*Applications oflhc Theory of Elliptic Functions to the Theory of Numbers
by
P. S. Nazimoff
Translated From Russian
Arnold E. Ross
[Formerly Arnold Chaimovilch]
Cfhe original was published in Moscow in 1884
UNIVERSITY OF CHICAGO BOOKSTORE	^ »
5302 aus AVENUE	*
CHICAGO -	- ILLINOIS.APPLICATIONS OP THE THEORY OP ELLIPTIC FUNCTIONS TO THE THEORY OP NUMBERS
N.AZIMOFF
CHAIMOVITCH
Prepared by 0* Si, Brown
Northwestern University 3v sn3ton, IllinoisPREFACE TO THE AM ER I CAN ' ED I TI ON
H15 a
In presenting this translation of the MORE ORIGINAL Pi of Nazimoff* s remarkable book to those who do not read Rossi IT is the hope of the editors that not only will the individu
RESULTS OF THE WORK BECOME MORE WIDELY KNOWN, AND THEREFORE PI VENT USELESS DUPLICATION, BUT ALSO THAT THE INGENIOUS AND POW.E FUL METHODS .1 NTRODUCED BY NAZIMOFF WILL PASS INTO THE COMMON Ai SENAL OF WORKING ARITHMETICIANS. OUTSIDE OF RUSSIA, NaZIMOFF’*S WORK HAS HAD PRACTICALLY NO INFLUENCE ON THE DEVELOPMENT OF THO PARTS OF ARITHMETIC TO WHICH IT IS DEVOTED. FOR THIS UNDESERVED NEGLECT THE AUTHOR HIMSELF IS LARGELY RESPONSIBLE, AS HIS OWN
French abstract in the Journal de l'Ecole Polytechnique does Bp
SCANT JUSTICE TO THE BOOK. THE MEAGRE ABSTRACT IN THE FORTSCH.ftj ' IS BASED ON THE FRENCH ACCOUNT, NOT ON THE RUSSIAN.
Certain of the topics treated here have inherited an augmented INTEREST SINCE NAZIMOFF W<FOT£. HIS NUMEROUS THEOREMS ON NUMBERS OF REPRESENTATIONS IN C Si IT AI N QUADRATIC FORMS IN THE EAR I ER CHAPTERS, FOR EXAMPLE, WILL BE oF GREAT VALUE TO WORKERS ON WHAT DICKSON HAS CALLED UNIVERSAL FORMS (THOSE THAT REPRESENT AL INTEGERS, OR ALL POSITIVE INTEGERS). AGAIN, THE ARITHMETICAL PROOF: FOR CERTAIN OF LlOUVILLE’S GENERAL FORMULAS IN Th£ THEORY OF NUMBERS WILL DOUBTLESS STIMULATE RESEARCH IN THESE POWERFUL METHODS.
Although many of Nazimoff's isolated results have been obtained
OTHERWISE BY LATER WRITERS, THERE iS A SUFFICIENT RESIDUE OF GENERALITY AND ORIGINALITY IN EaCH OF THE CHAPTERS TO MAKE ALL WORTH OF CLOSE ATTENTION.
The THEME OF THE BOOK IS ARITHMETIC, PARTICULARLY THE BROAD DISCUSSION OF WHAT IS ATTAINABLE IN THE THEORY OF QUADRATIC FORMS BY USE OF ELLIPTIC FUNCTIONS. ITS SPIRIT IS ALGEBRAIC-. SOME WILL PREFER TO USE STRICTLY ELEMENTARY METHODS To REACH THESE OR SIMILAR RESULTS, BUT FEW WILL DENY, FROM THE EVIDENCE PRESENTED HERE
that Nazimoff's algebraic attack is one of great power in discovery, WHATEVER MEANS MAY BE SELECTED FOR THE FINAL PRESENTATION OF THE PROOFS. TO ALL WHO PREFER ALGEBRA AND ANALYSIS TO PURE ARITHMETIC AS TOOLS FOR EXPLORATION, THIS BOOK WILL PROVE INVALUABLE.
* -	*	*	.*	;	*	; *	.	*	X
The copy of Nazimoff* s book from which this translation was
made IS PROBABLY THE ONLY ONE IN THE’UNITED STATES; IT WAS GIVEN TO ME BY P RO F E S SO R t) AM E S V. OUSPENSKY AT THE TORONTO CONGRESS IN 1924. To professor L. E. Dickson, at whose instigation the translation W AS UNDER TAKEN, AND TO HI S PUPILS WHO HAVE MADE' THE TRANSLATION AND PREPARED THE STENCILS, ALL FRIENDS OF ARITHMETIC OWE THEIR THANKS FOR RENDERiING NaZIMCFF'S WORK aT LAST ACCESSIBLE TO American and English readers.
B. T. Bell..
California -Institute of Teohnoloiy, Pasadena., 710850TRANSLATOR’S PREFACE
This translation was undertaken by me at the suggestion of Prof. l_.E.Dick son and Pro f. E. T. Bell and was originally intended for the private use of a few of their students. The work was done at
THE CLOSE OF MY SENIOR YEAR AT THE UNIVERSITY IN WHAT SPARE TIME I HAD. iT WAS LATER DECIDED TO MIMIEOGRAPH THE TRANSLATION FOR USE
in Prof. Bell’s class during the coming summer and offer it to the
INTERESTED ONES OUTSIDE OF THE UNIVERISTY. IT IS VERY UNFORTUNATE
that neither Prof. Dickson nor Prof. Bell had the opportunity to read the manuscript before stencils were cut.
Mr. 0. E. Brown, who cut the stencils, has read each page
ONCE BUT MORE THOROUGH PROOF READING COULD NOT BE ARRANGED. tTIS KIS PLAN TO DISTRIBUTE A LIST OF CORRECTIONS LATER AND HE HOPES THAT THE READER WILL NOTE AND REPORT ALL MISTAKES FOUND.
IN HIS INTRODUCTION (WHICH WAS NOT TRANSLATED) NAZIM OFF INTRODUCES THE FOLLOWING NOTATIONS WHICH HE EMPLOYS THROUGHOUT THE
book:	x
i v'U -x*)(l -k*x*)'	1 l
‘ ■ /;
dx
J /( 1 - x B) [ 1 - ( 1-k"*) Xs]
ng-’
q = e K , k 1 = /1 - "i<8
M x, q) = S(x) = 3 - 2qcos2'x + 2q4eos4x - 2q9cosSx ♦ . . .
^3(x>q) = S3(x) = 1 + 2qcos2x + 2q4cos4x + 2q9 ec s3x + . . .
1	2	£&.	40
q) - Sj( x) = 2q4sin x- 2q4sin3x + 2q 4 sin5x - 2q ^sln7x ♦
IS	£2	42
$2( x> q) = V x) =2q4ecs x ♦ 2q4cos3x ♦ 2q 4 co s5x + 2q 4 cos7x +
/2Kx \
sin am r-~^-,qI = Si n aic
{-f’j = si
sin am
2rCx _ J_ ^(x) n Vlt 9( x) '
I 3K x \	/ 2Zx t \	2Z-x
cosami—-~,qj = co s am i —k I = cos am----

’ o rT v X
,	. — -	.	hsx \	2K x
à am	qj = à am ~,kl = A am—~ =
?(*) *
University of Chicago, May 1928.
Arnold C’laimouitdiCHAPTER I
fyg FIRS? METHOD OF JACOBI. APPLICATION OF THIS METHOD TO THE
STUDY OF OU ADR ATI 0 FO R:- 3.
3	1. WE HAVE ALREADY EXPLAINED IN THE INTRODUCTION THAT THE
method of Jacobi employs trigonometric series representing elliptic or Jacobi functions. Take an elliptic function and expan
IT INTO A TRIGONOMETRIC SERIES; LET US GIVE TO THE ARGUMENT OF THE ELLIPTIC FUNCTION A PARTICULAR VALUE. I N PLACE OF THE ELLIPTIC FUNCTION WE SHALL OBTAIN A FUNCTION OF If AND kj IN PLACE OF THE TR T GO NOME TRIO n ERIE S '1E SHALL OBTAIN A SERIES IN WHICH THERE IS ONE INDETERMINANT . iUT THE FUNCTION OF X AND k THAT
WE OBTAIN MAY GIVE A PARTICULAR VALUE OF A FUNCTION OF 1,
Sg, AND S3; IN THAT CASE THIS FUNCTION OF X AND k MAY BE REPRESENTED BY SOME OTHER SERIES ¡IN THE INDETERMINANT C). COMPARISON OF THE COEFFICIENTS OF EQUAL POWERS OF Q WILL GIVE AN ARITHMETICAL THEOREM.
LET US TAKE, FOR EXAMPLE, THE FOLLOWING SERIES WELL KNOWN IN THE THEORY OF ELLIPTIC FUNCTIONS:
Or,
r/	O r/ rr
JA.	J	-,
— Aa-a—— =* 1 + n n
4 o
A ^
4o
4q
-—=-kcos2x + -z—^rcoslx + ,	»
x + q2	1 + q 4 •	1+Q6
co sSx + ...
2Xk
----S111 •;ii---
n	n
'■X	2;Cx
— ; sin am------
n	n
3Kx ii/o .	4i/q*^ .	0	. 4t/q"®
- ;sinx + -—^sin3x + r-^-^sinbx .+ ...,,
1— q	l-o"	1-q
4oq
¿q .
+	— smx + .
X— ,j	i—o
.	-	4a°	.
—jSiriAx +-—-^sinox +
1— *'
smx ^“*4
Let us put in the first equality x = 0, in the second x =
AND IN THE THIRD X “ *| . WE OBTAIN THEN
oo r
^= 1 **L tAt = 1 fiT_ °(n) = z ■<
r=i	n= l
oo
(1)
(2)
(3)
nX1 i 4K
(4)
OO
Zj^L
4
oo
r= i#	‘	a-1
3	5	7
4
5	7
q q
^ u	ç ^
o3?+1	q3p+3	q8p+5	q8p+7
(5)
+ .1(3)
In the above formulas ^ is an odd divisor of n; it may be equal to 1,
To FORM p(n) WE MUST TAKE all possible ODD DIVISORS OF THE NUMBER n.
On the OTHER HAND WE HAVE THE FOLLOWING FORMULAS:9
SR _
T ~ ~s
+ 00 +00
io)- i r« ' .
y^-oo y~-oo
(7)
,,(pp;(o) -■ ■£ f i4'"*' y’1) !,
xP-qd y®!
+ 00	+ oo
>1 -i. \^i + k* * £8(0, q)&8(0, q2) =	2_ q
X 2 + 2 y 2
X~-QD y^-OD
(3)
(9)
COMPARING THE ''COEFFICIENTS OF THE SAM £ • POWERS OF Cj IN THE FORMULAS (4).and (7),	(5) and (3), (6) and (9) and denoting by n and
0 INTEGERS, WE OBTAIN THE FOLLOWING THREE'THEOREMS.
Thsohsu 1.' The number of integral so lutions of t.ie. equation x*+y2 = n is equal to four times the difference between the number of divisors of n of tlxe form 4p+l and the number of divisors of n of the form 4p+3,
Theorem 2 „The number of integral solutions of the equation X’«+4y2 = 2n+) is equal to twice tie difference between the number of divisors of 2n+l of the form 4p+l and the number of divisors of the form 4p+3,
"Theorem "3 ,• The number of integral solutions of. the equation x2+3ys r n.is equal' to twice the number of divisors of n of the'formSp+1 and 3p + 3 minus twice the number of divisors of n of the forms Sp + 5 and 8p+7,
§ 2, I N §§ 3 AND 5 OF THIS.WORK WE GIVE..FOUR. EXAMPLES' OF THE APPLICATION OF THE METHOD OF 4 ACO SI. ALL ARE RELATED TO THE DE-TERM'INATION OF THE NUMBER OF INTEGRAL SOLUTIONS OF CERTAIN I N~ DETERMINANT QUADRATIC EQUATIONS, IT IS INTERESTING TO MAKE CERTAIN GENERAL REMARKS CONCERNING THE DETERMINATION, OF THE NUMBER OF SOLUTIONS OF A GIVEN QUADRATIC EQUATION.
IF Wf HAVE GIVEN AN INDETERMINANT QUADRATIC EQUATION
2 2
ax + bxy + cy + dx + *?y = n
IN TWO VARIABLES,, WE CAN EASILY REDUCE IT TO THE FORM
2 2
hk + 2bxy .+ cy = n
WHERE H, 0, C, AND n ARE I N'T EGER S,
TO DETERMINE THE NUMBER OF INTEGRAL SOLUTIONS OF THE LAST EQUATION FOR POSITIVE tt, LET US CONSIDER THE SUM
X- - oo
+ 00 „ * r~ ax f +Sb x.y + c y 2
L q
b 2)y2
(10)	Let us	GIVE	TO y VALUES at,		r.t + 1	, a t +	9,. . . ,, at + a—l wher	E t
1 s	AN ARBI	TR A R Y	1 N T E G E R J	THEN	3 X +	by =	a(x+bt) + b^ for	
y 35	at + A.	1 F W	E GIVE TO	X ALL	PO SS 1 8L 6		VALUES FROM -CD TO	
+ 00	, THEN	x + bt	= K W 1 L L	AL SO	HAVE	ALL POSSIBLE 1NTEGRAL		VALUES
FROM -00 TO +00			. Let us	ASSUME	NOW	THAT	a and ac-b2 (and	THERE-
FORE c) ARE POSITIVE INTEGERS. THEN IN PLAGE OF (10) WE OBTAIN
v* i(ac*bX)2
L L L “	«
A =s 0 Trm t = ***nn
a
-(at+M
^=La~i x2
j	^(bMlgq, q ) 9g [ ( ac-02 )Xil gq,
À=o
a(ac-b2).
q	J .
(11)
Very often the expression (11) may be simplified considerably Thus if ax£+2bxy+cy9 is not a reduced form, it is easily seen
HOW TO REPLACE THIS FORM BY A REDUCED ONE.
From what precedes we see that in case a, c, and ac-b2 are
POSITIVE THE INFINITE SUM (10) MAY ALWAYS BE EXPRESSED AS A FINITE SUM OF PRODUCTS OF (JACOBI FUNCTIONS. SUCH PRODUCTS OF
■ Jacobi functions are equal to 3j{/n, multiplied by certain functions OF THE MODULUS k. THE SAME FUNCTION OF k IS EVIDENTLY A PARTICULAR VALUE OF A CERTAIN ELLIPTIC FUNCTION. THE ELLIPTIC FUNCTION MUST BE CHOSEN SO THAT ONLY RATIONAL FUNCTIONS OF q AND NOT THE FUNCTIONS OF THE MODULUS k WILL ENTER AS FACTORS IN THE EXPANSION OF THIS ELLIPTIC FUNCTION IN A TRIGONOMETRIC SERIES.
§ 3., I F a
THE SUM (11) THIS CASE WE
A.ND aC-b 2 ARE POWERS OF 2 IT IS VERY EASY AS AN EXPANSION INVOLVING FUNCTIONS OF k. MUST DEAL WITH PRODUCTS OF EXPRESSIONS OF WHERE n AND 0 ARE INTEGERS.
TO WRITE
For in
THE FORM
WE MAY DETERMINE THESE EXPRESSIONS WITH LOWING EQUALITIES WELL KNOWN IN THE THEORY
*'4(x) * Sg(0)Ss(9x) +'»g(0)V2x) + ^g(x) = 9g(0)M3x) + Sg(0)V3x) ~ r>4(x) = ^(0)93(2x) - 9g(0)V9x) +
THE HELP OF OF ELLIPTIC
93(0)S(2x),
*3(0m2x):,
98(0)S(2x),
■9 s(mil gq) = q m c>g (0),
9(fflilgq) = ( —1 )ma""™ 2 S(0 ):, 92 (rail gq) = q ” 9g (0) ,
>
THE FOL-FUNCTIONS.
> (12)
(13)
By MEANS OF FORMULAS (12) AND (13),
CAN BE EX-4
PRESSED :|N TERMS OF A RATIONAL FUNCTION OF q (.WHERE t IS A fraction) AND IN TERMS OF A FUNCTION OF 53(0,q),	9(0,q), SgCO^q),
Square roots enter into the last function. In other words we ob-
TAIN AN EXPRESSION IN WHICH	'IS MULTIPLIED BY A RATIONAL F U N C-
t	,
Til ON OF q AND A FUNCTION OF k CONTAINING ONLY SQUARE ROOTS.
Let us recall, however, that the modulii q in each of the
PAIRS OF FACTORS OF THE PRODUCTS (H) WILL BE THE SAME ONLY IF
ac-b2 = 1; but we assumed that ac-b2 = 2™. In order to express
THE PRODUCT OF BOTH JACOBI FUNCTIONS IN TERMS OF THE SAME MODULUS k, WE APPLY lANDEN TRANSFORMATIONS BY MEANS OF WHICH SgCO^q2 ), VO,,**)-,	ETC. ARE EXPRESSED IN TERMS OF
Ss(0,q)-, 9(0, q), and 9^(0,q).
Performing these computations in the case of arguments i'(3m+l) ilgq an d 2m + l) il 1q, I found that Tacobi functions of such arguments are expressible linearly in terms of .Jacobi functions WITH THE ARGUMENT ZERO. THUS
But the products occurring in formula (33) may be written
~ °, q) - 9 3 (- ■	, o') . Therefore for p not exceeding 3, the
2p
EXPRESSION (11) IS DECOMPOSED INTO TERMS OF THE FORM
93 (0, q) 93 (0, qsm ), 9g (0, q) 9g (0, q2“ ), etc. (For p~ 3 there may
OCCUR ALSO TERMS & \ THE FORM 9 g (O’, q) 9 g (0 f q 2“ ) ETC.)
The method described is somewhat complicated. I
THE COMPUTATION MAY BE CONSIDERABLY SIMPLIFIED.
I SHALL CLOSE THE CO N S ID (E R A T I 0 N OF ThE CASE 3=2’ . 2	0 n
3C-b 55 * WITH TWO SIMPLE EXAMPLES.
MANY CASES
AND
Example 1, To find the number of solutions of the equation
:2	= n.
V 2yi*S+4y* = 93(0,q^)93(0,q4) = |9g(0,q)(l+/^)(l+/kr)
= — (l+Zk+A^+’/ik7) . n
Making use of the formulas on pagedc4 of the treatise of Briot and Bouqy et which are deduced with the help of well known
FORMULAS, ON DIVISION OF THE ARGUMENT OF THE ELLIPTIC FUNCTION 6.Y 2, WE MAY WRITE THE PRECEDING EXPRESSION AS FOLLOWS:
1 2kL ..	. K' i . X	K-K’.i\
n - ltC sla ;ua--7T~ + a ,+ e 4 s cos am —s— 1.
A \	'j	u	£, j
We SHALL NOT STOP TO DEDUCE THE ARITHMETICAL THEOREM ITSELF.
*
Theorie des Fonotions Blliptique par Briot et Bouquet, 1375.SXAWPLE 2, TO DETERMINE THE NUMBER OF SOLUTIONS OF THE
equation 4xS + 4xy + 3ys = n.
Z Z** Z Z^***’'’'4’" -VWWtf»VWV.i>.
The second term of the right member we compute immediately with
THE HELP OF LaNDEN’s FORMULAS.
1.2,.	, ¿rrrs 1 2K	v
5S (0, q )S8(0,q*)*. ^gi>a(0,q)/R? * .sin arn~ ..
g	4
TO COMPUTE THE FIRST EXPRESSION WE S H AL L DERIVE 9g (0 # q ' ) 9g (0 , q ) AND IN THE RESULT REPLACE q 8 Y /q.
^!>*(0,qHl-/ST)/I=irr
		^2	(0,	q?)92	CO,	4) -q ) “
		1	2;<	i k		k/F
		2/2	T*	1/1+k	1	/l+k»
§	4.	Let	US	CONS!	DER	NOW
n AND	m	ARE	ODD	AND	P *	S AN
E QU A T	» 0 N	1 S	REDUCED		TO	THE P
DINARY INTEGER. IN THIS CASE THE
duct $3 (0, q Va(0, qB) . Reg ar dl ess of
THE PARITY OF ffl AND U THE THEORY OF ELLIPTIC FUNCTIONS GIVES VERY SIMPLE EXPRESSIONS FOR §3(0,q ) AND ^C^q““) IN TERMS OF ^3(0, q) AND ELLIPTIC FUNCTIONS OF EXPRRSSIONS"^ ANO^V
WE SHALL CONSIDER ONLY THE CASE WHEN BOTh n AND ffl ARE ODD.
, (0	)
J s'J > T ■	3	/Tim Tt \ n m /
1	2
_ 2K ■
/ni n
Let us r É.-P l a c e
sm am ■
2hK
h~
m -1
2
sin am
2hK
(2h-l)K
m
s1nam*
2K
BY
n
2Kx
11=1
sin am
(2h-l)X’
(14)
m
AND ASSUME THAT n AND ill ARE RELATIVELY
n m	X
PRIME. If WE CONSIDER X AS INDETERMINANT, THE SECOND PART OF EQUATION (14) WILL BECOME A DOUBLY PERIODIC FUNCTION WITH PERIODS 2X AND 2K'i, AND WITH INFINITIES OF THE FIRST ORDER. THEREFORE WE CAN APPLY TO THIS FUNCTION THE THEOREM ON PAGE 2$7 OE
Eriot and Bouquet and express thepreceding function as the sum

do)
WHERE «h is
VALUE OF X, TAKEN INSIDE AN ELEMENTARY PARALLELOGRAM, FOR WHICH ThE FUNCTION BECOMES INFINITE, THE SUM BEING EXTENDED TO ALL	IN THIS CaSE A IS IN GENERAL A FUNCTION OF
ft
THE MODULUS K,' |N THOSE RARE CASES WHEN ALL A’S ARE CONSTANTS
OUR PROBLEM IS SOLVED.
IT WILL BE USEFUL TO MENTION CERTA!N THEOREMS WHICH INDICATEW HEN THE PROBLEM MAY BE SOLVED IN THIS WAY. To DO SO WE MUST CONSIDER THE WAY IN WHICI «h FOR X IN THE EQUALITY
CONSIDER THE WAY IN WHICH A. IS DETERMINED. LET US SUBSTITUTE
n
!K
lim
x=tt,
{ F(x)( x-a. )} = A. lim
■ s! (x-a )( x-a )
—1---a---= a
V'-V h'
The infinities of F(x) are of two widely different type.
1)	Let one of the cosines occurring in the function, for example-
COS am	BE EQUAL TO ZERO. IF A L SO NONE OF THE ELLIPTIC F U N C-
TT
TIONS IN THE NUMERATOR OF F(x) IS EQUAL TO ZERO
2X	2X ,	.	2pXah	,2pX«h	2DK«h
—A =	——F( a. ) co s am —-: sin am------A am-----—,
TT	p 7X “	K	Tt	Tt
2)	Let cos am- oo. Durege on page 29 gives
TT
2pKxA2pKx sn~—A—*■—
It	TT
sin am

cos am
2pXx
cos ami + X + K' ij
2pK«h _
TT
= 2nK + ( 2m+1) K' i; cos am
2pKa	\
vJ‘*x + x’V * °-
From Tihis we conclude that
2pK«
sm Hm —n
6 p A O'
2K	2K
i~\ = +^F(ah)cosam n
k. 0,- „ ITL-Aa a,n
Now WE COME TO THE DESIRED THEOREMS.
2 q X a
Theorem 1.; If cos am —a = 0 and p is odd then
2hXa
hyjlt sin am n
* A
2hK«
a:n-
ii
h~ i
ThxT
= +1.
cos am
In case p is even (the numerator of the preceding expression is not equal to ¿ero) then t'lis expression is equal to either +1 or -1.
Proo f:
2pK a
PrK«,
cos am
2sKah
h. -
2( r + s)K«,
co s am
CS-
•cs-
TT
sir
n	re
2rl(ah 2sXah 2rX«h 2sXah
n
sn-
■tr
•A-
-IT
1 - k sn
,2 22rXah 2
------_sn
2sXat
= 0
The ASSUMPTION THAT THE NUMERATOR OF THE PRECEDING FRACTION
IS EQUAL TO ZERO IS POSSIBLE, BUT THE ASSUMPTION THAT THE DENON <*
INATOR IS EQUAL TO INFINITY IS NOT POSSIBLE. FOR THE LAST WOULiIMPLY THAT
7
2X% _ 211X i (3ffi + l_)K'i T~ ~ T’*	r
BUT THIS EXPRESSION IS EQUAL TO
3 nX C2m+l)X'i
OR ------- +
s
(2n+ 1)X . 2,nX'i
WHICH IS :INC0MPATI6LE WITH THE FIRST EQUALITY. THEREFORE
o „ tr i
3 r.X oc 2 sK«
h	h	n __
cs—ft—cs—— - sn——-sn—jr
•'jrfia 2sXcc 2rX«. 2sKa
k:\—-------------
n
rt
AND I F p IS ODD, THEN
, p X o.
, pX a.
oK«.
P. ‘ h _	2*"* h* S *	h
c g ---•* - s n ---“A am----“,
rt	?t	n
Or-	2Xp a	,
i hsoh2M 3, Jr cos -i a—^— ~ -J, t ten
2Xra 3:<Cp + r)« *3XraA2X(j
sn*------SU-
rt
2Xra 3X(o+r)tt c s_---CS---r—.. —
rt	rt
THEOREM 3, If C S“ _2 " - OD t’T(3fT

2Xr.a 3X c P-r) a A 2X r« A 2X (o-r) « sn------sn-----------A------A-
■srr n	tc
u
3K r <x 3K(o-r)<x
;s—~--Cg-
- ±1.
Proof, From the equality
2Xr<*	3X( p-r) oc 2Xr<* 2X(dt) a 3Xr<* A2X(o-r) oc
C3----os--— --*— - sn*-----sir--^—■—-A-----
rt
rt
^	, p p2Kr<* r,3K(p-r)oc
1 - k sn rrsn^—r-----
CD
IT FOLLOWS THAT
sin a
3K(p~r)oi , ,	. 3;{rot
i'---—...~ 38. ±1: *c sm am-----,
n	n '
2K ( DT ) CX . A
cos am--------~ - ±i A am---—: k sin an
k	- n
, 2K(p-r)a .	2-{rOc
■Aam—------- ~ ±i co i:l am---.
TX	Tt
T'HE REMAINING PART OF THE PROOF IS CLEAR.
Theorem 4. If cs~^~ = oo, then
TC±1
SK roc 2K(p>r) oc 2Kroc 2K (p + r) <*
———on---g——A------A--— ----
n
sn-..-~sn
n
n
n
2Kr<> 2K(p + r)oc n	n
TChSORBM
_	'	^Kpot
5, If cs—- 00 i/zen
n
pK oc oK oc sn4—
n 7t
	pX<x c S'—— rt	+i,	O 1	or	±1.	
Using these	THEOREMS, it	1 s	EASY TO	SHOW THAT BY THE		PREVIOUS
METHOD WE CAN	DETERM INE THE	NUMBER OF		SOLUTIONS OF THE		EQUATIONS
2 2 x +3yZ	2 , c 2 _ - n, x +5y -	n,	? ? X '+7y		2 2 n, x +xy+y =	n,
	3x + 2xy+cy	=s	n, x2+xy+2		2 _ y - n	
AND SEVERAL OTHERS.						
$ 5. LET US	CONS 1 DER SOME	0 F	THE CITED		examples.	
ExaufIìS 3.						
a. . *	L L	Lq*2 + 3y3			-	
O r/
o A
1 2f< Slu aa 3
îua-
/3 n
2î-r
cos
„	2Kx
. I d 1 < cos ara——
-4, —^------------i-
/j/ n	dx
X=g
But we have
^ 1	2Kx
dl s cos a<a- “ ^	n
dx
THEREFORE
tix
r
q
l+(-q)r
sinSrx;
1 +
r= 1
___'i___
3/ l+(-q)r
00
=	1 + 2^ H(2n-1)
n- l
1 + 3q4( 2n_l)+3
1 e( Sn-l) 0 0 4(;2n- 1 )
1	+ 3q	+
3
where w) =0, +1, OR -1 AC CO ROING AS r ~ 3k, -3<C+1, OR 3k+2 . AND H(2n-l)-DENOTES THE DIFFERENCE BETWEEN THE NUM8ER OF DIVISORS OF v-n-1 OF THE FORM 6r+l AND THE NUMBER OF DIVISORS OF 2n*-l OF THE FORM or^S,
?o determine the numbur of solutions of the equation*>	^ g	-	.
x'+3y' = 2nrl me must subtract tic number of divisors of 3n-l of the form or+5 from the number of divisors of 2n-l of the form 3r+l and double the difference obtained,
p	2 OC— 1
The equation x +3y	= 2‘	(3n-l) for a ?-I has no solutions.
2	2	2 (X
The equation x '+3y' = 2	(2n~l) f'or as 1 has three times as
1 2 2
solutions as has x~ + 3y' = 2n-l.
Example .4 .
1
L
5 y2
JL
/5 rt
22 . 4PC,	2.2	42
r.r sin ^-'!*rsin -¡m-ri	—a as ~
2K	42
cos	—rcos --m —*
■"> >
Here F(x) will become infinite only if cos bio % =. 0 OR 00,
FOR IF COS 8'®
2Xx _
u or oo sin m
42 x _
infinities:
r ■{ v
tx w
O >•/
V
U , »‘HERE ARE IN ALL 6
3K' i
r> 1	O -r f • rr f -•
AWV _	+ r7 f *	' Y rr f *	‘•4	x	x ^
—	O 7 r\ 7 o	^ / . O ■	-1	7 r> 7	o' T'~ $ rf	* ' 7	o
¿V	.o	O	j	j	f.i	i
+ 2;
AT THE SAME TIME
22a
sin b ¡a —-A a ru ■ K
: c -i a m •
•'U'
HAS THE FOLLOWING VALUES:
	+1, -1,	+ 1 X f	-i, +i,	-i, + i, -i
1 N A	G R E E ME N T WITH	T H 1	S THE VALUES Of THE	
FORMULA	(15) JILL BE	A S	F 0 L L 0 *v s :	
	J, +i,		♦ 2, *i,	_i, + i, .i
	2 9/	~o9	o * o'	o' O' O O j ,'.j
But	1 N THE THEORY	OF	ELLIPTIC	FUNCTIONS
	^\(x')		00 r~-	‘ 2 r U
	Mx) '	cot	^ L	~	grsin-r^ 1-q r
			II h*	
Using this formula and having in mind formula (15), we obtain
THE FOLLOWING RESULT
l l 2s‘*j!
VBisinlS"

, ♦1;	i] - Ç^hf->]}•
Noting that10
^ = i + ^Tt^r; = i ♦ aT{-d1-1-!!!!!-
n	¿-i+qSr	¿_	l-q2r_1
AND THAT COS^?"“ + CO S”^”“ =	IF r IS DIVISIBLE BY 5, WE OBTAIN
O	3	o
2r-l
.2 r~l>
11.-’-' ■ ■ • 1(9*7 • ri-’-Mii
WHERE |t| AND (-'r.-—j ARE EQUAL TOO, +1, OR -1 ACCORDING AS r AND Or-1 ARE CONGRUENT TO-0,	±1 OR ±2 MODULO 5.
If we put n - 5ak = Ea.2^d5, where a and Y =? 0, d and 6 are
ODD INTEGERS, THEN WE MAY WRITE (iF k IS NOT DIVISIBLE BY E)
Ei---’’ ■ >	* or«-»
n~ 1
d-l
2
c//’
WHEDE. THE INNER SUM IS EXTENDED TO ALL POSSIBLE d's FOR A GIVEN n.
ot oc Y
The number or' solutions of' the equation x? + 5,ys~ £ k=, 5 2 d5, where d and 6.are odd a and y 5 0 and k is not divisible by 5 is ¿iuen by the formula
d- 1
From this general theorem it is easy to deduce many interesting
„ _ „	. a „ 3
corollaries. We see, thus, that the equation x?+0y2 = o.4 n has the same number of solutions as the equation x2+5y! = n. Also the FACTOR f +1 INDICATES ThAT THE E QU A T I 0NS X ?+5,y 2 “ 20h+3, 30h+3, 20h+7-, 30h+13, 20h+17, 20h + 13 have no solutions. Noting that
d- 1
d6-l
5-i
(-1)
T *• (-D
(-1)

we see that the equations x2+5y2 = 20’n+ll, 30h+19 also have no solutions. In general, of all numbers not divisible by E or 4, only the numbers of the form 20h+l, 40h+6, 20h+9 and 40h+lv4 are
REPRESENTABLE BY THE FORM Xi+Sy2, WHICH CAN BE PROVED ALSO IN
another way (see Chelyshev, Tneory of Congruences)-
The COMPUTATION IS CONSIDERABLY SIMPLIFIED IF WE MAKE USE OF THEFORMULA
cos am (u+v) + sin am u sin am v A am (u+v) = cos am u cos am v ;
Then
sm am'
Fv
A T/
.	A A
: i n am —f * n
cr
u
4r/
6K
c,
u
4K
2K
AT*
i?i~~	- cos	am *tt	cos am	—
F.	F	q
O'	* >
X
- cos
^ *
2K
sin am-r s in am ~ A am’ -
^	------¿nr
2K
am ■
co - a
~ A am
ss
1 +
cos am
lco s am1.1
We also have the problem of the number of integral solutions of the equation 3x2+2xy+3y2 = n, FOR this equation may be written:. (2^y)2 + 5y2 = 3n.
§6. The sa‘«e method will give the number of solutions of the equations x£+7y8 - n and x2+xy + 3y2 = n. For if we consider
sin am-
F(x) "
2Kx	.	4ax	. oXx	~Kx	4Rx	. bXx
— sir am  sin am —— A am--------A am -— A am ——
n	n	n	n	n	n
cos am
3Xx
n
ccs am
cc s am•
ôKx
THE INFINITIES MAY BE OF SIX KINDS, BUT IN EACH CASE THESE INFINITIES ARE OF THE FIRST ORDER, AND THE COEFFICIENTS A*S ARE CONSTANTS. LET US CONSIDER THE INFINITIES:
IRx-.	,	.	4X4}c	^	A	6Kx
— =	0;	then	-sin	am ——	-	0	and cos	am ——	= 0
n	n	n
1)
co s am
AND THE reST OF THE ELLIPTIC FUNCTIONS TAKE ON EITHER ZERO OR INFINITE VALUES. AS FOR A, IT IS DETERMINED AS FOLLOWS:
A *
2.X
2X* . 2Ka sn —- A —— A
n
n
n
cc oT-r oc 6K& - sn —f- A —i—
7t	K
( x-oc)-
sn
c s
4Xa
■ lim-
4Kx
n
2Xx ÔKx x=.a cs—“f cs“
2
+-
3*
n
T?
3}
3Xx	6Xx	. 4Rx
cos am—- = oo; then cos am —=. co and sm am— = 0;
n

n
A =	+ —	-
1	3
4 K x	1
■ 3) cos am — = 0; using Theorem 2 page 7 we obtain a -	+ -
K	■	O *
4kx	i	i
4) cos am - oo ; A = either + or “
,v	6Kx _	2Xx , .
o;- cos am — =. 0; but cos ^m— r 0, therefore n	n	'
cos am---- f 0 and t oo.
n
Theorem 1 of page 6 gives us A =	- - ,
3) cos
am
6Kx			2Xx
—*			= oo; BUT	cos	A? iß ■■■■»"«—
n	1	1	n
A -	► - or - C	3*	
Thus all AVs are constants.12
r-.-.
I SHALL. NOT DERIVE HERE THE SERIES REPRESENTING Ziq	,
SINCE THE DERIVATION IS LONG AND NOT DIFFICULT.
The method described above has only limited application.
For IF WE WISHED TO DETERMINE ThE number of solutions of the equation x2+9y2 = n or x8+llys = n, then some of the ,.Ah would turn out to be functions of k. This does not mean that these problems cannot be solved with the help of elliptic, functions.
The number of solutions of the equation x,s+9y2 =. n may be found by means of the following simple considerations. The square of every number divisible by 3 IS divisible by 9, AND the square of every other number has the form 3k+l. Therefore the equality is impossible if ii - 3k+2 or if n is divisible by 3 but not by 9. If n is divisible by 9 then the problem is reduced to that of determining the number of solutions of the equation x>-**y2 - §. IF n - 3k-+1, then either x or y in the equation x?+y8 - 3^+3 WILL be divisible by 3, AND therefore the number of integral solutions of the EQUATION X* + Sy2 = 3k+l WILL BE equal TO ONE HALF of THE NUMBER OF SOLUTIONS OF X 2+y 2 - 3k+l,;
§7. let us pass now to indeterminant quadratic equations in four and six variables. To consider equations in four variables
APART FROM THOSE IN SIX VARIABLES WOULD NOT BE WISE, SINCE THE METHODS OF DETERMINATION OF THE NUMBER OF SOLUTIONS ARE THE SAME FOR THE TWO CASES. IT IS 8ETTER TO BASE THE GROUPING EQUATIONS ON OTHER CONSIDERATIONS. FIRST OF ALL WE SHALL DIVIDE ALL THE EQUATIONS CONSIDERED INTO ThE FOLLOWING TWO CLASSES! 1ST THE CLASS OF EQUATIONS FOR WHICH THE NUMBER OF SOLUTIONS MAY BE DETERMINED BY MEANS OF LaNDEN's FORMULAS; 2ND ALL OTHER EQUATIONS.
WE SHALL FIRST CONSIDER EQUATIONS OF THE 1st CLASS IN FOUR
variA8LES. Among these equations it is interesting to note the
EQUATIONS OF THE FORM X y2 + 3* ( 3 s+t 8) = u , T H E DETERMINATION OF THE NUMBER OF SOLUTIONS OF THESE EQUATIONS IS OFTEN VERY CLOSELY RELATED TO THE PROBLEM OF THE DETERMINATION OF THE NUMBER OF SOLUTIONS OF THE EQUATION x £+ 2 m y 2 = n, ASSUME THAT THE NUMBER OF SOLUTIONS OF THE EQUATION Xs+3my2 » n IS DETERMINED WITH THE HELP OF THE SERIES REPRESENTING Tff(X,X'), WHERE f(X,X'.) IS A CERTAIN ELLIPTIC FUNCTION WHOSE ARGUMENT DEPENDS UPON X AND X' «
Putting it more clearly^ let us assume that S-3(0, q)-S3(0, q2m) is transformable into • X, X')..; It^is evident that we can determine the NUMBER OF SOLUTIONS OF THE EQUATION X î+y E+3® ( Z 2+t s)= n, IF WE KNOW THE SERIES IN THE I NDETERMI N A NT q, REPRESENTING
f S(X,K'.). Let	K*,) be a particular value of %^cp(x)13
AN ELLIPTIC FUNCTION WHOSE INFINITIES ARE OF THE
97/
WE ASSUME THAT	IS REPRESENTED BY A SER-
'V H E R E ÎP ( X ) IS FIRST ORDER.
I ES ARRANGED ACCORDING TO POWERS OF q OF DIFFERENT POWERS OF ‘.j
IS A PARTICULAR VALUE OF
AND
THAT THE COEFFICIENTS
4X *
ARE NUMBERS.
l.T7 2
IT IS CL EAR THAT
•P 2|
( X, X
The INFINITIES OF 9 s(x) MAY BE OF SECOND ORDER. THESE I N F I N-
«TIES MAY BE EITHER SUCH THAT -*i7*9S(x)- IS REPRESENTED IN A N E i
Tt 2
BORHOOD OF INFINITY BY A FORMULA 'i~l WHERE Z IS INFINITELY SMALL
,,	\	ZS A 3
OR SUCH THAT ip X/ IS GIVEN BY’	'	+ “ .' I N THE FIRST CASE, IF
z* * z
A IS A NUMBER THE PROBLEM ‘IS SOLVED WITH THE HELP OF THE EXPRESS'.
4K if \ n	' S
-^■jr9 8v.x/, But if at least one of the i nf i n i t i e sa of the second k i t
OR IF ONE OF THE A',S ISA FUNCTION OF k, WE MUST SEARCH FOR OTHER METHODS OF SOLUTION OF OUR PROBLEM.
IT IS CLEAR FROM THE PRECEDING THAT THE METHOD JUST CONSIDERED IS FAR FROM BEING GENERAL; BUT EVEN IN THE CASES WHEN THIS METHOD DOES NOT APPLY METHODS OF DETERMINATION OF THE NUMBER OF SOLUTION OFT HE EQUATIONS
8
X
9»	2	_
8 . X +
y
2miz2 + t2) =
n
' y “ n and ARE SIMILAR IN MANY RESPECTS.
Together with the equations x i + y 2 + 3m( z2+t8) =. n we must
CONSIDER ALSO CERTAIN EQUATIONS ;IN WHICH THE COEFFICIENTS ARE POWERS OF 2 AND TWO OF THE COEFFICIENTS ARE THE SAME.
TO MAKE IT MORE CLEAR I SHALL WRITE OUT THOSE FEW QUADRATIC FORMS WHICH LlOUVILLS HAS CONSIDERED IN H I S JO U R N AL , ( I N THE FIRST TWO VOLUMES OF THE SECOND SERIES) AND WHICH ARE OF THE TYPE WE CONSIDER:
2 X	+ /■ +	2(z2 + t2>,	2 X	+	y2 + 4(z2	+ t2 )
2 X	*	+ 2% + 4t2,	2 X	+	2 . 0 2 y + ¿z	+ 3t2
2 X	+ ,8 +	16( z2 + t2 ),	2 X	4	4y* + 3 ( z	2 + t2)
2 X	+ 4y2 +	16(z2 + t2),	2 X		3y8 + 3z2	+ 16t2
LlOUVILLE ALSO LIMITS HIMSELF TO THE DETERMINATION OF THE NUMBER OF REPRESENTATIONS OF n BY THESE FORMS,
IN THE SOLUTION OF THESE PROBLEMS WITH THE HELP OF ELLIPTIC FUNCTIONS, THESE FORMS ARE GROUPED ABOUT SEVERAL MAIN ONES.
TO DETERMINE THE NUMBER OF SOLUTIONS OF THE EQUATION
2 . 2 _ / 2
x + y~ + 2(z	+
WE MUST KNOW THE SERIES REPRESENTING
4IC
t2) V
il
(1+kV)	=
L-‘(l - AEHm|) .1 4
The number of solutions of the equation x2+2y2 + 8z2 + 4t'2 * 81*,
IS EQUAL TO THE NUMBER OF SOLUTIONS OF THE EQUATION
x2 + y2+2(z2 + t8) 3 n, since x in the first equation is necessarily even.: The number of solutions of x8+2ya + 3zs+4t'2 - 2m+l is equal to one half of the number of solutions of the equation x 8+y2+2(z2 + t8) - 2m+l, since in the second equation either or y is odd and when x is odd y is even and when y is odd x is even.
To determine the number of solutions of the equation x2+y2+2(z2+t2) 3 n we must arrange the expression
=	^(0,q) U + k'Hl+k)
2K2/. *■ *8 ^	,2.2 K'i ^ .,2	2 K+K'i\
3 —r*ll + 4 am“ - t: sin am—^— ■+ ik cos am—-—] .
Il* \	,r'j	cj	t.j J
ACCORDING TO POWERS OF q .
The equations x9 + ya + 8z8+3t* - n, x2 + 4.y2 + 3z8 + 8t2 3 n and x2+0z8+8ts+16y8 3 n form one group.; For, Th£ equation x2 + 4y8+8z8+318 3 n is impossible if n is of the form 4m+8. or 4m+3; if however n3 4m, then the equation is reduced to the equation xa+y2+2 ( z2 + t8) =. m; finally, if n = 4ia+l, then the number of solutions is equal to half the number of solutions of the
EQUATION X2 + y2+8z2+3 t,2 3 4,r,+1 ,
Also x2+8y 8+3 z'2+13t8 is possible only if n is of the form ‘ln, or 8m+l. In the first case the problem is reouced to the equation x2+2y8+8z2 +4t2 3 m; in the second case the number of solutions IS equal to one half of the number of solutions if the equation x2 + y2+3z2+312 - '8:n + l,
In determining the NUMBER of solutions of the equation x s+y s+3 z s+8ta 3 n it is also convenient to consider separately several cases. Cases n 3 4®+3 and n - 8m+6 are impossible. If
(X	_
n = 2 m, o; 15 2, ThE problem is reduced to determining the number
OF SOLUTIONS OF THE EQUATION X2 +y2+2Z2+2t2 3 2K “m.
Let n 3 Bm+2. In this case we must consider the expression
O, q) (D, q 2)	•*, ~ &8 O, q) (k+k * i)
i j
2K9 i • 2 . 2 K'i ... 2	2 K+K’ il
sin	+ iri cos am4-......;£■*-*> .
n*8 \ *
+ IX COS
We shall consider x2+ya + 3z8+81;8 3 4n+l in greater detail. Here either x or y is odd, and therefore
2.\0,q)S
o'
~ l
30,q)s*«,q*)■ » sJ(0,q)(/5+k'/E)
D,q) = SOfq)»e(°.î>V3,q>.
iK

