arts. 13-21. To the second criterion we have
already referred in this Report (Art. 24, and in the additions to Part I., printed at the end
of Part II.); the first is more elementary, and is inferred from the equation p = (4+ I) 2 + 8 6 s ,
in which p is a prime of the form 8k+\. Raising each side of the congruence
86 2 = (4a+l) a , mod p,
to the power ^ (/>!), and observing that
2*"-> = (})=!, <-!)*>- = 1, we find
But if b=2 y ft, where ft is uneven,
) (1) =
(?) -<)-'
because p = (4 a + I )*, mod ft ; and
(!ft!),(_.
Hence 2'( /> ~ 1 ) = ( l) a , modp, which is Gauss's first criterion.
iH
! = (-!)".
312 REPORT ON THE THEORY OF NUMBERS. [Art. 128.
Is even, or (since a + b + a is even by virtue of the congruence
(4a + l) 2 + 8& 2 = (4a + l) 2 , mod 16)
that a + )8 + b is even. The result is thus generalized by Jacobi :
'For any number P the sum 2( 1)*, i.e., the excess of the number of
solutions of the equation P = (4a + l) 2 + 8& 2 in which b is even above the
number of solutions in which b is uneven, is equal to the sum 2( l) a +0, i.e.
to the excess of the number of solutions of the equation P = (4a + l) 2 + 16/3 2 ,
in which a + /8 is even above the number of solutions in which a + j8 is uneven.'
The generalized theorem is expressed analytically by the equation
2(-l) n 5 (4m + 1)2 + 8n3 = 2( l) m + n 5 (4m+1)2+16B ", ..... (2)
hi which the summations extend to all values of ra and n from oo to +00.
But this equation is an elliptic formula ; for, on dividing by q, and writing
q for q 9 , it becomes
2 ( - 1) 5 3 x 2 5 2 3 + m = 2 ( - 1) g 2 " 2 2 ( - l) n q 2n > + ,
which is included in the equations (28) of Art. 125, and is therefore a corol-
lary from the fundamental property of the Theta functions expressed in equa-
tion (7) of Art. 124. We infer at the same time, from the equations (28),
that either of the sums
or
is equal to the infinite product
We thus arrive at an analytical proof of Jacobi's theorem, including, as a
particular case, a proof of the identity of Gauss's two criteria. But the con-
tinuation of Jacobi's memoir was intended to contain direct arithmetical
demonstrations (which, however, have never been published) of the theorems
of which the equation 2( 1)* = 2( l) a+ is an example. He says, 'Though
these arithmetical demonstrations of results obtained analytically present no
essential difficulty, yet they are sometimes of a complicated character, and
require peculiar classifications of numbers which perhaps may be of use in other
researches. We have here a certain amount of freedom in the choice of
methods, so that the proofs can easily be varied ' *. Probably one of these
methods was that employed by Dirichlet in his earliest arithmetical memoir,
* Hathematische Werke, vol. ii. p. 73.
Art. 128.] REPORT ON THE THEORY OP NUMBERS. 313
to which Jacobi expressly refers. In this memoir* (written when only the
enunciations of Gauss's criteria for the biquadratic character of 2 had been
published) Dirichlet gives a demonstration of the first criterion, which does not
differ from that subsequently given by Gauss (Theor. Res. Biq. Comm. prima,
Art. 13), and then deduces the second criterion, as follows, from the first.
Since
we have . [4 (a + ) + !] x [4(a-/3) + l] = (4a + l) 2 - 8& 2 .
No common divisor of 4 a + 1 and b can also be a common divisor of
4(a + /3) + l and 4(a-/3) + l,
i.e. of 4a + 1 and 6 ; for p is not divisible by any square. The greatest common
divisor of (4a + l) 2 and 6 2 must therefore be a product of two relatively prime
aeven squares S 2 and ($'*, dividing 4(a + ^) + l and 4(a-/3) + l respectively;
- is thus a divisor of the quadratic form x 2 8y 2 , in which x and y
are relatively prime ; it is, consequently, itself of that quadratic form, and
4(a + j8) + l = l, mod 8;
this congruence implies that a + /3 = 0, mod 2, or, which comes to the same
thing, that 6 = a + /8, mod 2. It will be seen that this demonstration of the
congruence b = a + /3, mod 2, applies to any two representations of any number
P by the forms /= (4a + l) 2 + 8& 2 and < = (4a + l) 2 + 16/3 2 ,
provided that in the two representations the four numbers 4a + l, 4a + l, 6, /3
ive no common divisor. To prove, for every uneven value of P, the truth of
Jacobi's equation 2 ( 1) = 2 ( l) a + , we observe, first of all, that the equation
evidently true if P is not = 1, mod 8, or if P contain an uneven power of a
prime of the linear form 8k + 7 ; for in these cases there are no representations
of P by either form. We may therefore suppose that P is of the linear form
8 k + 1 ; then the equation is true if P contains an uneven power of any prime
of either of the linear forms 8 + 3 ; thus if P=p 2v + 1 P', where P' is prime
p, and p = P'=3, mod 8, there are no representations of P by 0, so that
2 ( - 1)"+ = ; let the equations
denote generally those representations of p 2v + l and P' by the form (1, 0, 2)
* Crelle's Journal, vol. iii. p. 35.
S 8
314 REPORT ON THE THEORY OF NUMBERS. [Art, 128.
in which the first indeterminate is = 1, mod 4 ; then the representations of
P=p 2y + l xP' by /will be comprised in the formula
but of the two numbers ^(yX + xY), ^(yX xY) (both being values of the
second indeterminate), one is uneven and the other even ; whence
Similarly if P=p* v + l P', where P is prime to p, and p = P'=5, mod 8, there
are no representations of P by/, and it may be shown that
2(-l)"+0 = = Z(-l) 6 .
We may therefore confine ourselves to the case in which P is composed of any
powers of primes of the linear form 8k + 1, and of even powers of primes of the
forms 8k + 3, 5,7. If, on this supposition, P = P'xP", where P / and P" are
relatively prime, and each is = 1, mod 8, the sums 2(-l) a and 2( l) a+ ^,
relative to P, are the products of the corresponding sums relative to P' and P".
This may be proved by observing that the representations of P by / [or <]
may be obtained by compounding the representations of P f and P" by that
form, and that each representation of P has the character of an even or uneven
b [or a + /8] according as the representations of P' and P" of which it is com-
pounded agree or differ in respect of that character. Thus it is sufficient to
consider the four cases in which
(1) P=p v , p = l, mod 8; (2) P=p* v , p = 3, mod 8 ;
(3) P =p 2 ", p = 5, mod 8 ; (4) P =p*>, p = 7, mod 8.
In the last of these cases it is evident that
in the others, the proof is supplied by Dirichlet's method. (i) If P=p",
p=l, mod 8, there are two primitive and i> 1 derived representations of P by
each form ; and the application of Dirichlet's method shows that, for every
representation of P by <, (-l) a+0 has the same value as (-1) 6 in either
primitive representation of P by / and, conversely, that for every representation
of P by/ ( I) 6 has the same value as ( l) a+e in either primitive representation
of P by <p ; whence the units ( - I) 6 and ( - l) a+ ^ have all the same value, and
* For 2yY xX = 1, mod 4 ; and the representations comprised in the formula are all different,
their number being equal to the number of sets of representations of P by (1, 0, 8).
Art. 128.] REPORT ON THE THEORY OF NUMBERS. 315
The ambiguous sign is that of ( 1)* in the primitive representation of P by f,
and will be found (by reasoning similar to that which establishes Gauss's first
criterion) to coincide with ( 1)5(^-1) e ", where e is the unit satisfying the
congruence 2i (p - 1) = e, mod p. (ii) If Pp iv , p = S, mod 8, there is but one
representation of P by <f>, and 2( l) a+e = ( 1)"; there are 2i/ + l repre-
sentations of P by f, of which two are primitive, 2 (v 1) are derived from the
primitive representations of p z , p 4 , ...,p 2(v ~ l \ and in the remaining one 6 = 0.
Applying Dirichlet's method to the equation
in which <r = l, 2, ..., v, /3 = 0, 4a + l = ( T^} a <f, and the representation by
/ is primitive), we find ( 1) & = ( 1)" ; whence inasmuch as the character
( 1)* is the same in a derived representation, and in the representation
from which it is derived
i
(iii) Lastly, if P=p zv , p = 5, mod 8, there is but one representation by/, and
2( !)*=+!; there are 2j/ + l representations by fy. Applying Dirichlet's
method as in the preceding case, we find that for any primitive representation
of an even power of p by (f>, ( l) a + /3 = +1; whence, for a derived repre-
sentation in which the greatest common divisor of the indeterminates is q",
( - i )a + e = ( _ i )*. Consequently
K-l
This completes the demonstration of Jacobi's theorem.