S * N C EFurthermore
1 5
+ oo. r=+oo
X(D,'-:)	=. 'I £	¿ (-X) S + rq
s^.-oo ra 1
2	(r-i)2,^	..
q	I2r-1)
». s^¿ U(3k*ij jC57*1.- X^(3k*s) vP^};
whe'Re u)(2n+l) = 2(-l) r(^r-l) . The last sum is extended to all >ODD- AND '>OSI TI VE ‘ SOLUTIONS X = 2 »*—1 OF THE EQUATION X's+.4y-8= 2»+--CORRESPONDI NG TO POSITI'VE, NEGATIVE OR 2 ERO SOLUTIONS y.
4'.'2
• As FOR —i” k, IT IS E A S Y TO COMPUTE 7t s	-
*■	- ■ ■ i
kco s	A ?m
p/ f
-- /k + k/k,
(V i \ /íf i \
ikcos Em |-0r. .+ . *í A -r¡ú |-0- + ¡(I
/s - k/k.
GIVES
- €■
(4r+1) \j
r*\
1 - -4r+1
^ (ir* 3)
3r+f
%
4 r ♦ 3
0	.-.'4	• r=o
! SHALL DENOTE BY Ci(n) THE SUM OF ALL DIVISORS OF n. AS A RESULT OF THE PRECEDING' DISCUSSION WE HAVE THE FOLLOWING THEOREM:
The equation x*2+y2 + 8zp+3tA =, 8k+l has 8C1 () ■ + 2to(8k+l) solutions and the equation x-8 + y* +8z2+312 = 8k+5 has 2Ci(8k+5)- 2w(8k+5.)- solutions-. •
IT IS SUPERFLUOUS TO SPEAK 0F\ THE E QU A T I 0 N S X2+ y8 + lSC^+t2) = n, AND X2+4y2+16z9+lot8 - n BECAUSE THE METHOD OF SOLUTION REMAINS THE SAME FOR THESE .E Q U A T I 0 N S . T H E SECOND ONE COULD EVEN BE ADJOINED TO THE GROUPS J U S T "CO N S I D E R E D IN ALL CASES EXCEPT WHEN
n = 4k+l. In this case it would be reducible to the fi:rst one, that is To x8+y2+lo{3S+ta) = n,
§8, I SHALL PASS NOW TO MORE COMPLICATED PROBLEMS OF THE FIRST CLASS WITH FOUR UNKNOWNS. I SHALL WRITE OUT, FOR THE READER'S SAKE, ALL THE FORMS WHICH WERE INVESTIGATED BY LiOUVILLE IN HIS JOURNAL, RETAINING THE ORDER IN WHICH THEY WERE PRINTED.x® + y® + ?ß + 8 t®, Xs + y2 + 3s + 4t®, x® + 2y® + 4z® + 4t®, x® + y® + 2z® + 4t®, x*+2/*+2z* + 8t®, Xs + y2 + 4z® + ISt®, x® + y® + 2z® + 16t®,
x® + 2(y® + z® + t®), x® + 4y® + -4z® + 4 t®, X® + 2y®+ 8z® + 3t®, x® + y® + 4z® + St®, X® + 3y® + 4z® + lot®, x® + 2y® + 2z® + 15t®, x®+ 8y®+ 8z® +64t®,
x® + y® + z® + 8t®, x® + 0(y® + z® + t®}, x® + 8y® + 16z® + lot®, x®+4y®+8z® + 16t®, x® + 2y® + 8z® + 13t®, x® + y® + z® + lot®, 5!® + 8y® + 16z® + 64t®,
x® + 2y® + 4z® + 8t®, x® + 4y® + 4z® + 8 t®, x® + lo(y® + z® + t®) -, x® + 2y® + 16 Z *+ 16t®, x® + y® + 2z® + 8t®, x® + y® + 8z® + 16t®, x® + 8y® + 64z® + 34 t®, ;
In solving these equations by elliptic functions they are
GROUPED SO THAT IT SUFFICES TO SOLVE ONLY A FEW IN DETAIL. I CANNOT EVEN ALLOW MYSELF TO CONSIDER ALL THESE FEW FUNDAMENTAL EXAMPLES; BUT IT IS NOT NECESSARY SINCE ThE METHODS OF SOLUTION OF ALL THESE PROBLEMS ARE SIMILAR. THEREFORE WE SHALL LIMIT OURSELVES TO TWO OR THREE EXAMPLES. LET US NOTE ALSO THAT IN SOLVING THE PREVIOUS PROBLEMS WE EMPLOYED EXCLUSIVELY ELLIPTIC FUNCTIONS OF ARGUMENTS OF THE FORM IB&t.OK. .1 (• N 0 W WE SHALL EMPLOY ALSO ARGU-
mX+nX’i
MENTS OF THE FORM
To DETERMINE THE NUMBER OF SOLUTIONS OF THE EQUATIONS
x® + y® + z® + 2t® = n, x® + 2y® + 2z®, + 2t® =, n and some others
IT IS IMPORTANT TO KNOW THAT VALUES OF ^rf (k1 )/T+kT WHERE f(k') IS A RATIONAL FUNCTION OF /k7. LET US DERIVE
A SERIES GIVING ONE
OF THESE VALUES.
A® am
X' i
cos am
X' i
+ A am
K’ i
»
"o
K' i
1 + cos am
(l+k) Cl+/k) — (k+/k) /l.+ k;
(WkVPk
/k + /l’+k
A® am1
3XM
4
cos® am'
x' i
sin® am1
K' i
ir *
cos am
K
i , * K' i ~ + A am —
' t	Ki
- 1 +
cos am
K' i
(l+k)(l+/k) + (k+/k) /l+k;
4X®
n®
A® am
3X'i
4
A®
8X®
n®
(k+/k) /l+k
In THE LAST FORMULA LET US REPLACE q BY q4; THEN WIJH THE HELP OF LANDEN’S FORMULAS IT IS EASY TO SHOW THAT THE EXPRESSION
SK®
n ®
Ck+/k)Æ*
Xs
-ci-k '-ï/t+P.
WILL BE REPLACED BY.17
Let us perform the same operations on the series representing
1/ * -i	O V I
1/ ' n	9 v * •?
A*	and A2am-iUi~. We have
4
,	x	oo r
- A2am^]>	=	8 Y	co s3rx- cosSryJ ;
■re* 1 tc	n >	4^i-q^r	* >
j*-® 1
4KMA2 3R'i	K'O _
n2 1	4	4 >
l~	1__2 r	'
^C-kO^Tk'
n2
oo / r
V r(q ~
w
3r 5 r 7r,
I - q + q )
r- 1
1-q
8 r
(16)
(17)
Problem 1. To determine the number of solutions of the equatio
2 . . 8 .	2 . 0.S
x + y + 2 + 3 t
n.
It is necessary in the solution of this problem to find a series IN POWERS of q FOR THE EXPRESSION
	)S8(0,q*) -	
TO SEE THAT	' ^ - --^kcos a2	
k * l/l+lp =2		
	/r- «=sfe..ow>*.	
(18)
As A CONSEQUENCE OF THIS WE OBTAIN;
(13) •
9 r/*	r___	XT^
r “ 0
r 2 + r ,
(2r+l)qSr + 1	r(qr-q3r-q5r+q7r)
—---- -----— + O '	“*—“—'—'—1—---------
l-U2^1
I
r - 1
l—’Q® r
Theorem 1. If n denotes a number of the form 8k* 1, and it we
decompose it in all possible ways into products n= ¿6 of two
r5g-i
(-1)' ^ d, then the number of solu-tions of the equation x2 + y2+z8+3t2 = n v)ill be 6Wi(n), If, however, n is a number of the form 8k±3 then there will be 10wi(n) so luiions.
Pisb'op:- The 'last sum in the formula (19) gives 3ut(n) fo r d au a
EVERY ODD NUMBER, AND THE FIRST SUM GIVES -3u(n) 3 ”2/	(“1)	^ d ,
Therefore there are altogether 8wt(n)— 2w(n) solutions.
Now let d have successively the forms 8ai+1, 3ni+3, 8m+5,
O 3i + 7 J f H E NIS
FOR n	* 3k+l	Ô	HAS	T HE	FORM	8m+l,	3m+3,	8m + 5,	8m+7,
" n	= 8k+3	5	»»	»»	tt	8m+3,	8m+1,	8m+7,	8m + 5,
" n	= 8k + 5	Ò	ti	it	»i	8n+ 5,	8m+7,	8m + l,	3m+3,
" n	= 8k+7	Ô	tt	tt	n	8m+7,	8m+5,	8m+3,	8n+l.
Hence I	1 CONCLUDE		THAT	FOR	n *	8k±l			
d2-i	6 2-i
(-1) 9	= (-l)~8 ,
AND FOR n - Sk±3
dg-i	62-i
(-1) 3 = -(-1) 8 ,
WHICH PROVES THE THEOREM.
oc
Theorem 2, If n “ 3 m, where m is an odd number, then the number of solutions of the equation x2 + u2 + z2+2t2 = n will be
3^“*£ -
where tot(m) is the numerical function which occurs in the oredo us theorem.,
For, the first sum gives the same result as that for odd n.
In the second sum we may take the fraction only for r = 2 s,
<x	9
WHERE S IS ODD, AND THAT GIVES 8*2 W1(m),
Problem 2, To find the number of solutions of the equation
x + Py^ + 16z2 + lSt.2 = 8k+3,
To SOLVE this problem we must express as a series
-±
x=-l
The last term gives the sum 22(-1) 2 r, where r runs through ALL POSITIVE SOLUTIONS OF THE EQUATION X2+2y2 = 8k + 3, AS THE COEFFICIENT OF q8k+a. THE EXPRESSION
iv°.<î> v3^2> 03(0,0	ssO',q)(. = ^/£/I=kT.U*lcM
Replacing q by q2 in this expression we transform the preceding
EXPRESSION I N TO
1 K
2
r ( 1 + Æ* - k1.~ k* /k~' ) /l-k1
3 KZ/2
V2if
I SHALL REPLACE q BY V* :,N THE FORMULA (16);
r- ( £r	2r	J. 2Tr \
\ r ( q - <r “ q + or )
^-|(l*-k')/T^ki + (l+k')Æk4 = 4P
1 - q4r13
WITH THE HELP OF EQUALITY (17)	| OBTAIN
(20
DETERMINE THE EXPRESSION
■«	-	-	r,2 /r
IJ / t
Zjl-JL
a A	K.	K . K'A . /STT . /fT (2r-l)a 8
~“C0S ,.-A	- -Jj-K A-c - A-Z.—]—5TTÎ
cos
(2r-l)n
Sx.U
8 r ♦ 3
8 r + 5
8 r + 7
rJOr+l);).»	(3 r+3) o 2	(8r+5)q 2	(3r+7)q	\
L\ 1 - q9rM	" 1 - q8r-3	~ 1 - .f r ♦ 5	+	1 - ^8r*7
WITH THE HELP OF EQUATIONS (20) AND (21) WE SHALL FIND THE SERIES FOR -"T-Zl-k', AND ~S-k' /l~k'	|F WE REPLACE 0 BY “q IN
n--	't2	‘	4^2	___2
THESE SERIES WE OBTAIN THE SERIES FOR ’“T’k ’/P/J “k1 AND FOR dr2	n
n ■
■/T’/wr. iN THIS WAT 1 CO 1-1P U TE D
(0, q)02 (0-, f) (2.3(0, q) + 9*0, q) }
V (8r+l) q
4(8 r + l)
,16r + 2
I-
(Sr+3) :^(9r + a) t" (Sr+ 5)q"(8r + 5)
L	1	- g 1 8 r + 10
I	_	,16r+6
-L	.4
V (3rt7).,'i<3r‘,)
~	4	_	016 r + 1 4
bUMMING UP THE PREVIOUS DISCUSSION WE ARRIVE AT THE FOLLOWING
Theorem:, fhe equation x8 + 3y 2 + 16-(z,8+t8) =. 3 ¿c+3 has
à2 -1	.—	r-i
"Z_	'
v' — -L	...
2_(-l)	8 d + 2) (-1) 2
solutions inhere d, Ô u, and r are all cositive numbers satis-fyin% the equations dô = 8k+3 and r2+2u2 = 8k+3,
§9.: The method that we have used so far mat, at times, be replaced PROFITABLY BY OTHER METHODS, AS AN EXAMPLE OF CONSIDERATIONS SOMEWHAT DIFFERENT FROiM THOSE ABOVE I SHALL CITE ONE OF L I OU V I LL E1 S PROBLEMS.
Problem 3. To determine the number of solutions of the equatio
x
+ 3y2 + 3 z^ + 34t2 3k+ 1.
K(0,q)!>'(0,<I8){9,<!W) ♦H0>«')l * s^CO.qHl+k’H-Wk'Vk ♦
^9,0, /) q’(9,,)
K
LOi M*
O').

1
8*
4
/
0, '
~3
C‘>,
•q)
|s;O.:)“3C0,o)
Iti
>i(0,
q)S3(0:#-q) +
q2)S3i0,q)(l+k')îî^kr|Ç
8rj
4r{ 2
WE HAVE DETERMINED “~"/k ON PAGE 1$; THE SECOND TERM OF
ft z
PRECEDING SUM IS OBTAINED FROM THE FIRST BY REPLACING q BY THUS WE EASILY OBTAIN
THE
l
o
j
^(0, q)(l+k')( l+ZT'J/k »
8 r »1
(3rH)q 4
1 _01 6r +2
1
Jir + 3
(8 r + 3 )q 4
i-qlfir+6
+
5ÌXi5
(3r+5)q	4
1—ql6 r + 10
73.x*1 .
(3r+7)q	4 X
1« ql6r+14 j
si I(-i,vi(i,p*1)!‘*Wi) ♦ ££;(-ir,,^8r*I',**,(8nn).-
i»=0 s®—.oo
lu u;
r=0 e*-®
There remains auso the expression
. (0> Cj2 ) »3 (0,1 ) ( l*k ' ) y'C l-k ' ryip"
Heretofore we have always endeavoured to express theta funs-tions with different arguments in terms of &3(0, q) and functions of k AND k* Now we shall do the reverse, that is we shall os-tain expressions in terms of theta functions and their derivatives, OF ARGUMENT , FROM FORMULAS (12) WE FIND
lia
rt
x~2
cos amx
COS X
= lira
n L
x-
ccs2 avnxf ;sin2 amx11 amx 1 * k?'sin4amx'
2Kx
4: 0
WHERE X* =
lira
co s amx
re cos x
sin amx1 ¿amx1 sin x
cos8x- sinsx) ,
2Kk'
On REPLACING ELLIPTIC FUNCTIONS BY THETA FUNCTIONS WE HAVE21
/rt>
1 / n:\
surtul + 5i
4 K k,'/k Vt
/ ir£ .
na /!5
Now REPLACE q BY ~q IN THE FORMULA (22):^
4K*/£ MT7 '1\4/~®V4/	' tc2
From the equalities (22) and (23) we deduce
■S(o,q)(i+k')
Mi
*/5
’•'a'
^¡-.,(0,^553(0,,)(l+k')
1 Mflv0»«**
/3
(22. (2 2-;
(24
+ cc
- /2
I I <-»■
(2r+l)
4(:2r*l ) 2 +2
co st2r*l)~;
4
rs'0

Correlating the two derived formulas, we obtain the Theorem: The equation
x2 .+ «By® + 8$ + 6 4t2 = 8k + l
has
FOLLOWIN''
so lutions, where r and h run through all p.ositiu.e numbers satisfying the equalities r’8+4s2 = 8k+l, h8+2u8 .= 8k+l,where valuet r = h are to he counted if they correspond to unequal values of s, u and d runs through the divisors of 8k+l.
§10. In volumes IX AND X OF the second series OF HIS journal Liouville determines the number of integral solutions for the
SIMPLEST QUADRATIC EQUATIONS IN 6 VARIABLES, THE COEFFICIENTS OF WHICH ARE POWERS OF 2. Py APPLYING THE THEORY OF ELLIPTIC FUNCTIONS PROBLEMS ARE SOLVED WITH THE HELP OF LaNDEn’s FORMULAS.
Without dividing these problems into groups I shall cite them
IN THE SAME ORDER IN WHICH THAY ARE PRINTED BY L I O.Ü V I L L E ,22
xa + y3‘ + z2 + t3 + u2 + 2ra, x2 + 2(y2 + z£+t3 + u2 + r2), xs + y2 + z2 + t2+2(us + r3),	x2+y2+2Czs+t3 + u2 + ra), xa + 2(y*+z3+t3 + u2)-+
4r3,
+ 4(y 3 + Z2 + t 2
2 + r3), x2 + j 3 + 4(z3+t 2+u2+r2), x2 +
2
» 2 ) ^ V ^ T
2j.' 2 + 2z 2 + X t 3 + u 3 + r 2), x 3 - j 2 + .z 3 + 4( t 2z* + 212 + 4U* + ir2, x3 + ^2+.z3+t3 + ±us + 4r2, x 2 + y 2 + z3 + 2t3 + 2u2 + 4r 2, x2 + ,, 2 + z3 + ta + u2 + lr 2, x3+4(ya+.z2 + t3 + u2)+3Gr2.
But the question of the number of decompositions of a given
NUMBER INTO THE SUM OF SIX SQUARES MUST OF COURSE PRECEDE THE DISCUSSION OF THESE PROBLEMS. I T I S EVIDENT THAT MANY OF THE ABOVE PROBLEMS OF L I 0 U V I L L E ' H A V 6 CLOSE CONNECTION WITH THE PROBLEM OF SIX SQUARES. THUS, ONE OF THE PROBLEMS REQUIRES THE DETERMINATION OF THE NUMBER OF THOSE DECOMPOSITIONS INTO 6 SQUARES IN WHICH 5 OF THE SQUARES ARE EVEN; ANOTHER THE NUMBER OF THOSE IN WHICH 2 OF THE SQUARES ARE EVEN; ETC..
The METHODS OF SOLUTION OF THESE PROBLEMS DIFFER FROM THE ME-
THODS EMPLOYED BEFORE IN THAT WE MUST SFARCH FOR ELLIPTIC FUNOTIONS WHICH HAVE INFINITIES OF THE THIRD ORDER ONLY, IF THE P RO-3 L E M 1 IS NOT SOLVABLE WITH THE HELP OF JACOBI FUNCTIONS ALONE. THE ARGUMENTS OF THE ELLIPTIC FUNCTIONS 'WILL, AS BEFORE, BE I SHALL LIMIT MYSELF TO THREE EXAMPLES ONLY.
.aK+.nft.U.
4
§ 11. Problem l.To find the number of solutions of the equation xp+yi + zfc+tf+u£+r'~ a, either indicating or not indicating
WHICH OF THE VARIABLES MUST BE NECESSARIL Y ODD. To SOLVE THIS PRO 6 L E M W E ivl U S T CONSIDER SUMS OF THE FOR M
6 — m
c)*(a,q) $3' to, q4)
__L_of;c
- ra 3
co,q)u+/rr)r'~\
WE SHALL DIFFERENTIATE, ThE TRIGONOMETRIC SERIES FOR A 2am'
2Kx
		07v A			
A N 0	THOSE	for ¿i am	 twice; tx	WF	OBTAIN THEN	
(2K\ \ K 1	\S 2 1 tc sin	2r/v> °,r/' am	cos am	a am rc n	-vuA ft		-V r2jj 0L.1-A*r
sin ■ irx.
2k\? e
2Xx
2.X :<
co s “ a or
‘P ¡7
rN <J.4X
si a*' a n“" A am
ft	ft
ACZ_ 17^7 cos ijrx^
ft	rr tr r/ »
\ ft /	ft
n r.K' i
In THE FIRST OF THESE FORMULAS WE PUT X ~ 7 AND THEN X ~
'*	’*	¿-A
IN THE SECOND ONE WE PUT X = AND IN THEREEULT REPLACE Q BY "Cj .
Weobt.ainthus fOUR formulas:
r=l
q
4r-2
(25)23
, x_	*	,_..24r-2
fcti t..'*:-*) = ? ♦ 8y (-!)■• Ji>rlLa__________
^	^ j	+ o v v> J.;	4	,
£- 1 — q
r* 1	■*
? 9
—	(1-k’ )
i n /
ipr rV 10 Z_ TT7-''
r “ 1
/2X\ 3
n
q
r 2 r
•«•v, - »ri^.
r-l ^
(25.
(27.
(23:
If in formulas (25) - (28) wf= replace q by qs, we obtain onl
TWO DISTINCT FORMULAS*.
/ov\ 3	0
U+k' ) /F =
oe
V ,
-	-^-TT
r5* 1
ro 1 \ ^	^ r~ ^
q
4r-2	'
,2 4r-2
l~l)r/ 2r
r=t	^	r»l	.
Finally let us take the following expression:
3
kk". -	V°/<i?
(29)
(30)
■foe
= 4
i
r. s 58 0
2>
l)r*,(2rn)(2s*l}q'!iir*t)'*4(E-'1>i_
(31)
The seven equalities (25) - (51) solve our problem if we REPL ACE q BY q E IN (31) .
for/\3	0
(l-v )/k'
■foe
= lô£' £ (-3)r + 8(2r.+l)(2s+l)q
|(ürn) *+£(28+i) *
.	(31)
r, 8
» 0
For, these seven equalities make it possible to determine the
SEVEN SERIES REPRESENTINO
AND THEREFORE THE SERIES FOR THE PRODUCT
We SEE THAT THE NUMBER OF SOLUTIONS IS GIVEN BY NUMERICALFUNCTIONS OF TWO KINDS; IN SOME OF THE FUNCTIONS THERE IS TAKEN AN ALGEBRAIC SUM OF THE SQUARES OF DIVISORS OF THE GIVEN NUMBER; AND IN FUNCTIONS OF THE OTHER KIND THERE IS TAKEN AN ALGEBRAIC SUM OF THE PRO DUCTS OF ALL POSSIBLE SOLUTIONS OF THE EQUATION
(2r+l)	+ (3s+l)2	= 2n (or x + y = 4n>+2) ,
All OF THE FUNCTIONS, OF WHATEVER KIND THEY MAY BE ARE IN THE MAIN OBTAINED BY THE HELP OF THE SAME ALGORITHM:	THAT OF THE DE-
COMPOSITION OF THE NUMBER INTO PRIME FACTORS AND THE COMBINING OF THESE FACTORS INTO THE DIFFERENT DIVISORS OF THE GIVEN NUMBER.
Only for the functions of the first kind are these prime divisors
REAL, AND FOR THE FUNCTIONS OF THE SRCOND KIND THAY ARE GAUSS
primes. (The theory of Gauss numbers is given below.)
To DEDUCE THE NUMBER OF SOLUTIONS OF EACH ONE OF THE SIX
equati ons:
x2 +	y3 + z 2	+	t8 + ua + ra	“	n,	xfl + y2 + za + ta+u a+4r 2	-	n,
x8 +	y 8 + z 2	+	t8 + 4u.2 + 4r 2	=	n,	x2 + y 3 + z8+ 4t a+4ua+4r2	=	n,
xa +	^ 2 + 4z2	+	2 + 4u2 + 4r 2	-	n,	xs+4y a+4z2'1+4t2+4ua+4r3	“	n
IS NECESSARY; WE SHALL COMPLETE THE PROBLEM ONLY FOR THE FIRST
two equations:
>3
=	1+4
I'
(-1) (2r~l) q 1 - q2 r”1
2 2 r- 1
* “I-
r gqr
1 + a2r
.(32)
Let
AND
!Q = d&
CSU)
v~
L.
6-i
(-1) 2 d8
#
WHERE WE- ASSUME THAT 111 IS ODD AND EXTEND THE SUM TO ALL THE DIVISORS OF THE NUMBER m.
Ci	‘
Let n - 2 m. the coefficient of q in the second sum of (32) WILL BE 16'28aCg (m); THE COEFFICIENT OF qn IN THE fIRST sum will
be -4C2(m) for m = 4h+l and +4C* (m) for m = 4h+3. Wg have thus
THE FOLLOWING
oc
Theohem: The equation xt+y8+.z,+‘t*+ut+ra‘ -2 (2n+l), where « is
S 0( ♦ s
either zero or a positive integer, has 4{2	—(-1)n}C2(Sn+l)
solutions.
The number of integral solutions of the equation x*+y»+28+t*+u2+4v2 = n may be derived from the Equality2 5
2 2r-l
4I{'',. x	„ r, T*	C2r*l ) q	+ oX~ r 4r
pra’-'P) - 1 **L----------------i"-"q2~~>------ * 8L
„V C-l)r(2r-l)2q4r*2	„ y l-l) W'
* -L--------1 * ,'r-S---------8L—1 tq*r
Wg shall employ for brevity the following notation:
6-i
(2n+1) *	<M2n+D = ¿J-l)
Q(8 n+2)	». ^^(-l)r+s(2r+l)(2s+l),
d-i
I”.«
EXTENDING THE SUMMATIONS TO ALL THE POSITIVE INTEGRAL VALUES OF
p	p
the equations d6 =. 2n+l, (2r+l)	+ (2s+l); = 3n+2. let us note
that <pg(2n + l) = (-1) n£g (2n + l). The first sum of (35) gives -2<p{,
AS THE COEFFICIENT OF qn FOR n ODD, ANO IN GENERAL FOR n=2a(2m + ' THE COEFFICIENT OF qR IN THE FIRST SUM WILL BE -2<pg(2M+l). Let
n = 2a(2ai+l)The second sum gives 2sa+3£2(2m+l) as the coeffici of qn The third for a = 1 gives +2<pg (2m+D and for a > 1,
-2<pg (Sm + l). The fourth for a=l gives +3£g(2m+l), and for a> 1, -8-22a-2ig(2rai+l) .
'Theorem.; The num.be'' of integral solutions of the equation x *+y s+z 2+t 8 + u s+4v2 = n for n=4h+l is equal to 62^ (4h+l) + 4Q.(j9hf2); for n~ 4 h+3 is equal to 10Co(4h+3); for n - 4h+2 it is 40ig(2h+D for n = 2 (2h+l) , where a > l,it is equal to (3*2^	- 4(-l)hlÇg(2h+l.
Problem 2,To determine the number of integral solutions of THE EQUATION x 2+y 2+z 2 + t2+2 u 2+2 v 8 = n consider the expression
3
S3(0,q)93(0,q2)	». ~TT ^ + k')	.=
1 + 4
r-
(-l)r(2r-D%4r~g
1 - q4 r“2
WHICH IS DEDUCED F-ROM FORMULAS (2 6) AND (27).
4
OF
§12.Problew 3, To determine x2 + y2+z2+t 2 + u*+2v2 = n. Let
THE NUMBER OF INTEGRAL US CONSI DER
/2K\3 /T+y
\nj /2
SOL U T I 0 N S
n	2Xk . 2Rx
Differentiating the expression ---- sin am—* twice and the ex-
n	nPRESSION
ONCE WE GET
4K ,2 2 Kx
K‘
n
3
ksm am
2Kx í, s 2Kx

amir* + k cos am'

2 r.l
,V (2rl) 2q *	.	, x
2_--1 , Jr-i--Sln
3
g . 2Kx 2Kxa 2Kx
, ir sin îin“-“Cos aœ~rA ara	=	8)~n2rx.
K J	7t	K	TI	£— 1 — n ?r
,	* . nK'i
In the first equality we shall put x 3 - + —rr- and in the
nX'i SnKVi	4 2K
SECOND -rr AND —T——; ADDING THE LAST TWO RESULTS WE OBTAIN:
8K
8K '
2 3 r+ 1

(8r>*3) . q
2 9 r + 3
r=*0
2 8r*8
q'* -	+ -Tz-jn**'
fa r\ii *rT'o f _NS 8r*7,
(8r+£) q	(8r+7) q (
—qg-T5	1 qA"r ♦7-
Vw	1	- qZr
r-i
In THE last FORMULA WE REPLACÉ q BY q4!
2K\3_¿5k::	S) , f
/Z	*	l-Q»'
/2X\S /l+k7	.	,	2/2^-(2r*l) q
\Tj ~~7fT~ - 1 ' —¿-"Y--??
2 Sr+1 . C2r+l)n
J—
+ 32y-rg(qr+qar-q5*‘-q?>)
3 fin	‘
Let n * 2aC.'2m+1) AND d AND 8 BE POSITIVE SOLUTIONS OF THE
equation d6 3 2m+l. Noting that
fit
(-1) 8
AND THAT
THE FIRST SUM GIVES
fi) ■ ( - ifiyifik
-2 \/-2\
WE SEE THAT
AS THE COEFFICIENT OF q&/ WHILE THE SECOND SUM GIVESWg shall write for brevity
<?(2m + l) =
Thbohem. The number of integral solutions of the equation
a ■
i<_'ru_'r‘^v" - ^ (2m+l)	(a^O) is equal to
x 8 + y8+z8+t8+u 8+2 v 8 =
2 f a+g r\4
-p 0
o”T ><rt2in + l_) .
3Î	2a + l j
§13. !n volumes 4, 5, 6, S, 9, 10, and 11 of the second seri;
Of HIS JOURNAL LlOUVILLE CONSIDERS, AGAIN FROM THE SAME POINT Cf VIEW, THE SIMPLEST FORMS IN 4 AND 6 VARIABLES, IN WHICH THE COEFFICIENTS ARE COMPOSED OF 2,	3, AND l ONLY. ! GIVE HERE A LIST
OF THESE FORMS. FOR THE SAKE OF BREVITY T I O N S
ShALL EMPLOY THE NOT:
			ax2 + by	2 - cz2	e + dt	s		(a, b, c, d)				
	a	O XK ^	o 2 by + cz	+ dt2 +	o eu '	f V		= (a,b,c	, d, e	,f1	»	
	Vol .	4:	(1,1, Î , 5) .									
	Vol.	5:	U,l, 3, 3) .									
	Vol.	f\ •	(1,3,4,12).									
	Vo.L .	S:	U,l, 1,3),	x' s+y s+2	z 8+2 zt+	\> A ihi 0	» ,	x8+y8+za +	zt+t	S >		
•U	,1,2,	»),	(1,2,2,3),	(1,2,4,	6) , x a+	xy	+ y	2+z8+ztn8	, U	,1,	1	, ia,
(1	,2,2,	12) ,	(1,1,4,12)	, U,4,	4,12),	(3	,4	,4,4), (1,	1,3,	4),		U,3.
4,	4), (	2,2,	3,4), (1,4,	12,13),	d,3,6	> w	),	(2,3,3,6)	, (1	,3,	3	,3),
2 x	8+2xy	+ 2y8 + 3(z8+t8),		x 8+xy+y	2+3( z2+	A 8 u	>,	(3, 3, 3, 4)	, (3	,3,	4	,12),
(3	,4,12	,12)	, (.1,3,3,12	), (1,3	12 12) P >	p	(1	,12, 12, 12)	, (3	,4,	1	2, 46 )
x 8+ xy + y		a+2(	z'+ifH*) .									
	Vol.	9:	U, 1,1,3),	x 8+y 8+2	y s + 2 v t + /j U Kf	3t	8 >	(1,0,ô,ô)	, 2*	2+ 2xy + 3y		
o z	8 +5t 8	, (1	r,1,1,1,31	, d,3,	3, 3,3, 3) ,		X	2 + y2+z2*ta	+2( u	a + u V		♦v*>,
2(	x 8 + y a+ xy )		+ 3( za+18+u s+v a), x		8+ya+2(	z 8	+ z	i* 18)+3u8+	3v8,	X 8+		y 8 + 2yz +
2 a	8+3 {,*	, x2+xy + y 8+6( z a+zt+t 8)			, 2(x*+	xy	+ y	2) +3(za+zt	+ i*>	, X	£	+ xy+
y 2+3( z8		+ zt+	l8), d,2,3	, 6) .								
	Vol .	10:	d,l, 5, 5),	2x 8+2xy+3y ®+2		7 ü *-J		z t + 31 8, (1	,1,9	,9)	P	O v 8 + ^ A T
2xy+5y*+2z8+2zt + 3t8, (1,1, 0,6), (2, 2, 3, 3).
Vol, Ht x8 + 2y2+2ya+2z8+lôt 8, 2x 8+2xy+3( y 8+z 8+1 8), 3x8+oy8+ 10 ( z 8+ zt + t 8), 2x£+2xy+3yai-15z8+lota, x s+2 y 8+2yz+2 z 8+6 t *, (1,2, 3,3), 2x8+3y8+4zs+4zt + 4t8, (1,2,6,6).
We must note however, that not for all of THESE FORMS COULD LlOUVILLE FIND THE NUMBER OF REPRESENTATIONS OF A NUMBER 0 BY
the given form. For example Liouville could not find the exact
number OF SOLUTIONS OF THE EQUATION X 8 + 2 y 8+3 z S+6 t 8 = n IN CASE23
of an odd n. Solving these problems by elliptic functions, I also
MET WITH DIFFICULTIES AND t CANNOT SAY AT PRESENT WHETHER OR NOT THEY CAN BE REMOVED, THE SOLUTION OF THE PROBLEMS ENUMERATED A-8 0 VE IS REDUCED TO THE EXPRESSION OF CERTAIN PRODUCTS OF THETA-FUNCTIONS BY SERIES ARRANGED ACCORDING TO POWERS OF q. THE THETA-FU NOTIONS HAVE DIFFERENT MODULII q, q*,	q8,	q*,	q8,***.
In THE PROBLEMS IN WHICH THE COEFFICIENTS OF THE FORMS WERE POWERS OF 2, WE SOUGHT THESE SERIES, TR Y I N G, M A I I L Y, TO EXPRESS THE PRODUCTS OF THETA-FUNCTIONS IN TERMS OF FUNCTIONS OF k AND k' AND SEEKING AN ELLIPTIC FUNCTION WHOSE PARTICULAR VALUE WAS THE EXPRESSION CONSIDERED. We WOULD DEVIATE FROM THIS METHOD ONLY IF THE PROBLEM COULD BE SOLVED WITH THE HELP OF SPECIAL FUNCTIONS, ADMITTED BY LlOUVILLE AND DEPENDING ON THE DETERMINATION OF ALL THE SOLUTIONS OF THE EQUATIONS OF THE FORM X2+2ffiy2 = n.
(iD MAY BE EQUAL TO ZERO.) In SOLVING THOSE PROBLEMS WE USED LaN-DEN?S FORMULAS. IN SOLVING THE PROBLEMS UNDER CONSIDERATION, WE MUST FIRST OF ALL TURN TO FORMULAS OF TRANSFORMATION. THESE FORMULAS WILL MAKE IT POSSIBLE TO REPRESENT CERTAIN EXPRESSIONS IN TERMS OF FUNCTIONS OF THE SAME MODULUS. THEN AGAIN WE MAY SEEK AN ELLIPTIC FUNCTION, A PARTICULAR VALUE OF WHICH GIVES THE EXPRESSION CONSIDERED. THIS ELLIPTIC FUNCTION MUST BE OF A CERTAIN kind:	WHEN AN EQUATION IN FOUR VARIABLES IS CONSIDERED THE FUNC-
TION MUST HADE ONLY INFINITIES OF THE SECOND ORDER AND ONLY OF THE FORM A/Z*, WHERE A IS A CONSTANT AND Z IS AN INFINITESIMAL;
IN CASE THE EQUATION CONSIDERED IS IN S.l X VARIABLES, THE INFINITIES MUST BE OF THE THIRD ORDER, OF THE FORM A/z8.
The FINDING OF THE PROPER ELLIPTIC FUNCTION IS THE MAIN DIFFICULTY IN THE SOLUTION OF THE PROBLEM. In CERTAIN RARE CASES THIS DIFFICULTY IS OVERCOME VERY EASILY. For, IN THE TRANSFORMATION FORMULAS A FUNCTION OF THE MODULUS k IS USUALLY EXPRESSED THROUGH AN ELLIPTIC FUNCTION OF A PARTICULAR ARGUMENT. SOMETIMES
IT IS SUFFICIENT TO REPLACE THIS PARTICULAR ARGUMENT BY AN INDETERMINANT ONE AS, FOR EXAMPLE, IN THE CASE OF THE FORM X*+y*+3zS*
3-t*; SOMETIMES BY CHOOSING PROPERLY A FUNCTION OF k, NOT ENTERING DIRECTLY INTO THE PROCESS OF SOLUTION, IT IS SUFFICIENT TO REPLACE THE PERTI CULAR ARGUMENT BY AN I NDETERMINANT ONE IN THIS, AT FIRST SIGHT, ACCESSORY FUNCTION. (THE CASE OF X 8 +y a+Z 8+5 t *) .;•
In ANY CASE THE PRECEDING CONSIDERATIONS MUST NOT BE LOOKED UPON AS A DESCRIPTION OF A METHOO WHICH IS ALWAYS APPLICABLE.
IN ORDER TO BE ABLE TO SOLVE ALL OF THE EXAMPLES CITED ABOVE IT IS SUFFICIENT TO CONSIDER ONLY SEVERAL F U N D AM E N T AL P RO BL EM S; • HOWEVER THE NUMBER OF THESE IS TO GREAT TO CONSIDER ALL OF THEM HERE, AND WE SHALL LIMIT OURSELVES TO SIX EXAMPLES.29
§14. Problem 1. To determine the number of integral solution
OF THE EQUATION X 8+y2+ 3 (z'*+1 8)	= ll, CONSIDER THE EXPRESSION
1 *	P 2’i 2	2Z.	?	2'\
q)sin am-T-A am-r*:.cos ‘am-r“.
'* d	3d	3
s2(o,q)s|u>,q*i
The elliptic function
1 :r8
4rC“	2 2Kx .2 2Kx 2 2Kx
—r-sin am~—û am“T“ :.cos am——“ ns	n	n	n
WILL FURNISH the SOLUTION FOR THE PROBLEM. FOR
. p 2Kx 2 2Kx
...2 sin aia~r & am“—“ 4;{	n	n
.2	2:\x
.2 Û am------
4K	n
4S

r ^
? x cc s* a;n~n“
cos2am
2Kx
n
.g— o am——
nz	rc
45
n2
k 2 sin 2 am
0	4, |7 *	r/ f \	45 a2
n	+ i\ *	A7	O ^ äÜD n2 -
♦ St2	■<1 » i	♦!î	r nqn (1 L 1- (- q) n \
2?Cx
n
co s*
2jvrç'
3 ,
Theorem.« The number of solutions of the equation x£+ys+3zi+ + 3t8 = 2 .3 (3n±D, in which a and P * 0t is equal to 4§C6nil) in case a = Oand 4(2ft+1 - 3)§(6n±D in case « > 0. Sere $Cn) denotes the sujn of all the divisors of n.
P nO B ii .¿iii 4. >V h A T . I S THE NUMBER OF SULliTIUNS uF THE EQUATION
x*+xy + y8+z*+at+t8 = n?
V”4x^+4xy+4y2+4 z^+4 z t + 41 ^
_	^ ^ J J ^(2x*3)2+3y2*(gz»t)g+3t2
■	q’ Ss<-0,	. 3(0,q) 3C0,q»)}2	=	q) S3(3, ,•>
* ]S'«,-,)S^0,-,") ♦ ¡U3<3.q> V0,q’>«0,q>!K0,q»).
The first two products are determined in the first problem. If in the express ion
Ao,q)S2(0,q»>	-	»^0,q)S2lO,q»)n,lt,
WHERE 1* FOR q® IS THE SAME AS *' FOR q, W E REPLACE q BY q8 AND OBTAIN33
-^3’,i,S^'i,K1*k'mn,)‘TTF	3 98«»0>4(0,q,'-/i7Tr
= SaO.'jlKC:,,,) V0,q»>S(3,,s).
I CONSIDER IT UNNECESSARY TO CONTINUE THIS PROBLEM,
§15,* Problem 3. To determine the number of solutions of the equation x8+y8+28+ot8 = n.
This problem stands in very close relation to the problem of determining the number of integral solutions of the equation x s+oy8 + oza + ct* = n. If we set
2L
§2(0 r) s 21	A2(0,Qs) -
^3'J> Q'	n *	'3	n
O-u
JL
M
O V	A Z* a 9 7 r> A 7
p h*1 A	p	^	^	*A
sio am—si n* am—c am—o
9 7	¿3"'
p	p ^
co 3 am— co s am—
In order TO USE the formulas of addition and subtraction of elliptic functions, let us determine r. - 1:
— - 1 = -
cos a
25	6KL	,2.2 2K . 2 45\
;t t co a sis t a * « suv aarr si o aiorr I o________b \	_________o_______o /
O -7	¿5 -r
p	A	o -A
co s ' am—cc s ' am—
o	o
/ I	2	.	2 2X . g 4K\
U * i sm aiaTSin air—* J J______________o_______0_/_
25	4S
co s air. ~cos am ~
(f - ‘h#
25	** Or* -a: A		45/	, , 2 .	•9	2 4[{
*r-si n	a in — £ am — u	am		1 ” k si	o am—	sin am—
£	5 5		rs \		o	5
	2	2[\	2	4S		
	co s am	— cos		am—		
		0		o		
r \ 2
sin2 am1
/I r* - iv
sin2 air
t
2 45
& air
4 ^	by
p	p	p	— A
k cos aups*sia anr~r
o	o
2 2
cos au*rcos aij.*r*
/ . 2	45	, o 25\/- .
I si n ain-r - si n arc-r* Ilk ‘
/<*
A 7 * *
g	'MIL	p	4,*.
co s * air,—co s am—
•;> v
)■31
. 2	. 2		i ^	. 2
	sin \	aio-r - 0	k	s i n am~ o / ?7
:r 4* :<1	x)*	p 9 k ’ccs“	am	T * '
WE OBTAIN THUS
¡103 2cA jzoiz \n ~ nly n*
4K?
.2 2
4.'{
0'T o ' a
4i\ k . *	.	.
----r— \ sin ara~r ~ sin"am
n2	\	o	o /
1 \
if *\ . &	97,	. 2 7 [{
ksin am~r~ si n 'a:-““ o	o
I =	8
Z_l±
r q*
2 r ft	4 r rc\
COS—T“ ~C0;T~r~
1
1	1	» V ejr I 7pn	9rn\
— * —- * >
,7ft
-,8 ft
p" •'	- -*■- q
3x0'10 31° 10	r_1
10L
n
rr\ / A .r
(4i
n/ V n2
* .V rqr [ c\ t \	(-1)
4 *	’ 4 4,“
r i-i)Tv^r/c\
(34)
The symbol (■?] = 0 for r ■- 0 (mo i 5), = +i for r = *1 (mod 5)
and = ~1 for r = ±2 (mol 5).
WE NEED TO DETERMINE
23 . /43'K
Ty~
2S /4K3
A N C-V/-T SEPARATELY, ANO THERE—
nun1
FORE WE MUST DERIVE ONE MORE E Q U AL I T Y. L E T 3' BE RELATED TO qS IN THE SAME WAY THAT K’ IS RELATED TO Q.. THEN WE HAVE
n /231	)	?	(431 A 2	/ 2 31 A 2	/43r A
.«t»■ a* j	1 ) 1 3,1 (t-1 j
M
or*
.	9	'A . p
3i n - am—r~“sin «
n
am |	'21/ ^ 5	, 1 CO 3 ^	
El r » x aJ	-A2	9T 1 o j. A < a m“*"		a	AT ■» >f x fn-— -
or *at
O ***u x	9	jl
cc 3 am-—r— co s am—r~
(4 - i) -L
MAY BE OBTAINED FROM
It is now easy to see that
(1 \ 1
\M **• /-/fp IF We u ePL ACfc" * BY 1» the argument K by 3' i, and K' i
BY 3, AND CHANGE THE SIGN OF ALL THE TERMS OF THE RIGHT HAND MEMBER OF THE EQUAUTV*. We OBTAIN THUS
M' " 1/^
°T *
1 2	‘	2	' ■>	1 9 a 2
1 sm air:	1 sin *
43'1
2	2 f4u,;i	X
1 cos are 1	— + :i i + l>J - 1
a us"
2 2 ¡21/;! ,.
co s am\	:	+ u i
\ o
+ r
^he change in sign results from, extracting the square root of Mf •32
(2'X ' 2LÀ pZL
[T " TjyiT
4y r( g r~ q3 r~ q7- r* q9 r)
L	1 - ql0r
r* 1	4
- 4
I-
r*l
c-l) rr(q2r-q4r- q6r+q9r )^
1 - a10r
(35)
Let us multiply she equality (35) by 5 and add the result to (34):
0 7	/ a *t r
-6 a	/ 4alj
Tf -tr v + — + ac
*»♦„»♦**
7^ • 2JL2.L«
x, y, a, t-“®
x •♦y “«-a ♦ 5t
• EtHMD
r-1	'
y (-1). rgg^, (c\	-y_c$£zj£;.
k 1 - s2r W /£-	1.
p(qr-q3r-q7-r + q9r)
q10r
,T~ (-l)rr(q2r-q4r-qer + oSr)
b<L---------1 - ol«ï-----:--
(36)
Let n be odd oe the form c id, where id is not divisible by o. Let to - i6. The first sum of (36) will evidently give the coefficient of qn independent of 5, namely 2(^)d = (f) ^	» The
thiro sum.will give a coefficient depending on 5, for r must
NECESSARILY BE TAKEN SO THAT IT IS DIVISIBLE BY 5^. THIS COEFFli
p 5	_ p	i
cient will be 5f 2(-g)l. Thus, if n is odd and is equal to 5ru, then the equation considered has
INTEGRAL SOLUTIONS.;
ot p
Now let o = 2 5 in, where id is odd and not a multiple of 5, P § 0, a 5 1. The first sum gives us
The second sum gives us a series of terms, namely33
Therefore both sums give
The third sum gives 2a5^' + 12(g) d, the fourth one
,	, v ot. 1_
+ (-l)	2.
(-1)
a +1) _p ♦ i
H
Therefore the third and fourth sums give
6 1
1 Um , , va Si..p +1 3 T - (-1! 6/°
Thus, if n - 2a5^ifl t<he coefficient of qn in (3&) is
Theorem. If m - 10k±l or 10k ± 3 and it a and P may also be C the equation x,+ y* + zt + 5l* = 20t5^'ffl has
!)}{.- -	I), .
integral solutions, assuming that 6 and d are all positive integral solutions of the equation dô =m possible.
Problem 4. To determine the number op solutions of the equation
x2 + o(y2 + z2+t2)--n.
To oo that we shall add the equations (34) and (35):
(37)34
Theorem. The number of integral solutions of the equation
a p
x8 + öCy•+ z8+ t*)■ = 2 5 in, where m is a number not divisible by
2 and by 5 is given by the formula 1 oa <
S1'1’“ l1
!5P * (-!)*(?
*°t*1 - (-l)a5
I-
where jô = .m.i
In particular^ if n = 5g ±3, then there are no solutions for
p(	oc »
-1, (-1) (^f) s (-R) - -1. so that zero will enter into the ex-pression as a factor.
§ 1£. Problem 5. To determine the number of solutions of the
EQUATION X8+y8+Z*+t8+U, + 3v* =	0.
It must be expected that this problem is closely related to
THE PROBLEM of THE NUMBER OF SOLUTIONS OF THE EQUATION X* + 3y*
+3z8 ♦ 3t# + 3a8 + 3v* = n. Therefore we take
- 1
M
O *t JI	O
*	^	4 ¿i rv
sin aur^A aio-^
9 :/
4 w A
cos affl^
- 1
. * 2K P 4X. 2 2X 2	4;{	p 2K p 4X
sin 'aift^sin am~û aia~û ara-j ” cos am-^cos am-^*
4	3i\
co s aiD-rr-
d
o*r
¿j a
co s am ~ co s am
2x|l - k2 3si
2 2X . 2 4K\2 si o am-^r si n - am~^ j
n -t
4	^
co s amrg~"
- k2sin4a!C~j
3	^
cos am~
Ö
In these and other formulas employed in this problem the letters
l\, L, M, AND Ml DENOTE THE FOLLOWING!
2S n
V°,o\		Or ■ Tt	- V°>q*’	J. = iL 1 r JL ’ M K * Ml L'*
1	1		2X. sin ain~T-a d	2ZL , ? . 4 2K\Z ■ ai5—|l - k am amyl
uz/if	/r			M ¿J I V cos^anr^3o
sin amp
4K 4K [	2 . ? 2X\S
a>r~ U - k^sin
2K
co s am~
0'r ¿»a
2 sin am-c-A a id
<r> *■?
2K /. 2	2'C	5, . 2	2
~ IA a,D~3" " It sin as—cos ait
f)
0 V *0 A cos' aiDT" O		
o r* ¿j a	Or*-	v 2;:
si n am -^*co s am	3	UoT
+ 2k2 sin a id {r^\ +	1 003 aw j“T^+ •"'* i
A arc
Let us make use of the formula already employed:
“ ( t * v;) •
O *7	.	O .
------A air,-----------	= lb
n	n
r2 qr
a \ i h	•	-**>
bL~l- qSr-“" “•*;
2Qr
Vr, (38)
m - ^h]¥ ■ °
r~1	r ~t
r a	r-
WHERE (^) IS EQUAL TO ZERO FOR T A MULTIPLE OF 3 AND TO LEGENDRE* S SYMBOL FOR r NOT A MULTIPLE OF 3.
We shall make a few remarks concerning the coefficients of
o“oP
DIFFERENT POWERS OF q. LET THE EXPONENT B E n * 2 3'	• WHERE ID
IS NOT DIVISIBLE BY EITHER 2 OR 3. If a^O AND P=0 THEN THE COEFFICIENT OF qn IN THE FIRST SUM IS 8 * £2 012 (- 1 ) ft (<|) d S> W H E R E
id = i6. The second sum will give the coefficient only if a>0
AND THIS COEFF I Cl ENT WILL BE
92a-2 + 22«-4 _	^	^ - (-1) a4 - (-1)“}.
Therefore for a=0 the coefficient of qn is
l)1?
AND FOR Ot > 0 it I

Both formulasmay be combined into one:
(39)36
It is required to find another equation IN ADDITION TO, (38); WE SHALL TURN FOR THIS TO THE FACTOR USED IN GOING FROM IT TO
2K
Mt2
- 1
, 2LMa4 2L'(i sin ¿am	am—g—
sin am
2t I .* *a jl
C> T • ; x A ¿u x
-A am
2L* d
- 1'
cos am*
co s am
rs •* f «
1

3	*	-- 3
After calculations exactly similar to those above we obtain
• ¿W-<*
•= 1
'*1
q
, oP.
Let m be not divisible by either 2 or 3.. If n s 3fm, then the
COEFFICIENT of qn IN (40) WILL BE

8.3
Ms
- M8
d .
a p
If n * 2 3 m and «>0^ then the coefficient of qn will be
eV|«« - 4“-‘ ♦ 4“-2	C-lA - W)'l[| d»
8 P <c ♦ i	a t— / 5\ _
■ Y9 “	- {-iy 91 Ms)1
The cases of o even and odd may be combined:
0a
To determine —s „ n* V rc
f(-l>2V{4“*1 - 1-lAl
4S2 /4SL
(41)
WE MUST MULTIPLY EQUATION (40) BY 9 AND ADD THE RESULT TO (38); THEN DIVIDE THE RESULT OF THE ADDITION by 8. Therefore the coefficient of q* in the series representing	WILL BE
5'-’!Vm • '-■•*(!)} {•*■' • '■‘'■'ir®*'-
... ot. 6
Theorem. The equation x *+y £+z s+t fi+u *+3v * = 2 3 m ;Jas
1 2af
61-1-' i9
2"'-fUl	* l-u‘9
!)*•
solutions.
Problem 8. To determine the number of solutions of the equa-T I QN X* +■ 3(y 8 + z * * t * + U * + V *) s 0, WE MUST DETERMINE
4L' Ml
To do this we must add equations (38) and (40) and divide the result by 8, Therefore to determine the number of solu-TIONS WE MUST ADD (39) AND (41) AND DIVIDE THE RESULT BY 8,
«, j?
Theorem. The equation xi+3(y's+zt+li + a,+ vI) = 2 3 m has
''' 2
-w°y	(¡u
o	L	\3/J
solutions where 1 and & are given by the equation 35 in.
§ 17. To : I NVEST16 ATe THE PROBLEM FOR QUADRATIC FORMS IN A GREATER NUMBER OF VARIABLES. The COURSE OF SUCH AN I NV6STI GATI 0
was made Clear by too great a number of examples. One question I shall not, however, pass over in silence; one which has alway occupied the interest of mathematicians. This question is. the determination of the number of decompositions of a given number into the sum of 3, 10, 12, ETC.... squares.
The problem of the number of decompositions into 8 squares IS SOLVED very easily, it suffices to differentiate the SECOND DISPLAYED EQUALITY OF PROBLEM 1 PAGE 22. LET U =	(2Kx)''7l
. <c
a ainu
. 2	. Z
K sin^amu
A*
a amu
ic4sin2aiDU cos2 amu}
Lrsqr
"l- ■g;'cca3nt.
r*.l
Let us put in this formula first x=3, and then x
(42)
n	n K1 ii
2	2 K
On adding: /2X
\ rti
16
r 3qr
1- q*r>
1 >+ 16
Y~(-l)-rr 3q 2r
1— r
4
1
+
16
V r 3q r
Z_ 1 - q2 *•
♦
16
V'(-1)rr3q2r
L 1 - q8r““*
*
Theorem. If n is odd, then the number of its representations by the sum of eight squares is equal to sixteen times the sum of the cubes of all the divisors of this number. If however n = 2°m
e=o
sp-(-l)rrRq
L i - r
r rn$! i -co a—7—(k-e)
£L
Slim
g=0
1) rr3qt€ + 10
Y~(-l)rr3q2r
L 1- qp r
But
r
L-
C-D^qf6
. _ifL, _i£L _j£_.
«•* •/)= tl.f)8 U+i)9" (1*
THEREFOREæ
where u Is odd and a & 1 and if cr 3 C m ? * 2ia, where d are all possible divisors of the number id, then the number of integral solutions of the equation
§ 18. The question of representation of a number by a sum of
GREATER NUMBER OF SQUARES, AS FAR AS • KNOW, CANNOT BE CONSIDERED COMPLETELY SOLVED. (N VOL,. 39 OF CrelLe's JOURNAL (p.130) E(SEN-STEIN SAYS THAT THE NUMBER OF DECOMPOSITIONS INTO 2, 4, 6, 8 SQUARES DEPENDS ON THE SUM OF THE POWERS OF DIVISORS OF THE NUMBER, BUT THAT THE DECOMPOSITION INTO 10 SQUARES ONLY PARTLY DEPENDS ON SUCH A SUM. L | 0 U V I L L E IN VOLUME IX OF HIS JOURNAL (a LETTER TO BeRGE, SECOND SERIES, P.29b) ANNOUNCES THAT HE KNOWS
the number of decompositions into 12 squares of an even numser
BUT NOT THAT of AN ODD NUM8ER. LEAVING ASIDE THE QUESTION OF 10 SQUARES, WE SHALL CONSIDER THE QUESTION OF 12 SQUARES.
Differentiating the equality (42) we get
CC- 1
will be
0L+ 1
10} o' 3 ( m )
,4.4	,2	6.2	4	.	, 6	.	4	g
+ k sxn amu u ara u - k sin amu cos amu + x sin araueos a:cu
Let us put into this formula first x~0, then x
n n X ' i
-	+ ——r-
FlNALLY TO THESE TWO EQUALITIES WE ADD THE THIRD ONE:

(2r+ D(2g+ l)(2t+ 1)(2U + l)q'
*|[(2r+l) '*♦ (2s+l)9 + (2t+l)a + (2u+l) *)39
From these three equalities it is easy to derive
1
+
1
8
L
(-1) r5q2r
1 - q2 r
4
ELXT" (-iyr+a+t+U(2r+ i)(2s+ i)(3t+i)(2u+
4f(2r+l):
(2s+l) 8 4 (2t+l)8 + (2U+1>
In solving the problem we encounter two functions
0 3 ( II))
YV
¿J J
?<>)
r + 8 + t + u
( .2 r 4 1 ) ( 28 + 1) (2t + l) ( 2u+i?
WHERE 13 IS AN ODD NUMBER, i EVERY DIVISOR OF !B , AND 2T+1, 2S + 1, 2't + l, AND 2U4l ARE ALL POSITIVE ODD NUMBERS THE SUM OF WHOSE SQUARES I S EQUAL TO 4111.
IF m is odd and n = 2am, then qn will occur in the first sum
WITH THE COEFFICIENT 2 5<X + 3(7 8 ( m) .• * N D I N ; THF SECOND Wl TH THE COEFF ICI ENT '
501-2	5®-7	8.	,	.	_ 3	.	.
0	4 ...	4 2 }^s(ra) + 2 <?s(in)
- {2
_ 5® + 8
31
+ 2"<7S(®')
0 A	-	9
5® + 3
31
■&s(w).
Theorem. If o is even and is equal to 2 in where ic is odd, and if a8(ic) equals the sum of the fifth powers of all the divisors of m, then the number of representations of n as a sum of 12 squares is viven by the fornula
24
31
{21
+ o"
50C+1

If, however, a is odd, the number of representations is
8^5(0) 4 16?(n).
On page 29S of the yth volume of the second series of Journal de U athémat iques purés et appliquees L'ouville says.* "It seems
TO ME THAT IT IS DIFFICULT TO FIND ANYTHING ENTIRELY SATISFACTORY CONCERNING THE NUMBER OF DECOMPOSITIONS OF AN ODD NUMBER INTO THE SUM OF 12 SQUARES". SINCE, HOWEVER, HE IN HIS SOLUTIONS A DEMITS FUNCTIONS DEPENDING ON THE DETERMINATION OF SOLUTIONS OF ThE EQUATIONS X *4y * = n AND X*4 2y * =	0, THE ANSWER GIVEN ABOVE
INVOLVING THE FUNCTION ?{o), MUST BE CONSIDERED ENTIRELY SATISFACTORY. FOR, IN ORDER TO DETERMINE SOLUTIONS OF THE EQUATION X'* 4 y* + z * 4 f* s n WE MUST ONLY 8E ABLE TO SOLVF THE EQUATIONS
xs4y*= n, and x£4t*= n - ni, where n § fli sQ. i shall recall
the EXISTENCE OF A VERY PRACTICAL METHOD OF SOLUTION OF THE EQUATION X8 4 y8 = p FOR p A PRIME. THIS METHOD IS GIVEN BY GAUSS
in § 265 of his Disquisitions arithmetieae. It consists in form-40
IMG THE PERIOD OF REDUCED FORMS EQUIVALENT TO THE FORM ~X*+pyB.
Thus we can easily find solutions of the equation x* + y8 + z8 + t* = 8 n+4.
§ 19. So far, we have always spoken of the number of all integral solutions of a given indeterminant equation; these solutions COULD BE POSITIVE AS WELL AS NEGATIVE. IT MAY HAPPEN THAT IT IS REQUIRED TO DETERMINE NOT THE NUMBER OF ALL THE SOLUTIONS BUT THE NUMBER OF THOSE SOLUTIONS THAT SATISFY CERTAIN CONDITIONS
Thus, for certain equations, Liouville determines the number of
SOLUTIONS IN WHICH ALL THE UNKNOWNS OF THE PROBLEM DO NOT HAVE A COMMON DIVISOR; IN ONE OF HIS PAPERS 'HE DETERMINES THE NUMBER OF ALL ODO AND POSITIVE SOLUTIONS OF THE EQUATION X8 + y8 + Z8 +
O't 8 = n; FINALLY IN CERTAIN CASES HE DETERMINES THE NUMBER OF ALL ODD AND AND POSITIVE SOLUTIONS IN WHICH ALL THE ODD UNKNOWNS DO NOT HAVE COMMON FACTORS.
IF THE NUMBER OF ALL THE SOLUTIONS IS KNOWN, THE FIRST PROBLEM IS SOLVED VERY EASILY. LET N(n) BE THE NUMBER OF ALL SOLUTIONS AND M(il) THE NUMBER OF proper SOLUTIONS AS LIOUVILLE CALLS THOSE SOLUTIONS IN WHICH THE UNKNOWNS BO NOT HAVE A COM-MON FACTOR. LET 0 = a“ bPcYd*. . . , WHERE a, b, C, i, .	. ARE
PRIME NUMBERS, IT IS CLEAR THAT T H"E NUMBER OF SOLUTIONS IN WHICH
THERE IS NO COMMON FACTOR a, IS Mi(fl) = N(tt) - N(-%•). THE NUMBER
3
OF SOLUTIONS IN WHICH THERE OCCUR AS A COMMON FACTOR NEITHER A
nor b is Ma(n) = N(n) - N(^s) - N(j^) +	In general,
FOLLOWING THIS ARGUMENT WE OBTAIN
The SECOND QUESTION will be answered if we consider the PRO-DU CT
^s3(0,q)Ss(0,q').
I CONSIDER IT UNNECESSARY TO GO INTO DETAILS.chapter It
THE SB CO.HD METHOD OF J ACOBI. J ACOBP S INVESTIGATIONS OF SERIES IN WHICH EXPONENTS ARB EXPRESSED BY TWO QUADRATIC FORMS.,
§	20. IT IS KNOWN THAT ELLIPTIC FUNCTIONS CAN BE EXPRESSED
AS INFINITE PRODUCTS; THE SAME ELLIPTIC FUNCTIONS ARE EQUAL TO THE QUOTIENT OF TWO TRIGONOMETRIC SERIES, IN WHICH POWERS OF Th MODULUS q ARE ARRANGED SO THAT THEIR EXPONENTS FORM A CERTAIN PROGRESSION OF SECOND ORDER. |F WE GIVE TO THE ARGUMENT OF AN-ELLIPTIC FUNCTION A PARTICULAR VALUE, THEN IT WILL APPEAR THAT A CERTAIN FRACTION IN WHICH THE NUMERATOR AND DENOMINATOR ARE INFINITE PRODUCTS OF EXPRESSIONS OF THE TYPE l+q® OR 1 - q® (WHERE 10 IS A POSITIVE NUMBER), IS EQUAL TO A FRACTION IN WHICH THE NUMERATOR AND DENOMINATOR ARE INFINITE SERIES AftftANGED ACCORDING TO POWERS OF q WHICH FORM A PROGRESSION OF THE SECOND OaOER.. BUT SOMETIMES THE SAME EXPRESSION MAY BE WRITTEN IN DIFFERENT ways as an elliptic product (as Jacobi calls those fractions IN WHICH NUMERATORS AND DENOMINATORS REPRESENT JACOBI
functions). Therefore we obtain equality of two ratios of sums. Consideration of these equalities gives numerical theorems.
§ 21. In volume 37 of Crelle’s journal ..Mcobi derives with
THE HELP OF THIS METHOD, A CONSIDERABLE NUMBER OF EQUALITIES OF DOUBLE SUMS; JACOBI CARRIED OUT HIS INVESTIGATIONS SYSTEMATICALLY, DESIRING TO EXHAUST THE PROBLEM. THE DIMENSION AND PURPOSE OF OUR WORK DOES NOT PERMIT US TO REPEAT THESE INVESTIGATIONS FULLY.' WE SHALL LIMIT OURSELVES TO ONLY THE DERIVATION OF SEVERAL THEOREMS, MAINLY TO ILLUSTRATE JACOBI’S METHODS. THESE METHODS ARE SO ELEGANT AND INTERESTING THAT WE SHALL DERIVE, F0LLOWI NG JAC08 I , FUNDAMENTAL FORMULAS OF HIS PAPER, FORMULAS WHICH COULD BE EASILY DEDUCED FROM SERIES AND PRODUCTS KNOWN IN THE THEORY OF ELLIPTIC FUNCTIONS BY SUBSTITUTION OF PARTICULAR VALUES FOR THE ARGUMENTS.
Jacobi’s starting point is the equality;
«	CC	OC
ryd-^^yjo- q2r_1z) ]J(1-	=	1- q(z + z“1)
1	r* i	r» i
♦ q4(z8 + z"? )*-■ q9 (z3 + z"3) + q16(z4 + fc"4)
(1)
These are different expressions for $(x) in which exl is replaced by z. The equality (l) holds for every value of z and for values
OF q WHOSE MODULUS IS LESS THAN 1.42
L'ET US REPLACE q_ BY qa (iD A POSITIVE NUMBER), AND 2 EITHER
BY +q±n, OR BY -q±n; we obtain then two equalities
n { ( 1 - q2®i+® + r.)	_ q2mi <n»-n)	- q2mi+2m)|
i ai ♦ni
s 2(-l) <3
n {( 1 ♦	♦ q2«i*« + a)(1 - qa«i + s»^
= Zq
niSni
(2)
(3)
In these formulas i ranges over 0> 1,2> . •.>« in the product n AND OVER 0,±l,±2,±3/-.-»±ee *N THE SUM S.
Giving to ia and u particular values
3	l
1) a = 1, n = 0,	2) m = 2, n » 1,	3) ic=-, n = ->
we obtain five formulas:
n{ ( 1	-	q2i + 1)S(l	-	q2i*2)}
n{ (1	+	• q21*1) (1	-	q2i + 2)}:
n{ (1	-	q2i+1 ) ( 1	-	q4i+4)}
il{ (1	+	q21*1 ) ( 1	-	q4i +4) j;
n(l - q1*1)
zC-nV*,
2 q1*>
s(-l)lq2i%1,
2 q2i
2C-D- q^
These formulas differ from (2) and (3) in that in them either TWO products are equal or two products are combined into one or
ALL THREE ARE COMBINED INTO ONE.
§ 22.The rest of the procedure consists in expressing a cer tain infinite product as the quotient of two of the products ( 2) i (3), or (4), and therefore as the quotient of two sums.. Since the product can be expressed as a quotient in different ways, equalities of double sums are obtained.
Jacobi takes first the product
(l + q) {1 + q2) (1 ♦ q8) (1 + q4)‘ * * . Since (1 + q) (l - q) = 1 - q*, we have
(1 + q)(l + q2) ( 1 + q3)*
(1 - q2) (1 - q4) (1 - q6) (1 - q8)* • • (1 - q) ( 1 - q2)(l- q3) ( 1 “ q4)*“
_________ 1
( 1 - q)-(l - q3)(l - q5)* ' *
(5)
The first one of the fractions (5) is an elliptic product by(6)
43
THE LAST OF THE EQUALITIES (4). THUS
7 ( — Tl q3i * ^
(1 * q)(l + q£) (1 + q3)(l ♦ q4)' * * =  1:1----. ’i'/liTT!'
Formulas (5) and (6) were known to Euler who employed them in
THE DEDUCTION OF CERTAIN ARITHMETICAL THEOREMS. EUT THE SAME PRODUCT MAY BE REPRESENTED 8Y 6 FRACTIONS, IN WHICH NUMERATORS AND DENOMINATORS ARE PRODUCTS OF THE FORM (4). TERMS OF THE FORM l+qn DO NOT OCCUR IN THE D £ NO M I N A TO RS 0 F THESE FRACTIONS; IN THE NUMERATORS, HOWEVER, TERMS OF THE FORM 1 + q® ANO 1 - qr‘ MAY BOTH OCCUR.
Thus, for e xample, if we take the product
(1 - q) ( 1 - q3) ( 1 - q4)•( 1- - q5)(l- q^O - q3) ( 1 - q9) (1 - <?*) 1 - q12)*-*
AS THE DENOMINATOR, THE NUMERATOR WILL 8E OBTAINED BY MULTIPLYING
this expression by n(l + qm). Since (l~q)(l + q)(l + q*) = (l -* q4) , (1 - q 4 ) ( 1 + q4) = 1 - q9, (l - qB)(l + qs) (l + q* = 1 - q 1 *, ETC. we
have
n(l + qi+1)
II ( 1 “ q4i+4)	S(-l)iqet**2i
n(1 - iUl)(i - q4i + 4)	*
(7)
COMPARISON OF (6) AND (7) GIVES
2Z(-l)**V1,*“Sh'*h - 22(-l)1*Vi,‘2i^(3''**’’).	(8)
In ancient times special numbers called polygonal numbers were
MUCH STUDIED. PY DEFINITION OF DlOPHANTUS, AN ID- SIDED POLYGONAL
NUMBER IS THE SUM OF SEVERAL TERMS OF AN ARITHMETICAL PROGRESSIO
THE FIRST TERM OF WHICH IS 1 AND THE DIFFERENCE OF WHICH IS THE
POSITIVE NUMBER IQ — 3. IT FOLLOWS FROM THE DEFINITION THAT ID*» SIDED
NUMBERS HAVE THE FO RM 0 [ (iD - 2) (n - !)• + 21/2; AND THEREFORE TRIANGULAR NUMBERS HAVE THE FORM ( i**l)/2, PENTAGONAL (31*-i)/2
HEPTAGONAL (oiS”3i^,ETC. LET US GENERALIZE THIS D E F I N I T I 0 N ‘ 8 Y ADMITTING NEGATIVE VALUES OF 1 . THE NUMBERS 0 AND 1 ARE CALLED THE SIDES OF THE POLYGONAL NUMBER 0 [ (ID - 2 ) ( 0 “ 1 } ♦ 2]/2.
Taking in to consideration what was stated above we see that
■(8) GIVES US THE
Theorem ».If a number can be decomposed into the sum of a hexagonal number and a double of a pentagonal, and also into the sum of a pentagonal and i times a pentagonal then the excess of the number of decompositions of the first hind in which the sides of the components are of the same parity over the number of¡those of the first hind in which the sides of the components are of44
different parity is equal to the similar excess for decomposi tions of the second hind.
§ 23. The SAME PRODUCT n(l+qi + 1) GIVES SEVERAL OTHER THEOREMS.
'Similar theorems may be deduced with the help of the square and
THE CUBE OF THIS PRODUCT.
LET US CONSIDER n(l + qi + 1)'\ SINCE the theorem obtained WILL BE OF A SOMEWHAT DIFFERENT CHARACTER FROM THE ONE OUST CONSIDERED. IT FOLLOWS FROM FORMULA (5) THAT
H(1 ♦ qi+1)2 ==	1:11(1 » q 4 i + 4) 8 >
AND ALSO
n(1 + qi+1)
n(l + qS i ♦ 1 ).( 1 - q ri ♦ 4).
n(l + q141)3	*
Il (1 - qsi+2) n(l 4 qgi4l)(l - q41*4)
n(l - q2i*i)2(i - q2i*2)
But if we take the formula
A
^(x)	= 2q4sinxn(l - q2r)n(l- 2qSrcos2x + q4r)
AND DETERMINE
?n2i #+i
-rrr- W
SC-1Ì q*
lim
x«0
!il£-
sin x
«1	A	$
» 2q4II (l-q8r) S -	(0)	=	2S(-1) r*1q4( 2r“l) (2r-l)
THEN REPLACING q BY q, WE OBTAIN
oc	+ 0C
7f (1 - qiM)3 = -T-Ui + Dq

1=0
n (1 4 qi + 1):
n(l - q2i♦ 2)
J^2r *»oo
3
2(4‘i + 1) q
4 i +2 i
n(l - qi4i)	2(4i + l)q2i +i
From formulas (9) and (10) we derive
■22(41 + Dq2i ' + i + 2h%h = 22(-l) h(41 + l)q4l*+2i+h\
(10)
(ID-
Let us note that for h even, 161 * ♦ •81 + 1 + 4fc * is of the form 8n+l, and for It odd it is of the form 8N +-5.
Theorem. If we decompose a number of the form SN +1 into a square of an even number and a square of a number of the form 41+1 (admitting negative solutions), and the number 16N + 2 into the sum of squares of numbers of the form 4i +1 and in the and in the latter agree not to count as different decompositions45
b*+a* and a8+bs, then the aliébrate sum of the odd solutions is the same in both oases. if, however, we decompose numbers 3N + 5 and 16N +10, the previous aliébrate sums have the same absolute values but are of opposite siins.
§ 24. Other very interesting formulas follow from the relations DETERMINING K AND VZ. FROM 'WELL KNOWN FORMULAS WE 0 E-
^ . 3	v? .	us)
<	(1 + q2i + M2	1 1 l+qñi + 1
Noting that l-q4i + 4 s (1 + q2i + 2) (i - q^1 * 2) f w6 may write
(1 + q2i*l)(1 - q4i♦4)
vor *
(1+ qEi*l)‘(l-qal+8) Also, by (5)» we have t
*_ ^VTT(l“q2i + 1K'l*q4iM^ Æ =	/2 vq

,2q
Si *+i
T> i
¿J vj
(13)
* i £>i +i « _2(-q
(14)
(l-q4i48)^i-q4i*4)	2(-l)	q?1 '
From equalities (13) and (14), replacing q by q®, we deduce'
r-, ni 16i6+(4h*l)*	. i ( 4i ♦ l);S*8 h *.
¿¿.(-1) q	-	¿.¿(-1) q	(is)
Thsohsm. For a éiven number ? the excess of the number of those solutions of the equation P = (4us+l)	+ 16n * in which n
is even over the number of those in which n is odd is equal to the excess of those solutions of the equation ? = (4a*+ 1) +
8o' * in which m* is even over those in which a’1 is odd.
IF, NOW, P IS PRIME, THEN DECOMPOSITIONS INTO TWO SQUARES AND INTO A SQUARE AND THE DOUBLE OF A SQUARE ARE POSSIBLE IN ONLY ONE WAY (PAGE 2),* THEREFORE 1 IS OF THE SAME PARITY IN BOTH SUMS (15).
Tfitaop.aw. If a prime number 8N + 2 is represented by both forms a6+2b£ and c s + i 8 vjhere c is odd, then in case a is of the form 8m ±1, i will be of the form 81/ and if a is of the form 3m +3, à will be of the form 81+ 4.
This theorem was proveo by Gauss; it was this theorem that ,	the
INDUCED .lACOBI TO M A K E A I NVESTI GATION PUBLISHED IN VOL.37 OF CRELLE'S JOURNAL.
§ 25. IT IS POSSIBLE TO DEDUCE SEVERAL OTHER THEOREMS BY MEANS OF DIFFERENT REPRESENTATIONS OF ¿'IT AND \ÆT. J A CO 8 I DERIVED FOUR FRACTIONS FOR EACH ONE. WE MAY ALSO DEDUCE SEVERAL THEOREMS BY ANOTHER, SOMEWHAT MORE COMPLICATED, METHOD.If in formula (2) we put first id - ^ and n
n - %, w e o B T A I N
n-2.