Let P be any uneven number and x (P) the positive numerical transcendent
defined by the equation
X 2 (P) x ^ (P) = 2 ( - !)*-'> x 2 ( - l)iW 2 -" x 2 ( - i )*<<*-!) + tw-o,
where ^(P) is the number of divisors of P, and d is any divisor of P. It
will be seen that xCP) = 0, except when P is capable of representation both
by < and /: when P is capable of such simultaneous representation, let
P = (4a + l) 2 + 16/3 2 be a representation of P by (p in which the greatest
common divisor of 4a + l and ft is the least possible; let ra- = 4a + l-|-4i/3, and
let I represent the quadratic character (Art. 27) of 1 + i with respect
tow; the equation , . 6 _. 1 x a+fl f 1 + ^ "]
-i- 1 ; -^i- 1 ; -["^j
882
316 REPORT ON THE THEORY OF NUMBERS. [Art. 129.
will hold in each of the cases considered separately above ; but the numerical
functions occurring in this equation satisfy the condition
where P l} P 2 are relatively prime ; the equation is therefore universally true
for every uneven number P, and implies the identity
From the nature of the identity (2) it is evident that we may substitute
any function whatever (which renders the two series convergent) for the
exponential of q. Thus, for example, we find
[(4m + 1) 2 + 8 TO 2 ] 1 - 1 -? [(4m + l) 2 + 16% 2 ] 1 + i>
-nL- n * n
where Pi,p 2 ,p 3 are primes of the forms 8n + l; Sn + 3; 8n + 5 or 7, respect-
ively ; and n is a positive or negative unit determined by the congruence
It would seem that the method of Dirichlet which we have here described
may be employed to prove all the theorems of Jacobi's memoir in which the
two forms compared have different determinants. Those in which the two
forms compared have the same determinant, or determinants differing only
by a square factor, are of a more elementary character, and are capable of
immediate verification. But Dirichlet's method may also be extended to cases
in which one or both of the forms compared has a positive determinant. One
example will suffice. If P = (2 a + 1) 2 + 8 6 2 = (2 o + 1) 2 - 8 /3 2 , we have
<j> representing any function whatever which renders the series convergent,
and the limits of m and n in the first sum being 0, <x>, and oo, + oo ; in
the second sum 0, oo, and 1, ^(2m + l).
129. We proceed to indicate very briefly the origin of the principal
formulae in Jacobi's memoir. Three of them are distinguished from the rest
as general, being deduced from the equation (7) of Art. 124, without any
specialization. If in that formula we write successively + z and z, for v, and
multiply the results together, the left-hand member becomes 2 ( l) m q m * + " 2 z m + " ;
Art. 129.] REPORT ON THE THEORY OF NUMBERS.
the right-hand member may be written in the form
n (1 -g*-^ (I-q*>) x 5 (1 -2 4 '") (1 -g 4m - 2 z 2 ) (1 -2* m - 2 z- 2 ),
where the second infinite product, by the equation (7), is equal to
2 ( - 1) q* m * z 2m , and the first to 2 ( - l) m q* m \
317
Hence Z(-l)Fjf*+**+-Z(-.l)P+"jf* llp+1 iP i ' ...... (A).
which is one of Jacobi's general formulae. The other two general formulae,
and most of the special ones, are obtained in like manner by considering infinite
products which are capable of being expressed in more ways than one as the
product of two Theta functions. To arrive at his special formulae, Jacobi
transforms the equation (7) by writing q a , where a is positive, for q, and + q b
for v. He thus obtains the equations
(1 -
&\ /1 /-,2ma a + 6\ /1 xy2ma\ V / 1 \m.-,am 2
M 1 -^ J^ 1 -? ;~*l 1 /2
00
n
i
Any infinite product of either of the types occurring in these equations he
calls an elliptic product ; and every infinite product which can be formed in
more ways than one by the multiplication of two elliptic products, leads directly
to one of his special formulae. The five following elliptic products are of great
importance in the theory ; they correspond to the suppositions
a = l, 6 = 0; a = 2, 6 = 1; a = |, 6 = .
n (i -
n (i - 2 2m ) = 2 ( - i) m q m \
(1 + g*-i) n (1 - 2 2m ) = 2 q m \
i
00
_ V .-,2m 2 + m
(B)
U(l-q m ) =2(-l) m
1
the first two are the equations (19 and (20) of Art. 124 ; the last is a celebrated
formula due to Euler.
The infinite products in the numerators and denominators of the fractions
318 BEPORT ON THE THEORY OF NUMBERS. [Art. 129.
equal to u and u' (equations 27 and 28, Art. 125) are all elliptic products of
one or other of these five types, in some cases with q 2 , or q 3 , or q substituted
for q. Hence a comparison of any two of the fractions equal to u or to u' gives
immediately one of Jacobi's special formulae. The demonstration of the fonnula
(2) in Art. 128 will serve as an example of this process.
Again, Jacobi has shown that the Eulerian product (Art. 124, E.)
which Euler had himself represented by the fraction
oo 1 _ fjZm y\ ( _ -I V/
rr ? _ _ ^v L t ** _ rn\
- " - m < 3 ' 2+m >'
can be represented by six other fractions of which both the numerators and
denominators are elliptic products ; either the numerator or denominator, or
both, being of one of the types (B). Thus, for example,
Here again a comparison of any two of the seven equal fractions gives one
of the special formulae : thus writing g 24 for q in the two fractions (C) and (D),
We find
2( 1)"
which, however, is only a particular case of the general formula (A).
The Eulerian product is also of importance in the theory of the partition
of numbers. If it be developed in a series proceeding by powers of q, the
coefficient C(m) of the mth power of q in the development, expresses the number
of ways in which m can be composed by the addition of unequal numbers, or
by the addition of equal or unequal uneven numbers. Euler observed that his
fractional expression of the product furnishes a recurring formula for the
calculation of C(m), and the same thing is true of each of Jacobi's fractions ;
the simplest of the seven recurring formulae being that arising from the
fraction (D), viz., 2 ( - 1)' C(m - 3 s 2 ) = e ,
the summation extending to all positive or negative values of s for which
m 3s z is not negative, and f representing 1, or 0, according as m is or is not
of the form
Art. 129.] REPORT ON THE THEORY OF NUMBERS. 319
00 +00
The equation Tl(l-q m ) = 2 (-l) m gz< 3ml! + m > is memorable historically as
the earliest example of the introduction of a Theta function into analysis *.
It expresses the theorem
' The excess of the number of ways in which a given number can be com-
posed by the addition of an even number of unequal numbers above the number
of ways in which it can be composed by the addition of an uneven number of
unequal numbers is ( - l) m or 0, according as the given number is, or is not,
of the form ^(3m 2 + m).'
Of this theorem Jacobi has given an arithmetical demonstration, repro-
ducing Euler's proof of the analytical formula.
OO 1
The logarithmic differential of II (1 - q m ~) is 2 $ (m) q m , where $ (m), as
in Art. 125, is the sum of the divisors of m : Euler thus obtained the equation
+ 00 +00
2 ( iygz(3m 2 + f) _ 1 2 ( '
which supplies a recurring formula for the calculation of $ (m), viz.,
2 ( - 1)' * (m - 1 (3s 2 + )) = E (m),
the summation extending to all positive or negative values of s for which
m ^(3s 2 + s) is positive, and E(m) representing ( l) 8+1 m, or 0, according as m
is, or is not, of the form ^(3s 2 + s)f.
The cube of the Eulerian product is equal to the series
i2(-l)">(
(Art. 124, equation 22) ; so that
* In the year 1750 or 1751. Nov. Comm. Petropol., vol. iii. p. 155.
t On the equations
and their connexion with the partitions and divisors of numbers, see Euler, Nov. Comm. Petropol.
vol. iii. p. 125, vol. v. p. 59 and p. 75 ; Acta Petropol. vol. iv. Part I. p. 47 and p. 56 (or Commenta-
tiones Arithmeticse Collectse, Nos. IX., XI., XVI., L.; the first memoir in vol. iv. of the Acta is
omitted in the collection) ; Introductio in Analygin Infinitorum, part 4. cap. 1 6 ; "Waring, Philo-
sophical Transactions for 1788, p. 388 ; Legendre, Theorie des Nombres, ed. 3, vol. ii. p. 128 ; Jacobi,
Fundamenta Nova, p. 185, Crelle, vol. xxxii. p. 164, vol. xvxvii. pp. 67, 73 (or Mathematische Werke,
vol. i. p. 345, vol. ii. pp. 73, 79).
320 REPORT ON THE THEORY OF NUMBERS. [Art. 129.
a result which in an earlier memoir (Crelle, vol. xxi. p. 13, or translated in
Liouville, First Series, vol. vii. p. 85) Jacobi describes as ' hitherto unparalleled
in analysis.' Writing q 2 * for q, and multiplying by q 3 , it becomes
the summations 2 t and 2 2 extending respectively to all positive uneven numbers,
and to all positive uneven numbers prime to 3. In this form it expresses the
theorem
O
' The sum 2 ( - ) extended to all compositions of any number N by
Va^Os/
the addition of three uneven squares a\, of, o|, all of which are prime to 3, is
( _ l)J(-i>7/j, or according as N is or is not the triple of an uneven square.'
Differentiating logarithmically, we find
Z, (-!)*- ()&(<*'-&)) a*
V Cv'
= 0.
This equation (in which all the exponents have the same quadratic form)
admits of immediate verification, elementary considerations sufficing to show
O
that the sum Z( l)z (6-1) (-) &(a 2 6 2 ) extended to every solution of the
^Cx/'
equation N=a 2 + 3b 2 , is zero. Jacobi thus obtains a direct arithmetical proof
of the formula (F). (Crelle, vol. xxi. pp. 15-18.)
The square and the cube of the Eulerian product can also each of them be
represented in two different ways as the quotient of two elliptic products.
Other formulae of Jacobi's are inferred from the fundamental equation (7)
in a somewhat more complicated way. Replacing v in that equation by certain
roots of unity, and multiplying two or more of the results together, Jacobi
obtains products which can be expressed in more than one way by means of
elliptic products ; the formulae thus deduced are remarkable chiefly because
they lead to equations, not between two, but between three or more series,
the exponents of which have certain quadratic forms.
Lastly, a few additional equalities are derived not from the fundamental
equation, but from the modular equations of the third and seventh orders.
The modular equation of the third order was brought by Legendre into the
form ^//c'X' + +/K\ = 1 ; whence evidently
2 () (f) 2 (3) + \p () \p (3 ) = 1 ;
writing for the functions < 2 and \p their values given by equation (14), Art. 124,
Art. 130.] REPORT ON THE THEORY OF NUMBERS. 321
and changing q into q*, we find
2 1
The equation of the seventh order, in the form in which it has been put by
M. Gutzlaff*,
admits of similar treatment, and furnishes as many as seven formulae on account
of the variety of expressions which the equations (27) and (28) allow us to
substitute for (p and ^ in the equation
It is only necessary to observe that we must choose for < (w) and ^ (a>), and
similarly for cp (7 to) and ^ (7 to), expressions having the same denominator.
At the beginning of his memoir Jacobi says that the formulae to which it
relates are probably finite in number. It would seem that when he expressed
himself thus, he had not yet found his three general formulae, each of which
contains an infinite number of equations between series having their exponents
contained in the same quadratic form. But it is certainly very unlikely that
equations between series whose exponents are contained in different quadratic
forms, exist for any but a few of the simplest forms, or for them in infinite
number.
130. The Formulae of M. Kronecker. We now come to an important series
of results, discovered within the last few years by M. Kronecker, which form
a memorable accession to our knowledge of quadratic forms, and which have
opened an entirely new field of arithmetical inquiry. Their demonstration
requires considerations of a very complicated kind ; and as they are certainly
among the most interesting, so also they must be reckoned among the most
abstruse of arithmetical truths. Unfortunately, in the brief notices f which
* Crelle's Journal, vol. xii. p. 173.
t The following are the memoirs of M. Kronecker on the application of the theory of elliptic
functions to quadratic forms.
(1) 'Ueber elliptische Functionen und Zahlen-Theorie,' Monatsberichte, Oct. 29, 1857; and
translated in Liouville, New Series, vol. iii. p. 265.
(2) ' Ueber die Anzahl der verschiedenen Klassen von quadratischen Formen von negative!-
Determinante,' Crelle, vol. Ivii. p. 248 ; and translated in Liouville, vol. v. p. 289.
(3) ' Ueber eine neue Eigenschaft der quadratischen Formen von negativer Determinante,'
Monatsberichte, May 26, 1862.
(4) 'Ueber die complexe Multiplication der elliptischer Functionen,' Ibid, June 26, 1862.
(5) ' Auflb'sung der Pellschen Gleichung mittelst elliptischer Functionem,' Ibid, Jan. 22, 1863.
Tt
322 REPORT ON THE THEORY OF NUMBERS. [Art. 130.
M. Kronecker has given of his investigations, his methods are indicated only
in a very general manner ; and, notwithstanding the light which has been
thrown on them in the subsequent memoirs of MM. Hermite and Joubert *,
it is occasionally difficult to rediscover them. Nevertheless, as a mere enu-
meration of formulae, unaccompanied by any explanation of the methods by
which they have been obtained, would be of little use to the reader, we shall
attempt in the next article a complete demonstration of one or two of them,
which may serve as specimens of the rest.
The following (with an unimportant change in the notation) are the eight
equations given by M. Kronecker (Crelle, vol. Ivii. p. 248 ; Liouville, New
Series, vol. v. p. 289).
= 2$ (m) + $ (2' 1 - 2 m
II.
III.
= 0.
IV. 3G(m) +
= $ (m) + 3 * (m)
V.
VI.
VII.
VIII.
In these formulae m is any positive uneven number ; in the 1st, / is
* M. Hermite, ' Sur la the'orie des equations Modulaires ' ; 31. Joubert, ' Sur la Theorie des
Fouctions Elliptiques et son application a la Theorie des Nombres,' already cited in the note
on Art. 125.
Art. 130.]
EEPORT ON THE THEORY OF NUMBERS.
323
in the 7th, m is = - 1, mod 8 ; in the 8th, m is = + 1, mod 8, and the summa-
tion extends to all values of s for which ^ (m - s 2 ) is integral and not negative ;
similarly, the series in the first seven formulae are to be continued until the
numbers affected with the signs F and G become negative. If n is any positive
number, even or uneven, (n) is the sum of the divisors of n, * (n) the
excess of those divisors of n which surpass */n above those divisors which
are surpassed by .Jn; $' (n) is the sum 2 (-^ d extended to all the divisors
of n ; -V (n) the excess of the sum 2 (-^) d extended to all the divisors of n
which surpass *Jn, above the sum 2 (-=) d extended to all divisors of n which
are surpassed by Jn. Lastly, F(n) is the number of uneven classes, G(n) the
whole number of classes, of forms of determinant n ; the classes (1, 0, 1),
(1, 1, 1), and their derived classes, being counted as ^ and ^ respectively; to
.F(O) we attribute the value 0, to G(0) the value -^*.
The arithmetical functions F (n) and G (n) satisfy the equations
F(4n) = 2F(n); G (4 n) = F(4n) + G(n); G (n) = F(n), if n = 1, or 2, mod 4 ;
G(n) = 2F(n), if n = 7, mod 8 ; (?(n) = |i' T (n), if w = 3, mod 8.
With the help of these relations (which may be demonstrated by elementary
considerations [see Art. 113 of this Report], but which may also be inferred
from the theory of elliptic functions) the formulae I. VIII. may be transformed
and combined in various ways, so as to afford new and interesting results.
Of these derived formulae M. Kronecker has given two,
IX. F(n) +
X.
* The right-hand members of the formulae I. VIII. are rendered simpler by this conventional
estimation of a class of det. 1 as J, and of an improperly primitive class of det. 3 as . "We have
already seen that this convention is a natural one (Art. 126, note); it is, however, less easy to
interpret the assumption 6 (0) = ^. M. Kronecker has given his formulae in their complete
expression when these conventional estimations are disregarded ; in his subsequent notes, however,
he seema to prefer the simpler form, which we have adopted in the text.
T t 2
324 REPORT ON THE THEORY OF NUMBERS. [Art. 130.
where n represents any positive integer, X(n) the sum of its uneven divisors,
and E(n) = 2F(n) G (n), so that E(n) is a function satisfying the equations
E(4n) = E(n); E(n) = 0, if n = 7, mod 8 ; E (n) = $F(n), if n = 3, mod 8.
The first of these formulae is obtained by subtracting VI. from V. ; for
F (n - k . FFT) = F (4 n + 1 - [2k + 1] 2 ).
The other, if n is uneven, coincides with [V.] ^ [IV.] ; and with II. if n is
unevenly even. If n is the quadruple of an uneven number, its left-hand
member may be written in the form
the sum of the first of these series is f X(^n) = f X (n) ; the second series,
coinciding with f [I.] f [V.], has for its sum $X (n) ; the two series together
are therefore equal to 2X(n). Lastly, the formula, if true for any even number,
is also true for its quadruple ; for if n = 0, mod 8,
and every term of the second series is zero, because the numbers n I 2 , n 3 2 , ...
are all = 7, mod 8.
Of the preceding formulae those which contain the functions ^ and &' are
to be regarded as of a more abstruse character than those which only contain
X, <3>, and $'. The latter, in fact, are deducible from known theorems of
arithmetic. Thus, if we multiply the formula X. by 12, the right-hand member
becomes 8 [2 + ( - 1) M ] X (n), or the number of representations of n as a sum of
four squares (see Art. 127). Consequently 12 E (n) is the number of representa-
tions of n as a sum of three squares ; for, assuming that this is so for 1 , 2,
3, ..., n 1, we may infer from the formula X. that it is so for n. Thus the
celebrated theorem of Gauss*, which connects the number of representations
of a number n as a sum of three squares, with the number of classes of quadratic
forms of det. n, is contained in the formula X. ; and conversely, that formula
is itself deducible from the theorem of Gauss combined with the other and more
* Disq. Arith. Art. 291. Legendre had discovered particular cases of the theorem by induction.
Hist, de 1'Ac. de Paris, 1785, p. 530 sqq. Theorie des Nombres, ed. 3, vol. i. troisieme partie.
Art. 131.]
REPORT ON THE THEORY OP NUMBERS.
325
elementary theorem, which connects the number of representations of n as a sum
of four squares with the sum of the uneven divisors of n.
131. Demonstration of the Formulae of M. Kronecker. We shall first
demonstrate the formula V. For this purpose, we consider the equation
f g (x, lx) = 0, obtained by writing x for .*-, and 1 x for X 2 in the modular
equation f s ( 2 , X 2 ) = of an uneven order n (Art. 125). We shall determine
the order of this equation by two different methods ; first, by ascertaining the
dimensions of f s (x, 1 x), when x is increased without limit ; secondly, by
assigning its roots, and the multiplicity of each of them ; a comparison of these
two determinations will give the formula V.
(i.) Let x = < 8 (0) ; then
because (Art. 125)
the sign of multiplication II extending to every system of values of 7, 7', and k.
fi
Let 6 = 1 + -, a- representing a real positive quantity ; we obtain
o
(Table A, vi. Art. 125). Again, if S is the greatest common divisor of
and 7', and if a, b, S* are determined by the equations
a=
rhile c and d are two numbers, of which x is uneven, satisfying the equation
bc = 1, we find
c + d
_ Sa-i + dy ^
a + b-
t Since $ 8 (<o + 2) = < 8 (<o) (equation i. Table A, Art. 125), the systems of values represented by
is/y
and q> (
are identical, k denoting any term of a complete system of residues, mod y .