THEN tP ® ~ AND
H(l- q5i+1)(l - q5i + 4)(l - q5i + 5)	=	X(-l) q
i *(5i8+3i)
n(l - q5i+2)(l - q5i+3)(l - q5i*5)	=	2(-1) 1 q2C 51 * + °.
Multiplying these equalities together we obtain
Uiw,	» i ♦ H s(5i ®+3i + 5k** h)
n(l - q1 + 1)(l - q5l>5)	=	2S(*-lV q2
But the last of these equalities (4) gives
n(l-'qi + 1)(l- 45i+R) = . 22(-l)->hq2<?i‘+i + 15hS+BhV.
We obtain thus the formula:
Theorem. The excess of Vie number of those solutions of Vie
2 £
equation (101 +3)	* (I0h+1)	= ION (where N is a «icon number)
in which i + h is even over the number of the solutions in which i ♦ h is odd is equal to the excess of the number of those solu-tions of the equation (6ij +1)	+ 5(6ht +1)	= 6N, in which
i* + hi is even over that of those in uihici i1 + ht is odd.
§ 26. In all of the previous theorems we were considering
TWO EQUATIONS, THE FIRST MEMBERS OF WHICH WERE TWO QUADRATIC FORMS EITHER WITH THE SAME OR WITH DIFFERENT DETERMINANTS. IT • IS CLFAR THAT THE SAME IS TRUE WITH REGARD TO EVERY THEOREM DEDUCED BY THE METHOD GIVEN ABOVE, FOR THE EQUALITIES DEDUCED IN THAT WAY ALWAYS CONNECT DOUBLE SUMS OF THE FORM
*2"*	a h 8 + m ’ i 8 + n h + n * i
L L * »
IN WHICH 2lQ ANO 2s’ ARE POSITIVE INTEGERS. THE CHARACTER OF THE FORM WHOSE VALUES ARE THE EXPONENTS OF THE PREVIOUS SUM DEPENDS MAINLY ON THE NUMBER OBTAINED THROUGH ELIMINATION OF ALL THE SQUARE FACTORS FROM 4®^’. THE CHARACTER OF THE EQUALITY DEPENDS ON TWO SUCH NUMBERS U AND V. uACOBI THEREFORE DIVIDES ALL THESE
equalities into cl asses, denoting each class by the symbol (u,v). By methods indicated above equalities can be deduced belonging TO THE FOLLOWING TEN CLASSES:	(1,1),	(2,3),	(3,3),
(1,2), (1,3), (1,6), (2,3), (2,6), (3,6), (1,5).
ij A CO B 1 DERIVED ALTOGETHER 30 SUCH EQUALITIES. ALL OF THEM FOLLOW FROM THE FUNDAMENTAL FORMUAL (1).47
§ 27. Besides the equalities deducible by the method indica'i
ABOVE, THERE ARE OTHERS, MORE COMPLICATED, CONSISTING OF 3 OR MORE DOUBLE SUMS. SUCH EQUALITIES MAY BE OBTAINED IN THE FOLLOWING W A Y.
Equality (1) may be given .3 forms:
II{ (1 - qgi + 8)(l + q2i * 1 z) ( 1 + q2i + 1z’ ( z+ 7“1) fl {(1- q1 +1 )■ (1 .+ q1 + 1 zg ) (1 + q1 * 1 z“2 )} (a-if1) il{ (l“qi+1)(l-qi+12pV(l-qi+12_ii)}	=	2(-l^
*i(i?+i) Si+1.
q z ,
i V
8+i) Pi+t 2 >
(17'! (13) (19)
WHERE, AGAIN, THE MULTIPLICATION IS EXTENDED TO ALL THE INTEGRAL VALUES OF 1 FROM 0 TO +«*, AND THE SUMMATION TO INTEGRAL VALUES
of i from *-«= to +«. Formula (18) is obtained from (17) by the
SUBSTITUTION OF i/~q AND Z Vq" FOR q AND Z RESPECTIVELY AND MULTIPLICATION OF TRF RESULT BY Z.; FORMULA (19) IS OBTAINED BY THE SUBSTITUTION OF z/“l FOR Z AND DIVISION BY V-l.
If in These formulas we replace z by a root of unity of the 3rd, 3th, 8th, 1:2th, 16th, and ¡24th degrees respectively we obtain EXPRESSIONS OF CERTAIN PRODUCTS IN TERMS OF SUMS. |F NOW WE EMPLOY NEW EXPRESSIONS OBTAINED FROM (17),	(IS), AND (19),
FOR REPRESENTATION OF THE INFINITE PRODUCTS THAT OCCURRED BEFORE THEN SOMF TIMES WE OBTAIN 5 I THER EQUALITIES WHICH IMPLY SEVERAL EQUALITIES DEDUCIBLE BY THE METHODS DESCRIBED ABOVE OR. SOME NEW
equal I ti g$. ■ .Jacob i explains the appearance of such new equalitif
WHICH MAY BE DEDUCED FROM THE SIMPLEST ONES, BY THE FOLLOWING REMARKABLE PROPERTY OF THE ELLIPTIC TRAMSCENDENTAlSJ WHEN A ROOT OF UNITY IS SUBSTITUTED FOR Z, AN ELLIPTIC TRANSCENDENTAL DECOMPOSES INTO SEVERAL SERIES WHICH MAY BE OBTAINED FROM THE ORIGINAL TRANSCENDENTAL BY REPLACING q AND Z BY CERTAIN POWERS OF q.
LET US APPLY THE P R E C £ D ! N G C 0 N S I DERATIONS TO AN EXAMPLE. LET US SUBSTITUTE INTO EQUATION (19) FIRST C, A CERTAIN IMAGINARY ROOT OF UNITY OF THE FIFTH DEGREE, AND THEN <7*. FOR THAT PURPOSE WE.WRITE EQUATION (19) IN THE FORM
(z-Z’-Oili (l-qi4‘1)(l~qi'flz2) (l-qi + 1z“2)}	*	2(-i) lq2iSi*e)<8i*S>z10i'
2 (-1)
q2	(z
tOi+1
-tOi-l. *"■ 2 )
,	. i |(5i + l)( 51+2), -101-S .lOi+E
(-1) q 2	(z - a
Since i varies from -°c to +*, it is always possible to rep lac
1 BY -1-1 AND THE TERM (- 1) 1 Z 10 1 + 58 Y - (~ 1 ) 1 Z~ * » THEREFORE THE FIRST SUM MAY BE WHITTEN48
1 „ f .4 |( 5i + 2) (,f?i +3)>	101+5
- >. (-1 ; q8	- z	) .
NOW REPLACING Z B Y <7 A N D <7 5 W E OBTAIN
n{(l - qi + 1)0 - qi+1cr2)(l - qi + 1<7~2)}
.	. i |i(5i+i)	,	-U „ ,	„ , i 4(:Bi*l)(Bi+2>
= ' 3 (-1) q2	+ (7 + J ) i (-1) q*	,
n{(1 - qi+1)(l -	qi+l7)(] -	qi+17-1)}
=	2 (-l'/q®1*51*0 4 (<72 ♦ a’?)!(-l)V«(5l*l):(iii*2).
Multiplying these two results together we OBTAIN
II{(1 - q1 * 11 (1 - qB1 *-’5)} ' =	(S(-nlq"1<5“”lS
- S(-l)Iq^(BiM)£(-U1q*"1MH5I*e>
- (S(-l)lq"(61*1,i:51*2>tS	-	2ï(-l)l*hq‘
The last sum is taken from the last of relations (4). Therefore
♦ h \{ Si £ + i + 15h**5h)
+h^(;3i 5h 8+i *5h)	_	£?(-!) i + h fi ( 51*1) ♦ <£*(6**1)
( C4 i n \	/ S'
^i+hJi(5i + l)^|(5h+1)(5h+2) ^	^ i* h J ( 5i ♦ lK 5i+2) ♦ | ( 5h+1) ( Sii+2) , ^
WE MAY DECOMPOSE EQUALITY (20) INTO SEVERAL SIMPLER EQUALITIES For this purpose collect those powers of /q which on division by 5 give 0, 2, 4 as residues.The three double sums of the right
hand SIDE ARE RESPECTIVELY CONGRUENT TO 0,	2,	4 MODULO $. IF IN
THE EXPRESSION “(3i*+i) WE REPLACE X SUCCESSIVELY BY ol,“(¿X+2),
¿J
-(5X+l),	5x+l, AND oi+2 we OBTAIN
;i(151+1), ~ (3i +1 ) (oi *2), ~:i (31+1. ) +1,
gi(151+7) + 2, |(3i+2)(5i+l) + 2.
Thus after some simple operations we obtain from (20) three equalities
-,v/ iNi+h i( Si a+5h8+i *h)	i + h( ■>(i5i*+3h^: <h)	■|((isi+l) {6i+2)+3hS+h) . .
-S(-l) q	= 24-x)	{q^	+ <r	},(2i)
;x(-i)t““q®ei**8h**t*41“ - 2ïC-1)14V*
i' + 3h	___, "Xï+'a ( 3i 2 * i )	( 3 h 2 + h)
(33)
„.i + h*£(5i*+5hS+3i*3ii	, .i+h £(l5iS+3h2+7i+h)	|( (3i+2) (5i+l)+3h2*h) N
¿21-1) q6	= 22(-l) l4	- q*'	}.U3)49
The second one of these formulas may be obtained by the me
THOOS DISCUSSED IN THE BEGINNING OF THE CHAPTER.
LET US NO W NOTE THAT
¿(15i8+i),	<-(3i + l) (51+2),	S(l5i8+7i)
V.	A	Zj
2,
-(31+2)(51+1)
ARE PENTAGONAL NUMBERS IN WHICH 1 IS REPLACED BY 51,	-(51+2),
51+1,	51+2; AND §(oia+l) IS A TRIANGULAR NUMBER WHOSE SIDE IS o'
Theorsh 1. Tf a number of the form 5N is decomposable into the sum of two triangular numbers whose sides are divisible by o, and also into the sum of 5 times a pentagonal number and a pentagonal divisible by 5, then the excess of the number of such solutions in which the sum of the sides is even over the number of those in which the sum of the sides is odd is the same in both cases.
Theorem 2. Tf a number N is decomposable into the sum of a pentagonal number and five times a pentaional, and the number 5N into five times a heptagonal and a triangular with side divisible by 5, then the excess of the number of those solutions in which the sum of the sides is even over the number of those in which the sum of the sides is odd is the same in both cases.
Theorem 3. If a number N is decomposable into two heptagonal numbers, and 5N +2 into a pentaional which is congruent to 2 modulo 5 and 5 times a pentagonal, then the (¿xcess of the number of those solutions in which the sum of the sides is even over that of those in which the it is odd in both cases are equal in absolute value but are opposite in sign.
§ 28. Finally we note that a number of equamities between
DOUBLE SUMS CAN BE OBTAINED FROM THE MODULAR EQUATIONS OF THE 3rd AND 7th GRADES. IT IS KNOWN THAT MODULAR EQUATIONS OF THESE GRADES ARE
/k! ♦ /FT7 = i, Vkl + Vk^v = i,
WHERE k IS THE ORIGINAL AND 1 THE TRANSFORMED MODULUS. IF WE SELECT THOSE EXPRESSIONS FOR /k AND Vk' IN THE SECOND ONE, WHICH HAVE THE SAME DENOMINATORS, AND DO THE SAME FOR /T AND /3T OR i/T AND yu, THEN UPON MULTIPLICATION BY COMMON DENOMINATOR WE OBTAIN EQUALITIES CONNECTING USUALLY THREE, BUT SOME TIMES ONLY TWO, DOUBLE SUMS.
Thus if we take50
/k =
4"--	^ i + 2 i
2/q .
qz, q
. 2
2(-l)~q
. a *
/r =
q8S q
12i + 6i
2 q3i *
✓ff =
s(-iV q
i Si8
i q
Si
2 '
WE OBTAIN
22 [ 1 - (-l)i+h]qii+3hS
422q
4i 8 + 2i +1 2h * ♦ 6h ♦ l
(24)
Thsorbm. The number of decompositions of 4N into the sum of an odd square and three times a square is equal to twice the number of decompositions of N into a square and three times a square if N is odd.
TO OBTAIN EQUALITIES CONTAINING THREE SUMS, LET US TAKE-EQUALITIES (13) AND (14) AND ADD TO THEM EXPRESSIONS FOR'/iP" AND YV't HAVING THE SAME DENOMINATORS AS ?*k AND ?T,

r(-l)1
s
2 i
«
- qgi**\ _ TTCl - q4i*s)S(l
♦ q£i + V '	11(1+	q 2i + 1) 2 (i
""T( 1 - q2i ■*■!) 2( i ^ ■ qgi + g)
I d - q'ii + 2)S(l	-	q4i + 4)
q4i +4)
■-q~2TT2)
2 (-1)1 q1 8 ‘
2(-l)V1* *
NOW USING MODULAR EQUATIONS OF THE SEVENTH GRADE WE OBTAIN SEVERAL EQUALITIES ONE OF WHICH WE WRITE DOWN:
222(-l)Via+14h8+i+7h+1
:2(-i).ViW
-2S(-l)i+hq
. i 8+14h*
(25)CHAPTER III
HERMITS',S METHOD. KRONBCKER' jS AND GIERSTER'.S FORMULAS OF THE THIRD DEGREE.. PROBLEMS ON THE NUMBER OF DECOMPOSITIONS INTO THREE SQUARES AND ON THE NUMBER OF REPRESENTATIONS BY FORMS
IN THREE VARIABLES.
§29. IT WAS EXPLAINED IN THE INTRODUCTION, THAT THE METHOO OF HERMITE CONSISTS OF AN APPLICATION OF TRIGONOMETRIC SERIES REPRESENTING PRODUCTS OF ELLIPTIC AND JACOBI FUNCTIONS. THIS METHOD WE SHALL APPLY TO THE DEDUCTION OF THE FORMULAS OF K RO — NECKER; AT THE SAME TIME WE SHALL FOLLOW ONE OF THE SUGGESTIONS MADE BY KRONECKER, BECAUSE THE METHOD INDICATED 8Y HIM REQUIRES THE DEDUCTION OF CONSIDERABLY FEWER FORMULAS THAN DOES THE EXPOSITION OF HERMI TF.
Let us multiply together, member by member, two well known equal I ti es:
2Sic . 2Xx
—*•—sin am-----
n	n
r—	V72r“1
4	-g'r'-r Sin(2r-1) x,
r* 1
(n
Sg(x):
2^q4^2r ^ cos(2r-l)x
r».l
(2)
Performing the multiplication we obtain terms of the form
4 8
1 - q
? 3- 1
{sin2(r+s-l)x + sin2( s- r) x} .
Collecting the terms with sin2nx as a factor we have, as the coefficient of sin2nx, except for the factor 4, the expression
«	\ (2 s-1) ga-Sn-l) 8	°»	4(2s-l)+4(2a+2n-l)8
- - P-
L
8 = n +1
1- q88"1
8*1
1 - q88”1
. r- q
+ L-
«* 1
|{2s-l) >4(2n-2s+l) 8
1 - q28*1
-n( 2s-l)
- q
n(2s- l) •
s* 1
- 5(28-1)
■s (2a-l)
- q2
}
51f- .»nap)'} Li*-1#*-1*
2_ i 2 it
S=* 1
23~1
♦ .. . • + q
2
2 s^i
+ q
«* . .
( 2 n-l) ( 2s-l.l
. . + q
8*1
n*f-L a* (-1)* + q	)
-|r(s + l)S ( s-2) *T	r(sfn-l)8 ( *-n)
+ q ? |q	+ q J + . . . + q 2 [q	* q j|
{l ♦ 2£,a*}q"*{,-i i*
8*1
9
?
+	+ q
-<*¥*> |
/ST »ef -4-1 -f	^ -(^‘1
*. w— q<q4 + q4*q,-*...*q	f-
Hence it is clear that the result of mulitplying (1) by (2) is
-	x) k slo am
4 V n =
->rr o
6AA
Tt
oc
= ^Tqn sin2nx ^q“1 + ^q~9 ♦ . . .
+ V q
-(?n-l)
(3)
n* 1	^
Multiplying both members of (1) by ecsx, we obtain
2Kk
COSX 310 a ID'
7.Xx	V~ VqSn_i	Vq®»+1	, .
--- ~ 2 ) ~—:—5——r ♦  -----r—rr sin20X. (4)
rc	¿_ 1- q2n-l	1 ~ qgn* 1
n = l
IF WE MULTIPLY TOGETHER (3) AND (4) THE PRODUCT OF THE TWO SERIES WILL CONTAIN A TERM INDEPENDENT OF TRIGONOMETRIC FJNC-TIONS. LET US FIND THAT TERM.
k2X/!T(* . p 2K'x . ,
—T"*.-""	sin^ara-—.>*(x)-ccsx *ix
nVS*i-.	n *
L' Tf
1	v/q 2n'-
n* 1
¿n<~° .. *
, 2 n~ 1
n- 1

1- q2n + l n- 1
F* . .. ♦	*
B= 1
1 - q2n
JTiM1 ♦«** * 4'** .»:♦ q-"0"-15;
4 r V r~- f n + n-1+( 2n-l) b-a + a	n + n + ( 2 n♦ 1) *>- a *+ a
q4 )	)	) sq	■ + q
L l Li-
n*1	b«0	5*1
.(o)53
IT IS EASILY SEEN THAT THE EXPONENTS IN THE LAST PARENTHESIS
ARE TWICE- THE NEGATIVES OF ALL THE POSSIBLE DETERMINANTS ('WITH EXCEPTION OF ' CERTAIN -PART I CULAR- FORMS :v»HI CH .'WE' SHALL CONSIDER
later) of reduced forms a1 x8+2bf xy+ cf y3 =■ ( af, b*, e*) wi th’negative DETERMINANTS, . 1 N WHICH AT LEAST-ONE. COEFFICIENT IS ODD. r
For, if b <■a- a, then the expression n 8 + a - 1 + (2n - 1)b - a 8+ *
MAY BE GIVEN ThE FOLLOWING FORM:
(n + a + b) (.0 r.a +b + 1) - (b + 1)2,
AND THIS LAST. EXPRESSION MAY BE LOOKED UPON AS THE NEGATIVE OF
THE determinant OF THE FORM
2 2 ( n - a + b + 1) x + 2(b + 1) xy + (n + a + b)y .
Giving different values of b, n, a, ws ostain all possible reduced FORMS IN WHICH ONE OF THE EXTREME COEFFICIENTS MS EVEN,
THE OTHER MS ODD ANC THE MIDDLE CO E F F t C I E N T . I S POSITIVE. TO OBTAIN - A L L.T H E S E FORMS WE CHOOSE b TO BE AN ARBITRARY POSITIVE I N — TFGF.R OR ZERO, A - a TO BE A NUMBER GREATER THAN b, AND A + a TO
BE A NUMBER, ALSO ARBITRARY, BUT GREATER THAN A-a + 1 AND OF
BE
DIFFERENT PARITY THAN THE LATTER. IN THIS CASE 0 AND a WILLACERTAIN POSITIVE NUMBERS WITH a < A .
1 r,	HO WEVER	, b A — a, WE WR 1 TE			
A2 + A	- 1 + (	2 a - 1) b - a2 + a *	(2o-l)(b+n-a	+ 1) -	(n- a)2
If b + a	- 3 + 1	IS TAKEN TO BE EVEN	AND A, a, AND b	are at	VE N OIF-
F E REN T	VALUES,	then we obtain the	SAME FORMS WHICH	WERE	CON SI DE R-
ED ABOVE. IF HOWEVER b + A-a+1 IS ODD, IT.IS EASILY SEEN THAT EACH FORM WILL OCCUR TWICE (MIDDLE COEFFICIENT BEING STILL POSITIVE). For 2 a - 1 may be either > b + n - a + 1 or < b + n - a + 1.
_	,	n3+n-1 + ( 2n-1 ) b - a 8 + a
Summing up the expression q	we ostain.as
EXPONENTS TWICE THE NEGATIVES OF THE DETERMINANTS OF ALL POSSIBLi REDUCED FORMS IN WHICH THE MIDDLE COEFFICIENT IS POSITIVE AND AT LEAST ONE EXTREME COEFFICIENT ODQ.
But we have not yet considered the limiting cases: 1) when
TWO EXTREME COEFFICIENTS ARE EQUAL; 2) WHEN THE FIRST COEFFICIEN IS TWICE THE SECOND (THAT IS 3t = 2b' IF THE FORM IS 3,XI+2b,Xy+ C'y); AND 5) WHEN THE MIDDLE COEFFICIENT IS ZERO.
1)	Each of the forms a'x8+2b,xy+a'y4 will occur once when 2n-l = b + n- 3 + 1. The same is true of the forms (a1,0, a *) =
* ’ x * + a' y s; i n t h e l a st c a s l i = a .
2)	If b< n- a the forms (2b1, b*, a*) will occur when b + l
= a - a. Ii b? n - a the FORM,2b'x8+2b,xy+a,yi cannot occur, forhat would imply b + 1 = fl - 3 WHICH contradicts the hypothesis.
3) For a = n the forms a 'x * + b 'y8 wi ll occur once if a’ and b'
RE OF DIFFERENT PARITY AND TWICE IF a' AND b' ARE OF THE SAME
ariity (form a'x* + a' y * being an exception).
Let us turn now to the exponent n* + n + (2n + l)b- a8 + a.
If n - a + 1 g b we can write this exponent in the form
( n + a + b) (n - a + b + l) - b 8.
ARYING a, b , AND 0, WE OBTAIN THE NEGATIVES OF THE DETERMINANTS F ALL POSSIBLE REDUCED FORMS WITH ONE EXTREME COEFFICIENT ODD HE OTHER EVEN AND THE MIDDLE COEFFICIENT POSITIVE, AND BACH JUST
jnce* ThE'FOrms 2b'x8+2b'xy+ a'y8 will also occur once. Forms ,j'x5*b'y8 wiLL occur once if a' and b' are different in parity and IF a1 <AND b' are of the same parity the forms :will not occur
IN THE SUM AT ALL. FINALLY, FORMS (a',^,*’) AND ( 3f , 0, a ' ) WILL ALSO NOT OCCUR AT ALL.
If n - a + 1 < b, we write
;2 + o + (2n + 1) b - a2 + a =	(2n + l) (b + n - a + 1) - (n-a + 1) S.
/ARYING a, b, AND 0 WE OBTAIN ONCE EACH .OF THE NEGATIVES OF THE 6 T ERM I N AN T S 0 F ALL POSSIBLE REDUCED FORMS WITH ONE EXTR-EME COEFFICIENT EVEN AND THE OTHER ODD AND TWICE EACH OF THE FORMS WITH I WO EXTREME COEFFICIENTS ODD. FORMS (2b',b',a') WILL NOT OCCUR, V/HILB FORMS (a',b(,3*) WILL OCCUR ONCE. SINCE 3 CANNOT BE EQUAL TO 0+1, THE FORMS ( a'' , 0 , a'' ) AND IN GENERAL ( 3 ' , 0 , b ' ) WILL NOT OCCUR AT ALL.
SO FAR WE HAVE BEEN CONSIDERING ONLY THOSE FORMS WITH A POSITIVE MIDDLE COEFFICIENT. IF OUT OF THE FOUR TIMES THAT FORM
.' x 8 + 2b' xy + tf' y * occurs we take the middle coefficient two times
AS NEGATIVE AND		TWO	TIMES	AS POSITIVE WE SHALL			SEE	THAT ALL FORMS
EXCEPT	(2b', b' , a		(a',b‘	, a ' )	and (a',0,a	') OCCUR		TWICE, 6UT THE
FORMS	(2b',b',a'	>,	(2b',-b		ARE EQU1 VAL ENT		A S	ALSO ARE Th€
F 0 R M S	cr &	AND	( a',-b	',a')	; therefore	ALL	NON	— EQU1 VAL ENT RE-
u U C E 0	FO RMS WITH	AT	LEAST	ONE ODD COEFF1 Cl		ENT	WILL	OCCUR T WICE,
ilTh THE SINGLE EXCEPTION OF THE FORMS (3',0,3*) WHICH WILL OCCUR J U s t once.
Thus, if we denote by F(n) the number of all non-equivalent
SEDUCED FORMS WITH AT LEAST ONE EXTREME COEFFICIENT ODD AND WITH DETERMINANT — (1, THEN
!<
1—
4n

r
si n^ aic-
5g(x)cosx ix
2 ^F(n)qn
n« 1
I
( 2n-l)
1
n= 1
(6)Oü
L € T US NOW TAKE, APTES, H E R M I T E , TWO SERIES
,.Sn-l
* _ 2 y ________i
n
l< 2K 2 .	2S x
si cl am
n2
90(x)co SK
- 2 Z
nq‘
L- (1 - q2“-1)8	" ¿-1 - q2n
n» 1	n-1
cos2ox, (6')
*	- r a* r>
1 + ( 1 + q2) cos2x + . ».	■ ( qr "“r + q ) cos2rx ♦ .,
MULTIPLYING TOGETHER THE TWO EQUALITIES AND INTEGRATING WE OBTAIN THE FOLLOW ING EQUATION NUMBER ( 7 )
2X2 ( .	5, 2XXft , s	a	^V" Oqn	Oqn i n*+n nS-n,
* 3Lir^ - 2_7^Hq tq
n q4 O	n=l	n=1
From equalities (6) and (7) follows
,rw.s „ if	. 1 Ff 0 T.il’tn . n'-n)	,„v
2_f(oI<j»--^o	3 z'WsLTrr^if -3^ 7-	'85
n= 1	n= 1	n- I
From formula (3) we may deduce several of Kronecker’s formulas. r-^7, To this effect multiply both members of the equality byv	. Denoting by cf( a) the difference between the number of div-
\ K
sors	of of the form 4b+ I and the number of divisors of
the form 4h + 3, we obtain
T { F( 0) v+ 2F( n - l2) »+ 2F( 0 - 22) ♦ ... } qn - | Y <p(2o - Dq2"'1
»<■*1—0	t'j 1	I-- -
n=* 1	n« 1
^(2o-l)q4»-8
n- 1
n = l
In the first sum the arguments of F in the coefficient of qn
MUST 86 DIMINISHED UNTIL NEGATIVE VALUES RESULT.
Equating the coefficients of equal powers of q in equality
(9l, WE IMMEDIATELY OBTAIN THREE OF KRONECKER'S FORMULAS:	1)
FOR ODD POWERS; 2) FOR POWERS THAT ARE DOUBLE AN ODD,* 3) FOR POWERS THAT ARE DOUBLE AN EVEN.
In THIS CHAPTER WE SHALL DENOTE ODD NUMBERS BY IS. THEN EVERY NUMBER WHICH IS THE DOUBLE OF AN ODD WILL 8E EXPRESSED BY 2«3 AND THE ONE WHICH IS DOUBLE AN EVEN BY 4n.
Let us now arrange the right hand member of equality (9) according TO POWERS OF q AND LET US FIND THE FUNCTIONS WHICH ARE THE COEFFICIENTS of the DIFFERENT POWERS. FIRST WE HAVE ThE SMM
.LET $(m) DENOTE THE SUM OF ALL DIVISORS OF A NUMBER IS.T'hEN EVIDENTLY
ôO
2§(œ){qm + 2q2“ * 4q4“ + 8 q8* +
.} .
If i(n) denotes the sum or all divisors of a, X(n) the sum of all oad divisors of n, and if n * 2am, then
Ha) =	$(m)f	$(0)	=	+2+l)$(!i!),
•$(“)	=	2'a_1{$(o) ♦ X(n)}.
AE MAY THEREFORE WRITE
2$(æ)qŒ + 22#( to) qÊB1 ♦ 22{ *( n) + X(n)} q4n.
Consider now the other sum of equality (9):
HfJr- a«P - sr(,'‘*2,*2,'<■”«> ♦
IN THE fcAST PARENTHESIS THE 0 th POWER OF q' ' W I L L L 0 C C 0 R I I F T IS A
civrsoR of n, smaller than /n and of the same parity with o/r. 'The coefrioient of qn will be, in this case, 2r. The quantity qn
WILL ALSO APPEAR :IN THE PARENTHESIS IN CASE 0- CS. LET ¥( 0 ) be
the difference between the sum of those divisors of n which are treater than /a and the sum of those that are smaller than /a. LET US C0N9T DER THE EXPRESSIONS	“ ï(m) AND 2$(ll) “ 2f(lO.
In THE FIRST EXPRESSION WE 'HAVE TWICE THE SUM OF THOSE DIVISORS OF ID .WHICH ARE SMALLER THAN /¡fl, AND TO THIS SUM ¡IS ADDED /it IF
IT is rational. In the expression 2$(n) - 2^(o) we have twice
THE SUM OF ALL THOSE EVFN DIVISORS OF 4n WHICH ARE SMALLER THAN V^4n AND SUCH THAT THE COMPLEMENTARY DIVISOR IS EVEN; IN CASE 0 IS AN EXACT SQUARE, 2/0 IS ADDED TO THIS SUM. CONSEQUENTLY THESE
\ u
TWO EXPRESSION'S REPRESENT THE COEFFICIENTS OF THE o'*1' POWER OF q IN THE LAST SUM. TAKING INTO CONSIDERATION THE PRECEDING RE-MARKS'WE MAY WRITE
2{2F(o) + 4F( n *- 12) + 4F(n-22) + .-..}qn	=	2{§(ic) + ?(m)}q*
+ 42$(io)q2ra ♦ ZliUa)■* 2X(n) + $(<0} q4n ♦ 2<p(2n - 1) q2n_1
+ 32<P(2o -	(10)
WHERE THE SUMMATIONS EXTEND OVER ALL POSITIVE INTEGERS 0 AND POSITIVE ODD INTEGERS !5.
Comparing the coefficients of the same powers of q we obtain
2F(m) +4FU-12) + 4F(iE - 22) + ...	= $(u>) ♦ ’J'(m) + q>(o),	(11)67
F(2in) + 2F( 2m-l2) + 2F(2m-28) + ... * 2$(m) +	(12)
F (4 n) + 2F(4n - 12) + 2 F(4n - 2 2) + • * - = $(n)+3 X(n)+?(n)	(13)
§ 30. For the derivation oe the other formulas of Kronecker
WE MAY USE THE SAME EQUALITY (6) REARRANGING SOMEWHAT ITS FIRST
member. Noting that
2aX
A 2ax . n $2(x)	k	k 2ax
k sin^aa----9„(x)	= —£-~..-Sax)	- -—	sm am—— co a	am—-9	(x).
*	y £ *	re	jt 1
n
> (x)
WE MAY WRITE EQUALITY (6) As FOLLOWS
n
v— i sin atn-^cos ajn::^49, (x)cosx ix - 2SF(fi)qn- 2q^2n-1-	(14)
n	n 1
rt2 Vq/2^k1
Let us multiply together the following two equalities and integrate the result:
2 .2
a &
n*
2K x
rsid aiG -—cos am
2K x
n
o V-..*3___3ir,rx
u L-l +. 0?r	’
r= 1
i	-1	5
q 491(x)ccsx =	^(-l)r* sin2rx(qr r -
2
r ♦ r v
q J;
r » 1
p o Tt
X2;< n Yq 0
a
2 a x
:> 7 „
A A
8iD a
a is •—— co 3 am —— 9, ( x) co sx Jx n	n 1
r"(-l) rrq
1—	1 + q 2 r
L *	c
/ ^ r 4*r .r - r>
a q - q / •
r= 1
From (14) and (15) follow
3SF( n)qn - Sq( 2n_1;
^7~r / **(-q)
U' £_
r * qr ~ q~ r cr + cfr'
(16)
(16)
22{f(n) - 2F(n- l2) + 2F(n -2s) - 2F(-n -3s) +	qn
- 2p(2n- l)q2l>*1 - 2F©(3n- l)q 4n“2+ £r(-q)r {-1+ 2if r-2q4r + 2q6r-2q9r+ . . .] , (17) 1) In order that the power of q (to odd) enter into the
expression
f(“q)r {-1 + 2q2r - 2q4r
/J q
6 T _
* * * •} >
C MUST BE A DIVISOR OF ID NOT GREATER THAN /¡D , If p IS CONGRUENT TO ITS COMPLEMENTARY DIVISOR MODULO 4, THEN qm WILL OCCUR WITH THE + SIGN, AND IF p IS NOT CONGRUENT TO ITS COMPLEMENTARY D I V-tSOR- MODULO 4, if WILL OCCUR WITH THE “ SION. THE FIRST WILL BE TRUE WHENEVER ID IS OF THE FORM 4p + 1, THE LAST IF ID IS OF THE PORM 4p+3. NOW ONE CAN EASILY SEE THAT THE COEFFICIENT OF THE m 4 POWER OF q.lN THE LAST SUM OF EQUAL I T Y ( 17) IS
(-1) *' ra~1^ (id) -$(id)}.
x u
2) 2lE0U POWERS OF q DO NOT ENTER INTO THE LAST SUM.
WE OBTAIN TWO MORE OF KRNOECKER’S FORMULAS:
2F(m) - 4FU-12) + 4F(id - 22) - 4F(id-32)’ + 4F(id-42) - . . . .
*	(-1) ^	(id) -	+ <p(m),	'	(13)
F(2id) - 2F(2m- l2) + 2F(2m - 2s) - 2F(2m-32) +	-<p(m). (19)
WE SHALL REPLACE THE EQUALITY FOR
F(4n) + 22(-l)pF(4n - p2),
3 Y A MORE ELEMENTARY ONE.
§ 31. CoLL EC TI NG : I N FORMULAS (9) AND (17) ALL POWERS OF q WHOSE EXPONENTS ARE DIVISIBLE BY 4 WE OBTAIN
23g(0, q4)XF(4n)q4n + 42q(2n“1) #2F(4'rt + 3) q4n + 8
- n V~ nq4K . , y nq4n8(l ♦ g4n)
8 L. 1 - q«n ~ *—	1 - q4n >
2^g(0, q4)2F(4fi)q4n - 43q( 2n_l) S?.F(4'n + 3)q4n+3
V" 4“* q2R - q"2n
= 2 ) Oq —4--------——— •
L—	q2n + q-2n
Adding these equalities we obtain
5”—'"	_	4 ft
4^g(0, q4)2?(4n)q4n	-	8
4 r. 1 * q
4	^	1 -• q® n *
If in this equality we replace q* by q, then the last two sums will be transferred into similar sums of equality (b) WE MAY WRITE THEREFORE
2F(4o)qn
22F(n)qn
„	( 2 n-1)
¿•q
(20)
From equaliit/ (20 odd square
WE CONCLUDE THAT FOR EVERY 0 NOT AN EXACT
F(4n^
(21)
AND FOR EVERY m A PERFECT SQUARE
2 F ( o) ;These equalities make it possible to determine
F(4n) + 2Z(-l)PF(4n - p2) .
§ 32. Let us make one more transformation of equality (6): 2K x,
si n2 ais-
U*)
1 x la 2X x . . ^„(x) - rsA am-----&„( x)
22’
1	1	9fr y	9 Tf y
=	—5^o(x) - -~-^.cos a!5~i	( x) .
k2 2	k/x	rc	n a
Consequently equality (6) may be replaced by the following ore:
/ v n	_	( 2n~1 ) 8
22F( n) q - 2qv n
- ¡Jl
n V2n
K/ak r5
■y ■ _ ■	| CO S f?!D'
2.'Ix
'qn V2 n
\ a,r2_ll^ ( x) co 3X j x ,	( 23)
n	n 3
Up to the present we have considered only function F(n). Now
WE SHALL INTRODUCE A NEW FUNCTION G(n) DEFINED AS THE NUMBER OF ALL REDUCED N0N-SQUI V ALENT FORMS WITH DETERMINANT -0. THEN THE NUMBER OF NON-EQUIVALENT FORMS WITH DETERMINANT - 0 AND WITH TWO EXTREME COEFFICIENTS EVEN WILL BE GIVEN BY 0 (ll) -	F(n),.
Let us examine the integral of equality (23) in the same manner that WE EX ami NED.-:.THE INTEGRAL OF EQUALITY (6). WE HAVE
2Tk
'CO s a ID
o " V
■ 4Lrr^003(2r-15
$3(x)	=	1 + 22q8 eos2sx.
(24)
(25)
Multiplying (24) and (25) term by term we obtain terms of the form
„ a /q2r-"l
4q‘
r^j~{ cos(2s + 2r - 1) x + co s(2s - 2r «■ l) x} .
1 + q2r'
Collecting all -the terms having cos(2o+l)x:
E“ /q 2 r - i r
i"."08r-i
r~ 1
j/q2r-i r (v„n„x) *	(*♦»)
FFriSq	+ q
■}
4cos(2n + l)xy q
r= 1
(n + l)2 + (r-i) * vq
_ / -{ 2 n +1) ( 2 r- 1)	/(:gn + 1) ( 2r-l)
Vq	+Vq
4cos(2o + 1) x(-l) nq^	-
/^r+i +
=	2(-l)Vn+2) cos(2n+ l)x{l - 2q~x + 2q*4 - 2q"9 + . . . + 2(-l)nq"n* VO)..ThERF.FORE THE MULTIPLICATION OP (24) AMD (25) GIVES
1 c—-	a a
5 \	\	-	. n+a n +n-a
1— L~	^ eos(2n + 3)x.	(26)
¡2Zk. , .	2Xx
----9- ( x) CO 3 à IQ-
V n 8	n
n=0 a=-n
2Xx
From the series for A am------ we deduce
n
2X 2Sx	f~ Vf r( 8b + i) .	_	..
— A am-—co sx = cosx + 2^	) (-1 ) q	{ocs(2r+l)x +
Tt	Tt	. ®r"r ,	■
r=1 b = 0
co s(2r - 1) x} .
From the equalities (2:6) and (27) we obtain
(27)
n
VSk f	2a X 2a X
] cos am-- A am--9 (x)cosx ax =
re2	q
oc
Il l I (-]1
1 1
- +
4
4 n
n ♦ a + b n * ♦ n ) 2b + g)«*a5
n- 1 b = 0 a-~n
+ 2.^	^ jp ( |'jn'fa+bqna+n','(;n+1)(;21:, + 1)“al
(28)
n=0' b = 0: a=-n
WE ASSUME THAT a IS POSITIVE AND WILL EXPRESS THE EXPONENT OF q IN THE FIRST TWO FORMS.
(n+b + a+ l)(n + b-a+l) - (b+1)2	=	2n(n-*a+b+l) - (n-a)2.
Ipb+lsn-a, then
2 2 (n+b-a+l)x + 2 ( b + .1 ) xy + (n + b + a+ l)y
WILL BE A REDUCED FOtFM WITH MIDDLE COEFFICIENT POSITIVE AND WITH THE TWO EXTREME COEFFICIENTS OF THE SAME PARITY. I F WE VARY 0, b, AND d, THEN, SINCE A CHANGE OF SIGN OF a WILL ONLY INTERCHANGE THE EXTREME COEFFICIENTS OF THE FORM, EVERY FORM (WITH THE EXCEPTION OF CERTAIN FORMS ABOUT WHICH WE SHALL SPEAK LATER) WILL OCCUR TWICE.
• ' I f b + 1 > n-a, then
2nx2 + 2(n-a)xy+(n-a+b + l)y2, (n-a+b + l)x2+2(n- a)xy+2ny2
REPRESENTS ALL REDUCED FORMS WITH ONE EXTREME COEFFICIENT EVEN AND THE MIDDLE COEFFICIENT POSITIVE. IF A FORM HAS ONE ODD COEFFICIENT, THEN IT WILL OCCUR TWICE, OTHERWISE IT WILL OCCUR FOUR TIMES.
(_l)n+a+b $HQWS THAT TQ THE FORMS WITH TWO EXTREME COEFFICIENTS EVEN THERE CORRESPONDS -1, AND TO OTHERS +1.61
From this we conclude that the COEFFICIENT of qn IN the first sum WILL BE
-{F(fO - 3M(n)}' + correction
where	id bhe number of re due si non-equivalent forme with
two extreme coefficients event and the correction depends on
FORMS OF SPECIAL TYPE.
WE SHALL EXPRESS THE EXPONENT OF q IN THE FOLLOWING TWO FORMS
(n + i) + a + 1) (ft + a- a + 1/ - b ~	=	( 3o + 2)(n + 0 - a + 1) - ( n - & + 1) 2.
IT IS EASILY SEEN THAT IN THIS CASE THE SECOND ONE WILL GIVE THE SAME RESULT AS THE FIRST ONE EXCEPT FOR THE CORRFCTI 0N.
Thus, both sums will give
4F(nV- 3G(n) + correction
A 3 THE COEFFICIENT OF q',
1) Forms of the t yp e ( 2c, c, i) = 2cx* + 2exy + ¿y*: w i ll occur
half AS often AS THOSE NOT OF PARTICULAR TYPE, IF WE ASSUME THAT THE MIDDLE COEFFICIENTS ARE POSITIVE. SUT FORMS (2c,C,d) AND ( 2C, -C, i)- ARE EQUIVALENT; THE R E FO R E THERE WILL.BE NO COR RECT I 0 M FOR THESE FORMS.
2)	Forms (c,0,i) with two coefficients odd will occur, twice
IN THE SECOND SUM IF b * 0, THOSE WITH ONE COEFFICIENT ODD WILL OCCUR TWICE IN THE FIRST SUM IF U = ± 0, THOSE WITH TWO- COEFFICIENTS EVEN WILL OCCUR TWICE IN THE SECOND SUM IF b 3 AND IN
the first sum of a-±n,; Thus these forms also require no correction.
3) Forms of the type (c,0,c) with c odd are possible only in
THE SECOND SUM IE b-0 AND	AND THOSE WITH C EVEN IN THE
SECOND SUM IF 6 = 0, A -3 AND IN THE FIRST SUM IF A- ±0, b ♦ 1 * 3fl
The sum 4P(n) - ?G(n)i qn must then be give* a correction
1 V1 . ( 2 n - 1 ) *	'3	4 n
O /	*	+ O )	*
4) The form (2c,c,2c) occurs twice in the first sum if a = 0,
n = 0 + 1, AND twice in the second sum if s~ 0, b = fi + 1. CON SE-
a
QUENTLY WE MUST-INTRODUCE A CORRECTION +22q3n	.
n = l	n* 1	n»tIF WE COMPARE EQUALITIES (23) AND (29), W E SHALL SOLVE THE ROSLEM OE THE NUMBER OE DECOMPOSITIONS OF A GIVEN INTEGER INTO HE SUM OF THREE SQUARES. SINCE THIS THEOREM W4LL MAKE CERTAIN EDUCTIONS EASIER, I SHALL, FOR THE MOMENT, DISCONTINUE CALCU-ATIONS FOR ITS SAKE. FROM (2?) AND (29) IT IS EASY TO DEDUCE
- + 0*
¿.v	¡¿x
ioti
n V n
£ TV *J *■ * i * iJV" « 8^,
Sn
L
x, y, z ~ **oc
( P.n- t) •
q
+ 12S{2F( n) - G(o)}=q..
fl i MUST ALSO REMEMBER THAT
2F(n) - G(n) *. F(n) - M(nO,.
(30)
Thsorbm. Me number of solutions of tiie equation x8 +y* + z8 * n n’iere n Is neither an exact square nor three times an exact square, is equal to 12 times the difference between the number of reduced non-equivalent forms with at least one extreme coefficient odd and the number of such forms with both extreme coefficients even; the determinant of all such forms is -n. If n is an even square, then to 12 times the difference just mentioned we must add ft; if n is an odd square we must subtract 6; if o is 3 times an exact square we add 4.
§ 33. Formula (50) may be employed for the deduction of a formula SIMILAR TO (11),	(12),	(15),	(IS), AND (19), BUT RELATED TO
THE FUNCTION G(ll). FOR T HAT PURPOSE LET US MULTIPLY BOTH MEMBERS OF THE EQUALITY BY 9gO»q):.
i29g(0, q)2 {2F( fl) - G(n)} qn
,.1 v
rfciV
- 9.(0, q) + *39 (0, q)2q
(gn-i)
- 69.(0, q)2q4n - e.9Q(0,q)2q