326 REPORT ON THE THEORY OF NUMBERS. [Art. 131
whence (Table A, iii.)
if 2l + l = dy, mod <T. If we give to y, y', and k in succession all the <(/i)
/ r (i+i)+2*\
systems of values of which they are susceptible, < 8 1 - - - 1 will acquire
in succession <J> (n) values, which (except for particular values of o-) are all
different; $~ B (- ^ -) will therefore also acquire the same number of
different values ; i.e. S will represent in succession every divisor of n, and 2Z+ 1
every residue of its conjugate divisor <T. We thus obtain the equation
the sign of multiplication extending to every combination of the values of S, *,
to/ *V
and 21 + 1. Now let <r increase without limit, so that x= TJ-> v increases
without limit, and is of the same dimensions as e"' (Art. 125). Observing that
the factor
I" .
has a finite ratio to e?*, if 1 ^ v , and to e 8 ' , if 1 ^ ^ , we see that the product
o-i + 2Z+l\-|
? )J
has a finite ratio to
6 , viictu IS T<0
Hence f s (x, 1 x) + x* ^
is finite, when x increases without limit, or^ (x, 1 x) is of the order
<fr (n) + ^r (n).
(ii.) Neither nor 1 is a root of the equation f & (x, 1 x) = 0; for f % (0, 1) = 1,
j^ (1, 0) = 1, therefore (Art. 125) any one of its roots can be represented by <$>* (<o),
u> denoting an imaginary quantity, in which the coefficient of { is different from
zero and positive. But if x = (p B (w),
f (x 1 x) = II
Art. 131.]
REPORT ON THE THEORY OF NUMBERS.
327
hence the supposition that <p 9 (&>) is a root of the equation f g (x, 1 x) =
implies that
=
for one system (at least) of values of 7, 7', and &; i.e. (Art. 125) that there
exists a unit matrix
and the congruence
a, b
c, d
satisfying the equation
yw + 2k
a, b
c, d
0,1
1,0
,mod2 (A')
Thus, if (f) s (to) is a root of f s (x, 1 x) = 0, w is the root of a quadratic equation
whose extreme coefficients are both uneven, and whose determinant, if
is <r 2 n, a number necessarily negative, because <a is imaginary. Conversely,
if w is the root of a quadratic equation, of which the extreme coefficients are
both uneven, and of which the determinant is negative and included in the
formula er 2 n, < 8 () is a root of f a (x, 1 x) = 0. Or, more precisely, if is the
root of a properly primitive quadratic equation, of which the determinant A is
negative and the extreme coefficients are both uneven, and if n can be repre-
sented by the form (1, 0, A) with a positive and uneven value of the second
indeterminate, < 8 () will be a root of f s (x, 1 x) = 0, and the multiplicity of
this root will be equal to the number of such representations of n *. To establish
this, we shall show (a) that w annuls as many of the factors
as there are representations of n ; (/3) that f s (x, 1 x) is divisible by x <f> 8 (u>)
as often as there are factors annulled by w. (a) Let A + 2J3oo+ Ca> 2 = be the
equation satisfied by w, and let n = <r 2 + AT 2 ; A, C, and T being positive and
uneven ; the four equations
2ak cy'=rA,
ly'=rC,
' The method by which the multiplicity of the roots of the equation f a (x, lx) = is here
determined is chiefly taken from M. Joubert's Memoir, ' Sur la Theorie des Fonctions Elliptiques &c.',
pp. 22-24.
REPORT ON THE THEORY OF NUMBERS.
[Art. 131.
will supply one, and only one, system of values for 7, 7', and k, and one, and only
one, unit-matrix
a, b
c, d
satisfying the equation (A) and congruence (A'). For the
equations ay = rB + <r, by = -rC, show that 7 is the greatest common divisor of
rB + <r and rC; this common divisor is a divisor of n, because
n = <T 2 + Ar 2 = (er - Br) (a- + BT) + A CT* ;
L
thus a, 6, 7, and y = - are determined. Again, the congruences
2ak = rA
2bk = rB-<
determine the value of k, because 2a and 26 have no common divisor with the
modulus, while the determinant
2 ( - raB + rbA + era) = - [<r 2 - rlB 2 + r z AC] = 2 - = 2y
is divisible by it ; when k is determined, the equations
C7'=2a^-T^, dy=2bk-(rB-<r),
will supply integral values of c and d ; the matrix
unit matrix, because
1
it also satisfies the congruence
a, b
c, d
thus obtained is a
-
it/
a, 6
c, d
0,1
1,0
, mod 2, because, from the equa-
tions (B), taken as congruences for the modulus 2, we find 6 = c = l, mod 2,
a = d, mod 2 ; but also ad = 0, mod 2, so that a=d = 0, mod 2 ; lastly, the
equation -, = = is satisfied by virtue of the first three of the equa-
7 a + bta
tions (B). Thus to each representation of n there corresponds one, and only
one, evanescent factor ; conversely to each evanescent factor there corresponds
one, and only one, representation of n. For, if <a annuls the factor
a, b
c, d
the equation (A) and congruence (A') are satisfied by a urut matrix
in which b > 0, but, even if A = - 1, by only one such matrix : so that the
equations (B) determine the values of a- and T without ambiguity. The number
Art. 131.]
KEPOBT ON THE THEORY OF NUMBERS.
329
of factors annulled by w is therefore equal to the number of representations of n.
(/3) Writing < 8 (0) for x, we have
where A is the coefficient of the highest power of x in f s (x, 1 x), and Uj
extends to every root </> 8 (&>) of f a (x, 1 - x) = 0, each root having its proper
multiplicity. M. Joubert has proved that, if \p (0) < 8 (- -, ) vanish when
I Q f/\\ _i a / f yv-T-K\
v ( )-< ( -> ;
= ), lim , > _ . > is neither infinite nor zero. For this limit is,
by the usual rule,
" d /yO + 2k\
de"P (~y)
-1-
j y = !
where M is the multiplier appertaining to the transformation <a =
since (equation 31, Art. 125),
n
n
dd'
d
s\
The determination of M is effected as follows : from the equation
70) + 2k c
or
y'c - 2ka + (yd -2kb) <a
it appears (Art. 126) that the multiplier corresponding to the compounded
transformations
7,
-2k, y
and
a, b
c, d
, applied to , is
- 1 ; while
that corresponding to the second of these transformations is simply ( l)z (i) lj i
(Table A, n.); therefore = -(<r + TVA) 2 , and the limit above written
u u
330 REPORT ON THE THEORY OF NUMBERS. [Art. 131.
becomes 1 + r . , which is certainly neither infinite nor zero. Hence
f & (x, 1 x) is divisible by x </> 8 () precisely as often as there are factors
^"(0) <b* ( - -. ") which vanish when = w; i.e. the multiplicity of the
\ y J
root x = < 8 (<o) in / g (aj, 1 - a) = is precisely equal to the number of solutions
of the equation 7i = o- 2 + AT 2 , T being positive and uneven. It remains to
assign all the values of w, which, annulling one or more of the factors
>P(0) <p* (- ; ), give different values to $ 8 (>). It is evident that values
of w, arising from equations associated to properly primitive forms of different
determinants, or of the same determinant and different classes, give different
values to ( 8 (<o) ; again, of the six subclasses, contained in any one class, the
extreme coefficients are uneven in only two ; so that from each class we obtain
two and only two values of < 8 (to). In the particular case in which the deter-
minant is 1, there are but three subclasses, and but one subclass in which
the extreme coefficients are uneven ; so that to such a class there corresponds
but one value of <p s (w). Denoting then by h (A) the number of properly
primitive classes of determinant A (we count -^ instead of 1 for a class of
determinant 1), and by (n, A) the number of solutions of the equation
n = <r 2 + Ar 2 , in which T is positive and uneven, we have, for the number of
unequal roots of^ (x, 1 x) = 0, the expression
22 A (A),
and for the whole number of its roots, when each root is reckoned with its
proper multiplicity, 2 2 (n, A) h (A),
the summations in each case extending to every value of A, for which the
equation n = <r* + &T 2 is resoluble with an uneven value of T.
We have now obtained the equation *
2 2 (n, A) h (A) = $ (n) + * (n),
of which the left-hand member may be written in another form. Instead of
counting the roots which appertain to the same value of A, and then summing
with respect to A, we may count the roots which appertain to the same value
of <r, and then sum with respect to <r. If F(N) is the number of uneven
* M. Kronecker (Crelle, vol. Ivii. p. 250) has exhibited each of the equations I. VIII. in
a similar form.
Ait. 132.]
REPORT ON THE THEORY OF NUMBERS.
331
classes (primitive or derived) of determinant N (we again count ^ for a
class of det. 1, or for a class derived from such a class), 2F(n a- 2 ) will be
the number of roots appertaining to either of the values + cr or <r. We
thus obtain, finally,
2 F(n) + 4 F(n - I 2 ) + 4 F(n - 2 2 ) + 4 F(n - 3 2 ) + . . . = * (%) + * ( n ),
which is the formula V.
132. We shall also demonstrate (but with less detail) the formula VII.
Writing x for u, and - for v, in the function f(n, u, v), where n= I, mod 8,
x
and multiply ing by x*^ n \ we obtain an integral function of order 2$(w), which
we shall designate by f(x). This function is not divisible by x, for /(O) = 1 ;
but we shall now show that it is divisible by
and that the quotient is prime to x 8 1. For this purpose we shall first
r f(%) i
determine the index \, for which lim , v ' . , x = 1 is finite and different
L(x 1)* J
from zero. Let x = $ () , so that
and let the positive quantity <r increase without limit ; then, ultimately,
1 fr\ Q VI H 1 ^ O* -
,VUly Ctlld X tA/
Also, if $ is the greatest common divisor of 16 k and 7', and if a, b, ' are
determined by the equations
IGk 7' n
a= -
while c and c? are two numbers (of which d is divisible by 2) satisfying the
equation ad be = 1, we find
r _ 4
c + d -
7_
i
a + b 7 ^
7
U U 2
332 REPORT ON THE THEORY OF NUMBERS. [Art. 132.
whence solving for ; , and applying the formula n. of Table A,
H6&\
(2\ i /dy + Si(r\ /2\ . /Si<r + 2l\
c)n- -) = (c)^(- -)'
if 2Z = e < y, mod <T. But ( ) = (j), because bc= 1, mod 8, and
.&>
2
- r ^ - ( ^ - (-^
l) " V<$ 2 6/ " Vy'^/ " V 7< $/'
because 77' = 1, mod 8 ; so that
ultimately; and
,x ,+ x
"~~/ = d) *( y~ ) = (l
,. axl .
hm . v ' = 8 X hm
the sign of multiplication extending to every divisor S of w, and to every term
I of a complete system of residues of its conjugate divisor (T. Observing that
every factor of the numerator, in which f-j\ = 1, is finite, and that every
factor, in which = + 1, is evanescent, and is of the same dimensions as
_
e~" or e *'", according as ^>^, or S<S', we see that, in order to obtain a finite
ft x \
value for lim T"-J?S , we must take for X twice the sum of those divisors of n
(x-1)*'
which satisfy simultaneously the equation (r) = +l> an( ^ ^ ie inequality
$ < ^/n, so that we shall have
X = [* (n) - * (n) + $' (n) - V ()].
Further, if 17 is any eighth root of unity, it will be found that, when
n = - 1, mod 8, / (- , vi) =f(u, v),
M '
whence f () =f( x )> r f( x ) contains only powers of x having exponents
Art. 132.]
REPORT ON THE THEORY OF NUMBERS.
333
divisible by 8. Consequently f(x) is divisible by
( 30 ~ X ) .
and the quotient is prime to X s I.
Representing, as we may now do, any root of the equation . ^ ' = by
(f> (co), we find that must satisfy the equation
for one system at least of values of y, y', and k ; that is (Table A, in. Art. 125),
<o satisfies a quadratic equation of the form
C-\-dw yw-
a + b
a>
or
where
a, b
c, d
1,1
0,1
, mod 2; c = 0, mod 8 ; (-
The determinant of this equation, if cr = ^(ay + dy') 8kb, is <r 2 n. For
brevity, let us suppose that n= 1, mod 16; we shall now prove that in this
case <r = 0, mod 8. Since ad = 1, mod 8, it follows that a = d, mod 8 ; let
a = 8a + fj., d = SS + fj.; substituting these values in the equation ad &c = l,
considered as a congruence for the modulus 16, we infer that
8(a + S) = c + fjt. 2 -l, mod 16.
Again, since yy'= 1, mod 8, let y = 86 + v, y' = 86' v; substituting these
values in the congruence yy' = 1, mod 16, we find
8(0 + 0') = 2 -l, mod 16.
But 2o- = ay + dy'= 8 [a + S + 6 + O'] =c + /J. 2 - 1 + v* - 1 = 0, mod 16,
because (
Therefore a- is divisible by 8, and the quantity o> is the root of an equation
in which A is even, C uneven, and of which the determinant is included
in the series of negative numbers,
-n, -n
An application (which we need not here repeat) of the method already employed
to prove the formula V. will show that every quadratic equation satisfying these
334 REPORT ON THE THEORY OF NUMBERS. [Art. 133.
conditions supplies a value of <p (<o). Sixteen different values of (to) will be
obtained from the quadratic equations associated to the forms of any uneven
class of a determinant included in the above series ; because the conditions with
respect to the extreme coefficients are satisfied in only two of the six subclasses
contained in each class, and because each of these two subclasses supplies
(Art. 126) eight values of < (). Lastly, the multiplicity of the root (f> (<a) of the
ffa)
equation . g u = is ascertained, by an application of the method of M. Joubert
(x 1)
(which also we need not here repeat), to be equal to the number of factors
which are annulled by ; or, which comes to the same thing, to the number of
representations of n by the form a- 2 + A r 2 , the first indeterminate being divisible
by 8, the second being uneven and positive, and A representing the determi-
nant of the primitive equation by which <a is determined. Denoting by (n, A)
the number of such representations, we have for the whole number of roots of
fi x \
the equation , ^ ^ = 0, each root being taken with its proper multiplicity, the
(x L)
expression 162(, A) h (A) ; whence, by a transformation already employed,
16 F(n) + 32 F(n - 8 2 ) + 32 F(n - 16 2 ) + . . .
(A)
= 2 $ (n) - 4 [* (n) - * (n) + $' (n) - *' (n)] . j
Considering, instead of the function /(x), the function x^^ffx, V we
obtain, by reasoning precisely similar, the formula
n-ivn-vn+v n -v n
whence, by subtraction,
in accordance with M. Kronecker's formula VII.
If we had supposed n = 7, mod 16, the left-hand members of the formulae
(A) and (B) would have been interchanged, and the right-hand member of
the formula resulting from them by subtraction would consequently become
133. We shall only indicate the origin of the remaining formulae. Of these,
the formula I. requires the simultaneous consideration of the modular equations
Art. 133.] REPORT ON THE THEORY OF NUMBERS. 335
u\ v 8 ) = 0, and / 8 (n, 1 - u s , v s ) = (Art. 125). Writing x for v 8 , and
eliminating -w 2 dialytically from these two equations, we obtain a resultant
R(x) of the order 2^ + 1 $() in #, as appears from the theory of elimination.
Writing < 8 () for x, and observing that the coefficient of the highest power of u
in f(u 2 , v s ) is v 2 <* + 1 , and in / 8 (1 -tt 8 , v 8 ) is unity, we find
an equation which expresses the resultant in terms of the roots of the two
equations, and in which the sign of multiplication II extends to every combi-
nation of two roots. Since all the roots of f s (1 u s , 1) = are zero, while
none of the roots of f(u 2 , 1) are zero, no root of the equation R (x) = is a
positive unit. But the equations f & (1 tt 8 , 0) = 0, / (u 2 , 0) = have common
roots ; so that x = is a root of R (x) = 0. To determine its multiplicity, write
cri for eo in the expression of R(x), and increase <r without limit. The quantity
o
<p* (- ) which occurs in $ (n) of the factors of R(x) is equal to $~ 8 (2^),
and therefore increases without limit ; but since lim [< 8 (f^)] 2 '' x [<~ 8 (2^ a- i)~\ = 1,
these 3>(?i) factors are cancelled by the initial factor [0 8 (o^')] 2 ''* w . Evaluating
the remaining factors by the method of Art. 131, we find that
lim *
is finite when x diminishes without limit ; so that the order of R (x), after
division by the highest power of x contained in it, is
2*+i $ (n) - 2 * (2>>- 2 n) + 2 ^(^-^n),
or, since (2''- 1 - 1) $() = $ (2' 1 - 2 n),
4 $ (n) + 2 4> (2*- 2 ft) + 2 * (2^- 2 w).
The formulae II. and III. are obtained by successively combining with the
equation f t (n, u*, v*) = (Art. 125), the equations
v*u* + v* + u* 1 = 0, and v* u* v* + u* + 1 = 0,
the first of which is equivalent to the system v l = <p* (>), tt 4 = ^* (^ w) ; the
second to the system v* = 4 (w), t' 4 = ' \f/ 4 (^ w). The resultant of the elimi-
nation of u* is, in each case, an equation of the order 2 <E> (n) in x = v 4 , and is not
divisible by cc, x 1, or cc+ 1.
The following Table indicates the highest powers of x and of the divisors of
a; 8 1 by which the functions specified in it are divisible.
336
REPORT ON THE THEORY OP NUMBERS.
TABLE B.
[Art. 133.
Function.
Order.
Extraneous Factor.
Order after Division.
f ()
*() + *()
(ar 8 )*<)-*<")
3 *()-*(*)
f t (x, l-x)
*() + *()
*() + *()
/.(*,])
2*(w)
(a;_ !)*(")-*()
*() + *()
(1 ;r\*( n ) f fa- - ^
2 * ()
24>()
7 V8^i fZj;
^ 1\fn) f (y. X \
2 * ()
jj,*^)-*^)
4> (n) + * ()
l > SoC*' x -i)
x *Mf ( x a; 1 \
24>()
2 * ()
S\> x )
/.{*,)
2* (a)
(a; 2 !)*()-*()
2*()
x*^fi(x, -i)
2*()
2 4> (n)
n = 3, mod 4
* w /,(, l ~)
2*(rc)
a,* !)*()-*()
4*()-2*()
= 3, mod 4
/,(% -s)
2 * (M)
2*()
n=l, mod 4
^w /f (^ 1)
2*()
(* 2 i)w-*()
2*()
= 1, mod 4
*/.(. -i)
2*()
(^4- !)*()-*()
8*()
w== 1, mod 8
()/ (a,, -\
OS
2*()
( 8 1)* [()-*()+'()-*' ()]
n = 1, mod 8
** w /(*, -5)
2*()
(a; 8 1)* C*()-*()-*'()+*'(H
= 1, mod 8
/(*,")
*()+*()
.,.(n)-*(n) /,j,8 _ j\4 [(R)-*(II)+*'()-*'()]
= 1, mod 8
/(*, -*)
*() + *()
.j*^)-*^) ^8 _ j Y [*()-*()+*'(")-'(
Art. 133.] REPORT ON THE THEORY OF NUMBERS. 337
The formulae in this Table relating to f s (x, x), f(x, x), and f(x, x)
require a certain modification, when n is a perfect square *. If, on this hypo-
thesis, we represent by f 8 (x, x) the function obtained by writing x = v 8 = u* in
f (U S 17 8 )
5-j-, f 8 (x, x) is of the order < (n) + ^ (n) 1, and is divisible by
(^-z)*-*- 1 .