(31)
Let us equate coefficients of q where ra is odd.Using a certain FORMULA*, WE SEE THAT T H E CO E F F I C I E N T O F q13 IN (4'\e)/rta IS
That is the formula
* 1 +
-	1 + ~j
_ - 4 _.9 _ 16 , 3q + 3q ♦ 2q + 2q + . .	
8q + 16q2 + 24£	"K? v4 •JVj »i «f ■ «»*»■! fim
1-q l+q2 1-qS	W
•-. g	o 0
ü‘t_ , . ü9 ,	üq
(1- q)2 (1+ q8)2	(!->)’
c;q>
= 1 + 82p(p){(f + 3if» + 3q4p + fa1*9'* 3qsep +-------}
WHICH THE'; ' .THOR QUOTE.: IN HIS INTRODUCTION FROM A PAPER feY JAC08I IN OSELLE1S JOURNAL
v.3, p.30y.‘ Translator's note,8$(in), Let us denote by H'C'c) the difference between the number of divisors of w of the form 3k + 1 and the number* of divisors of rc of the form 3k- 1. The results of page 8 show that the
O	*
coefficient of qm in the e xp ft e s s i o n 8 $ g( 9, ;q) 2 q	is EQUAL to
8 ( m ")	IN CASE B) IS NOT AN EXACT SQUARE.'		ÎHU S : 1 F IB	IF NOT AN	E X*
ACT SQUARE WE HAVE					
oF(ia)	+ 122 F (in - p2) - 3 G ( to ) -	82G(m - p 2)	- 24(l0)	- 2V (in),	\ i
WHERE	■IN THESE SUMMATIONS p,	AS BEFORE,	TAKES ALL	INTEGRAL	V AL-
U E S FOR 'WHICH M - p 8 > 0 .
| F, HOWEVER, It! I S AN EXACT SQUARE WE MUST ADD 3 TO THE SECOND MEMBER OF (3 2). ÇUT IT SUFFICES TO DEFINE G(0) = 1''2 IN ORDER THAT THE FORMULA CONTINUE TO HOLD EVEN IF IB IS AN EXACT SQUARE,
Let us then, from now on, tut G(0 ) = 1/ 3.
Multiplying (11) by 3 and subtracting (32) from the result w. obtain
30(is) + 6G(>n -l8) + 30 (m -2*) + 3G(œ -3*) + oG(ra - 4 ) + . ...
3	4(rc) + 3!{(in) + 39(10) + 2f(io).	(33)
•§ 34. LET us differentiate the equality (24). Replacing q BY
q IN the result and using the notations
„	O',	0
K(0,q’)	-
b2(0,qs)
O' I
¿J *J i.
a
AND IB AND P for ODD NUMBERS, WE OBTAIN
4Gel
AX	3LX
si it arc----- 1 air,----
4 ) -----r-sin ic x;
L-1 + q2m
$2(x, q) sln2x =	4^^”	^ ~ 4^* + mj>s;i-rt m
sin ara
A arc
^g(x>4)
sio2x ix
m- 1
1 - q3"‘
1 + q2 m
(34)
The right hand member of equality (34) may be written
2I,!(qT’‘ .	. . . , i
Let s be a number of the form 3N + 1. Let us denote by 4'(rc) the sum of all	divisors of	rc of	the form 3It ± 1 diminished by
the sum of the	diuisors of	is of	the form 8Lt ± 3; by ?* (rc) let	us
denote the-sum	of divisors	of it	of the form 3b ±1 which are
¿.renter then /IB and divisors of » of the form 3ii ± 3 which are smaller than YTt, diminished by the sum of divisors of ty e form 3b ±1 smaller than /£ and divisors of t he form 3b ±3 greater, than /liT.'The expression ^' Cm) - fr(sr.) is twice the sum of the divisors *
OF III OF THE FORMOli±i WHICH ARE SMALLER THAN v'TT, DIMINISHED BY TWICE THE SUM OF THE DIVISORS OF ffi OF THE FORM S 11 3 WHICH ARE SMALLER THAN vTn . [ I F 10 = (8ii ± l)2 THEN i>'(tn) -	' (i£ ) , CONTAINS 8l1±.l
once; :if a a (80 ± 3)2 then $'(id) -	(a) contains -(30.t 3) once.
1) Let s5=18N+l = iS, is 6, If i = 8n ± 1, then 5 = d (mod 16) and
i WILL ENTER.INTO THE EXPRESSION OF THE COEFFICIENT OF q8/4 IN THE SUM (34) WITH THE + SIGN. I F i = 8h ± 3, THEN 5 ^ d + 8 (iflOi 16) AND d WILL ENTER INTO THE EXPRESSION FOR THIS COEFFICIENT WITH THE- SIGN.
The coefficient of qa^4 in the right hand member of (34) is, thus,
2) Let s = 16N + 9 =• iô, 15 6. If i=3h±3, then ô 5 i (mod 16) and
i WILL ENTER INTO THE EXPRESSION FOR THE COEFFICIENT OF q8^4 IN THE SUM (34) WITH THE + SIGN. |F 1 = 8 ¡1 ± 1, THEN Ô^i+8 ( îflO i 18)
IT IS REQUIRED TO EXPRESS THIS INTEGRAL SO THAT THE CO £FFIC I ENTS WILL INVOLVE ONLY F(o) AND u(o).
(s) - 2iI,,(s).;
Equality (34) may therefore be written
2, n
o
(35)
n
sin am
LOOK FOR THE COEFFICIENT OF 3in2llX IN THE LAST SERIES:
oc
m» l
1) If n is even we obt.ain
+
2) If a IS ODD WE obtain- OF
'(3'J 'P)	=
(O'j'F)V»
ZÌ
zRo + fix}?. + axp
SNOIXVION 9NIM0VI03 3HX ‘ S S 3 N 3 3 I ¡3 S 30 3MVS 3HX «03 ‘ 3 S il MON 3fó
*_(* -11) - (* -q+ü)(I +Ö2) = c, - (p -q+u)(i+e>q +u) = (b Vu)**
o	3
' (I -* -u) -{? -q +w)(X -US) = _(t +Q) -(p -q + u)(i + * +q +0) = (* V^1)8*
(«&)<
V»-u) - -q + u)u£ = q - (e +q + u)(q + e + u) = (®Vu)su
3	3
' (I -» -o) - (e -q+u)(s-ug) =	+q) - -q+u)(e+q+u) = (B'q'u) *
3	ü
:o«3Z a o 3 a ti i s od si ?	i V h i y niwns sv	‘ in « o 3 9nimotio3 3 hx
NI SXN3N0dX3 3 H X	30 S«3XHVflÒ	3S3H1	SS3«dX3	T1VHS 3 AA ’INS03 N IV X
—«30 V 30 XNVNIWH3X30 3 H X S 0 N I IN SV XN3N0dX3 H0V 3	30 HX«003	3N0
«3aiSN00 nVHS 3,V, ,*i? AS 319ISIAI0 3dV (S'¿ ) AXi:"lVnÇ>3 30 N 0 I S -S3«dX3 X3X0VHS 3 H X NI b 30 S«3M0d 3 H X 30 SXN3N0dX3 3 H X HV
*1 - ,tI + «S) - (I + qS)(Z + n) + ufr = (8#q'u)*y*
V	G
‘I - (T + ÇS) - (T + qS)(S-wt) + • gufr = -(B'q't>)8 u*
'gB - (t +	qS)u +	ci - gU	=	(e'q'u)3u
•b	- (1+ q 2 ) ( I	~u) +	u - u	= (B'q'u)Tu
3	o
r	«-=® o=q x=«
(3t)	I 1S *
I+Xl-=B :0=q I=u
(«
b (H 7 77s- b (H 7 7 7
'q'«)«U* qU .-A A -A (*-q*ti)^ qu ; __i A -A
I- U ao ao I+u-s* O q 3 sài
^ 7
s + r
I -U
t-u B « 09
(¿2)
r
ó
u		b g u
-	 üiÇ ü TS i		
xqc		IJI*
	u	3
		T -4
guts^ )	..xj» cr cr +	-Is
	43	00
(9g)
u	u
XgUTS =	XgUTS —— MB y
w •	• y r- r?
* ¿O
'IS
x(S - u>)uts [
<XÜÎ?UTS f	, b£ +
S ■ l8 (ï-«3)-
• » »
bg
6-
T -
t>S]
\xp
l =U
b7 *
(X--U	-
(b'x)
3S
li
&£ + I]
aie u T.s
(b 'o)3çs
V rr<y	* Trf<7
* JC?	L iw*»
oo
ACCEPTING THESE NOTATIONS WE SHALL DETERMINE THE COEFFICIENTS
A
of certain powers of q (for example q ) in the four sums:
2S2(-1) * q1*3 ^

i

(40)
IN- WHICH THE SUMMATION IS EXTENDED OVER THE SAME LIMITS AS'IN.E-QUAL I TY (38) .	*
1) If b+1 s n-a-1, take ni(n,b,a) the first of the expan-
0I VE
SIONS (39), OTHERWISE WE TAKE THE SECOND ONE. THE FIRST EXPANSION A FORMS EITHER WITH TWO EVEN EXTREME COEFFICIENTS, OR WITH TWO (TDD EXTREME COEFFICIENTS;- THE SECOND EXPANSION GIVES FORMS EITHER WITH TWO EVEN EXTREME COEFFICIENTS OR WITH ONE OF THE COEFFICIENTS EVEN
AND THE OTHER ONE ODD. AT THE SAME TIME, IF ONE OR BOTH OF THE EX-
b	\
TREME COEFFICIENTS ARE ODD, THEN (“1) * (“1) . IF BOTH ARE EVEN
then (-1)*3 = -(-1)A, Taking into consideration the fact that in THE SUMS (40) a may be negative and that the forms (i,f,c) AND
(i,-f,c) ARE NOT ALWAYS EQUIVALENT, IT CAN BE SEEN THAT THE COEFFICIENT OF q^ • IN THE FIRST OF THE SUMS (40) WILL BE
(-1)^{4F(A) - 30(a)}: + correction
WHERE THE CORRECTION DEPENDS ON CERTAIN FORMS OF A PARTICULAR TYPE.
2) The coefficient in the second one of the sums (40) will be
GIVEN BY THE SAME FORMULA.
3)	If n-a-1 ë b+1, then the first representation in (39) of n8(n,b,a) will give reduced forms with one of the-extreme coefficients ODD AND THE OTHER ONE EVEN; IN THIS CASE ALSO (-1)^ =
-(-1)^. If, however, n-a-1 < b+1 from the second representation
IN (39) WE OBTAIN REDUCED FORMS WITH ONE OF THE EXTREME COEFFI-. CIENTS ODD AND THE OTHER ONE EVEN, OCCURRING ONCE AND THE FORMS WITH BOTH EXTREM.E COEFFICIENTS ODD OCCURRING TWICE; IN THIS CASE
(-1)° = ->(-l)A, Therefore the coefficient of q'^ in the third of THE SUMS (40) WILL BE
- 2(-l)^F(û) + correction.
j \ -r	A
4)	The coefficient of q in the fourth sum will be given by
THE SAME FORMULA AS THAT IN THE THIRD SUM.
The RIGHT HAND MEMBER OF EQUALITY (33) MAY THUS BE WRITTEN
^g(0,q)<l + 4 ^ (-1)^{2F(A) - 3G(A)}	+ correction^-. (41)
1	A»i	J
For the determination of the correction we must consider forms
OF A PARTICULAR TYPE.:'7
1) Forms (c,f/,e). If c is odd then in the sum (41) there if a term + 45^vC ’ 0,c'\ But, from the sums (40) taken twice wf
CAN OBTAIN ONLY *	, Fc R IN THE S. CCO H C SUM THERE OCCUi
A TERM	WHICH IS OBTAINED FOR a=D, b~Q; AND IN THE
THIRD SUM ♦ 2'q^0>® >°\ WHICH IS OBTAINED FOR a = 0 “ 1, b L“ 2 0 I' AND a = -0, b = 2rt“2. WE MUST, THEREFORE, CORRECT THE SUM (41) SUBTRACT IN 6 FROMIT
0-4	. or.3 6 , o.jOO .	.-,,19 6 *
IF C J S even, them in. the SUM (41) THERE OCCURS The term
-,	4A (c, 0 f C)
, 4à(c,0,c)
“ 3 4
But from the sum (40) taken twice we obtain only namely: -2q/*~v"0:
’ from	THF FIRST SUM FOR	à ~
	» „ 4A(c,0,c)	
, b - 2n -	3 ; A N D - q *	F RO
W E MUST,	therefore, add to	(41)
S{q
4* 4
4 ♦ 16	4*36 .	, of 4*4 L 4*16^	4*36-
q +q	- 2 {q + q + q +
• • ».
LET US NOW CONSIDER THE EXPRESSION
Sg(0,q) + 4Sg(0,q)	(-1)U{2?(A.)
A= l
HQ
(b)}q * + correction. (41* )
It is clear from the preceding that a part of the correction arising from the forms ( c, 0, c ) will be
WHERE X IS ANY ODD POSITIVE NUMBEh, AND 'J IS ANY POSITIVE NUMBER. We shall denote by cp'(s) half of the number of all solution o f the equation x# + 64y8 = s, counting also negative values of x and y, where s is a number of the form 81' +1. Then the fourth
(the LAST) SUM MAY BE WRITTEN AS FOLLOWS:
is
22 -fT'(s) + 4$'Cs) q4 - '3^f,(3, q) .
2) The form (2c,c,3c)- occurs in the sum (41) with the coefficient +12 WHEN 0 IS ODD AND WITH THE COEFFICIENT -12 WHEN C IS EVEN. In FACT IT CAN OCCUR ONLY IN THE FIRST ONE OF THE SUMS (40
for a = 0, n = b+2 and in the second for a * 0, n=b. Let us dene t by y’(s) half of bhe number of the solutions of the equation x8 + 3*,64y2=s. The correction from the form (2e,c,2e) in (41') will be (x is odd)
<er V~ ¿(x#+48ys)	_ 3
lb ;	/ -q	+ 3q4
4( x * + 19 2y 8)
x- 1 y= 1X &
*	-4$2(0, q) ♦ 82{- ■?( s) * 2v|/f { s) }=q4 ..
COMPAM N3 EQUALITIES (35) AND (33), WE OBTAIN
X	f-	\ a
(3) - ?<(s^qi
S * 1
“3^g(0, q) -	^fp(s)q4'
s>* 1
*	i	“	I	“	i
4 J%’(s)q33 - 4^t(3)q48 ♦ 3^~4''(s)q381
3=* 1
S- 1
oc	N
+ 29g(0,q)	(-D!‘i2F(a) - 3CU)^q4\	(42.)
A=1	L
To EQUATE THE COEFFICIENTS OF THE SAME POWERS OF Q, WE MUST l) MULTIPLY THE LAST SUM BY $ (0,q)- AND 2) EXPRESS A IN TERMS OF ¡3, WHERE A IS ANY POSITIVE NUMBER, AND 3 A POSITIVE NUMBER OF THE FORM 8b + 1.
3V0,<,)S(-1)4$W(A) - 33<a4-i4,:' - <7 7	- saiiiK44*^
■ A -D	^
LET 4A + &- =	THEN A 9 --7-r-.; SINCE A IS AN INTEGER, II! IS NOT
an Arbitrary positive number:	if s=16t+l, then id =? 1, 7, 3, lo,
17,,,..; if s * 18‘t + 9 then ib = 8 n ± 3. Taking into consideration ALL THAT PRECEDES AND PUTTING a(0) = 4, WE OBTAIN
2*
— tnS

S “10 2 )
- 3G —■r- ^
1 O )
=	(-1)^* 1^{$,(s)“ Ï1 ( s)} + <p ( s) + 4\{/( s). - 4cp * ( s) - 8'f/’ ( s),	(43)
WHERE ID VARIES SO THAT 3 “ ID8 5 0 ANQ “-j-r-- IS AN INTEGER.
§ 35.Finally, for the deduction of the last of the eight fundamental RQRM^AS OF KRONECKER WE SHALL TAKE THE FOLLOWING EQUALITIES ;
~	y—nw.	r
rc/q3
. 2a x , 2Kx •¥~^rrS3.o a|8**r*~ A aiû-r—<
n2	s!i n	n
T ■
•sin lax,
» *
.1 + q*
P£i	il
&1(2x, q *)p03x	=	sin(3p + l)x + si n ( 2p ^ l) x q”^”,
WHERE IS. ANQ 9 ARE ODD NUMBERS. F'ROM THESE EQUALITIES IT FOLLOWS THAT
J *
ns
o:r v -a X
2Kx.
4iv ^ C .	->»**.0	vu*». v> >	o,
J sin aw 4	(Sx, q *) co sx ix
o
o Y"ÎÊ£_LllSÎ^ * Z_ 1+q8»*l
n= 069
	oe Ip <r~ von + o ) q3	O	OC r~	(3n	+ 5)		Ow - o ri!
	+ * ¿_ i + q3~3		¿_	**1 +	qBn^B		" ¿- i
	n=*0		n«0				n-0
WHERE							
\ =	(8n + l)2 + 6(3 o +	1)	+	1,	p	« .	(3o + 3) 2
cr ~	(8n ♦ 5)2 + 6(8n +	o)		1a	T		(8o + 7 )2
,8 n + 7
■,	(44)
uenotini by r a number of the form 8o + 7, we may write (44) as fol lows:.
2 n 4K k
4S"k (	. 2rU ^	21x	,0	8.
—r-	Sis', aia- à am---SA'2x, a^icosx ix
w J	n	n 1
n "?~q
OC
r (-1)^ ^(r) - r( c)\^r.
L
T- n
( 441 )
For the determination op the integral by a nother method we take the equality
^;r	o:c y	m-1 t_ *
— :Ui— S,(3x, qO =	ast-iT»“«*“ titò
ix	n 1
wx
+ 4^	2	sin2( ti + ìc) x - sin2(n - id) x} v
WE DENOTE BY t A POSITIVE INTEGER. LET US DETERMINE THE COEFFICIENT of sin2lx. WE. MUST DISTINGUISH TWO CASES. THE CASE WHEN t IS ODD AND THE CASE WHEN t .IS EVEN.-
1) IF t IS ODD THEN THE COEFFICIENT OF SÌn2tX WILL BE
2(-D^MtS - 4T,*(-i)n’^'it+i>:<j^:s!n“'tv*+8B
..-	1	^ (j
n~ l
After some computation the preceding coefficient may be written:
).
	2-5(0, q2)q^	& (1 +	ij Q
2)	IF t IS F V E N,	THEN	T HE
AS BEFORE WE DENOTE		BY ID	AN i
	OG		
-i(t-i)8
- 4^~ a+m
1	+ CSm
«=* 1
1	2	1	Q	% 5	1/4.	\ ®
=	4$(0,q2)q2	( q‘2 + q~2 +	2 + . . , q E(:	} ).
q*)	=	2S(0, q8)
|( 2n*i) 8-Sa ‘	.
sin(4n + 2)x70
« a~n-l
gna-|( 2 a +1) *	.
' aL T. qK°	(44")
n=,l &= 0
Let us multiply this equality member by member by the equality:

2 rU n

oe	oc
^q2'2 ~ m {si n(m+l) x + si n(in-l) x} .
m* l b* l
Having performed the multiplication, let us consider the iintegral
n
¿aX
(45)
4X2k ( . 2Kx A	_ 2N
—-- i sin a is-- 6 am-—0.(2x, q )eos x ix
na^q J0nn
= 29(0f f (qJP>(».-...) . qiP»(n.t,.),
C» ■»-*■	MVt.-	V
nsO b*l a = ~ n
.	3f	f	yi(,ipa<">b'*>■,
fcl »■»»■»	i	y
,	n~0	b^O	a=0
P1(n,b,a) = lon? + .32 nb + 6b - 16a2 - 1,
P2(n,\*,a) = 16n2 + 32nb ♦ 24b - 16a2 - 9,
p^(rt>b,a) = lSns + 16n+32nb + 8b-16n2-16a~l,
P4(n,b,a) = 16n2 + 16n+32nb-8b-16as-16a-9.
Now let us	express pt(n,b,a),	P*(n,b,a), p3(n#b,a), and p4(n,b,a)
AS THE NEGATIVES of the determinants of quadratic FORMS:
Pi(n>b,a) » (4n + 4b + 4a) (4n + 4b •* 4a) - (4‘b-l)2 = (3n +• 2) (4o + 4b - 4 a) - (4o - 4a + l)2,
Pa(n,b, a) = (4<n + 4b + 4a)( 4n + 4b - 4a) - (4b - 3)S = (3n + 6) (4o + 4b - 4a) - (i«n - 4a + 3)2,’
P3(n,b,a) « (4n + 4b + 4a+4) (4n + 4b - 4a) - (4b+-l)2 = (8n + 2) (4n + 4b - 4a) - (4n-4a-l)2,
p4(n,b,a) = (4n+4b + 4a+4) (4n+4b - 4 a) - (4b+ 3)2 * (8n - 2) (4n + 4b - 4a) - (4n*-4a-3)2,
llN THE EXPRESSIONS Pi(fl>b,a) AND P’*(fl,b,a), a MUST BE REPLACED BY -a ! I N CASE a IS NEGATIVE.
On CAREFULLY EXAMINING THESE EXPRESSIONS, WE SEE .THAT THE SECOND MEMBER OF EQUALITY (45) CAN BE WRITTEN
29(0,qe)2(r)q^r.71
FOR, WITH n,b AND a VARYING WITHIN THE LIMITS INDICATED IN E UALITY (45), Pi(o,b,a) GIVES US FORMS OF TWO TYPES:	1) FORMS IN
WHICH THE MIDDLE COEFFICIENT IS OF THE FORM ±(4n' +3) AND BOTH E TREME COEFFICIENTS ARE DIVISIBLE 0V 4 AND ALSO ARE CONGRUENT MO ULO 8;	2) FORMS W I T.H THE -MIDDLE COEFFICIENT OF ThE FORM ±(4n’+l
ONE OF THE EXTREME COEFFICIENTS 8fl*+2 AND THE' OTHER DIVISIBLE 6 > 4. The EXPONENT p8(n,b,a) GIVES US ALSO FORMS OF; TWO TYPES:	1)
FORMS I N WHICH - THE MIDDLE 'COEFFICIENT IS OF THE FORM ±(40,+l) AN BOTH OF THE EXTREME COEFFICIENTS ARE DIVISIBLE BY 4 AND ALSO ARE CONGRUENT MODULO 8;	2) FORMS IN WHICH THE MIDDLE COEFFICIENT IS
OF THE FORM ±(4o'+3), ONE OF THE EXTREME COEFFICIENTS IS DIVISIBLE BY 4, AND THE OTHER ONE IS OF THE FORM 8n’ + 6. THE EXPRESSION
P3(n,b,a) also gives forms of the two types: 1) forms in which t
MIDDLE COEFFICIENT IS OF THE FORM ±(4n'+l) AND BOTH OF THE EXTRE'. COEFFICIENTS ARE DIVISIBLE BY 4 BUT NOT CONGRUENT MODULO 8;	2)
FORMS WITH THE MIDDLE COEFFICIENT OF THE FORM +(4o'+3), ONE OF Ï EXTREME COEFFICIENTS OF THE FORM 8 0 ’+2 AND THE OTHER EXTREME COEFFICIENT A MULTIPLE OF 4. FINALLY P*(i1,b,a) GIVES FORMS IN WHIC THE MIDDLE COEFFICIENT IS OF THE FORM ±(4n’+3) AND 80TH OF THE E TREME COEFFICIENTS ARE DIVISIBLE BY 4 BUT NOT CONGRUENT MODULO 8 AND FORMS WITH THE MIDDLE COEFFICIENT OF THE FORM ±(4nf+l), ONE EXTREME COEFFICIENT OF THE FORM 8rt*♦6 AND ThE OTHER EXTREME COEFFICIENT A MULTIPLE OF 4. EVERY NON EQUIVALENT REOUCED FORM WITH BOTH EXTREME COEFFICIENTS SVEN WILL HAVE, THUS, ONE AND ONLY ONE
representative among the exponenes p i( n, b, a”), p a ( o, b, a), p3(n,b,a AND p4(n,b,a) OF THE expression enclosed in the parenthesis of equality (45) and therefore the right HAND member of (4$) may be written as follows:
2S>(0,q2)]T{a(r) - F(f)} q® r.
But numbers of the form 8h+7 cannot be decomposed into the
SUM OF THREE SQUARES AND THER EFORE FROM FORMULA (30) IT FOLLOWS
that G(r)=2F(r); consequently
2 TT
4a k f	2ax	‘^aX	i
Xrc-Ji sin aiD-^1— h	(2x, q 8) cc sx ix = 2$(0, q8) ZF( r) q8 r. (4A)
/an3-I	n	K
0
Let us compare (46) and (44’):
2S(0,q8)SF(r)qSr =	S(~l)ê(r"7){f,(r)'(r)} q^r*
2F(r) -4F(r-42) + 4F(r-82)-...
= (-l)^(r"7)
{$' (r) - Ï» (r)}
(47) ( 48 )
§36. Formulas (11), (12), (13), (18), (19), (35), (43), and
(48) ARE THE FUNDAMENTAL FORMULAS OF l< R P N E C K E R, GIVEN BY HIM IN THE ¡q7TH VOLUME OF QRELLE'S .JOURNAL. FOR CONVENIENCE I SHALLWRITE ThEM OUT IN A TABLE RECALLING THA'T IN THESE FORMULAS 0 DENOTES AN ARBITRARY POSITIVE INTEGER, ¡2 AN ODD POSITIVE INTEGER,
3 A NUMBER OF THE FORM 3 0+1, AND I* A NUMBER OF THE FORM 8o+ 7 :
F(4o)	+ 2F(4n - 1H) + 2F(4n - 22)	+ ... = .$( n)	+ 21 (a) +	9 (n),	(49)
F(2in)	+ 2F(2a - l2) + 2F(2m -	22) + . . . =	2$(a)	+	9(a),	(50)
F(2m)	- 2F(2m - l2) + 2F(2n> - 22) - .	. . =	-	<p(ic),	(51)
2F(a)	+ 4F(m - l2) + 4F(m-22)	+ ... = §(a)	+ ?(ra)	+	<p(a),	(52)
2F(m) - 4F(ffi-l2) + 4F(m-22) - 4F(a-32) + 4F(a - 42) - ....
3G (ia) +
=	(-i)i(m"'l) {$(!a) - ¥(m)} + . <p(®),
6G (a - 12) + 30(111-2*) + 6G(a-32) +■ 6G(m-42) =	$(m) + 3? Or) + 3<p(a) ► < 29(a),

(53)
(54)
2F(r) -4F(r -42) +4F(r-82)
(-1)*
( r-7)
{3>’(f) “	(£*)}, (55)
m
1	S?
ni®-“ ) s — ia2'
16
- 3G
s - a
A'
16
= (-1)	3-1 ^	:(s) -	(s) } + CP ( s) +49(3) - 49 1 (s) - 89’ ( s) .
WITH THE HELP OF THESE FORMULAS IT IS POSSIBLE TO COMPUTE THE NUMBER OF NON-EQUIVALENT CLASSES OF FORMS WITH NEGATIVE DETERMINANTS; Formulas (55) and (56) are expecially convenient for this
PURPOSE.
§ 37. Besides these formulas, Kronecker also deduced certain others. But these last can be derived from the ones given; they can be obtained either from (49) - (5 6), or by simple transformations OF SERIES FROM WHICH (49) -(56) WERE DERIVED. WE SHALL LIMIT OURSELVES TO THE DERIVATION OF ONLY THE MOST INTERESTING FORMULAS.
First of all we have a sequence of elementary formulas, that
IS, OF FOLMULAS WHICH CAN BE DEDUCED BY ELEMENTARY CONSIDERATIONS
IF 0 IS NOT AN	ODD SQUARE, THEN	F ( 4 n)	=*	2 F ( 0) J	(57)
If, however, jj-	is an odd square,	then	F(4m)	=	2F(a) -	1.	(53)
Fof every o	G(4n)	*	F(4n) +	G(n).(59)
If ns	1	or 2	(mod 4), then	G(n)=F(n)	(80)
If n	=	7	(mod	8), then	G(n)	=	2F(n)	(61)
If n	==	3	(mod	8) and not 5 times a square,then	3G(n)	=	4F(n);	(62)
If n	=	3	(mod	6) and 3 times a square,	then	3G(n)	=	4F(n) + 2..	(63).
For sake of brevity we shall give elementary proofs where POSSIBLEFormulas (57), (58) and (61) have already been proved.
Formula (60) shows that it is not possible to write a form o ThE type 2ax8+2bxy+2ey8 which has the negative of its determinant OF THE FORM 4N + 1 OR 4 N + 2 .
. Formula (59):	any form w i th- det erm i n ant -4n, •• i n which both
EXTREME COEFFICIENTS ARE DIVISIBLE BY 2 HAS THE FORM 2aX8fc4bxy + Cy8; THEREFORE THERE ARE AS MANY OF THESE NON-EQUIVALENT FORMS AS THERE ARE NON-EQUIVALENT FORMS WITH DETERMINANT -0 AND OF THE
form ax8± 2bxy+ cy8.
Finally to prove (62) and (63) we shall find expressions for
THE NUMBER OF DECOMPOSITIONS OF Sn + 3 INTO THREE SQUARES. IT IS EVIDENT THAT ALL THESE SQUARES ARE 0 D Di. FIRST OF ALL FROM FORMULA (30) FOLLOWS
Theorem. The number of decompositions of 8o+3 into three squares is equal to 24 F(8n ♦ 3' - 12G(8n ♦ 3),if 8o + 3 is not thre t imes an exact square; if, however, n + 3 is t'ree times an exact square, then the number of decompositions will be ,24F(8o +3) - 123 (Go +3) + 3.
TO OBTAIN A SECOND FORMULA FOR THE NUMBER OF REPRESENTATIONS BY THREE SQUARES, WE REPLACE 0 BY 2 0 + 1 IN EQUALITY (49) AND SUfcTRAC THE EQUALITY (52) FROM IT MEMBER BY MEMBER, PUTTING INTO THIS
last w = 2n+ 1. Taking into consideration formulas (57) and (58)
!»E OBTAIN
F(8n + S) + F(8n + 3-8) + F(8n+3-2i) + F(8n+3-48) +
*	§(2o + 1).
(64)
AND SUMMING
OC
oc
n=0
Let us put x = rc/2 in equality (6f)* Noting that
V" nqn
Z_ 1 -
cc
WE OBTAIN
2n+t
4n+27 4
8.) F(8n + 3)q
00
v
Theorem. The number of decompositions of 8n + 3 into three squares, counting also negative solutions, is equal to 8F(8n + 3).
Comparing this theorem and the preceding one we obtain the formulas (62) and (63)•
WE 'HAVE thus two theorems on the number of decompositions of a
GIVEN NUMBER INTO THREE SQUARES; ONE IS A GENERAL THEOREM; THE OTHER GIVES THE NUMBER OF DECOMPOSITIONS INTO THREE ODD SQUARES. TWO THEOREMS SIMILAR TO THE LAST ONE FOLLOW IMMEDIATELY FROM THE GENERAL THEOREM UPON APPLICATION OF FORMULA (60).
Thbohem. "U The number of decompositions of a number of the form 4n + 1 into three squares is equal to 12F(4n + l) or 12F(dn + 1) - 6, according as 4n + 1 is or is not an exact square.
Theorem 2. The number of decompositions of 4n +2 into three squares is equal to 12F(4n+2).
LET US NOTE ONE MORE FORMULA,DUE TO KRONECKER, WHICH CAN EASILY BE DEDUCED IF WE PUT in = 4rt + 1 IN THE EQUALITIES (52) AND (53) AND SUBTRACT THE 'SECOND ONE FROM THE FIRST. USING ALSO THE EQUATIONS ■(37) AND (38), WE OBTAIN
WHERE T(4n+1) IS THE NUMBER OF SOLUTIONS OF ThE EQUATION (2x+l)
2
* 4(2y + 1)	= 4n +1, counting all negative solutions, Consequently
for n odd x(4n + 1) = 2'p(4n+l), and for n even t(4n + 1) = 0.
§ 38. IN the preceding formulas there occurred functions representing THE SUMS OF DIVISORS OF DIFFERENT TYPES, OF A NUMBER n;
$(n), ?(n), $'(n) and ?'(n). assuming that we are able to decompose
NUMBERS INTO GAUSS COMPLEX INTEGERS, IT IS POSSIBLE TO DEDUCE OTHER FORMULAS, CONNECTING THE FUNCTIONS F AND G WITH NEW FUNCTIONS, DEPENDING ON THE SOLUTION OF INDETERMINANTE QUADRATIC EQUATIONS WITH
two unknowns. 'Thus kroneck.e-r determined
8
(67)
+2
(68)
h
* * 9 • >75
WITH THE HELP OF THE FUNCTION
(63' )
2(0) =■ ^c-n^-’V
m
WHERE 10 RANGES OVER ALL ODD POSITIVE NUMBERS SATISFYING THE EQUALITY 1' * + ID * = R IN WhICH 1 IS POSITIVE, NEGATIVE, OR ZERO. LET US NOTE THAT FOR R=3 (fflGi 4) T H 6 SUM (68) IS OBTAINED BY A SERIES OF SIMPLE COMPUTATIONS FROM FORMULAS (52),	(53)» AND (54).
TO DETERMINE THE SUM (68) IN CASE R =1 OR 2 (MOl 4) WE SHALL TRANSLATE THEOREMS 1 AND 2 OF THE PRECEDING SECTION (§37) INTO ALGEBRA I C LANGUAGE:
oc
4^r(4n + 2),n*2	,	^(0,q)9s(0,q),
n~ 0
oc
4 ¿~F(4 ri + l)qn + 4	=	S2(0, q) { S*(0, q) + 1}.
n»0-
Multiplying both equalities by 5(0,q) and recalling that 5^(0, q) “ ^2(0,q)5(0,q)5g(0,q) we obtain
2 2(-l)niP(4n + 2)qBl* + a*i
Hi n
m m ^
2 2 (-irF(4>n + 1 ) q
n Hi
« 1
n i 4-n + x
4 -r
m ni
2 (-1)* V1
u-
•K
* 1
mq
I*
WHERE 10 AND IB j — 1,3,0, O,. •»*, G * 0,1,2,3,»*., 01 “ Q , il, i.j, i 3 » , C0MPARING THE SAME POWERS IN BOTH EQUALITIES, WE OBTAIN
22(-l)hF(4n+ 2- 4h2) = Si-l)®* “"1) « = Û(4n+2),	(69)
h
22(-l) hF(4n + 1 - 4L2) = Q(4n+l) + (-1) \ (4n * 1) .	(70)
§ 39. From the eight fundamental theorems of Kronecker and from the equalities (57) - (63) many other interesting theorems can also be derived. These theorems have at times such unexpected form that at first they seem to have nothing in common with the formulas of Kronecker deduced by us. Liouville, for example, in the 14th volume of his journal gives three interesting formulas, which, in spite of their appearance of originality are corollaries of the theorems on page. 75.-
Sometimes, however, the deduction of these results as corollaries.is so complicated that it is preferable to derive them.inde-75
penDently, by the methoo of hermite, that is, by the method used IN the derivation of the eight fundamental formulas of kronecker. Apart from simplicity this method may lead to theorems entirely
INDEPENDENT OF THE EIGHT THEOREMS OF KRONECKER. TO THEOREMS OF THE LAST KIND BELONG THOSEM WHICH GIVE PROPERTIES OF FUNCTIONS MORE COMPLICATED THAN F(o) ANO G(o)^ SUCH MORE COMPLICATED FUNCTIONS ALSO GIVE THE NUMBER OF FORMS OF A CERTAIN TYPE WITH A GIVEN NEGATIVE DETERMINANT. To THE THEOREMS WHICH are NOT DEDUCED FROM THE EIGHT FORMULAS OF KRONECKER BELONG ALSO THEOREMS ESTABLISHING NEW PROPERTIES OF THE FUNCTIONS F(o), G(tl) AND	GlERSTER, FOR EX-
AMPLE, GAVE A SERIES OF SUCH NEW PROPERTIES. IT WILL THEREFORE BE PROPER TO MAKE A FEW GENERAL REMARKS CONCERNING THE METHODS USED IN THE DERI VATION OF KRONECKER* S FORMULAS.
In ,tHE PREVIOUS DERIVATIONS WE WERE DETERMINING CERTAIN COEFFICIENTS :IN THE TRIGONOMETRIC EXPANSION OF A FUNCTION GIVEN BY THE RATIO OF THE PRODUCT OF THREE .IaCOBI FUNCTIONS AND THE PRODUCT OF TWO OTHER SUCH FUNCTIONS. IN HIS TWO PAPERS ON APPLICATIONS OF ELLIPTIC FUNCTIONS TO THE THEORY OF NUMBERS, HER MITE ALWAYS DETERMINES ONLY THE CONSTANT TERM IN THE TRIGONOMETRIC EXPANSION, IT IS PROBABLY DUE TO THIS FACT THAT IN HIS PAPERS ONLY CERTAIN PARTICULAR FORMULAS WERE DEDUCED, AND ThAT ThE PROBLEM ON THREE SQUARES WAS NOT SOLVED BY HIM aT ONCE. IT IS KRONECKER TO WHOM WE MUST ASCRIBE THE IDEA OF DETERMINING THE COEFFICIENT OF COSX IN THE TRIGONOMETRIC EXPANSION, AS A CONSEQUENCE OE THIS THE CHOICE OF FORMULAS WAS CONSIDERABLE WIDENED.
IN THE PREVIOUS DERIVATIONS THE COEFFICIENT JUST MENTIONED IS DETERMINED BY TWO METHODS, AND THEREFORE EQUALITIES RESULT.
WE FIRST DETERMINE THE COEFFICIENT OBTAINED WHEN AN ELLIPTIC FUNCTION OF SECOND OEGREE IS MULTIPLIED BY A GAC08I FUNCTION. IN THIS WAY WE OBTAIN A SUM OF FRACTIONS IN WHICH THE NUMERATORS AND DENOMINATORS ARE FUNCTIONS OF THE MODULUS q. IF THIS EXPRESSION IS ARRANGED ACCORDING TO INCREASING POWERS OF q, THE COEFFICIENTS OF THE DIFFERENT POWERS OF q WILL BE CERTAIN ALGEBRAIC SUMS OF DIVISORS OF THE EXPON-ENTS OFq, /q, v" q ETC. We SAW, HOWEVER, THAT other functions are also POSSIBLE, FOR EXAMPLE THE FUNCTION Q(n).
This coefficient was determined in our deductions by another method, as follows. First we determined the trigonometric expansion of
A QUOTIENT OBTAINED FROM THE DIVISION OF TWO ilAOOBI FUNCTIONS BY A
third one. Such a quotient is no longer a doubly periodic function.
In it the coefficients of trigonometric functions are composed of
SUMS OF A FINITE NUMBER OF POWERS OF THE MODULUS q , FURTHER ON THE SERIES REPRESENTING THIS QUOTIENT WAS MULTIPLIED BY A DOUBLY PERIOD-77
•C FUNCTION — QUOTIENT OF' TWO JACOBI FUNCTIONS. In PERFORMING TH MULTIPLICATION OF THESE SERIES THE SAME COEFFICIENT'S SOUGHT FO AS IN THE FIRST CASE. IN THE PRESENT CASE THE COEFFICIENT WAS EX PRESSED IN TERMS OF q IN SUCH A WAY THAT THE POWERS OF q COULD 8 CONSIDERED AS NUMBERS PROPORTIONAL TO THE NEGATIVES OF DETERMINANTS OF CERTAIN FORMS.
In THE PROCESS INDICATED ABOVE CHANGES MAY BE MADE. HERMITE, FOR EXAMPLE, PROPOSES TO USE DERIVATIVES OF THE LOGARITHMS OF JACOBI FUNCTIONS IN PLACE OF THE DOUBLE PERIODIC FUNCTIONS.
LET US APPLY THEMETHOD JUST DESCRIBED	THE DEDUCTION OF SOME FORMULA-
DIFFERENT IN CHARACTER FROM FORMULAS DEDUCED BY KRONECKER IN HIS VARIOUS MEMOIRS.
EXAMPLE 1. LET US MULTIPLY MEMBER BY MEMBER FORMULA (3) PAGE 5? S Y THE EQUAL I T Y
°°	qn
4	(-1)	■—~sin2nx.	(71)
L—	1 - q * **
n* 1
Having multiplied we integrate and obtain:
2Xx
— “—' 7~i—¿x «	5g(x)
K'U)
V*1
4
3C	OC	CC
V" V	f	n6+n + 2nb- i(ga + i)C
L. L	L (_1) q
n=t b «0 a* — n On THE OTHER HAND WE HAVE

ndliaam—	»	^ 2n(sn-i)
- -------—-S.CxVix * 16 ) -----------
nj	ix	s	l—	1 - «4 n - g
n- l
i-l n ( n + 2b+ l) - 4 ( 2 »+ 1 )
4	4.
( 2 n- 1 ) ( 2 n + gb )
(72)
(73)
The EXPONENT OF /q in THE FIRST SUM may HERE ALSO BE EXPRESSE tN TWO FORMS
2o(2n + 2 + 4b) - (2a+l)% 2nQ«n + 4b - 4a) - (2n-2a-l)2.
If (2a + l)‘"^o , theai
2nx2 + 2(2a+l)xy + (2n + 2 * 4b)y2
REPRESENTS ALL POSSIBLE REDUCED FORMS WITH DETERMINANT 0^ THE7S
•ORM 1-8N AND WITH ONE OF ThE EXTREME COEFFICIENTS TWO TIMES AN ODD,
.NO ThE OTHER TWICE AN EVEN. |F THE FIRST COEFFICIENT IS TWICE AN DO, THEN (“l)" 1 =	+1; |F ThE LAST COEFFICIENT IS TWICE AN ODD, THEN
-i)0”1 - -1.
LET US DENOTE THE ABSOLU8E VALUE OF 2a+ 1 BY 2ai +1 AND LET US PUT ai > n, n § 2n +-2b - 2at; then
2nx + 2(2n - 2&x - i)xy + (4ti ♦ 4b - 4aa)y
WILL REPRESENT EITHER REDUCED FORMS WI T H DETERMINANTS OF THE FORM 1 - 3 N A NO.WITH BOTH EXTREME CO EFFICIENTS TWICE ODD NUMBERS (iN THIS CASE -1)	=	-i), OR REDUCED FORMS IN WHICH THE FIRST. CO E F F I C I E N T ' I S TW1ICE
xH ODD, AND' THE LAST TWICE AN EVEN (iM THIS CASE (-l)1*""1	= +1).
Finally let 2a1>n>2n-2a1+2b; then
4(n+b-ai)x2 ± 2(2o - 2at - l)xy + 2ny2
¿ILL REPRESENT REDUCED FORMS WITH TWO EXTREME COEFFICIENTS TWICE ODD ¡UMBERS, AND ALSO FORMS WJ T H THE THIRD COEFFICIENT TWICE AN ODD AND HE FIRST ONE TWICE AN EVEN. IN THE FIRST CASE (-l)**	*	“1; IN THE
iECOND CASE (-l)””1	= +1.
The triple sum of he left hand member of equality (73) is com-
4 ( ao-b *)	o 2
OSED OF MEMBERS OF THE FORM ±2Q4	, IN WHICH CS a, a ?? 4 b ' , a
AND C ARE POSITIVE, C IS NECESSARILY TWICE AN EVEN, AND b IS ODD; IF - IS TWICE AN EVEN, THEN THE TERM MUST BE TAKEN WITH THE SION-; IF,
0 NEVER, a IS TWICE AN ODD THE TERM MUST BE TAKEN WITH THE SIGN +.
let us denote by A(n) the number of all non-equivalent reduced forms it'i determinant -n and with two extreme coefficients which are twice ven numbers; by B(n) the number of ail non-equivalent reduced forms ■ith the first coefficient twice an odd and the last one twice an even; ind by Z(2n) the sum of those odd divisors of 2n which are smaller ■■hanV2n. We then obtain the
Theorem: B(3o**1) *• 8(8 a - 3♦ 8(8 o - 68) + ... + -B(8n - m2) + ...
- A (3 n — 1) - A (3n - 3R ) - A(3n~52) - ... = Z(2n),	(74)
Sxample 2. Multiplying the same equality (3) by
iioo
V— q R
4 ) ------r— si n,2 nx .
L- 1 - q2n
WS OBTAIN
n
BXk* r . 2Kx	2Xx , , , • N	v	.
' 3i n am—— cos	am —ix)dx	s 4^	£	£	q
oc. oc
n%n + gnb - J ( 2 a + 1) 8
frc^r
o
n~ l b=*0 a * - n79
oc	ae
G(8n -1) - F(8n -njqen_4 = 8
n=1	n=i
Multiplying both members of equality (75) by 9 (0)9g (0)	(0) we obtain:
9/(0)
(75)
,.,2. 2 4 a k
o
2;'x	2 A X » / ,
ats---cos am-—-9 (x)ix
n	n
f x
r* 1
'\— ,	•- r ♦ b ♦ 1 2 r *♦ r + 5? r b
y (-1) r q
b « 0
=	162(-1) * mq^“ 2F(8n - 1 )q
(76)
IN WHICH, AS USUAL, ID DENOTES AN ODD NUMBER.
Let 2n = i5, 6 > i. Assuming always that one of the numbers' 4 or 6 is odd, ‘.VE shall employ the following notation:
= *(2n).
(77)
WHERE Ì AND & TAKE ALL VALUES SATISFYING THE PRECEDING THREE CONDITI ons. Then
F(3n
Y_ r(-l)P(2P+DF(8o
»-1 p = 0