Again, if n = v* and (-) = !, we represent
,. f(u, V) r IIP/ \
lim J ^ ' \u = v = x 1 by f (x, x) ;
VU L J ^ '
this function is of the order $(?i) + ^ (n) 1, and is divisible by
if f ) = 1, we represent
= -?; = a;1 by f(x, -x),
v u
and this function is of the order $ (n) ^ (n) 1, and is divisible by
x *(n)+V()-l x ^8_ ^[4-W-^Wt^W-*'^)]-^
The formula IV. may be deduced from the equation
of which the order (after division by powers of x and x 1) is shown by the
Table to be 2 $ (n) + 6 * (n).
In proving the formula V., we might have employed the equation
x* W /8 (x, -) = instead of / 8 (x, 1 - x} = 0.
\ iC^
If, instead of the former equation, we employ the two
we obtain the formulae V. and VI. simultaneously,
Lastly, the formulae VIII. depends on the equations
f(x, x) = 0, and f(x, x) = 0.
* The necessity for a corresponding modification of the formulae IV. and VII. is obviated by
the assumption G (0) = ?.
X X
338 REPORT ON THE THEORY OF NUMBERS. [Art 134.
134. Connexion of the formulae of M. Kronecker with Elliptic series.
Researches of M. Hermite. M. Kronecker has given a remarkable analytical
expression of the formulae IX. and X. (Art. 130). He employs the identical
equations
x 2. E(n) q n
o
of which the first two are immediately verified by expanding their right-hand
members, the last two by multiplying together the series in their left-hand
members. Combining these identities with the formulae IX. and X., and
attending to the equation E(0~) = ^, we obtain
of which the second (Art. 127, equation 3) may be written in the form
...y, ...... (BO
in which it expresses the arithmetical theorem of Gauss, to which we have
already referred (Art. 130).
It appears from the equations (A) and (B) that the generating functions of
F(n) and E(n) are elliptic series; and M. Hermite, in two important memoirs
(Comptes Rendus, Aug. 5, 1861, or Liouville, New Series, vol. vii. p. 25, and
Comptes Rendus, July 7, 1862) suggested, as it would seem, by these equations,
has succeeded in deducing the second of them, and others of the same character,
from the general expansions of elliptic functions, without having occasion to
consider the special modules which admit of complex multiplication. He has
Art. 134.]
REPORT ON THE THEORY OF NUMBERS.
339
thus discovered a new and comparatively elementary method of arriving at the
formulae of M. Kronecker ; to whom indeed this method was already known, as
his expressions of the generating functions of E (n) and F (n) indicate, and as
he has himself expressly stated in a note published after the appearance of
M. Hermite's first memoir (Monatsberichte, May 26, 1862, pp. 307, 308). M.
Hermite's method is an extension of that employed by Jacobi (see Art. 127),
and depends on the developments of doubly periodic functions in series pro-
ceeding by sines or cosines of the multiples of the argument. To this set of
developments, however, M. Hermite adds a second obtained by dividing the
product of two Theta functions by a third. A series of the first set, and one
of the second (both alike containing only sines, or only cosines, and only even,
or only uneven multiples of the argument), are then multiplied together. The
non-periodic part of the product (or its integral, taken from the limit to TT)
is a function of q only, and if the product can be formed in more than one way,
we obtain different expressions of this function, a comparison of which supplies
hi each case an arithmetical formula. We take the following example from
M. Hermite's first memoir ; and, with him, we write for brevity, 0, 1( H, Hj for
2Kx\ . /2Kx\
9 (
V TT
Multiplying together the three pairs of series
" nq n cos 2 nx
8 2j < 57;
(i)
1= 2 q n "cos2nx,
*00f
A J. ^-J-t
A V V "
i ^ -tin fw L 1 W "\
e 2
o 1 +
f*& It T 1 * / '
2
oo
^ U
H =
2 2 ( - 1)
2i (2n - | - 1)2 sin(2?i + l)a;, J
*
H
4^9'
i)
sin (2 n + 1) x,
yi e
4Z/ 1 _n
9 ' 9
e t
ue,
CO
2 2 o (1%^*
1 + 1)2 sin (2 % + l)o;,
(iii
\
c\
where
Q n =
1+2^-1 +
2g-*+... + 2g- 2 ;
" The developments of elliptic functions, in series proceeding by sines or cosines of the multiples
of the argument, which are employed in this article, will be found in the Fundamenta Nova (sections
40-42), or in IT. Hermite's second memoir (Comptes Rendus, July 7, 1862).
X X 2
340
REPORT ON THE THEORY OF NUMBERS.
[Art. 134.
/It TT2
f-1 dx by -a-J, we obtain
i,J=
l- 2n
00
_ /v2n + l
Let A () represent the sum of those divisors of n whose conjugates are
uneven, and ^(n) the sum of those divisors of n which do not surpass ^/w,
and which are even or uneven, according as their conjugate divisors are uneven
or even ; we find immediately
also
+n oo
-
if F r (4 7i + 3) represent the number of solutions of the equation 4n + 3 = ac b 2 ,
in which a and c are positive and uneven, a is less than c, b is even, and less
in absolute magnitude than a. But M. Hermite has shown that
For .F'(4 + 3) is evidently the number of quadratic forms (a, b, c) of determi-
nant (4n + 3), in which the second coefficient is even, and less than either
extreme coefficient, and in which also the first coefficient is less than the third.
But each uneven reduced form is equivalent to one, and only to one, of the
forms (a, b, c). For the reducing transformation of a form (a, b, c) is neces-
sarily one of the five following :
1,0
0,1
1. 1
0, 1
1,1
-1,0
therefore, conversely, a reduced form can be transformed into a form (a, b, c)
only by one or more of the transformations,
1,0
0,1
1, +1
0, 1
o, -i
Art. 134.]
REPORT ON THE THEORY OF NUMBERS.
341
and upon trial it will be found that there is always one, and only one, among
them which applied to a reduced form, produces a form (a, b, c). The number
of forms (a, b, c) is therefore equal to the number of reduced forms of deter-
minant -(4 n + 3); i.e. F' (4% + 3) = F(4:n + 3). Eliminating J, we obtain the
first and third formulae of M. Hermite's memoir,
or, equating coefficients,
= A(n)-r i (n),
In his second memoir M. Hermite occupies himself with the demonstration
of the equation (B'). Multiplying together the two series,
HO
1
= tan a + 2 2 ( - 1)-' #_, o J sin 2nx,
and employing the formulae
/sin 2 nx cot xdx = -IT,
.
/sin 2nx tan xdx = ( I)*"- 1 * v,
finds
an expression which, by a detailed discussion, is shown to be equivalent to
342
REPORT ON THE THEORY OF NUMBERS.
[Art. 134.
In the note of May 26, 1862, to which we have already referred, M. Kro-
necker has given other examples of the use of this method. Multiplying
together the three pairs of series,
cos 2nx
" 7
cos
(i)
HHl
H cos x = ( - l) n
*] sin
= 42 ^tf" sin 2 no:,
i
where
/* H 2 H
* cos ccc^a; by TT I, we find
o
2 ~ n 2
Let F (n) represent the sum of those divisors of n which do not surpass ^/n,
and which are even, or uneven, according as their conjugate divisors are even or
uneven ; and let I" (n) represent the sum of the same divisors, each divisor being
taken positively or negatively according as the sum of itself and its conjugate is
unevenly or evenly even ; if n is a perfect square, we are to replace V n by ^ \/ n
in the sums T (n) and I" (n) ; we then obtain the expansions
,
n) 2 =
Art. 135.] REPORT ON THE THEORY OF NUMBERS. 343
because the coefficient of q n in the expansion of ^q~$ I is a sum containing
two units for every solution of the equation 4w = ac b 2 , in which, a, b, c
being positive and uneven, 6 < a < c, and one unit for every such solution in
which either b = a <c, or b <a = c; and because (by reasoning similar to that
already employed in this article) it is ascertained that this sum is equal to
the number of reduced forms of determinant in, i.e. to F(in). Eliminating
I, we obtain, finally,
0! 2 F(n) q n = 2 [A (n) - T (n)] q n , 6 2 F(n) q n = 2 F (n) q,
or XI.
XII.
These equations are equivalent to the formulae I., II., III., V., VI. of M.
Kronecker; these five, therefore (and with them, according to M. Kronecker,
the remaining three, IV., VII., VIII.), are deducible by analytical transfor-
mations from the single equation
0! , /* H 2 H,
5 r / Q 2 cos xdx = i X (n) q n .
& T <f Jo i
135. M. Kronecker asserts that the formulae I. VIII. are independent, i.e.,
that none of them can be deduced from the others by means of the elementary
equations satisfied by the functions F and G ; and that all the similar relations,
which are supplied by the theory of complex multiplication, may be obtained,
with the help of those elementary equations, by combining the eight formulae.
And it is certain that the system of the eight formulae does, in this sense, ex-
plicitly contain all the relations of similar form, which have been subsequently
given by MM. Hermite and Joubert. Thus, many of these relations are par-
ticular cases of the formulae XI. and XII., or of the combinations (XI.) + (XII.)
(in M. Joubert's memoir, the formulae 1, 2, 3, those of page 28, and the first
of page 29 ; also the first two formulae in M. Hermite's memoir (Liouville, New
Series, vol. vii. p. 25) are of this kind) ; others, again, are immediately deducible
from the two formulae
+ . . . = $ (n), n = 3, mod 8,
- 10 2 ) + ...=<& (n), n = 7, mod 8,
combined by addition or subtraction with V., VI., and VII. But each of these
two formulae results from the combination f (V.) + f (VI.) - 2 (IV.), simplified by
344 REPORT ON THE THEORY OF NUMBERS. [Art. 135.
means of the elementary equations satisfied by F and G. In this way the
formulae 4-9 of M. Joubert's memoir, and the third formula in M. Hermite's
memoir (Liouville, ibid. p. 36), may be obtained. Lastly, the equation
, n = l, mod 4,
(M. Joubert, p. 30) arises from the combination (IV.) | (V.).
M. Joubert's formulae, however, as they are given in his memoir, are not
immediately comparable to those of M. Kronecker. He rejects from the modular
equation of the uneven order n, the factors due to the square divisors of n (see
Art. 125 of this report), and, in consequence, those derived classes whose
coefficients have any common divisor with n are excluded from his enumerations.
At the same time, the numerical functions, depending on the divisors of n, which
occur in the right-hand members of his formulae, are rendered somewhat more
complicated than those of M. Kronecker. It is always possible to pass from one
of M. Joubert's formulae to the corresponding formulae of M. Kronecker, by an
elementary process, of which M. Joubert has himself given an example (at p. 25
of his memoir).
One formula, however, has been obtained by M. Hermite from his investi-
gation of the discriminant of the modular equation, which is entirely distinct in
form, and as it would seem in substance, from those of M. Kronecker. Taking a
modular equation of a prime order n, M. Hermite shows that its discriminant is
of the form /2\
where (u) is a reciprocal polynomial, prime to u and to 1 8 , containing no
equal factors, and of order -$(n 2 1) -g + (-) . From the nature of a
discriminant, if w renders two of the quantities ( J (p (- , -- ) equal to one
another, <p (o>) is a root of the equation 6 (li) = 0, and conversely. It is thus pos-
sible, by a method of which we have already given examples, to assign a system
of quadratic equations (or quadratic forms) having integral coefficients, which
shall correspond, one by one, to the roots of the equation 6 (u) = 0. Equating the
number of these quadratic forms to the index of the polynomial (u), M. Hermite
obtains a formula which is essentially limited to the case when n is a prime, and
which, translated into the notation of M. Kronecker, is as follows,
Art. 136.] EEPOET ON THE THEORY OF NUMBERS. 345
the summations 2^, 2 2 , 2 3 extending respectively to all values of S which give
positive values to the numbers
A = (8J-3)(-2J), A = 8J(-8a), A = <$(/<,- 16 <J).
The difference between these series of determinants, and those which occur
in M. Kronecker's formulae, is very remarkable.
136. Arithmetical Demonstrations of the Formulae of M. Kronecker. M.
Kronecker informs us that, when he had connected his formulae, in the manner
already described, with the expansions of certain elliptic functions, he directed
his attention to the process (Art. 127) by which Jacobi transformed the ana-
lytical proof of the ' theorem of four squares ' into an arithmetical one *. Apply-
ing a similar transformation to the analytical proof of the equation of
H 2 H
he succeeded, after many reductions, in obtaining a purely arithmetical proof
of the formulae I. II. and V. which are included in XI. This important investi-
gation has not yet been published : instead, M. Kronecker has given a remark-
able theorem which appears (as he observes) to contain the germ of another, and
very different, arithmetical demonstration of his formulae. He has enunciated
the theorem for prime numbers only, remarking, however, that it admits of
extension to composite numbers also. The result is simplest in the case of a
prime number p of the form 4 m + 3.
' Let (a, b, c) represent in succession every uneven reduced form of the
determinants p, p + 1 2 , p + 2 2 ,...; only, if a = c, let the reduced form
satisfy the special condition (Art. 92) b < 0, instead of b > ; the roots of the
congruences aw 2 + 26eo + c = 0, mod p,
of which roots the number is
are a complete system of residues for the modulus p.'
As it appears from the formula V. that the number of these congruence-
roots is equal to p, it is only necessary to prove that they are all different ; the
demonstration of this very difficult point M. Kronecker has effected by showing
that the contrary supposition is inconsistent with the inequalities satisfied by
the coefficients of the reduced forms. A proof, independent of the formula V.,
* Honatsberichte for 1862, p. 307.
vy
*
346 REPORT ON THE THEORY OF NUMBERS. [Art. 136.
that every residue of p is a root of one of the congruences, would of course
supply a direct arithmetical proof of that formula, for the case in which n is a
prime of the form 4 ra + 3.
Arithmetical demonstrations of the formulae of M. Kronecker have also
been obtained by M. Liouville. These demonstrations depend on the principles
introduced by him into arithmetic in the series of memoirs ' Sur quelques
formules ge'ne'rales qui peuvent etre utiles dans la thdorie des nombres ' *, and
originally suggested (as he himself informs us) by Jacobi's arithmetical proof of
the theorem of four squares. M. Liouville has given, as an example of his
method, a proof of the equation (XL) (XII.), or
*F(n- l*) + 4F(n- 3 2 ) + 4F(n-&) + ... = A(n) - T(n) - T'(n),
for the two cases in which n is unevenly, and evenly, even f. We shall con-
fine our attention to the latter and somewhat simpler case. It requires two
preliminary Lemmas, both included as very particular cases in M. Liouville's
general formulae.
I. Let m represent a given uneven number, a a given positive exponent
other than zero, f(x) an even function, so that f(x) =f( x); we have the
equation 2 [f(d f - d") -f(d' + d")] = 2*-*2d [/(O) -/( 2 <0] >
the summations in the left and right-hand members extending respectively to
all solutions of the equations
the indeterminates d', d", X, S" in the first equation, and d, S in the second,
being positive and uneven, and two solutions of either equation being regarded
as different, unless the indeterminates of the two solutions are the same and in
the same order.
To establish this equation, we consider the system
d' + d" =2/*, ( a )
in which M and v are given positive integers. The solutions of this system are
equal in number to the solutions of the system
(a)
d'-d" = -2n,
<r+r =2^.
* Liouville's Journal, vols. iii.-viii. (New Series). t Liouville, New Series, vol. vii. p. 44.
I
Art. 136.]
REPORT ON THE THEORY OF NUMBERS.
347
For, eliminating 8' and d", we find that (a) has as many solutions as
2 a ~' m = vd' + fj.8" has solutions in which d' <2^; and (a') has as many solu-
tions as the same equation has solutions in which $" < 2 v ; i. e. (a) has as many
solutions as (a'); inasmuch as to every solution of 2 a ~ l m = vd' + n$" in which
d' < 2 n, but 8" >2v, there corresponds a solution, in which d' > 2 p, but 8" <2v,
and vice versd; for example, if cf <2y, but S">2, let 2k v be the multiple
of 2 v next inferior to 8", then
2 a m = v (d' + 2 >) + M (<T - 2 Jb)
is a solution of the equation, in which d' + 2 km > 2 /x, but <$' 2 & i/ < 2 1/.
Similarly it will be seen that the solutions of the systems
d' + d" = 2,,,
J'-J" =-2,,
(b)
<f-(T = 2 M , :
a-+r =2,, )
are equal in number.
Also the number of solutions of either of the systems
d' + d" =2"
<r-cT =s,
( c )
= 2 a d, >
in each of which d, 8 are two given conjugate divisors of m, is 2 a ~ 1 d.
Let us now attribute to n, v, d, 8 in the systems (a), (b), (c), (a), (b 7 ), (c'),
all values, in succession, for which those systems are resoluble. We shall
evidently obtain the sum 2/(cT + d"}, which occurs in the equation to be proved,
by extending the summation, first, to all solutions of the various systems
(a), secondly, to all the solutions of the various systems (b), and lastly, to all
solutions of the various systems (c). Similarly, we shall obtain the sum
2/(r/-cT) by extending the summation to all solutions of the systems (a'),
(b 7 ), (e). But the terms f(d' + d") arising from any one of the systems (a)
or (b), are cancelled in the difference
Y y 2
348 REPORT ON THE THEORY OF NUMBERS. [Art. 136.
by the terms f(d' d") arising from the corresponding system (&) or (b').
That difference is, therefore, equal to the excess of 2/(cf - d"), extended to all
solutions of the systems (c'), above ~2f(d' + d") extended to all solutions of the
systems (c) ; so that, finally,
II. Let m be an uneven number, and f(x) an uneven function ; we have the
equation zf(tf + 2m') = Zf{i(d 1 + S 1 )},
the summations in the left- and right-hand members extending respectively to
all solutions of the equations m = 2 m' 2 + d' <T
2m = m\ + di S 1}
the indeterminates d', cT, d 1 , S lt m^ being positive and uneven, but m being even
or uneven, positive, negative, or zero.
If we write 2m' + d' = x, ' 2m' = y, 2m+d' S' = z, so that conversely
2m' =x y z, d' = y + z, X = x z, the equation m = 2m' 2 + d'S' becomes
2 m = x 2 + y z z 2 , the indeterminates being subject to the conditions
y + z > 0, z < x.