V” /-> \ 2n
£ K(2o)q ,
n* i
1) -3F(3n -32) + 5F(8n- o£) - 7F(8n - 7 2) + ... = k(2o).
(78)
On page 75 we mentioned three formulas of Liouville. Now we shall derive one of these formulas, that which contains the same functions of VARIOUS F(n) as that which occurs in the equality (78
Let us multiply both members of the equality (66) by 9	(0):
1672(-1)
-ni“-1’
lF(4m - 1) q
■♦iCi8-
l)
8 a n 3
WHERE 1 AND ID ARE ANY POSITIVE -ODD' NUMBERS. APPLYING FORMULA (31* ) ON PAGE 23 WE OBTAIN
2(-l)^(i*l)lF(4ii - i2) = 22(-l) r*a(2r + 1) (2s + 1)
= 22<a2-4b*),	(79)
where 1 is taken so THAT 4id - i ‘ > 0, (2r+l)' + (2s+0 = 2tn.
2r + 1 = ±(a+2b), 2s + 1 = ±(a-2b), a 2 + 4b 2 = in.
The last sum is extended over all positive a's and over all positive, NEGATIVE AND ZERO b * S.
§ 40. Sxampls 3. This example we shall devote to the derivation50
OF 0 I ERSTER* S FORMULAS OF THE THIRD D E 0 ft JF E. WE SHALL SAY A FEW WORDS ABOUT OTHER FORMULAS IN CHAPTER 7.
The followi NS RELATION IS E XT A 6L I S'HE D I N THE THEORY of ELLIPTIC
functions:
= lg(2q^si nx') - T"— -— --(2eos2rx + 1).
1	¿_r 1 - qr
From these series we obtain through differentiation
31gS,(x,/q)
dx
= .cotgx + 4
V" q1
L~1 -
;si n2rx,
dl,^ (x,/~q~)	1
1------3--------— S -	-
-■---^-----” 8 " I(l"-V)ii(3c0i2ni+ n-
Noting, howf.ver, that
fdl jS, (x,/q)j‘
i 1 £9 (x,/q)	„ dlg9,(x,/-q)
-X__ - o,—----------,
WE easily find
lllllÄlL^SX
X d x j
= cotg2x ♦ 8(1 - coä2rx) + 16 y-------ecs2rx,
¿~(l-qrr
dX *
2^ q“E + 4^§(n):qn + 20	?( n) qn.
n» l
n~ l
n* 1
(80)
(81)
(82)
In the last formula 4 (n) and ¥ ( n) have the same meanings as in the
:0 R M UL AS OF KRONECKER.
LET US DETERMINE ThE SAME INTEGRAL IN ANOTHER WAY;
gg(x)1L£^V2=	otjx_sy ,
dx
“	n* .	f- JL n8-(a-d)'
2^g(0)2_ q si n2ox + 4Sg(0)	2_q	sin2tix. (83)
nq sin2nx
—	n8-(a~g)
n* 1 a=-n+1 NOTING THAT
r.= 1 a=l
• If in a Jacobi function ;bhe modulus is omitted then the modulus q is understood.81
n	n n
- (sin2nx cot4xix = 1,	- 1 sin2px «io2nxdx = 0 or 1,
n J	■	n y
0 0
ACCORDING AS p IS NOT EQUAL OR IS EQUAL TO f), WE FIND
.2
“ f
nj
o

dx
* - 8
V* n •	OC oc
L** -	16z_nq
n» l	n » 1 b =* 1
<* n (o )f t	nMa-4)* q 2
n ( n ♦ b )
n — 1
♦ 2Sj(0) Y_ 1-	^

n* \ a*l
»».«>£ i
n*t a^-n+1 J*L 00 n	-
n i •
n* 1 b* 1	a=*~ n+ 1
ft
nb~ a
ft
(84)
n«l b = 1 a-l
Understanding by Ç A positive number or zero, let us consider t he expr ess ion 4n2 + 4nb - Ç2. If Cin, then 4n2 + 4nb ~ Ç 2= 2n(2n+2b) - Ç2 may be considered as the negative.of the determinant OF THE REDUCED FORMS:
2nx2 ± 2£xy + (2n+2b)y2.
Giving to n positive values, to b positive values and the value
ZERO, TO Ç ALL POSSIBLE VALUES NOT EXCEEDING n AND N'O T SMALLER THAN ZERO, WE OBTAIN ALL POSSIBLE REDUCED FORMS WITH TWO EXTREME COEFFICIENTS EVEN AND WITH NEGATIVE DETERMINANTS. IF 2n^C>n,
then 4n2 + 4nb - Ç2 = 2n(4n+2b-2C) - (2n —C)2 may be considered
AS THE NEGATIVE OF THE DETERMINANT OF REDUCED FORMS:
2nxs ± 2(2n- C)xy + (,4n + 2b - 2C)y2,
(4n + 2b — ?,X>) x2 ± 2(2n~Oxy + 2nys.
We shall denote by M(n) the number or all non-equlvalent reduced forms ith the determinant -n and two extreme coefficients even, counting every form 2px8 + 2py8 as one half and the form 2px2+2pxy +2py2 as one third (n and p are certain positive inte iers),
l& IS THEN SEEN THAT THE RIGHT "HAND MEMBER OF THE EQUALITY (34) MAY BE .WRITTEN
n-4
- 82$(n)qn + 8S?(n)qa + 1253(0)2M(4n)qn + 12S£(0)SM(4n- l)q V
Comparing the last formula with formula (32), we obtain an82
arithmet'cal theorem:
2M(4n~hs) = §(n) + f(n),	(85)
h = 0, ±1, ±2, ±3,.,>,hss 4n,
1
if we put M(0) = -
X
This theorem is a corollary to Kronecker’s formulas and its derivation IS GIVEN WITH THE SOLE PURPOSE OF MAKING IT EASIER TO UNDERSTAND DERIVATIONS WHICH FOLLOW.
LET US NOTE ONE PECULIARITY OF ThE EXPRESSION 2o(20 + 2n) — K * IN THE CASE WHEN IT IS CONGRUENT TO 1 MODULO 3» P U T £ = ±1 (iBOi 3); THEN
2n=±l(inoi 3), 2n + 2b = +1 (mod 3). I n this case, in spite of the
PREVIOUS LIMITATIONS IN CHANGING 0, b, AND C IN FORMULAS
2nx2 ± 2Cxy + 2(»+b)y2,	2nxs ±2(2n-C)xy + 2(2n + b-C)y8,
2(2n + b—C)x2 ± 2(2n- C)xy + 2ny2	(86)
WITHIN THE SAME LIMITS AS ON PAGE 81, WE 08TAIN ALL POSSIBLE REDUCED FORMS DETERMINANTS CONGRUENT OF -1 MODULO 3. FOR, IF Z =2H (mod 3)
THEN IN THE LAST TWO FORMULAS (86) WE HAVE ALL FORMS WITH THE MIDDLE COEFFICIENT DIVISIBLE BY 3 AND IN THE FIRST FORMULA ALL OF A FORM IN WHICH THE FIRST COEFFICIENT IS CONGRUENT MODULO 3 TO THE ABSOLUTE VALUE OF THE SECOND COEFFICIENT. AND IF 2ll £-C (mod 3)' , THEN + 2b = C, 4n + 2b = 2n — C (mod 3). In the first and second formulas the third
COEFFICIENT IS CONGRUENT TO THE ABSOLUTE VALUE OF THE SECOND ONE MO 0-
ulo 3* In varying n, b, and C every reduced form will be given twice BY THE FORMULAS (86)
LET US NOTE, BY THE WAY, THAT IN CASE C^O (mod 3) THE THREE FORMULAS (86) GIVE ALL REDUCED FORMS WITH DETERMINANTS CONGRUENT TO -1 MODULO 3, AND ONLY ONCE; NAMELY THE FIRST FORMULA GIVES ALL THOSE FORMS WHOSE MIDDLE COEFFICIENT IS DIVISIBLE BY 3,. THE SECOND FORMULA ALL THOSE FIRST COEFFICIENT IS CONGRUENT TO THE ABSOLUTE VALUE OF THE SECOND COEFFICIENT MODULO 3, AND THE THIRD. FORMULA GIVES FORMS WHOSE THIRD COEFFICIENT IS CONGRUENT TO THE ABSOLUTE VALUE OF THE SECOND MODULO 3. .
The PRECEDING CONSIDERATIONS LEAD TO THE FOLLOWING CONCLUSION:
IF IN THE LAST FOUR SUMS OF FORMULA (84) WE DROP ALL TERMS IN WHICH
a or 2 a —1 = 0 (mod 3) and select among the remaining terms all those
IN WHICH EXPONENTS OF fq ARE CONGRUENT TO 1 MODULO 3, THEN THE SELECTED EXPONENTS WILL FORM A SYSTEM OF REDUCED NON-EQUI V ALENT FORMS WITH
DETERMINANTS CONGRUENT TO -1 MODULO 3; AT THE SAME TIME THESE FORMS o
WILL OCCUR ^ TIMES AS OFTEN AS IN FORMULA (84).LET US NOW MULTIPLY THE RIGHT HAND MEMBER OF THE EQUALITY (33)
B Y
Q	f	r". u (i
4Ir^^3in(3p + 2)x + 4Ir^rrisin(3p+4)x*
ep + 4
(87)
p = o	p = 0
IF WE INTEGRATE THE RESULT OF THIS MULTIPLICATION, THEN ALL THE TERMS IN WHICH n6 IS DIVISIBLE BY 3 WILL DISAPPEAR. THE EXPRESS!«
p
4n(n + b)—C will be either hi (mod 3) for C = ±1 (mod 3) or hj (mod 3) for C =0 (mod 3). Multiplication by 5„(0) and 5„(0) will
give the formula 4n (n + b) - C* ♦ S>1a h 1 (mod 3) por 4n(n+b) - C8h (mod 3) and otherwise the formula 4n(n + b) — Z,2 * 9l2 = 2 (mod 3): the expression 4n(n + b) - f,2 + (31 ± 1) cannot be congruent to 1 modulo 3. Therefore if we select all terms in the definite integral OF THE PRODUCT (87) AND IN THE SECOND MEMBER OF EQUALITY (S3) IN WHICH EXPONENTS OF q ARE CONGRUENT TO 1 MODULO 3, WE OBTAIN
85.
oc	oc
¡(0,q#) LM(13n + 4)q3n+t + S5g(0,q*)^^(lSn+ 7)q3n + V (33)
n«0
n*0
LET US SELECT IN THE INTEGRAL
n
3P +1
4 r . dig5 (x,/q) (V~ q‘
-----{I—l^Sin(5p + 2)X
0
+ T
ep+4	,
sin(6p + 4) x>dx;
<T r	)
2 -
(89)
all those terms in which exponents of q are congruent to 1. For
THIS WE MUST COMPUTE TWO PRODUCTS; ONE WHICH IS MULTIPLIED BY Sg(x> q) “ 5s(3x, q ), AND ONE WHICH IS MULTIPLIED BY &3(3x, q ) BEFORE THE INTEGRATION IS PERFORMED.
That part of the first fro duct which we need may be written 8)	) -—-—x*rr^ -—2—^rT-cos(öf — 6p + 2)x
+ 8)	).-—2—;rr—r -—2—-T-rr-cos(6r -So - 2)x
-8r
L.
	q?p+l		q3 r + g
1	- q9 P ♦ 3	1	— q9 r + 6
	,, ep +4		q6r+2
i	- q9p + 6	1	- q9r + 3
	q 3 p +i		q 6 r + 2
1	__ q9 P + 3	i	_ q 9 r + 3*
	q 6p +4		q s r + 2
1	1 qöp+'e	1	- q'3r + 6
COs(òr + Op +4)x
We shall give here only THE COMPUTATION of the coeffioient of84
ios(6n + 2)x. This coefficient may se written
q3n+6P*3 + q6n+12p+s	q € n - 3 p
8 Z- TT-^P^ür- c^T^n^y 8 l_	_■ gn-Vp-Sl^
5*0	P"Q	■*	M
8
Us
3 n + 6 p * 3
1 2 n > 6P +• 6
, 6ti + 3p + 3
2 _ q9 n ♦ 3 {	) ^_q9 p + 3	2. - q9 n + 9 P + 6
P®0 s
6n*3p+8	1
2 — q9 p ♦ 3	^	n +$v + 6)
8
n-1
va
6 n + 6 P ♦ 6
6n~3p
1 — q9 n + 3	h _ q9p+ 6
p'*0
+ —
1 - q9 a -9 p
4
In the second and fourth fractions of the last two sums we replace
p 8 Y p —■ il, AND IN THE SIXTH p 8Y n “ p — 1J THEN IN PLACE OF THE SIX FRACTIONS WE MAY WRITE FOUR IN ALL OF WHICH THE SUMMATION IS EXTENDED OVER ALL VALUES OF p FROM 0 TO «. FINALLY, THE PRECEDING EXPRESSION MAYBE WRITTEN
8q<?	~	f ( 3J.+1 ) ( 3t + 1 )	( 3p + £) ( 3b + 2)'i
LL'q	■q	?
I— s;q
p,b =0 b * 0,5
8qSnH V" V" i < -3P*1>( 3b + 2)	( 3 p + 3 ) ( 3 b + 1} Î
• '+ i-q9»*3 z_ z_ r	“q	;
p, b * 0
T IS EVIDENT THAT THE SECOND SUM IS EQUAL TO ZERO
By THE SAME METHOD WE FIND T'hAT THE COEFFICIENT OF COs(ôll+4)x IS
q.to 00	«
8qrf " m (8g*l):(-3l>»i)	_ ( 3p+SK 3b + 2)\
■l ~q9n + fi zL zL r	q
0 0
/
Multiplying by §3 (x, q) - S3(3x,q9) and integrating we evidently
OBTAIN
n > b « 06
_ V“	5 (:3n + l ) ( 3n + 3b +3)	( 3n+ £>) ( 3n+ 3b + 3) !
8 L ¿_V	- q	Ï
n, b *0
pjb	_
x Y \	^ (Bp + l) ; 3b* l)
3? , b - 6 V
(3p+2)(3b*p)N(
LET >US NOTE THAT IN THE EXPONENTS OF THE FIRST SUM THE GREATER FAC-"OR IS DIVISIBLE BY 3 •
Turning to that part of the integral (S9) which is multiplied by
)?(3x,q9), BEFORE THE INTEGRATION IS PERFORMED, WE NOTE FIRST THAT ;c t,gx si n2 rx = cos.2rx + 2cos.3(r - l)x + 2cos2(r - 2)x + ,. . * 2co.s2s + 1;85
Therefore the expression which we wish to consider may se written IN THE FOLLOWING FORM:
6p+ 4
06	2E	Sp + 1	"	q'""*
4Z Ll-Vg"+~COS(6P~ 5r)X * A L- Ll-q«ir«C0S(6P ”3r)X
p-0	r=0
p , r = oc 9 r 4- 3
r~ q
p*0	r«0
3 P + 1
•* 8}_	■_^i;¥oo3(6p-ar)
p , r=0
.aft’-	9 r + 6 q	6P + 4 q
6L LT P , r =0	_ q9 r + 6	1 - q9f+6'
	9 r+ 6 q	3p * 1 q
3£- L i Sr r*0	_ q9 r ♦ 6	1 _q9P*6
- s? r	9 r + 3 q	6 P ♦ 4 q
aL Li p , r»0		q 9 r + 3	1 ~q9r + 3'
^cosCsp + 8r + 6)x.
The constant coefficient may be written:
*1
(1 4 q9p*3)q3g
(l-q3**8)8
4- 1
‘I
(1 4 q9P*«)qeg*4
(l~q®p + 6)2
p,b=*>
(2b +1)q
P , b =0
(Sp+l)(3b+l)
41_ X_i2b + 1)^
p, b=0
( 3P+ 2)'(;3b+ 2)
The coefficient of cosSnx falls into three parts: the first con
SISTING OF THE PRODUCTS OF FRACTIONS IN WHICH EXPONENTS' NAVE AS DIVISORS, BESIDES 3, NUMBERS OF THE FORM 3p +1; THE SECOND CONSIST ING OF PRODUCTS OF FRACTIONS IN WHICH DIVISORS OF THE EXPONENTS ARE OF THE FORM 3p+ 2; AND THE THIRD CONSISTING OF FRACTIONS IN WHICH DIVISORS OF THE EXPONENTS ARE OF DIFFERENT TYPES.. THE FIRST TWO SUMS ARE INFINITE AND THE LAST ONE IS FINITE.
The first sum may be writtem:
8
y____s
L.( i -
4p + l	12p-3n+4
q ♦ q______________
I> = n
(1 _.q0p+3)(l _ q9p-0n+3)
8
1 -q9n
:Srr
p* n
3 P * 1
.3P + ‘9 n4* 1
9 p —9 n+ 3
,9 P+ 3
3p+6n+ 1 ^	n4- s
3p + 6n+ l
„9 p
*36
IT IS EASILY SEEN.THAT THE SECOND CRACT I ON OF THE LAST SUM BECOMES EQUAL TO THE THIRD FRACTION IF IN IT '.V E REPLACE p. B~Y p.-nj- THEREFORE FROM THE SUMMATION OVER THESE FRACTIONS THERE WILL R.EMA'IN A FINITE SUM CONSISTING OF 11 FRACTIONS. ALSO, UPON REPLACING p BY p -n THE FOURTH FRACTION 8ECOMES EQUAL TO THE FIRST MULTIPLIED BY
q3n. Considering all this we express thev preced,i ng sum as follows.
n-1	O n + 1
8q 6n(l +qqn) V—i—-
---------st---} -] „ _ 9 p ♦ p
1 ~q9	^o1 q
.8q3n(l-q3n)^bf“
L'A
(Sp+l)v3b+l)
i-q
p, b=>0
By THE SAME REASONING WE CAN WRITE THE SECOND PART OF THE COEFFICIENT OF COSÔnX IN THE FORM
Sq3n(l + q6h)T“1	q'"“*1'3 Sq^Cl-q3“)
\J-	' 'j / \	M
-	9 n /	-	9 n-9 p- 3
1 — q —A - q
P-0
1-q
9«>
( 3 p + 2 )<3b*g)
P , b.= 0
The fi nal sum wi ll Then, be:
q9n-6p-g + ^en + Sp+'l
( 1 -* q	) ( 1 — q ) ’
p = o
REDUCTION TO A COMMON DENOMINATOR SHOWS THAT THE LAST SUM CANCELS WITH THE FIRST SUM OF THE OTHER TWO PARTS OF THE COEFFICIENT, SO THAT AFTER INTEGRATION WE OBTAIN FROM ALL THE COEFFICIENTS OF DIFFERENT cosônx
3n(3n+3p+2)
- q
oc oc
jin
b»0
(3p+l)(3b+l)
q-
(3p+2)(3b4-2){
q	( •
LET US NOTE THAT EVERY EXPONENT OF THE FIRST SUM F A C TO R.S I N TO T WO FACTORS OF WHICH THE SMALL.Eft IS DIVISIBLE BY 3, WHEREAS BEFORE WE 0 8 T A I NED THE SAME SUM BUT IN WHICH THE GREATER FACTOR IN THE EXPONENT WAS A MULTIPLE OF
|N VIEW OF WHAT HAS. BEEN SAID WE MAY WRITE THE EXPRESSION (88)
as follows:
293(0, q9)ZM(13Ji * ^q3”*1 * 39£(0, q9)2M(i2ll ♦ 7)q3“*ï
» |ss(3p .l)q(3r>1,(3b*” + |j2(3p
+
■ Sn(3p+l)
q
3n(3i + ?)
q
(3p+l)(3b+l)
(3p+2)(3b+2)
q
J. (80)
THE FIRST TWO SUMS OF ThE SECOND MEMBER OF (90) may be written as
■|s$(-3n + l)q?n + 1In ORDER TO TRANSFORM THE REMAINING PART OF EQUALITY (90) LET US REOALL A THEOREM ON PAGE 9 OF THIS BOOK. ACCODING TO THAT THEOREM THE NUMBER OF SOLUTIONS OF THE EQUATION a2 + 3y2 =, 4a(2n + 1 ) F 0 ' 0t=0 IS EQUAL TO 2H(2 0 + 1), AND FOR 0C>G IS THREE TIMES THAT, ' T H A ’
is, it is eqîual to SH(2n+l); and the equation x2 + 3y8 = 4°(4n + 2)
'HAS NO SOLUTIONS. |T IS EASILY SEEN THAT BOTH OF THE LAST SUMS OF EQUALTIY (90) MAY BE REPRESENTED WITH THE HELP OF THE FUNCTION
H(2n + 1), that ¡is, the function givim the difference between the
NUMBER OF DIVISORS OF 2n+l OF THE FORM 6p +1 AND THE NUMBER OF
p
DIVISORS OF THE FORM op +5.;- FOR, R E C ALL I N G T H A T ^	=-1, IT IS EAS
TO GIVE TO THE LAST PRODUCT OF EQUALITY (90) THE FORM
(1 ^	«V" ■'!/■/. o\f 6n~3	4 ( 6n- 3)	16(6n-8)	64(6n-3)
13 2L	+ 4	+ q	♦ q
n* 1
]}
:T
n»0*
H(6n + l)[q
6 a +1
4(en+1)
> + q
16(n+ 1)

(91)
Let us now consider the equation
(x!
¿y
) + (zV+St*') = öp + 1.
(92)
LET US PUT 6p+l = 3p j 4 P a WHERE 3p 4 ' I S ASSUMED TO BE REP RESENTED IN ALL POSSIBLE WAYS BY THE FORM X8 + 3yS, AND pa BY THE FORM Z* + 3tS. IF pt IS ODD, AND THEREFORE pa DIVISIBLE BY 4, THEN THE COEFFICIENT OF q3P1 IN THE FIRST OF THE TWO FACTORS IN (91) IS EQUAL TO THE NUMBER OF SOLUTIONS OF THE EQUATION X 8 + 3y 8 = 3pi, AND THE COEFFICIENT OF qP2 IN THE SECOND OF THE TWO FACTORS IN (91) IS EQUAL TO j OF THE NUMBER OF SOLUTIONS OF Z 8 + 3t * = pS. |F pi .IS DIVISIBLE BY 4, AND THEREFORE p2 IS ODD, THEN THE COEFFICIENT OF q3?1
EQUALS | OF THE NUMBER OF SOLUTIONS OF XS+3y8 =3pi , AND THAT OF
0 -
q3pl EQUALS £ OF THE	NUMBER OF	SOLUTIONS OF Z2 +	3l8 = p'a. IT IS	CLEA
THAT THE COEFFICIENT	OF q6P* 1 ' :	I N, THE P RO D U'C Tv (9	1) IS EQUAL TO	Q~
i c
THE NUMBER OF SOLUTIONS OF EQUATION (y2).
BY A THEOREM ON PAGE ¡29, THE NUMBER OF SOLUTIONS OF EQUATION
(92) IS EQUAL TO 4§(6p+l); THEREFORE THE COEFFICIENT OF q6g + 1 IN
THE PRODUCT (91) IS ^$(6p+l).
Q
‘Let us now put 12p + 10 = 3Pi + p2. In order that 3pi and pa both
BE REPRESENTABLE BY THE FORM X*+3y8, THEREFORE THE COEFFICIENT OF q3pl IS EQUAL TO THE	NUMBER OF	SOLUTIONS OF THE	EQUATION X 8 ♦ 3 y 8 ~
8ti A N 6 THE C.OÊ F FIX) I t N ? OF q:'8	. | « THE SECOND (J F	THE FACTORS ¡N	.(91)
IS EQUAL TO ONE HALF OF THE NUMBER OF SOLUTIONS OF Z2+3y8=p2.
'The coefficient of q12P+1° | thus equal to one quarter of the number of solutions of the equation88
Xs + 3y2 ♦ zg + 3te = 12o + 10,
THAT IS, ev THE THEOREM ON PAGE 29> <S EQUAL TO
$( 6p + 5) =	12p + 10) .
Let 3n +1 = 4a(5p +1) or 4a(12p +10), where a > 0, let 3n + 1 =
3pl+D2, AND LET 3p j. 8E AGAIN REPRESENTED BY THE FORM X8 + 3y8, AND p g BY THE form Z*+3v*. We SHALL SUBDIVIDE THE SOLUTIONS OF THE EQUATION
( x2 + 3y2) + (z2 + 312) = 3n + 1	(93)
INTO SEVERAL GROUPS. LET THE FIRST GROUP CONTAIN ALL SOLUTIONS FOR WHICH BOTH X* + 3y* AND Z 8 + 3 t 8 ARE ODD NUMBERS; WE SHALL DENOTE THE NUMBER OF SUCH SOLUTIONS BY u(a). LET THE SECOND GROUP BE COMPOSED OF ALL SOLUTIONS FOR WHICH 4 IS THE GREATEST POWER OF 2 DIVIDING x8+3ys and z8+3l8; let the number of such solutions be 9u(oe-l). Let the third group contain all solutions for which 16 is
THE GREATEST POWER OF 2 DIVIDING BOTH X4+3y2 AND Z8 + 3t8J LET THE NUMBER OF SUCH SOLUTIONS BY 9u(ct-2); ETC. WE MUST NOW NOTE THAT FOR
a > y > 0, u( y) >s the number of solutions of the equation x8+3y8 + z8 +3'l8 = 4Y(Sp + 1) or 4Y(12p + 10), for which both x8 + 3y8 and zs + 318 are odd. as for u(0), in the case of the equation x8 + 3y*
+ Z 8 + 31 * = 4 (8p ♦ 1) the number of solutions of the equation x8+ 3y8 + z8 + 318 = 5p + 1 is 3u(0), and in the case of x 8 + 3y 8 + z8 + 3t8 = 4 (12p +10) the number of solutions of the equation x8+3y8 + z8+3t8= 12p+10 is a(0); therefore in the first case 3u(0) = 4$(6p +1), and in the second 3u(0) = 4$(l2p+l0). Recal-
a
LING NOW THAT THE COEFFICIENT OF Q4 n IN EACH OF THE FACTORS OF (9I) IS EQUAL TO THE COEFFICIENT OF qn, WE SEE THAT THE COEFFICIENT OF q	IN THE PRODUCT (¿>1) WILL AL WAY'S BE GIVEN BY THE FORMULA
4 {p(°0 + p(d—l) + p. (ft—2) + p( ft—3) + . .. + p(2)+p(l)+p(0)}	(94)
•H-
LET US NOW ASSUME THAT THE FOLLOWING THEOREM HAS BEEN PROVED: FOR ALL a' S SMALLER THAN A CERTAIN PO S I T I V E I N T 6 G E R THE EXPRESSION
(94) is equal TO ^$(3n+ l) . Then
p(<x) + n(a-l) + ... + p(0) = |§(3n +1) = |(2Y+1 - l)$(2s + 1),
ASSUMING THAT 3b + 1 = 2^(2s + l) . BUT THE NUMBER OF ALL OF THE EQUATION X 8 + 3y * + Z * + 3 t 8 =. 2Y+£(2s+l) IS EQUAL 4(2 i'*8 - 3) $(2s + l); therefore
y(“ * 1) = (3Y‘8-3)*C3s *1) - |(u(«)	*
SOLUT tONS TO
4

+ p(l) +p(o) ];89
-;[y.(a + 1 ) + ia( °0 m.(« — 1 )	• . + Ii(l) + |i(0)3 = ('2"Y " - 3)$(2s + l)
4
r .. rtY*4 0-i
$(2s +1)
. . . + |j.( l) + ji(0 ) ] =	r Y+4 „-i _ Y+ 3 J. 8 2 -3 „ ~+ -
	L 3J
|[2Y*8 - l] #(2s + 1) =	|§(l2n + 4).
The assumed theorem holds true for a=0; it can be easily proved for a = l; therefore the theorem holds for every a. Thus in any case the coefficient of q3n+1 in the product (91) is equal to
■J$(3n+1).-
Taking into account all that has been said, we see that from
EQUALITY (90) FOLLOWS THE ARITHMETICAL THEOREM;
22M(12n + 4-9h2) = 4(3n+l>, h =. 0, ±1, ±3, ±39h2<12n+4.l
(95)
From equalities (85) and (95) immediately follows another formula of Gierster:
2v
42M(12n + 4 - hi ) =» *(:3n + 1) + 2?(3n + 1>,| in = 1, 2, 4, 5,..., 3p±l,...,	12o+4.i
Let us multiply both members of equality (33) 8Y
II
*»1(x,/qs')	1 ‘il ¿si(3x,/q®)
---L-------- - r .-----1-•------^ = CC t X — C0ti?3x
ix	3	ix
+4
Ir?
ix
9 n + 3
,9n»3
__ On*6
v Q
sin(6n + 2)x + 4	y—si n(6n + 4) x.
Since
n	n
- f sin6nxcot¿3x ix = 1,	- J si o( 6n ±2)xcotg3xix = 0,
(9 6)
(97)
THERE WILL REMAIN IN THE SECOND MEMBER OF EQUALITY (8 4) ONLY THOSE TERMS FOR WHICH nS±l()DCi 3) AND b=0 (mci 3).
In the EXPRESSION RESULTING from the integration of the product (83) AND (97) let us now select all terms in which the exponents of q ARE CONGRUENT TO 2 MODULO 3;
( 3h±l) *	( 3n±l) s- 9&*	( 3 nil) (3n±l*3b) -9 a *
4^q	[2iq	+ 2i2>.q	]
a-, |(6h.±l)*,	(3n±l)*-9(a-i)*
+ 8Sq4	[2^q	2	+
Sn±l)(3n±l+3b)-9(a-|)
1.(98)
In the last expression 3h±l, 6h±l, and 3n±l are assumed to be
POSITIVE AND THE SUMMATIONS WITH RESPECT TO a AND b ARE EXTENDEDOVER THE SAME RANGES AS IN FORMULA (84).
WE HAVE ALREADY NOTED ON PAGE 82 THAT IN CHANGING 0, b AND X, THE
expression 4n*+ I2nb —9z8 in which n is not divisible by 3, gives
ALL POSSIBLE NEGATIVES OF DETERMINANTS CONGRUENT TO 1 MODULO 3.
Hence it is clear that (98) may be written in ThE form;
82q(3h±l)*2M(12o +4)q3n+1 + SSq^(6h±1>*SM(12n + 7)q3n+*. (93f)
LET US S EL EC T, NO W, ALL THOSE TERMS IN THE INTEGRAL
Cil^(*,/?~) _ 1 i] *M3x,/qV( i]t.
nj3	dx	l ix	3	ix
0
WHOSE EXPONENTS ARE CONGRUENT TO 2 MODULO 3. TO DO THIS WE MUST
dx; (99)
COMPUTE TWO PRODUCTS: ONE WHICH BEFORE INTEGRATION IS MULTIPLIED BY &g(.3x,q9), ANO ANOTHER WHICH BEFORE INTEGRATION IS MULTIPLIED
by $3(x, q) - $ (3x, q9)-. The first product is obtained through multiplication OF THE EXPRESSION (97) BY
__	6 n♦2	3 n + 2
4 L	+ 2> * + 4h i':'q9'n*a8in(6nt4) x>
The SECOND PRODUCT.IS OBTAINED THROUGH MULTIPLICATION OF THE EXPRESSION (97) BY
q3n + 1	r— q6n + 4
4Li-i'9»^8in^n * 2U * 4j_'i"'-"q^6sio(6n * 4)x-
A PROCEDURE INTIRELY SIMILAR TO THAT EMPLOYED IN THE DEDUCTION OF FORMULA (90) GIVES US THE CONSTANT TERM OF THE FIRST PRODUCT IN ThE FORM
I
ep + 2
(2p + 1)q
9 p + 3
+ 4
I
3p «-2
(2p ♦ 1)q
9 p *6
1-q'""	“	1-q
The COEFFICIENT OF CCS6flX IN THE FIRST PRODUCT IS ZERO.
IN THE SECOND PRODUCT THE COEFFICIENT‘0F C03(60+2)x IS
9 n ♦ 3 Pi b «go ,
3 q	v V~ ) ( 3p «-1) ( :3b ♦ 1)	( 3p+2) ( Sb+2;?
“----TiTTa l ¿_1?	“ q	r
AND THAT OF C0s(6n + 4) X I S 9 n +6 P > b = *
8q
1 - q
9 n ♦ 6
V“	( ( 3P ♦ 2) ( 3b ♦
1 l{*
2)	(3p + l)(3b
- q