If in addition z + x<0, the conditions are satisfied by the two solutions
[x, y, z], [ x, y, z] ; which give rise, in the sum ^f(d" + 2 m), to two terms
which cancel one another. We need only therefore consider those solutions.
which satisfy the inequalities, y + z > 0, x + z > 0, z < x, or, which is the same
thing, if [z] represent the absolute value of z,
y + z>0, x>0, [z]<x ......... (d)
Again, if we write \ (d 1 + S 1 ) = x, m 1 = y, \ (d^ S^) z, the equation
2m = m\ + d l 8 l becomes 2 m, = x* + y 2 z 2 , the indeterminates being subject to
the conditions
y>0>
To establish the proposed equation it is now only necessary to show that
the equation 2m = x 2 + y 2 z 2 admits of equal numbers of solutions satisfying
the inequalities (d) and (d'}. But this is evident ; for if [a;, y, z] satisfy one
of the two sets of inequalities, but not both, [x, y, z] satisfies the other,
but not both.
By combining these two lemmas it may be proved that four times the
number of solutions of the equation
....... (A)
Art. 136.] REPORT ON THE THEORY OF NUMBERS. 349
(in which m is a given uneven number, and a a given exponent > 0) is
A (2" + 1 m) - T (2 0+1 m) - F (2 +1 m),
the indeterminates m^, d 2 , S 2> S 3 being all positive and uneven and d 2 S 2 being
evenly even. Eepresenting by m any number whatever, and by d', X, 3 positive
uneven numbers, let us consider the two equations
2"+*m-2d 3 S 3 = ml+d 2 S 2 , ........ (e)
2" m- d 3 S 3 = 2m' 2 + d'S', ........ (e')
and let f(x) be an even function, so that f(x d 3 )f(x + d.^) is an uneven
one, and may be used instead of f(x) in the second lemma. Applying that
lemma to the two equations (e) and (e 7 ), and afterwards summing for every
value of d 3 , we find
the summations extending to all solutions of (e) and (e 7 ) respectively. Observing
that if m' = 0, /(2TO' + a;) is an even function of x, and that if m is not =0,
f( 2m' + x)+f(2m' + x) is an even function of x, we transform the second
member by the first lemma, and we obtain
= 2T- 1 *Zd [/(2m') -f(2i d + 2 m 7 )],
the second summation extending to every solution of the equation
2 m- 2m' 2 = 2^dS,
d and S being positive and uneven, and 2? representing the highest power of
2 contained in 2 a m 2m' 2 . Let f(x) = 1, if x = 0, but let/(o;) = for every
other value of x ; the sum
will then represent the number of solutions of the equation (A) ; the sum
27- 1 'Zdf(2m f ) will become 2 a ~ l "Zd, the summation extending to all solutions
of the equation m = d 8; and the sum 2?- 1 '2df(2m+2Td) will become Z2T~ 1 d,
the summation extending to all solutions of the equation
2"- 1 TO = 2T- 1 d (2ir-i ^ + ^_
Of these sums 2 a ~ l '2d is evidently A (2"- 1 m) = A(2 a+1 m) ; and 22?-^ is the
sum of those divisors of 2"- 1 m, which are less than x /2 a - 1 m, and which are
not of the same parity as their conjugates, a sum which is identical with
350 REPORT ON THE THEORY OF NUMBERS. [Art. 137.
F (2 a + 1 m) + 1" (2 + 1 m) ; as may be seen by considering separately the cases
in which a 1, and a > 1.
A second determination of the number of solutions of the equation (A) is
obtained as follows. Write 20 + 1 for \ (d 2 + S 2 ) and 4 a for d 2 <J 2 ; it becomes
which is of the same form as that considered by M. Hermite (see Art. 134).
If then we attribute to w t any particular value, the number of solutions of the
equation (A) is F(2 a + 1 m m?) ; its whole number of solutions is therefore
equating this expression to that which we have already found, we obtain the
formula (XI.) (XII.).
M. Liouville tells us that, until M. Hermite's discussion of the equation
he had not observed that the number of solutions of the equation
2 a + ^m-ml = d 2 S 2 +(d 2 + S 2 )S 3 , d 2 = S 2 , mod 4,
is equal to the number of classes of quadratic forms of det .mf 2 a+1 m; but
that with this exception all the principles of the preceding demonstration were
in his possession ; so that he had already arrived at formulae identical with
those of M. Kronecker, but referring to the numbers of solutions of certain
indeterminate equations instead of to the numbers of quadratic forms of certain
determinants. We also learn from him that formulae exist, analogous to
those of M. Kronecker, in which the series of determinants are of the type
2 s- n, 3s 2 n, ... instead of s 2 n.
137. Equations satisfied by the Modules which admit of Complex Multipli-
cation. We have already observed (Art. 126) that the 6 G (A) values of < 8 (eo)
corresponding to the quadratic forms of det. A, are the roots of an equation
of that order, having rational coefficients. Several important properties of this
equation have been indicated by M. Kronecker ; but, notwithstanding their
intimate connexion with the theory of quadratic forms, we can only offer an
imperfect account of them.
We resume the notation of Art. 131 ; and we shall begin by showing that
if n is an uneven number, greater than 3, the values of $ 8 (o>), corresponding
to the properly primitive classes of det. n, satisfy one or other of three
equations, each of the order 2 h (n), and each having rational coefficients. We
have already seen in Art. 131, that every value of < 8 (w), corresponding to a
Art. 137.] REPORT ON THE THEORY OF NUMBERS. 351
form of which the extreme coefficients are uneven, and of which the determi-
nant is a- 2 n, is a root of the equation f s (x, 1 x) = 0, and that this equation
has no other roots. Again, if y^ \-^n , * 2 ) = is the equation satisfied by the
squares of the multipliers appertaining to the 3> (n) transformations of order n,
the equation X2 ( n , x) = will be satisfied by those roots of the equation
f s (x, 1x) = which correspond to quadratic forms of det. n, but not by
the other roots of that equation. For, if < 8 (to) is a root of f a (x, 1 x) = 0,
(j) s (w) is transformed into >p (to) by one of the $ (n) transformations of order n ;
and if < 8 (<o) corresponds to a quadratic form of det. n, the multiplier apper-
taining to this transformation is + -j= ; whereas if < 8 M corresponds to a
Jn
g.2 _ ft
quadratic form of det. A = , the multiplier is [T\/ A + io-]" 1 (see Art.
131). Forming then the greatest common divisor of the two functions
f a (x, 1-x) and X2 ( n , x ),
we obtain an equation of which the roots are, exclusively, those values of <f> B (w)
which correspond to quadratic forms of det. n*. Let ^ (n, x) represent this
greatest common divisor, and denoting by p lt p 2 , ... the different primes, of
which the squares are divisors of n, let us form the expression
If (A, B, (J) symbolize a system of quadratic forms, having their extreme
coefficients uneven, and representing the properly primitive classes of det. n,
the roots of the equation ^ (n, x) = are those values of < 8 (w) which corre-
spond to the systems of quadratic equations
Thus the order of the equation is 2 h (n) : if x = $* (<o) is a root, 1 x = (}>*( --
is also a root : the first coefficient is a power of 2, and the last coefficient is
* It is here assumed that if 7=. is not the multiplier appertaining to any of the trans-
v n
formations of order n by which x is changed into 1x, it is also not the multiplier appertaining
to any of the <b (n) transformations of the order n.
352 REPORT ON THE THEORY OF NUMBERS. [Art. 137.
unity, because ^ l (n, x) divides f^ (x, 1 x), of which the first and last coeffi-
cients are respectively a power of 2 and unity*. From the equation ^(n, cc 2 ) = 0,
we may deduce two others, Sk 2 (n, x 2 ) = 0, ^ 3 (n, x 3 ) = 0, by the substitutions
1 /v. ' . 1
l/ 2 0-3 i
these equations will have for their roots the 4 h (n) values of < 8 (o>) corresponding
to properly primitive forms of det. n, not included in the subclasses (A, B, C),
(C, B, A). The roots of the equation ^ 2 (n, x) = are the reciprocals of the
roots of ^ (n, x) = : its first coefficient is therefore unity and its last a power
of 2 ; the equation Sk 3 (n, x) = is a reciprocal equation, and its first and last
coefficients are units.
Each of the three functions i^ (n, x), ^ 2 (n, x), ^ 3 (n, x), can be decomposed
into two factors, of the order h(n), and containing no irrationality but >/ n.
_
If n = 3, mod 4, the value of - ~ - (Art. 125) corresponding to one of the
two forms (A, B, C), (C, B, A) is +Vn, and that corresponding to the
(_ 1)3(7-1)
other -/n; if n = l, mod 4, the values of /. corresponding to those
two forms are both \/ n or both -Jn, according as the generic character of
the two forms is ( \)W-^= +1, or ( 1)4^-!)= 1 ; in either case, therefore,
the decomposition of ^(n, x) into two factors, can be effected by comparing
it with the equations
a; = 0, ->* = 0, if
is the equation satisfied by the multipliers appertaining to the transformations
of order n.
But M. Kronecker has shown that ^ (n, x) admits of a more profound de-
composition, when n is a composite number. In fact, if v is the number of the
primes p lt p 2 , p a , ... dividing n, S^ (n, x) can be resolved into 2" factors, each of
the order 7^^- h (n), and containing no irrationalities but V pi, */p 2 , -Jpa,
If n = 3, mod 4, the order of each factor is precisely equal to the number of
classes in a properly primitive genus of 