P , b = 0
We note that for r positive
sin2rx cofc^3x = cos2rx + 2cos2(r-3)x + 2cos2(r*~6)x + . . .
+ U>91
-sinx
where n* 1 for r-0 (mod 3) n - sj"~'jx
{i =
sinx
sin'Sx
fos r
(mod 3)»
for r = 1 (mod 3), ano
Also the integral
n
2 f cos.8rx sinx
-i
nj
si n3x
dx
is equal to 1 for r?l (mod 3), to -1 for r= 2 (mod 3) and to 0 FOR r A multi ple OF 3.
sill X
The coef f i c i ent of
an* i
si n3x
6n ♦ 4	« °«
IN ThE SECOND PROOUCT WILL 8E
- 9
T~ _(SP*lV( 3b*l)	_( ap *2) ( 3b *8)
O n *6
1	9	1 ~ Q	0	0
The integral (99) gives thus
'Zr*r*n-£r*™-*LZ
" 1 ~q	*"*1 —Q	a. o
(2p + l)q
6p *2
, V ( "P +
4L i
1—q
v 4
(2p♦ l)q
r~ (2o +
3p +2
9p *6
„ (3n+iy* Ar. (3n*2) *
^	4Xq
06	OC
’ 31 It
n=* l b»i
(3n*l)(in*l*3b)
( 3n*2) ( 3n* 2*3b)-1 )r~ C" [" ( 3p ♦ i ) ( 3b ♦ 1 )	( 3p ♦ S) ( 3b ♦ 2) 1	.	.
- q	J )_ [$	-*■ q	J- (loo)
If to the expression (100) we add
4__	(3p*l)(3b*2)	4	(3p + 2)(3b*l)
- |22q	+ |SSq
IT MAY THEN SE WRITTEN IN THE FORM
8T (3p + l)q6p+2	8 \~ (3p +3) q3P + 1
"3 L~ , 9p+f + 3t_.	,	9 p ♦ 6
1 - q	1 - q
+ 4 [22(q
( 3p ♦ 1 ). ( 3b ♦ l)
( 3p *2) ( 3b + 2)
- q
(100*)
The FIRST TWO SUMS of THE EXPRESSION (lOO*) MAY 8 E WRITTEN AS
I^SOn +-)43n + 2*
n*0
The number 3p +2 cannot be represented by the form x8+3y8;
THEREFORE IF WE HAVE THE EQUATION X * + 3y 8 f Z8 + 3 l 8 = 3n ♦ 2, THEN
x* + 3y8 = 3nt + 1 and z,+3t8 = 3n# + 1. If 3fli + 1 ano 3ns + l are
ODD, THEN THE COEFFICIENTS OF q3“**1 AND q8”®*1 IN THE SUM 22£( n( 3p *l) (	♦!) _	( 3p*£) (3b*2) ^ARE eg U AL TO THE. number of SOLUTIONS OR the equations X8 + 3y2 =■ 3:U + 1 an d za + 3 t 8 = 3ns+l: if 3nx +1 and 3n?+l are each the double of
AN ODD OR OF THE FORM S201*1 (2p + l), THEN THESE COEFFICIENTS ARE E-
qual to zero; if 3nx + 1 and 3ns + 1 are of the form 4°t(2s+l), then
1
THESE COEFFICIENTS WILL 8 E EQUAL TO - OF ThE NUMBER OF SOLUTIONS OF THE EQUATIONS X8+3ÿ8= SlTi+1 AND z’* + 3t*= 3oa+l RESPECTIVELY. Therefore we may apply to the expression
4[22(q‘3P'1,<Sl*1) - T3"*3’'8"*31)!8
THE REASONING EMPLOYED IN THE DEMONSTRATION OF (90), THE ONLY DIFFERENCE BEING THAT HERE WE DO NOT DIVIDE THE RESULTS 8Y TWO AS WAS DONE THERE ON ACCOUNT OF THE DIFFERENCE BETWEEN THE SUMS. THE LAST EXPRESSION IS THUS.EQUAL TO
4_ . , _	3 r. + 2
“2$(3n + ki) q
Since the expressions (98') and (100’) are equal, we obtain
82q( 3h±l) *SM(l2n + 4)qSn+1	+ 82q^(6h±l) #2M( 12n + 7)q?n^
=	42$(3o+2)q
3 n a 2
AS A CONSEQUENCE OF THIS WE OBTAIN THE FOLLOWING FORMULA:
22M(12 o +3 - ] 2) «$(30 + 2),	(101)
1 = 1, 2, 4, 5,..., 3h ± 1,...., 1 2 < I2n + 8;
SM(12n + 6 - 9b2) = ?(3n+2),	(10 2)
b = 0, ±1, ±3, ±3, ±4, . . . , 9hZ < 12n +8.
The last formula is obtained by term by term subtraction of equality (101) FROM EQUALITY (8 5) IN WHICH 0 IS REPLACED BY 3fl + 2.
Let us multiply equality (85) term by term by the equality
1 digs Ox,/?)
= C0tg3x + 4) r—-—T—sin6nx;
L_j - n
9 n
(103)
3 i x	-	¿_ j - q*
AFTER PERFORMING THE MULTIPLICATION '.VE' FORM THE DEFINITE INTEGRAL OF BOTH SIDES AND OBTAIN AN EQUALITY, THE LEFT HAND MEMBER OF WHICH WE SHALL WRITE IN THE FOLLOWING FORM:
1 ?f 1	, .il^^ixOq) ilgiL(3x,\/?) n . .
“nJF3(x) lr--------------"~a—	- ^(O)cotgx cotg
0
gx co tg3x^>ix. (104)
Let us select in both members of the equality obtained all terms
IN WHICH THE exponents OF q ARE CONGRUENT TO 1 MODULO 3; THEN THE SECOND MEMBER OF THE EQUALITY WILL GIVEg JV" r“1	r— r— 9(n*+nb-»°)
2CS8(0).	I 1	+ 2Z_L Z_ «'
jn»l	n=l b*l a=-n+l
J
ofns_(Â4)‘] nV'-V’"SL 9Ln**nb-(ar|)'3l
q
*'> +
r
lit-LJ
n-1 b=l ««1
= 242q(3h±l) 8y M(4n)q9“ ♦ 242q*( 6h±l)‘ M( 4n - 1) q9 n_4 ,	(105)
n= 1
n=* 1
In the two sums in which the limits of summation are not indicated the summation must be extended to all positive integers of the DE-
SI ON ED FORMS.
If WE SELECT THE TERMS WITH EXPONENTS, CONGRUENT TO 1, IN THE INTEGRAL (104), THEN THE TOTALITY OF THESE TERMS MAY BE REPRESENTED BY THE INTEGRAL
(3h±l)
£
co s( 8b ± 2)x
11 j S, ( x,Sq*) 11 j ( 3x>/jr) lx	lx
- 2eotgx co'L^3x2q^3h;t1^ llx.	(10 6)
J
WE SHALL NOT STOP HERE TO COMPUTE THE INTEGRAL (106), BUT SHALL ONLY NOTE THE SERIES FOR THE FUNCTION
dl^Cxy^V 11 ,4^(3x,/?) s 11 ^.(x,/^) Cllj&.Cx^/g7)
ix	1 x	lx	t. lx
i x	lx	J
(107)
MAY BE MOST CONVENIENTLY OBTAINED BY THE METHOD GIVEN BY W. BRIOT
and M.Bouquet in the third chapter of the sixth book of their treatise*, having COMPUTED THE SERIES FOR THE EXPRESSION (107), AND HAVING TAKEN THE DEFINITE INTEGRAL, WE FIND THAT THE INTEGRAL (106) MAY 6 E WRITTEN IN THE FOLLOWING FORM:
22q(Sh±l)	+ ?2«(3n ♦ 1) q3a + 1 - |s[$(3n-Kl) - ?( 3n + 1) q3R+ 1
•3	3
- -^[1 + 627.(q3bi3p + l) - qSb(3p + 2))]S7(qi3P+l' i3b + l) - q( 3p+2) ' 3b+s5
= 2x,q
( 3 h± 1 >
3n +1) q
Sn+i
- 2Z$(3n + l)q
3 n+ 1
(108)
*	et gouguet. Théorie des Fonctions Doublement Périodiques et. en Particulier
il&S Fonctions EltlPtlCfUCS. Parig, 1899* Transiator* 3 nota#94
The x press ions (10 s) and (10 6) are equal; let us then equate co-
EPEICIEJTS OF THE SAME POWERS OF q, DENOTINO BY p ANY POSITIVE NUM-OF ZERO, WE OBTAIN
24[2M(4p) ♦ ZM(4p + 3) ) » 4?(3n + 1) - 3$(3n ♦ 1),	(109)
putting M(0) - - ~. In soth sums of (109) p must satisfy 9p < 3o +1. In the first sum p^and n are connected sy the equality
9p + (3h±i)2= 3n + 1;
and in the second sum they are connected by the equality
d{ p +	♦ -(8ft ± l)2 = 3n + 1.
The last two equalities may be written as follows;
,	12n + 4 - ( 5ft ± 2) 2	„	„	12n + 4 - (6h .t 1) 2
4p « ----------------4p +3 *---------------r--------.
0 0
Therefore equality (109) may be replaced by the equality t—	12 o + 4 - ft 2
1.2 M ----------: = 2?(3n + 1) - $(3n +1),	(110)
WHERE ll IS ANY POSITIVE NUMBER SATISFYING THE CONOITIONS ft2 S 12« + 4,
ft2 s-120 + 4 (mod 9) .
Formulas (110), (95), (96), (101), and (102) are given by Gier-
STER ON PAGE 49 OF VOL. 21 OF THE WATHEMATISHE ANNALEN. GlERSTER DORS NOT GIVE ANY OTHER FORMULAS OF THE THIRD DEGREE.
$ 41. We saw that kronecker’s formulas were closely related to
THE PROBLEM OF DETERMINING THE NUMBER OF DECOMPOSITIONS OF A GIVEN NUMBER INTO A SUM OF THREE SQUARES; WE EVEN DEOUCED SEVERAL FORMULAS WITH THE AID OF EQUALITIES SOLVING THAT PROBLEM. PREVIOUS FORMULAS NOT ONLY GIVE A SOLUTION OF ThE PROBLEM OF 'THREE SQUARES BUT ALSO SOLVE THE PROBLEM OF FINDING THE NUMBER OF SOLUTIONS OF THE EQUATION
x s + 2my 2 *• 2	= p
FOR SMALL VALUES OF ¡0 AND 0. IN THIS EQUALITY D IS A POSITIVE NUMBER AND tfl ANO G ARE POSITIVE OR 2ERO.
For THE SOLUTION OF SUCH P OBLEM WE ALREADY HAVE FOUR FORMULAS;
2K
n
1 ♦

1)
n n
q
2
* 3Zq*n +.12S[2F(n)
0(o)}qn, (111)
= 82F(8o + 3) q
2n + |
= 42F(4n+2)q
n + j
(112)
(113)95
¥*v
/2jik
n
3 4 S F(4n + 1)n 4.
BESIDES THESE WE'HAVÊ THE WELL KNOWN FORMULAS
2K /2Xkk'
n
= 22(-l)	( 2r - l)qr"2^
(114
(115,
BV THE HELP OF FORMULAS (57)	- (65) WE MAY WRITE (111) AS FOL-
LO ws:
~ \f^ = 1 “ 8Zqn# + 82F(3n + 3) {q8E + 8 + q4(8n + B) + q16(8n + 3) +
n y n
+ 12SF(4-n + 1) { q ♦ 122F(4n+ 2) {q
4 n ♦ 1	4(4n + 1 )	
	+ q	+ q
4n+2	4 ( 4 n + 2 )	
1	+ q	+ q
+ ...}
(116)
Since the sum of three odd squares is of the form 8n + 3, of two
ODO ONES AND ONE EVEN ONE 4tl + 2, OF ONE ODD AND TWO EVEN 4n + 1, W< EASILY DEDUCE FROM (116) THAT
OV h~)rr‘'s t
LdiXiK
n.
_	1	0-	( 2n-l) 8	4n*
= 1 - 22q	“ o^q
„ n , Q	8 n + S 4(8 n+ 3)	16(8n + 3)	,
+ SSF(8n + 3){-q + q	+ q	+
♦	42FC40 ♦ l)<q4'“1 ♦a,4“"“’ *3qie('4'“,,*...)
*	42FC4»»3)<-q4n‘E*3q4<4n*E’ ♦ aq1*«"“*’ «• .. . ) .	(117)
Replacing q by -q transforms
4 ¡2K* *■	. l2K EkmT
\hr> Mt> v~;
m a
/2a k’
I N TO
tx V n	V n
AND T HE R E FO RE F ROM THE PREVIOUS FORMULAS WE DEDUCE
2Xti' /2a!<'
= 1 - 62(-l) V
+ 122F(4n + 2){ q
2X.'k1, /Ü
lx
4 (8 n+ 3)	16(8 n+3l	+ . . .}
+ q	+ q	
4 ( 4 n + 1 )	1 6 ( 4 n + 1 )	+ . . . }
+ q	+ q	
4(4n+2)	1 6( 4 n+ 2)	+ . • . } ,
+ q	+ q	
-, OT, (2n-l)	4n‘
1 + 2^q	- o^q
(118)+
Ï? o
+ B 2F(8 n + 3) {q
8 n + $	4 ( 9 n ♦ 3 )	16{8n+3)
+ ü
/	_ x 4 n ♦ 1	»	4(4n*-l)	„	16(4n+l)	,
+ 42F(ien + 1) {-q	+ 3q	+ 3q
/	, 4n + g _ 4(4 n ♦ 2 )	_ 16(4n + 2)	/no\
+ 42F(4e + 2) {-q	+ 3q	+ 3q	+ .. .)=, (llfc >
in :* * f ~ J	X
n + A
\F^- = 42(-l) F (4n +2)q*,
n \j n
(120)
23k '
42(
l)nF(4n +1)q
«xi
n + 2
(121)
These formulas make it possible to determine the number of solutions of the equations:
**> X * *	y2	+	2z2 =		x2 + 2y2 +	2z2 = û,	X2	+ y'2 + Az2 = a,
x 2 +	y2	♦	8y2 =	o,	? ? x ♦ y +	1 oz2 = n,	o X	+ 4 y2 + 4z2 = o,
X2 +	By2		8 zs =	0»	x2 + 16y2 +	- , 2 I'OZ ~ up	2 X	O 0 + 4y"+ loz = n
ano certain others.
In THE &ETEHMI NATION OF THE NUMBER OF SOLUTIONS FOR THE EXAMPLES CITED, AS FOR OTHERS WHICH CAN BE SOLVED BY THE HELP OF EQUALITIES (111) -(121) AND REQUIRE THE DEDUCTION OF OTHER SERIES, LANDEN'S FORMULAS WILL INDICATE THE SERIES WHICH WILL GIVE THE SOLUTION OF THE PROBLEM. IT WILL BE PROFITABLE, HOWEVER, TO EMPLOY CERTAIN ELEMENTARY CONSIDERATIONS BSEORE THE APPLICATION OF LAN-DEN’S FORMULAS. THUS, THE PROBLEM OF THE DETERMINATION OF THE NUMBER OF SOLUTIONS OF THE EQUATION X 8 + y 8 + 3 Z 8 = 11 DEPENDS EITHER ON
THE DETERMINATION OF THE SERIES FOR $«(0 , q) ( 1 + '< ) , OR ON THE SER-
1 a	r—
1Es for ~9-g(0)(L + k ' ) ( 1 + vk ' ) . The- equation x8+2y8+2z8=n is con-^	1 a
NECTED WITH THE EXPRESSION “ S 2 ( 0 ) ( 1 + X 1 ) , XS + 4y* + 16z8= tl WITH
d
^9g(0)(-l +VT+'/iTf +/7i<') , X8+y 8 + 428 = 0 WTTH .^(0)(l + vXP),
<8 + y 8 + 8z 8 = 0 with (0 ) ( 1 + ¡0(1 + /o ), x8 + y 8 + 16y 8 = n with
o
P(0) (1 + v/T)2 (1 + v/k7), AND SO ON.
WE SHALL NOT DEDUCE HERE THE THEOREMS ON THE NUMBER OF SOLUTIONS OF THE EQUATIONS UUST MENTIONED SINCE THE METHOD OF DEDUCTION WAS MADE SUFFICIENTLY CLEAR IN THE FIRST CHAPTER. LET US MERELY NOTE -HAT THE PREVIOUS FORMULAS ARE SOMETIMES SUFFICIENT FOR THE DETERMINATION OF THE NUMBER OF SOLUTIONS OF T'hE EQUATION 3 X8 + b y 8 + C Z8 - 0 EVEN IN THE CASE WHEN a, b, AND C ARE NOT POWERS OF TWO. THUS, ON PACE 359 OF THE 14TH VOLUME OF HIS JOURNAL LlOUVILLE STATES THAT BHE NUMBER OF ALL SOLUTIONS OF THE EQUATION X8+2y* + 3z8 = OjJ, ± 1 = ID97
IS F(8ffl), WHERE x, y, AND 2 MAY 61? POSITIVE, NEGATIVE Oft ZERO. INDEED, TO EVERY SOLUTION OF THE EQUATION X8+2y8+3z8 =	8^+1 THE*
CO ft ft E CP 0 N D FOUR SOLUTIONS OF THE EQUATION X x £ + y i * + 4zt * =	6(-6li±l
WHICH ARE DETERMINED FROM THE EQUALITIES
±xa = x ± 2y ±3 2, ±yx = x±2y+3z,	±z4 = x ♦ y;
THIS 0 A N BE EASILY VERIFIED. CONVERSELY, FROM EVERY SOLUTION OF T H >-EQUATION Xx2 + y 1 8 + 4Zx * = o( 6ji * 1) , WITH THE HELP OF THE SAME E-
qualities, we may obtain just one solution of x8 + 2y8+3z8 = 6jn-±l
IF WE CONSIDER ONLY POSITIVE Oft ZERO VALUES. HENCE WE CONCLUDE THAT THE NUMBER OF SOLUTIONS OF THE EQUATION X 8 + 2y 8 +.3z8= SjA ± i IS ONE FOURTH OF THAT OF ThE EQUATION XxS + yi8 + 4Zx8 = 6 ( OfJ. .± l) ;
BUT BY FORMULS (113) THE NUMBER OF ALL SOLUTIONS OF THE LAST E-QUATION IS EQUAL TO 4F(8ffl)-.
The formulas (111) -(121) may all be deduced directly by the
METHOD OF HERMITE *, AS WAS (111). THIS DIRECT DEDUCTION MAY BE
MADE MOST EASILY BY T!.HE METHOD, USED BY' HERMITE.4-N DERIVATION OF
FORMULAS (111),	(112)’,	(113),	(114), AND (117). THE METHOD IS AS
FOLLOWS:	FIRST, TWO ELLIPTIC FUNCTIONS WHOSE PRODUCT IS EQUAL TO
2
uni ty, e.g. sin am 2£-2 and --------, are chosen. One of thesf
n	si n am 20.
FUNCTIONS IS MULTIPLIED BY A ^	.JACOB 1 FUNCTION. THEN A COEF-
FICIENT IN THE TRIGONOMETRIC EXPRESSION, OBTAINED AS A RESULT OF TWO SERIES:	ONE REPRESENTING ONE OF T’HE CHOSEN ELLIPTIC FUNCTIONS,
AND THE OTHER REPRESENTING THE PRODUCT OF THE JACOBI FUNCTION BY THE OTHER ELLIPTIC FUNCTION, IS DETERMINED. THE PROBLEM IS THUS SOLVED, FOR THE DETERMINATION OF THE COEFFICIENT IS MADE BY THE METHOD OF FOURIER. CONSEQUENTLY, ON ONE HAND W= OBTAIN A SERIES IN WHICH EXPONENTS OF q ARE PROPORTIONAL TO 3C“ b*, AND ON THE OTHER hAND, SINCE THE RECIPROCAL FUNCTIONS CANCEL ONE ANOTHER, WE OBTAIN A COEFFICIENT OF THE TRIGONOMETRIC EXPANSION OF THE JACOBI FUNCTION MULTIPLIED BY PARTICULAR VALUES OF THREE JACOB I FUNCTIONS.
These particular values result from the fact that the elliptic
FUNCTION IN THE EXPANSION INTO A TRIGONOMETRIC SERIES IS MULTIPLIED
by two Jacobi functions with argument zero, if the elliptic function DOES NOT HAVE INFINITIES OF HIGHER ORDERS; ANO THE PRODUCT OF THE OTHER ELLIPTIC FUNCTION BY THE JACUBI FUNCTION IS MULTIPLIED BY A PARTICULAR VALUE OF ONE JACOBI FUNCTION.
For EXAMPLE, WE SHALL DEDUCE BY THIS METHOD FORMULA (113). LET US TAKE TWO SERIES
j is	')	*	T	u
,	¿jwfk. *	.	*C	n ,	,	• »	«■» «%
— ksio ara--------x) = 4 / q 3in2nx( q 4 + q 4 + .
n	n z
n* i
* With, the exception of (tl5) of course.
+ q
|( 2n-l) 8^0 7	97*
— co 3 x; si n am-
n	n
r~ Q
= coUx + 2 > ~—
jgn-l
¿x 4 2 / ——-—~{sin2nx + sin(2n-2)xt. Ä L—\-
na 1
Multiplying these equalities together member by member and tak-
«
I NG THE DEFINITE INTEGRAL WE OBTAIN
rc
07 > /0'>	o:r ^ /orf 1	r— w »
t"\/t	* «£«' (’
0	n= 1
.1
A
-4(8n-l)'
4 + .	♦ 0 4^“-^	)
+ ,f n 8 / q2^1 + qgR^
4/_q	\1 - q£n-i	x - q8a+t/
n=* 1
Dividing both members of the last equality by q? we obtain.in
f(ao-o )	a
THE RIGHT HAND MEMBER TERMS OF THE FORM q4	, WHERE aC~D IS
even, afi>4’o% e> a, a and c are positive. A familiar argument 'w i ll lead us to the formula (113).
Being able to deduce formulas (111) -(121) directly we obtain,
IN A sense, a weapon for the deduction of different arithmetical properties. The more we modify the original method recommended by Hermite the more will t>hese properties differ from those already
KNOWN. WE INTRODUCED ONE SUCH MO 0\g 1 C A T10 N I N §§ 34,35 AND PARTLY IN 40. In THESE SECTIONS MOOULI I q OF DIFFERENT FUNCTIONS ENTERING UNDER THE SI GN OF THE DEFINITE INTEGR AL WERE DISTINCT, WHEREAS IN PAPERS OF HERMITE ALL THE PARTS OF THE INTEGRATED PRODUCT ALWAYS HAVE THE same modulus q.
LET US NOTE ALSO THAT WE MAY GIVE ANOTHER FORM TO THE FORMULAS (111) — (l-‘21) IF WE REPLACE THE FUNCTICNS F (11) AND G(n) BY THEIR VALUES GIVEN BY LEJEUNE-DIRICHLET ON PAGE 272 OF HIS TEXTBOOK. |N THAT CASE THE PROBLEM OF THE NUMBER OF SOLUTIONS OF A GIVEN EQUATION ACQUIRES GREATER COMPLETENESS, SINCE THE ANSWER WILL NOT DEPEND ON THE KNOWLEDGE OF THE NUMBER OF FORMS OF CERTAIN CHARACTER AND. WITH CERTAIN DETERMINANT.
r*.
-4< 2n-lV
§ 42. WE SHALL CLOSE THIS CHAPTER WITH THE SOLUTION OF THE PROBLEM PROPOSED BY DlRICHLET ON PAGE 407 OF THE THIRD VOLUME OF CRELLE’S
Journal: Let o be a prime number of the form 4N + 3. It is required
TO DETERMINE V FROM THE CONGRUENCE
1* 2‘ 3* 4*
o-l
11 a r +\'T /	■	\
= (-1) (¡noa n).
From the theorem of wilson .it follows that 1* 2* 3* *• • (n-1) + 1 is divisible by n, that is 1* 2’ 3* • * • (n-1) + 1 =0 (mod n). But n-1, •i-w, n-3, ...» (»+!)/2 may be replaced in the congruence by -1, -2,-3, ..., - ^-=r. Ip n is of the form 4N + 3, then the number of re-
¿J
PLACED NUMBERS IS ODD; THEREFORE
n - i 2
1* 2*-3’ 4* * * >—~	= 1 (mod a).
WE ASK.IN WHAT CASES 1* 2‘ 3‘ ‘	~ IS CONGRUENT TO +1 AND IN WHAT
TO -1; IN OTHER WORDS, IN WHAT CASES THE CONSIDERED PRODUCT IS A QUADRATIC RESIDUE AND IN WHAT CASES IT IS A QUADRATIC NON-RESIDUE MOCULO n.
LET THERE BE AMONG' THE NUMBERS 1,2,...	~ A RESIDUES AND B
non-residues; if B is ? ven then the product is a residue, otherwise IT IS A non-residue. Thus v = B. But Dirichlet in his text boo GIVES
F(o) = A - B =	- 2v.
Thus
V
0 - 1
.1
4:
1
f(n)
(122)
Let us add together the equalities (32) and (53) and in the re sult put o = is = 3 (iso d 4). assuming that n is prime, we have
?(o) = n - 1; AND THEREFORE
v = ■jh(n) - -F(c) = F (n - 22) + F(o - 42) + F(ii“6?) +
4:	¿J
„ *
Among the forms with determinant -n +45 and with one of the extreme COEFFICIENTS ODD FORMS OF THE TYPE 2bx8+2bxy + X.y 8 DO NOT occur. Therefore to every reduced form ax8+2bxy+cy8 in.which
b>0 THERE CORRESPONDS THE FORM dX8- 2bXy +Cy8 EQUIVALENT TO IT, AND THEREFORE
v = ?(n - 2 8) + ?( n - 4B) + ?(n-6a) + . .
>
WHERE ?(p) DENOTES THE NUMBER OF REDUCED FORMS WITh THE MIDDLE COEFFICIENT ZERO AND WITH THE DETERMINANT “p. BUT THE NUMBER OF SUCH FORMS WITH 'DETERMINANTS 2* “ tl, 4 8 - H, 6 8 - 0, . . . IS EQUAL TO THE NUMBER OF DECOMPOSITIONS OF C-2% 0 - 48, 0 “ 8 % . . . INTO TWO FACTORS a AND C OF WHICH a IS SMALLER THAN C.
Let o - 45	»
a p y
p q 3 .
, WHERE P,q, 3, ... ARE PRIME NUMBERS. THE
number of divisors of o n 45® is (a + 1) (p + l)(y + 1)* * * •; the number OF DIVISORS smaller than Vn “ 45 8 • I S EQUAL TO *5 (a + 1) ( £ + 1 ) Y + 1 )' * since n-458 s 3 (mod 4), a, p, y, ... cannot all be even. If the NUMBER OF ODD NUMBERS AMONG 0C, P, Y> . ■ • BE 2, 3, 4, 5, O, ... . , THEN ?>(a + 1)(P +1)(y +1)* * * will be even.• Therefore -attention must be
PAID ONLY TO THOSE 0“ 45 8 WHICH ARE EQUAL TO rSp2l+1, WilERE p IS100
1
PRIMS AN.® P I S HOT C l V 1 S .1 9 L g	p .	! F i IS ODD, THEN % ( & * 1 '> ( '' + l) ( Y + 1
is even; therefore we must pay attention only to those gases where
i I S E V E N. t F n - 4H
FOLLOWS THE THEOREM GIVEN BY KRONSCKER ON PAGE 260 OF THE THE 3RD
Q	2 4 1 ^ 1	r» /	, f* g \	...
2 = r p , then P(fl -4c ) .is odd. From this
VOLUME OF THE SECOND SERIES OF L I 0 U V I L L E1 S JOURNAL: V Is equal tO
the number of exoressions n -2% n - 4*, o - 6*,..., which are o f
,	g 4 i *f i
t ie form c p * 'where p is a crime, amt r is in ■general a cantos-ite not diuisible by p.
In THIS WAY THE SOLUTION OF THE PROBLEM IS RECUCTED TO THE DECOMPOSITION INTO PRIME FACTORS OF CERTAIN NUMBERS SMALLER THAN 0 THE NUMBER IF WHICH IS SMALLER THAN /ft.
From this example we see that the formulas of kronecker may have
NUMEROUS AND DIVERSE APPLICATIONS.CHAPTER !V
BUM BRIO AL THEOREMS OF l JO WILL E, RELATED TO GENERAL PROPERTIES OF ANALYTIC AMD ARBITRARY-. NUMERICAL FUNCTIONS.
§ 43. The trigonometric series representing elliptic functions
MAY 9 E WRITTEN IN a FORM SOMEWHAT DIFFERENT FROM THE ONE IN :WhICH THEY USUALLY APPEAR, IF WE ARRANGE THESE SERIES ACCORDING TO POWERS OF THE MODULUS q. |N THIS CASE THE CO EFFICIENT OF A CERTAIN POWER OF q WILL 8E A T R I GO N.OM ET R I C FUNCTION, DEPENDING ON THE DIFFERENT DIVISORS OF THE EXPONENT OF THE POWER OF q OR ON A CERTAIN ROOT OF q. WE HAVE THUS
2Kk
si n m
3Xx
sc
4-
n~ 1
gn-l
\
■ u
gn-i * SO
sin
(i)
2X*
•eo s d’s
2iv x
oe'
n= i
1	(-1) 2 cos
gn-i=i5
(2)
o :r *
¿J.i. ■ .	. A A
— b am--------------- = 1
ix	n .
♦ *fU'
> <q”	(“li cos2ixr,
n«"i .0=4(26-1)
(3)
slnsain~p =	Yl d(l - cc s2ix)| .	(4)
n= 1 n= ¿(25-1)	.	’
In THE FIRST TWO EQUALITIES THE SECOND SUMS ARE EXTENDED TO ALL POSITIVE INTEGRAL SOLUTIONS OF i6 = 2rt “ 1, IN WHICH 2(1 - 2 IS GIVEN, AND i AND 6 ARE UNKNOWNS. IN THE LAST TWO EQUALITIES THE SAME SUMS ARE EXTENBED TO ALL POSITIVE INTEGRAL SOLUTIONS OF THE EQUATION ¿(25-1) -0 IN WHICH n IS GIVEN AND 4 AND & ARE UNKNOWNS.
Many elliptic functions may be either immediately represented
BY A TRIGONOMETRIC SERIES OR MAY.66 EXPRESSED AS A CERTAIN FUNCTIO OF OTHER ELLIPTIC FUNCTIONS; SUCH FUNCTIONS ARE REPRESENTABLE IN MORE THAN ONE WAY BY TRIGONOMETRIC SERIES; AS A CONSEQUENCE OF THI FACT WE CAN OBTAIN EQUALITIES COMPOSED ENTIRELY OF TRIGONOMETRIC SERIES, ON WHICH DIFFERENT OPERATIONS ARE PERFORMED. THE COEFFICIENTS OF DIFFERENT POWERS OF q IN THESE EQUALITIES FORM EQUALITIE
In this way we obtain theorems concerning trigonometric functions;
INTO THESE THEOREMS THERE WILL ENTER SUMS EXTENDED TO ALL POSITIVE INTEGRAL SOLUTIONS OF CERTAIN I N DETERM I NAN;T£ E QU A T I 0 N $. ( f N THE EXPOSITION OF LlOUVILLE SOLUTIONS ARE SUBJECTED TO CERTAIN LIMITATIONS.)
10110 2
The majority of analytic functions are, within certain limitations, representable by trigonometric series. Giving to the variable in the trigonometric equality referred to above suitable particular VALUES AND SUMMING UP, WE MAY EXTEND A THEOREM ESTABLISHED FOR TRIGONOMETRIC FUNCTIONS TO ALL FUNCTIONS WHICH ARE, UNDER CER-T“AT N CONDITIONS, REPRESENTABLE BY CONVERGENT SERIES OF THE TRIGONOMETRIC FUNCTIONS FOR WHICH THE THEOREM WAS ESTABLISHED.
Since, however, the arguments of functions in such theorems are
ALWAYS INTEGERS, AND SINCE EVERY ANALYTIC OR DISCONTINUOUS FUNCTION
having definite finite values for integral values of the argument
MAY BE REPLACED BY ANOTHER ANALYTIC FUNCTION EQUAL TO IT FOR THE VALUES OF THE INDEPENDENT V AL t ABLE UNDER CONSIDERATION, SUCH THEOREMS ARE EXTENDED ALSO TO NUMERICAL FUNCTIONS OF A CERTAIN TYPE AND TO ANALYTIC FUNCTIONS WHICH ARE NOT REPRESENTABLE, WITHIN REQUIRED LIMITS, 8Y TRIGONOMETRIC SERIES.
THE ONLY CONDITION FOR THE APPLICATION OF THE THEOREM TO A GIVEN ANALYTIC Oft NUMERICAL FUNCTION ARE AS FOLLOWS; l) THE FUNCTION MUST HAVE DEFINITE FINITE VALUES ON THE RANGE OF ARGUMENT CONSIDERED,
AND 2) IT MUST SATISFY CERTAIN CONDITIONS BROUGHT ON BY THE FACT THAT THE THEOREM IS NOT TRUE FOR EVERY TRIGONOMETRIC FUNCTION; (FOR INSTANCE, IF THE THEOREM WAS PROVED FOR COSINES AND NOT FOR SINES THEN THE FUNCTION MUST BE EVEN).
LET US NOTE THAT WHAT HAS BEEN SAID ABOVE IS ALSO TRUE FOR CERTAIN OTHER NON-ELLIPTIC FUNCTIONS, USUALLY STUDIED TOGETHER WITH ELLIPTIC FUNCTIONS; WE HAVE IN MIND LOGARITHMS OF ,J aCOBI FUNCTIONS AND THEIR DERIVATIVES. WE HAVE ALREADY CONSIDERED, IN THE 3RD CHAPTER THE RELATION BETWEEN THE SERIES WHICH REPRESENT THESE LAST FUNCTIONS AND THE SERIF. S WHICH, WITHIN CERTAIN LIMITS, REPRESENT ELLIPTIC FUNCTIONS. |N THE 4TH EXAMPLE OF THIS CHAPTER WE SHALL, IN PAR TICULAft, EMPLOY SERIES REPRESENTING lgS^Cx) AND ls§(x). I THEREFORE GIVE THE SERIES FOR THESE FUNCTIONS IN THE FORM IN WHICH WE SHALL NEED THEM,
•N THE LAST SUMS OF EQUALITIES (5) AND (6) THE SUMMATION IS EXTENDED TO ALL POSITIVE INTEGRAL SOLUTIQNS OF THE EQUATION n = d(2&-l),
(5)
2 i xl.	(o)103
IN WHICH 0 IS A GIVEN INTEGER AND d AND S THE UNKNOWNS.
Having indicated the method of deduction of a certain type of
THEOREM FROM SERIES REPRESENTING ELLIPTIC FUNCTIONS, LOGARITHMS
of ,Jacobi functions, a kd the derivatives of these logaritms we
SHALL TRY TO MAKE IT MORE CLEAR BY THE FOLLOWING FOUR EXAMPLES.
§44. Sxampls 1. From equality (1) we easily obtain
- e £ {<,*£ [«..U--*»)* - co,(4* *	(7)
n= l
WHERE THE LAST SUM IS EXTENDED TO ALL POSITIVE ODD SOLUTIONS OF THE EQUATION
i' 5' + d»6" = 2n,
IN FOUR UNKNOWNS d',d",	Ô', AND 5".
2Y COMPARISON OF THE COEFFICIENTS OF THE ODD POWERS OF q IN EQUALITIES (4) AND (7) WE OBTAIN
2{cos(d' - d") x - cos(i' + df,)x} = 2d(l - co s2dx).	(3)
The sum in the right hand member of this equality is extended to
ALL POSITIVE INTEGRAL SOLUTIONS OF THE EQUATION
d5 = 2o - 1;
AND THE SUM IN THE LEFT IS EXTENDED TO ALL P0S1ITIVE ODD SOLUTIONS OF THE EQUATION
d'.5* + iM&" = 4n - 2.
Let f(y) be an even analytic function well defined and finite everywhere between the limits -h and + b; then
f(y) = Ia0 * Ajcc-Ç-
, 2yn
Agcos—
myu
♦ A CO 3“-—
m h
a»
A . -1
+ n
r
a )
-h
(y) co s-— iy.
b
Let us put y equal to 0, 2d, ¿’ - and df + d", successively in equality (9), and in equality(3) put
rc 2n 3n 4it	mn
* = 7/ h' b* h'“*’ h'***
Let us multiply both members of (s) by Ax, for x??, by A'2 for 2n	1
X	AND IN GENERAL, BY Am FOR X s-jT . IF WE ADD THE RESULTING
EQUALITIES AND TAKE INTO CONSIDERATION EQUALITY (9) WE EVIDENTLY OBTAIN	•	'104
sKU'-r') - fC-i'+r)} * 2«*{f(D) - f(2i)}.	do)
Equality (10) holds for all values of the arguments i' - d", df + d”,
AND 2d NOT EXCEEDING b IN ABSOLUTE VALUE.
NOW, LET f(y) BE AN EVEN ANALYTIC OR NUMERICAL FUNCTION WHICH HAS A DEFINITE FINITE VALUE FOR EVERY VALUE y » i' - d", d'+d”, 2d, FOR WHICH EQUALITY (10) HOLDS, AND FOR THE VALUE y = p . THEN WE CAN ALWAYS FIND AN ANALYTIC FUNCTION £i(y) EQUAL TO f(y) FOR EVERY VALUE y= i'-i", d'’*!'1, 2'1 FOR WHICH (10) HOLDS, AND SUCH THAT fi(y) IS REPRESENTABLE BY A COSINE SERIES WITHIN THE LIMITS -1» AND + h; h MAY BE CHOSEN GREATER THAN ALL THE ARGUMENTS IN (10). THEN EQUALITY (10) HOLDS FOR ¿x(y) AND THEREFORE FOR f(y).
Theorem 1. If d, 6, i', 6’, d" and 6" are odd positive integers satisfying the equalities
d6 = 2n - 1, d’S' ♦ i*b" = 4o - 2,
and if f(x) is an even analytic or numerical function taking definite finite values for Integral values of x occurring in the equality we are about to mention, then f(x) satisfies the equality
2{f(d»-d") - f( df ♦■*")}:* 2d{f(p) - f (2d)} .	(10)
From the same equalities (4) and (7) we obtain, by equating the
COEFFICIENTS OF EVEN POWERS OF q,
2°t“1Sdif(0) - f(20Cd)} = 2{ f( d* - d") - f(d'+d")},	(n)
WHERE d, 1	j!‘,	6,	5' AND 5fl ARE ODD POSITIVE NUMBERS SATISFYING
THE EQUALITIES
dS = 2n-l,	d’5' * d,,6w * 2a(2n-l);
IN WHICH n AND a ARE GIVEN POSITIVE INTEGERS; OC > 1.
In ORDER TO UNDERSTAND ONCE FOR ALL THE IMPORTANCE OF SUCH NyM'-SRICAL FORMULAS WE SHALL INTERRUPT OUR EXPOSITION TO CONSIDER W|fH LIOUVILLE VARIOUS APPLICATIONS OF FORMULA (10) (see Liouville's Journal series 2, vol.3* pp 145-150).
Let f( x) = x*; then equality (10) gives
2d'd" = 2d3	(12)
LET Cjin) BE THE SUM OF THE DIVISORS OF n AND C3(n) BE THE SUM OF THE THIRD POWERS OF THESE DIVISORS; THEN,IF m IS ODD,
f,3(rn) = Cl(l)C1(2m-l) +	(3)C1(2m-3) + ... + Ct(2m - 1)^(1). (12')
1) From the last equality it follows that the number of decom-POSITIONS OF 16n + 8 INTO THE sum OF EIGHT ODD SQUARES 13 EQUAL TU THE SUM OF THE CUBES OF THE DIVISORS OF gn + 1. FOR THE NUMBER OF DECOMPOSITIONS OF Sn + 4 INTO THE SUM OF FOUR ODO SQUARES IS EQUAL TO THE SUM OF THE DIVISORS OF gn+ 1.
2) LET m BE A PRIME NUMBER OF THE FORM 8*1 + 3. EQUALITY (12') MAY THEN BE EXPRESSED AS FOLLOWS:
^{C3(m) -	= ^(Dqcan - l) + £*(3)q(2n - 3) + • • • . . + C^m ♦ 2)Ct(m - 2).
The NUMBER 2( 8u + 3) IS NOT DECOMPOSABLE INTO THE SUM OF TWO SQUARES;
<x 8 y
THEREFORE D AND gm - p CANNOT BOTH BE SQUARES. IF p = a b C ...
(p	1S ASSUMED TO		BE ODD)	THEN	
	^(p) =*	(l ♦	a + a2 + .	. . + J	*a)(l+b+b2+...+b^)	
1 F	& > P ! Y>» • •	ARE	ALL EVEN	, THEN £ (p) IS ODD; BUT IF AT LEAST ONE	
OF	THE NUMBERS OC,		P# yp • •	. 1 S	ODD, THEN £ (p) 13 EVEN. THEREFORE,
SINCE p AND		2m - p	CANNOT	BOTH	BE SQUARES, ( p ) C ^ (2ffl - P ) *s EVEN.
But
C3(m) = 1 + (8* + 3)3, Cjim)2 = (3* +3) 8 + 2(8* + 3) + 1;
T herefore
-{ C3(m) - C1(m) }	= 2(8* + 3) 16*':* + 10* +1) =. twice an odd number.
Hence we conclude that among the numbers r, 1(p)C1(2ia-p) »n which p <iH, THERE is an odd number of numbers WHICH are doubles of odd NUMBERS. But in ORDER THAT
(1 + a + a2 +... + aa) (1 + b + b 2 + . .. + b'?’) (1 + c + c2 +....+ c Y). . . .
BE DIVISIBLE BY 2 BUT NOT BY 4 IT IS NECESSARY THAT ONLY ONE OF THE EXPONENTS (X, (3, . y, . . . BE ODD. THUS, IF C i ( p ) K l ( 2m - p ) IS TWICE AN ODD, THEN ONE OF THE NUMBERS p AND gffi —p MUST BE A SQUARE AND THE OTHER THE PRODUCT OF A SQUARE BY AN ODD POWER. OF A PRIME; THIS PRIM IS OF THE FORM 4X+1 AND ITS POWER HAS AN EXPONENT CONGRUENT TO 1 MODULO 4, SINCE OTHERWISE THE EXPRESSION 1 + a + a2 + . . . + a“"1 + a“, IN WHICH a IS THE PRIME NUMBER CONSIDERED, IS DIVISIBLE BY 4. HENCE
the theorem: The double of a prime number of the form 8* + 3 is always expressible as the sum of an odd square and the product of of another odd square by a prime number of the form e-v + 5 (8v +5 since 16* + 6- (2n + l)2 is of the form 8n + 5).
Putting f(x) = x*, f(x) = x4, we obtain other theorems, which are, however, not so elegant.
Finally we Put f(x) = cos xt, where t is a constant. Then equality (10) GIVES2{sind't sin d"t}
2d sin2 dt
(13)
IF WE PUT t = rt/2, WE OBTAIN
A'-1	d »-1
2{ 2(-l) 2 2(-l)	2 } = ^(¿n + l),	(14)
WHERE THE FIRST, SECOND AND THIRD SUMMATIONS ARE EXTENDED TO ALL POSITIVE ODD SOLUTIONS OF THE equations: in' + m" = 4n +2, m'. = d* 6 \, m" = d"6" respectively
BUT (-1)	2 IS THE NUMBER OF DECOMPOSITIONS OF TWICE A CERTAIN
ODD NUMBER INTO THE SUM OF TWO ODD SQUARES; THEREFORE (14) GIVES THE
following theorem (of .jacobi): ■? he number of deoompo sit ions of 8n + 4 into the sum of four odd squares is equal to the sum of the divisors of the number 2n +1.
LET US NO W PUT t*K/3 - IN (13):	WE GET
L
L
31 Il-
Sl n-
d"7i
si Ü
But by page 9
4 v” . d’n
7§z_slnir
IS THE NUMBER OF SOLUTIONS OF THE »NDETERM I N AN TE EQUATION
(15)
x2 + 3y2 - -in' + 1 s d* 6'.
Since the equation x2 + 3.y2 = 4n + 2 has no solutions, then the
EXPRESSION
iôr, . r>dn
3
13 v~ . ?i
~ ) d sm ■
ó L—~-
WILL GIVE THE NUMBER OF SOLUTIONS OF THE EQUATION
x2 + y2 + 3(z2 + t2) = 4n+2= 2d6..
Hence the theorem: ? he number of solutions of the equation x2 + y2 + 3(z8 + t8) *. 4n +2 is equal to four times the sum of those odd divisors of 4n +2 which are relatively prime to 3,
§45. Let us replace x by ~ - x in equality (i); we obtain
ij
2 Kx
cos a;fl-2 Ktc	n
à ara
2Xx
=,	4
n= 1
r (-1)
2n-l* d&
d- : 2*
CO ;
f
Multiplying together member by member equality ( 1)
AND THE LAST107
EQUALITY WE OBTAIN
2Kx
OTf •*
.o A A
2 p sin am----cos am —
4K k	«	n
Tt‘
A am
2£x
n
a« -1
= 3
^	^qn2(-l)	2 £sin(d' + d")x .♦ sin(d* -d”)x]|.	(16)
n» 1
The last sum is extended to all odd, positive solutions d', 6', d" 6" of the equation 2n = d* 6'+d"6", where n is an arbitrary positive
INTEGER CONSTANT FOR THE LAST SUM.
Subtracting member by member equality ($) from (6) we obtain
IgAam—^ = lg/k7 +4^^q2n_1 ^ £cos3d4*
n* 1	2n-i = d6	J
Differentiating the last equality we get
2Kx	2Kx
s sin am----cos am---
2Kk _________n_________
n	, 2K x
A am----
n
Multiplying this last equality member by member by the equality
t f r.,sinMx}
i3 1	sa^l = uo
2 K n
WE OBTAIN
2
rc
2Kx	2Kx
sin am----cos am----
n	n
3Xx
A am---
n
3 Iq^^Ssia 2dx|
n* 1
+
OC
n2cp(m2) sin 2dt
(17)
where <p(m2) i s the differencr between the number of divisors of OF the form 4H +1 and the number of divisors of ms OF THE FORM 4N +3. The last sum in equality (17) is taken as follows: the number n is to be decomposed in all possible ways according to the
FORMULA
a
n = !£ ! + 2 ID 2
in which idi and m8 denote odd positive numbers and a is either positive or zero; further id i and a* are to be decomposed in all108
POSSIBLE WAYS A CCO Fi DI NO TO THE FORMULA
IQ i =	115 j , !E g = d 2 5 g j
THE SUM CONSIDERED IS EXTENDED TO ALL ID 2 THUS OBTAINED AND xr> ALL ii CORRESPONDING TO A GIVEN 1D2.
A COMPARISON OF THE COEFFICIENTS OF THE ODD POWERS OF ^ IN (1 e) . N0 (17) GIVES THE EQUALITY
Y_	sio( i' + d")x + sin(d’ -d!,)x] = Y_ sill‘
gn^l’ô’ + d"6"	m-iô
*dx
+ 4
\
/
n ~	in 2
fp(ffi s) si n2i ix,
(13)
IN WHICH IE, >ifi2, it, bx, i‘,	5* , i% AND 6" ARE ODD POSITIVE NUMBERS
AND a > 0.
LET F(x) BE AN ODD ANALYTIC FUNCTION WELL DEFINED AND FINITE WITHIN THE LIIMITS “h,	+1', THEN
PU)
,	. n x
Ai sin-,“
ii
+ A 2 S1
+
n Tiy. n I
(19)
A . -1 n h	+ h J P(x)sio	flrtx , —- d X. n	
Let us replace x by different	h 1' + d", d'	- d':, 2d and 2dt	IN E Qt> AL 1 T Y
(19) AND IN EQUALITY (13) PUT	SUCCESSIVELY		
n 2n	3rc	an	
x = b » T*	^ > • • • 9 ii	h*		
Summing up the results of the	PO RE GO 1 NO	SUBSTUTI T ION S IN	(16), H A V 1 N
PREVIOUSLY MULTIPLIED THESE RESULTS BY At, A2> .	A n,... RESPECTIVELY,
WE OBTAIN THE EQUALITY
2(-i)	i» + 1") + F(d’ “ d")}	= SF(2d) ♦ 42<?(m 2)F(2ii), (2C)
IN WHICH THE FIRST, SECOND, AND THIRD SUMS ARE EXTENDED TO ALL SOLU-T10 NS OF THE EQUATIONS
1*6' + d!'5M = 2ic, 15 = m, di5x + 2aic2 = id;
IN THE LAST EQUATIONS ID {WHICH IS GIVEN) AND THE UNKNOWNS d',	5',	d",
5”, d, 6, di, and in2 are all odo and'posi ti ve, and the unknown a is
A POSIT1VE INTEGER.
Noting again that every function having definite finite values for
CERTAIN VALUES OF ThE V A DI ABLE MAY BE ALWAYS REPLACED BY A FUNCTION WHICH IS CONTINUOUS AND FINITE WITHIN CERTAIN LIMITS AND WHICH IS E-QUAL TO THE ORIGINAL FUNCTION FOR THE VALUES OF THE VARIABLE CONSID-ìoy
cftg o, we conclude the following: equality (20) holds for every fun tion Fix')/ which has definite rinite values for x equal to those i1 + i", i'-i", 21, and 2li, which occur in (20), orovidini that
F(-x) = -F( x)..
CoaOLLAKY. LET F( x) ■* X AND LET “ i (B) DENOTE THE SUM OF ALL THE VISORS OF ID. W E THEN OBTAIN
x~
L
ÌT1C îû- 1 ) rp ( m *' )
2’S-rn ' +D3 "
+ 4 ) 11 (ip 1 )ip (ts g) . ir.=in i+2aa a
If the number of odd posit Luo solutions of the equation
x2 + y2 + z2 + t2 + 2(u2 + Vs) = 16n +8 be denoted by A and the number of odd positive solutions of ' he &q-,
x2 + y2 + z8 + t8 + 2a(u 2 + v 2) = 8n + 4,
in which a>1 is variable by B, then
A = 43 + CjCOn +1).
§ 46.. Example 3. We shall as before denote ev m,mi,»*, 1,6,1', 6', i% 5", i"'f 6f", odd positive numbers and by a a positive integer.
On raising to the third power we get
3, 3
3 iv kî
2Xx
~-sia8a»~~ = -16)	{q 2 2 [ si n 11 * 1 ” + i *" ) x
m *1
+ sin(i,-i"-i,")x - sind'-j!, + l«”) x - sin ( V ♦ i"- ln ) x]}, (21)-
WHERE THE SECOND SUM IS EXTENDED TO ALL SOLUTIONS OF THE EQUATION
1*5» + d"6" + afM 6f,r a a..
DIFFERENTIATING EQUALITY (1) TWICE, WE OBTAIN
3X8k
.3, 3
/•1 ,	.	3i\ X	16a X . 0
(1 + K )sin aw-------- - -------r~-sin8a'?t
3	n
= 4	^212 sin lx| ; (22)
m* 1	J
n	"	n
The last sum is extended to all solutions of the equation From a formula from the introduction* it follows that
4X2
4KS-'' 8
n
, * 1 * 8SC1(»){q* tSq2* *3q4- *3q3* + 3q* 90 + . . . .
2	« ■ * SZ^UX-q" *3qs" *3q4* .3q3” ♦ 3q19'' ♦.¡,
n
n.
* The author refers to a quotation from Jaoobi’s paper in U at hem at i so ha Annalen v*$,p. in whioh Jaoo&i gives an expression for	Translator’s note*110
4K
V(l + k2) = 1 + 24SC,(«i){q‘+q8**q4,, + Q8“ - • . • . ),	(23)
7t
4K
n
2
~(l+k2) = 1 + 24^ H^i(m>qi
a-i
m,	(24)
m oc
WHERE « TAKES ON INTEGRAL VALUES FROM 1 TO ®.
Multiplying together member by member equalities (24) and (1) we
OBTAIN
8K9k
r">	o v	m	^.
(l+k^) sin	= 4/ {q^2sin cb$-+■ 93/	(iDg) sin c^x}
(25)
m* l	ra~ 1
WHERE IN THE SECOND SUM ffl= dO, AND IN THE FOURTH ONE ill = d, 6j ♦2°tna2.
From the equalities (21), (22), and (25) we deduce 32{sin(d' + d" + dm )x - sin(d"+d'" -dT)x - sin(d’ + d"' -d")x - sin(d‘ +d"-d’")x}	= 2(d2-l)sindx - 242C1(mg) sin dJx. (2o)
Repeating the arguments used in examples 1 and 2 we obtain the
FOLLOWING THEOREM
If F(-x) = — F(x) and if ra, m.9f dr, 6',... are odd positive numbers then
82{F(d' + d" + d) - F(d" + d,"-dl) - F(d* + d,H-d") - F(d* + d" - d,M )}
=	2(d2 -l)F(d) - 24SCl(o2)F(d1),	(.27)
where the sums, taKen in the order in which they are written are extended to alt the solutions of the equations
m = d*» 6 * + d" 6" + d,"6'", m = d6, a =	t*+sV*
respectively. In these equations a is a variable ranging over all positive integers.
Corollary. In (27) let us put F(x) = x!’, and let us denote by r^Cm)
THE SUM OF ALL the DIVISORS OF A NUMBER ill, BY £3(1») THE SUM OF THE CUBES OF ALL THE DIVISORS OF m, AND BY WgC®) THE SUM OF THE FIFTH POWERS OF ALL THE DIVISORS OF ffl . THEN
1922^(01' K1(m"K1(ra'" ) + 242$3(m1K1(!Bg) = C5(ra) - ^(m),
Ci
p u T T 1 n g	m =	raf. + in ” + m =	m + 2	m g.
If ra	is	an odd number,	G the	number	of decompositions of 4m into
tye sum of twelve odd squares, and H the number of decomposit ions of 3m into	the	sum of twelve	odd squares	according to the	formula
x2 +	y2 + z2 + t2 + us	+ v2 +	r 2 + s2	+ 201*1 (p2 + d2+ h2	+ o2), *
then the following equality holds
* It was proved, in the first example that the number of decompositions of ien+8 into the sun
of eight odd squares is equal to c,3( £n ♦ l).Ill
SG +
H =
(ni) -£3(m) ].
§ 47.Example 4. In the previous examples the complementary divisors 0’s DID NOT OCCUR IN THE COEFFICIENTS OF THE SUMS CO N SI D E ED. LET US SEE IN WHAT WAY THE FIRST POWERS OF THESE COMPLEMENTAF DIVISORS CAN ENTER INTO THESE SUMS. LET US NOTE THAT TERMS qn/d OCCUR IN THE EXPANSION OF THE LOGARITHMS OF .J ACOB I FUNCTIONS [FORMULAS (F) AND (6)3 AND ALSO IN THE SERIES REPRESENTING LOGARITHMS OF ELLIPTIC FUNCTIONS. If, NOW, WE DIFFERENTIATE THE TERM ^qn WITt ftESPECT TO q WE OBTAIN
J q	- o q
Making use of this remark we shall prove a theorem of liouvill (.Journal Liouville, tome 3,serie 2).
Thsohsm. If f(-x) = f(x), then
22[f(d' -
2ad") -
f (d*
2<Xd")
3 = 2(6 - d)f(d),
( 28)
where the first sum is extended to all the odd positive solution-$ d', O’, d", and 6” of the equation
d',6' + 2ad"0" = m
U is a given odd number, a is a variable taking on only positive
values), and the second sum to all positive odd solut ions of the
equation	„ e
.10 * m.
Proof


1
L doos
a - dô
(29)
WHERE IB IS ODD. OPERATING ON THIS EQUALITY WITH
WE OBTAIN
"jL d*~	, /* \ /X \
_qdq * ix8_	
Sut it is easy to deduce that
42[qms(0 - d)cosdx].	(30)
1 2
•J>	= 4‘2[qm2(0- d)co sdx]. ( 31)
Differentiating equality (29) with respect to x we obtain112
1

ix
42[qtt2sin ix 1.(32)
From the same equalities (3) and (6) we deduce
Multiplying together the equalities (32) and (33) and comparing the
RESULT WITH EQUALITY (31), WE 0 STAIN
2 T" ■ico s( 4” - 2ai”) x - cc j( d' + 2a d") xl = ^ (5 - i) co s ix .	(34)
.•D=l’l6» + 2cti"5"	a * d5
FROM EQUAL ITY (34) BY a SEQUENCE OF FAMILIAR ARGUMENTS ONE DEDUCES (28)-«
§ 48. IT IS SUPERFLUOUS TO CONTINUE THE DEDUCTION OF VARIOUS FORMULAS. IT WILL BE MORE TO THE POINT TO MAKE A FEW REMARKS CONCERNING THE CHARACTER OF THOSE NUMERICAL THEOREMS WHICH COULD BE OBTAINED BY ThE METHOD DESCRIBED ABOVE FROM THE SERIES RESP RESEN T ! N G JACOBI FUNCTIONS AND THEIR DERIVATIVES. |N THE TRIGONOMETRIC SERIES JUST MENTIONED EVERY TRIGONOMETRIC FUNCTION IS MULTIPLIED BY A RATIONAL FUNCTION OF q OH A CERTAIN ROOT OF q; THf COEFM Cl ENT OF THE ARGUMENT OF THE TRIGONOMETRIC FUNCTION IS USUALLY PROPORTIONAL TO THE EXPONENT OF q IN THE NUMERATOR OF THF. RATIONAL FUNCTION, THE DENOMINATOR OF THE FUNCTION IS USUALLY OF THE FORM 1 + q r , WHERE r Is ALSO PROPORTIONAL TO THE EXPONENT OF q IN THE NUMERATOR. |F NOW SUCH A SERIES BE ARRANGED ACCORDING TO THE POWERS OF q, THEN EACH POWER OF q WILL HAVE AS ITS COEFFICIENT A CERTAIN ALGEBRAIC SUM OF SIMPLE TRIGONOMETRIC
functions. These sums are extended to divisors of numbers proportional
TO THE EXPONENTS OF q. THAT IS SUCH FUNCTIONS AS
d- 1
¿sin ix, ¿cos ix, 2(~1) 2 cos ix , ¿i2cos ix, 2$eosix	etc.
If now we multiply together SEVERAL OF SUCH SERIES WE obtain sums SUCH AS
Ssin(d' + d" + d'" ) x, Sdcos( d + 2ad'.) x, ¿(&-d)eosdx,
IN WHICH THE SUMMATION IS EXTENDED TO A CERTAIN KIND (HERE ODD) OF INTEGRAL, POSIT I VF SOLUTION OF CERTAIN EQUATION; FOR THE SUMS JSUT CITED THESE EQUATIONS ARE RESPECTIVELY
d' 6’- + df'5,f + d,M 6'" = ¡3, d6 + 2ad'6' = !3, j& = a.
From the last sums we come without difficulty to the sums113
,a -
2F(i' + d" ♦ d'" ), 2df( i + 2~d’V, 2(6 - i).f(d)-, etc.
IN WHICH F(x) AND f(X) DENOTE ARBITRARY ODD AND EVEN FUNCTIONS RESPECTIVELY. THE EQUATIONS '.'/HI OH GIVE THE ARGUMENTS OF THE. FUNCTIONS IN THE SUMS ARE,	IN	GENERAL,	FORMED AS FOLLOWS: A	GIVEN Nu\
BER IS TO BE DECOMPOSED IN VARIOUS	WAYS, ACCORDING TO A	CERTAIN
LAW, INTO SEVERAL TERMS (THE NUMBER OF TERMS IN SOME CASES MAY BF EQUAL TO ONE, THAT IS	THE	NUMBER MAY NOT BE DECOMPOSED)	AND EACH
TERM IS FACTORED INTO	TWO	FACTORS;	OF THESE TWO FACTORS	THE ARGU-
MENT OF THE FUNCTION ALWAYS CONTAINS ONE. THERE ARE VAR.IOuS LAWS OF DECOMPOSITION OF THE NUMBER FIRST INTO THE SUM OF SEVERAL TERM AND THEN INTO FACTORS.
PUT IN THE THEORY OF ELLIPTIC FUNCTIONS WE MEET ALSO OTHER T R I ‘ 0 NO METRIC SERIES OF AN ENTIRELY DIFFERENT CHARACTER:	1) SERIES RE-
PRESENTING Jacobi functions, 2) series representing hermit.ian series. LET US CONSIDER HOW THE THEOREMS DEDUCED WITH THF HELP OF THESE SERIES WILL DIFFER FROM THE THEOhSMS DEDUCED ABOVE.
IN THE JACOBI FUNCTIONS THE ARGUMENTS OF THE TRIGONOMETRIC FUN TIONS FOLLOW A SIMPLE ARITHMETICAL PROGRESSION AND THE EXPONENTS OF THE MODULUS q FOLLOW AN ARITHMETICAL PROGRESSION OF THE SECOND ORDER. IF NOW .WE MULTIPLY TOGETHER SERIES OF WHICH ONE REPRESENTS
a Jacobi function and the other represents either an elliptic funo
TION Oft ANOTHER FUNCTION Or § 43, THEN THE EXPONENT OF q IN THE PRODUCT WILL BE GIVEN (tF WE LIMIT OURSELVES TO CONSIDERATION OF
Jacobi functions in a narrow sense) by the formula
aM 2 + b l6,
IN' WHICH a AND b ARE CERTAIN POSITIVE NUMBERS AND M, 'i, AND 6 ARE VARIABLES TAKING ON ONLY POSITIVE INTEGRAL VALUES OR THE VALUE ZERO. AT THE SAME TIME THE ARGUMENTS OF THE Sifl£ AND CO 31 0t3 IN THE PRODUCT WILL BE GIVEN BY THE FORMULA
( PM + Oil) X,
IN WHICH p AND >7 ARE CERTAIN NUMBERS - POSITIVE, NEGATIVE, Thus, FOR EXAiMPLE, MULTIPLYING TOGETHER THE EQUALITIES:
9"v
#0 A ft	& ¿V A
----si n aic ——
n	n
i 6
sin ix
*
OR 2 PRO.
IN WHICH i AND GER, W6 OBTAIN
2iCk . —- si n n
+ OC
rn
q ecs2nx,
n -
6 ARE ODD POSITIVE NUMBERS AND 0 AN ARBITRARY
■f oc	+ $c
ara~^9g(x)	n( i + 2rO x.
n=-*
I NTE-
(35)m
Arranging such a sum according to increasing powers of q, we obtain as THE COEFFICIENT OF EACH POWER A SUM OF TRIGONOMETRIC FUNCTIONS 'WITH ARGUMENTS OF THE FORM
(pfci + ci) x;	(36)
THIS SUM IS EXTENDED TO ALL THE SOLUTIONS,OF A CERTAIN KIND, OF AN IN06TERMINANTE EQUATION OF THE FORM
uH"	+ v .16 = o,	(37)
IN WHICH U, V, AND 0 ARE CERTAIN POSITIVE INTEGERS OR ZERO. THE NUMBER 0 IS PROPORTIONAL TO THE EXPONENT OF q. LET US NOTE THAT THE NUMBER U MAY ALSO BE NEGATIVE, BUT i AND 6 CANNOT BE NEGATI VE. THUS,
IN OUR EXAMPLE [EQUALITY (3$] WE HAVE
CC
— sin ais— %Ak) =4 Y <fqr “ ^2sin( i + 2n) A,	(36)
u	n 3	L__ ^	\
T* 1
|T IS CLEAR THAT SUCH SUMS, EXTEND ED TO ALL THE SOLUTIONS, OF A CERTAIN KIND, OF THE I NDETERM INANTE EQUATION (37), WILL OCCUR ALSO IN THOSE THEOREMS WHICH CAN BE DEDUCED FROM THE SERIES CONSIDERED BY THE METHODS SUGGESTED IN THE PREVIOUS SECTIONS; IT IS ALSO CLEAR THAT THE ARGUMENTS OF THE ARBITRARY FUNCTIONS WILL BE PROPORTIONAL TO (36) .
IF WE MULTIPLY TOGETHER TWO .JACOBI FUNCTIONS, THEN THE ARGUMENTS
of cosines and sines will be proportional to
n + Sn't	(39)
'.vhere H IS a numerical coefficint, and n and n' are variables taking ON CERTAIN INTEGRAL VALUES. IF WE ARRANGE THE PRODUCT ACCORDING TO POWERS OF q, THEN THE COEFFICIENT OF A CERTAIN POWER OF q WILL BE A SUM OF TRIGONOMETRIC FUNCTIONS. THIS SUM WILL BE EXTENDED TO ALL THE SOLUTIONS, OF A CERTAIN KINO, OF THE EQUATION
nn2 + ¡in’2	= N,	(40)
WHERE	N	IS	A	GIVEN	I N T F. G E R, POSITIVE	OR ZERO,	AND 0	AND	C	ARE	POSITIVE	INTEGERS,	CONSTANT	FOR A 0 1 V F N	PROBLEM.	IT IS	EVIDENT	THAT
(39) AND (40) WILL ALSO PLAY A ROLE (AND IT. IS CLEAR WHAT ROLE) IN A NUMERICAL THEOREM D E D U C E D T H RO U GH MUL T I PL I C AT I 0 N OF TWO JACOBI FUNCTIONS.
LET US NOW PASS ON TO HERMIT!AN SERIES REPRESENTING THE PRODUCT OF AN ELLIPTIC FUNCT IONAANO A JACOBI FUNCTION. FOR THE SAKE OF GREATER CLARITY IN THE FOLLOWING EXPOSITION WE SHALL CONSIDER THE HERMIT-IAN SERIES EMPLOYED IN THE DEDUCTIONS OF THE PREVIOUS CHAPTER, AND ALSO WE SHALL DEDUCE ONE SERIES WHICH IS CITED FOm THE SAKE OF EXAMPLES, SINCE I PREFER TO CHOOSE MY EXAMPLES FROM THOSE PROPOSED BY LIOUVILLE.
Mo ting that 4ng — m2 - ( 2n — m) (2n + ra) » d6 and that d + 6 = 4n, we116
CAN GIVE TO SERIES (3) OP THE THIRD CHAPTER THE FORM
m,, x. .	■ 2kx o\“Jn+i” •d+i> ^
«“«t * "M-1 *“T^
(43
n = 0
WHERE THE LAST SUM IS EXTENDED TO ALL POSITIVE INTEGRAL VALUES OF d AND 6 SATISFYING THE EQUATION
4n + 3 = d6.
(41!
NOTING THAT
n2 + n + - - a2 = 7(211 + 1 + 2a)(2n + 1-2a) *» 7<*6,
4	4	4
that n + a = ¿(d - 1) and that 2n + 1 = j~(d + 6), we may write equ al-
rj	¿t
•TY (26) OF the third chapter AS FOLLOWS:
/2Kk.	. ,	2KX V ' C r.+ 1 . . \	d + 6 ")	f *
\/—^(x)c°s am— =	4M**l) 2cos—(4.3)
R=»0
THE LAST SUM BEING AGAIN EXTENDED TO ALL P0SITt VE : I NTEGRAL SOLUTIONS OF THE EQUATION
d6 = An + 1.	(42')
IF AS BEFORE
, p.	¡2 1j	.	v> \	/2LI
5s(j,q ) * y—,	= \j~iT’
THEN EQUALITIES (36) AND (44") OF THE THIRD CHAPTER MAY BE WRITTEN
OQ
■^Ss(x«q) = ■" H ^?n*'3:siii(d + 6)x|,	(13)

•sin ara
n= O
27 2Kx
am
TtN/k»
OC
“St(.2x, q2) = .3 y^/qn+p2sirx(d + 6)xj;
n * 0'
(44)
WHERE THE INNER SUMS ARE EXTENDED TO ALL THE POSITIVE INTEGRAL VALUES OF d AND 6 WHICH SATISFY THE EQUATION
2n +1 = dó.	(45)
Let us now obtain hermitian series for
2 Kxft . sin am----$„(x).
71
For this purpose we multiply together member by member the equalm
+ OC
2Xk
n
-sm am
¥ - *EE-"
2r-l)(2b-i)	.
sin(2r - l)x,
r, b* lWE OBTAIN
*a(x)
118
+ «
s
co s2nx.
n=*
_oc
2K*t . 2Kx ( s
.---sin am--------
n	rc 3
+oc	+ 00
. 2y £ 5"q“ *i(sr-,,('*‘‘1,t*i«(2r «2« -IV» ♦ 3in(2r - 2n - 1) xl,
n* r, b = l
LET US DETERMINE THE COEFFICIENT OF si G ( 2« + l) X I
y- /q2r-1	„	(r-n-i)8	(r+n)	1
A \	----;---- IÛ	- a	I
q*
"¿L 1 - q2r-iLq	“ q
ra 1
|( 2r-l) *+i-( ?n + l) 9 Q
-( r-i) ( 2n + l)	( r-|)(2n + l)
= 4 2_ q4-	;	4
- q
r= 1
/ -2 r + 1	/*2 r- l
/q ** /q
= 4	„it«“*1-’ *{qi( S--IV* ,	qJ(«r-«>*,
r=s l

*	i,„
4(2n * l )
= 2S_(0> q)q5V ^“Ti;	[1 + 2q_1 + 2q“4 + . . . + 2q~“ 1
WE HAVE THUS
'-Llk . 2'vX . . si n am -—- S „( x) a *
^	Si
= 2^_q4(2n*l)	[ 1 + 2q~ 1 + .. . ¡-2q‘n ]sin(2o + l)x.
n=0-
LET US ARRANGE THIS SERIES ACCORDING TO THE POWERS OF q. ARGUMENTS SIMILAR TO THE PRECEEDING WILL GIVE US
.■•i i
4ti + l=15
From all the previous formulas we see that in such a series the
SUM OF TRIGONOMETRIC FUNCTIONS IS EXTENDED TO THE SOLUTIONS OF THE EQUATION i6=n. This is the same as in the series representing ELLIPTIC functions. But the argument of the trigonometric FUNCTION IS
ENTIRELY DIFFERENT FROM THE ARGUMENT IN THE SERIES OF § 43; HERE IT PROPORTIONAL TO THE SUM OF TWO COMPLEMENTARY DIVISORS.
1+5

(46)117
«■ IF Vi E MULTIPLY A HERMIT.IAN SERIES BY A SERIES REPRESENTING A
.Jacobi- function either for an i nde-te rmi n ante argument or for a
PARTICULAR VALUE OF THE ARGUMENT THEN THE EXPONENTS OF q WILL BE OF THE FORM SO2 + r| i 6, WHERE K AND T| ARE POSITIVE NUMERICAL QUANT I TIES; THE ARGUMENTS OF T H E T R I G 0 N 0 M E T R I C FUNCTIONS WILL BE PROPORTIONAL EITHER TO
, i-t&'+pn, or i + 5,	(47)
WHERE P IS A NUMERICAL CONSTANT; IF THE ARGUMENT OF THE .JACOBI FUNCTION IS I NDETERMI NA-NTE IT WILL BE THE FIRST ONE AND IF A PARTICULAR VALUE OF THE JACOBI FUNCTION IS CHOSEN IT WILL BE THE SECOND one. Therefore-the arguments of the functions in the theorems
WHICH ONE DEDUCES THROUGH SUCH MULTIPLICATION OF SERIES WILL 6E PROPORTIONAL TO (47); THE SUMMATIONS OF THE ARBITRARY FUNCTIONS WILL BE EXTENDED TO ALL THE SOLUTIONS,OF A CERTAIN KIND, OF THE EQUATION
		(u,n 8 +■ A i5 = ft	1,				(48)
WHERE H, X, AN-D	Ñ ARE	GIVEN ANO G, i,	AND		6	AR E	UNKNOWN.
Let us assume	THAT	A CERTAIN HERMITIAN			SERI ES		1S MULTIPLIED BY
A series of the	KIND	CON S 1 DEfi E D I N §	43	AND		LET	ASSUME USA TH AT BY A PAM-
ILAR ARGUMENT WE	D‘E D Ü	CED, MAKINS USE	0 F	THE		FO RE	GOING PRODUCT, AN
ARITHMETICAL THEOREM. |N THAT CASE, THERE WILL OCCUR IN OUR THEORE'-SUMS OF FUNCTIONS, WITH EACH OF THE SUMMATIONS EXTENDED TO ALL THE VALUES, OF THE ARGUMENT OF THE FUNCTIONS, SATISFYING A CERTAIN EQUATION
fió + ni' 5'	= u;	. (49)
THE ARGUMENTS OF THE SUMMANDS WILL BE PROPORTIONAL TO
u(i + Ô) + Xi' .	(50)
IN THE PRECEDING FORMULAS 0 IS A GIVEN POSITIVE INTEGER OR ZERO, f, H, |i AND X ARE INTEGERS, f- AND H BEING POSITIVE.
BY SIMILAR REASONING WE COULD CONSIDER WHAT H.A PP Ê N S - WH E N WE MULTIPLY TOGETHER TWO HERMIT I AN SERIES OR IF WE MULTIPLY TOGETHER THREE, FOUR Oft. MORE SERIES OF DIFFERENT FORM; BUT THE PRECEDING WORK MAKES CLEAR THE CHIEF CHARACTERISTIC PROPERTIES OF THE ARITHMETICAL THEOREMS DEPENDING ON PROPERTIES OF THE SERIES USED IN THE DEDUCTION OF THE THEOREM.
Before we pass to examples, let us note that the arguments of
the TRIGONOMETRIC FUNCTIONS IN THE EXPANSIONS OF
1 1 1 1
Vx)' Mx)> V*)
WILL BE PROPORTIONAL TO THE DIFFERENCES OF TWO COMPLEMEN T AR Y DIVI-118
SORE OF THE EXPONENTS OF q AND ■/q; NAMELY*.
»(*)	”L^	9
.	.i>8
n+%	*—
¿^i	i-5
+ 4} (qn*3 y (-1)~2 eos-^xj*, (51)
lm	{	L.- II	/J
r.= l
n*°:	4 n +1= i 6
8	n*l	n=»0	An+I-TA
st (0)
S. fxV
oc	i> 5
V' f.	Sn V~
L. v * "" - ^ .	j L.
n=i	2n=d&
40 + 1= i& 6
s;p)
Mx)
00	&	i + 6+1	a
= —J— + 4 iq2n Y	cos(d~5)xL (54)
• co a x ' L )	‘—	5
2n* i 6
n * 1
In the last two equalities 2n is always decomposable into two fac
TORS OF DIFFERENT PARITY,
Let us consider now some examples.
§ 49. Example) 5. let us take the equality occurring in the beginning OF THE second example and multiplying it by the trigonometric
EXPANSION FOR 9 (>:) ;
^H(x) - 4Z{q^xL (-l)~*cos(d-3n)xj.	(55)
m=l
!n equality (55) m is odd, n may be positive, negative, or 2ero,
even OR ODD, d AND £ POSITIVE AND ODD; THE LAST SUM IS EXTENDED TO ALL THE SOLUTIONS, SATISFYING THE CONDITIONS JUST MENTIONED OF THE t Q U A T/l 0 N
d£ + 2n* = m.	(55' )
On THE OTHER HAND WE HAVE (INTRODUCTION AND PAGES 1 AND 2), DENOTING BY !E i,	i', AND 5 ’ POSITIVE NUMBERS,
$g(x) - 5 2_ Cl^m 1 cos iii1x,
OK
n

h -	1 f~ llzi id'5'
-14T - 9 ¿_ (-1) 2 q->	;
d' 5 ’ = 1 ; •
d'-l
o Tf	r—	1	^ ¿%
•^/i{9g(x) - 4/ {q2 2(-l)	2 cosm.x}.
<X	'	w—	1
»*.1119
The last sum is extendeo to all solutions,of the character alreac
MENTIONED, OF THE EQUATION
nil,8 ♦ d r 6* = 2m.	(56*)
Comparison of the coefficients of the same powers of q in equa • ties (55) and (56) gi ves
d * ~ 1
S(- 1) " * co s(d*2 n) x * 2(-l)mtx.	(57)
Theorem. ; If m, Uj, d, 6, d', and 5' are odd positive numbers and n is an integer, if these quantities satisfy the equations
ib + 2n2 = m, d* 6* +mt2 = 2to,
for m constant, and if f(-x) = f(xV where x) is well defined for the integral values of x considered, then
d- 1	d » -1
SC-1) 2 f(d- 2n) « 2(-l) 2 f(mx).
§ 50.2 xamplb! 6. Let us multiply (46) by $g(0):
2Kk . 2rCx . .f f n*lr ■ d,+ô l
—31,1 “T s*w = 4z_iq L“’TT
n = 0
(56)
(59)
The last sumis extended to all positive odd solutions of the
EQ UATION
m2 + dô * 4n + 2.	(59')
Let us compare (59) with (38):
j*6
Ssin(d‘ +2n')x = Ssin——x.	(60)
«£»
The familiar reasoning gives us the Theorem:
If F(-x) = -F(x) and F(0) = 0, then
SFC 3* + 2n’ ) = 2F ,	( 61)
where the summations are extended to all the solutions m, i, 6, d’, 6', and o' of the equations
d' 5» + 2n* 2 = 2o ♦ 1, dô + æ2 = 40 + 2,
such that «,d,6,d',5» are positive and odd and n' is an integer.
§ 51. Example 7. let us multiply together the series (1), (46)
AND THE SERIES REPRESENTING 9g(D); WE OBTAIN THEN
4:\2k2 .	2IÏX
---r-s in2am---aQ(x)
it£ ■	n J120
OC
m* 1
L
g
q
oc-2
tt*2
d + ô
O
(J
ij x
3 + 6	,	)’
ces ~ + Si
(62)
The last summation is extended to all the solutions of a certain
EQUATION
IB!2 + ¿6 + 2ii&i * 2 IB,	(62*)
in which a and id are given and mi, d, 6, dt> and 6x are sought; in
THE EQUATION (62’)	® 2; 2 AND IS AN INTEGER, l&i, !fi, d, 6,	¿1, AND &i
ARE POSITIVE OOD NUMBERS. |T IS EVIDENT THAT d& IS OF THE FORM
Ot
4N + 1, FOR OTHERWISE 2 ID WOULD EE DIVISIBLE BY 2 BUT NOT BY 4.
ON THE OTHER HAND MULTIPLICATION OF SERIES (4) REPRESENTING ^(x) (PAGE 113), GIVES
,.,2: 2 44 K . 2 -—sin am
rt ^
24 x

6 Y THF series
oc
¿_Ÿ** 2_2 l8LCCs2n,iX - cos(2n' +2 i 2 ) x 1 ^ ;	(63)
n* 1
= O
THE LAST SUMMATION IS EXTENDED TO SOLUTIONS OF THE EQUATION
nT +2
a' -1
i sà.
ot - 2
n = 2 in,
( 83 ’ )
IN WHICH n IS A GIVEN POSITIVE NUMBER, Ot1 RANGES OVER POSITIVE INTEGERS, n‘ OVER INTEGERS, 4a AND 5a OVER POSITIVE ODD INTEGERS.
Comparison of the coefficients of the same powers of q in equalities (62) and (63) g « ves
cc s
d + 5
9

[co s2n'
x
co s(2o'
d8) xl
(64)
ïhüîoïîem. Let a, mi, j, 6, ii, ôj, d2, and bs be positive odd integers, and o’ an integer; let ot and a* be integers >0. If f(x) is an even function then
f(2 Ï;
( 65);
where the summations extend to all solutions of the equations
a+ l
« "m * » ic + dô +.2diô1#	2a-1’û * n':2 +2“'	*i8ô‘2i
in and ot are constants and ib x, d, b, di, blf nfa d2, and b2 are portables.1*1
CorLLL'ARY. Let f CO) = 1 AND f<x)=0 WHEN X IS AN INTEGER DIFFERENT FROM ZERO.	/ii + 6	\
Since d, 6, and dt are positive, f V+ .dtj = 0 always, and
f	- j. IS DIFFERENT FROM ZERO ONLY IF di = *( d + &) BUT IN THAT CAS!
O	1	»o
a +1
2 m
p	i+'ô/d*ô c \	/ i— 6\ ^
- iDt2 = d(-0 + bx) + bbx =	+ 2&xj - i-j-j .
I	J.
I T. -I S EASILY SEEN THAT ”( d + Ô) IS ODD, A N D ?,( d “ £>): IS EVEN, I F
'—T > (d-ò)2,
WE CAN LOOK ON
d + ô/d+6	\ /i-6js
—I— *26*] ■ “/
* \ * / \ -* /
* /
AS THE NEGATIVE OF THE DETERMINANT OF A CERTAIN REDUCED FORM WITH BOTH OUTER COEFFICIENTS ODD. THIS MINUS DETERMINANT IS OF THE FORM 4n + 3. I F, HOWEVER
i + b\z
m
< (d-sr, d> 6,
WE WRITE, NOTING THAT IN THAT CASE i > 36,
a + i	2	d+6, „e .. .
2 io ~ id i = — — ( 2b + 20x) “ 5 2,
*
AND WE HAVE IN THIS. EXPRESSION THE NEGATIVE OF THE DETERMINANT OF A REDUCED FORM WHICH HAS ONE OF THE OUTER COEFFICIENTS ODD AND THE OTHER ONE EVEN. IF, FINALLY
< (d “ 6) £,
d < 5,
«♦i-	2
2 m - m i
~^(2d ♦ 2ôx) - Is.
In ANY CASE IT IS CLEAR THAT IN THE LEFT HAND MEMBER OF EQUALITY (65) WS ARE TO CONSIDER THE SUM
F(4n-12) + F(4n -32) + F(4n - 5s) + . . . . ,
WHERE F(n) IS THE NUMBER OF CLASSES OF FORMS OF DETERMINANT -fl AND WITH ONE OUTER COEFFICIENT ODD.'
IN THE RIGHT HAND MEMBER OF EQUALITY (65) WE MUST, AT FIRST, PUTH,=0; THEN WE OBTAIN
Cl-1
Zuj i 2, IS = d 2 S2 .
This is evidently what was denoted in the third chapter (page f$)-1
X
00
ot- i , .
3	$(»). Secondly, we put
a-*1,	a-i	ot'-i	a'-i .
n ' = “2	ol s > 2	!D=/j d a ( & a + ^	¿a)*
! T . I S EVIDENT THAT	._____
«'“I	J 0-1	.
2 d* < V2 ffi ;
$ * - 1	OC"* 1
AND THEREFORE 22	d2 DENOTES THÉ SUM OF ALL THE DIVISORS OF 0 = 2 K>
SMALLER THAN v/TÏ AND WITH PARITY DIFFERENT FROM THAT OF THE COMPLEMENTARY divisor; let us call this function ¥t(n); then
?(4n-l2) + F(4n-32) + F^n-o^) ♦ ... * 2	$(») - ¥1(0).	(65)
Equality ..¡66) is a direct corollary of the Kronecker fundamental
EQUALITIES (49)	-	(56) OF THE PRECEDING CHAPTER. ThlS EQUALITY CAN
ALSO BE EASILY OBTAINED DIRECTLY BY HERMITE’S METHOD AS VMS 00 N E BY HERMITE ON PAGE 35 OF THE SEVENTH VOLUME OF THE SECOND SERIES OF I.IOUVILLE'S »JOURNAL. TO DO THAT WE MUST FIND FIRST, BY THE METHOD OF Fourier, the constant term of the product of two series representing
si n
AND

n
sin
am
2;rx
n '
AND THEN BY THE SAME METHOD FIND THE CONSTANT TERM OF THE PRODUCT OF TWO SERIES, REPRESENTING
A 7 2 ,, 2	T)?v
4:V <i .	_ 4i.h.X	.	.	*
---r— sin 2 am- and §_(x).
n2	n	3
This possibility of deriving kronecker*s formulas from liouville’s
FORMULAS MUST NOT SURPRISE US. IT IS EASY TO SEE THAT IN THE DERIVATION OF EQUALITY (66) WE OPERATED ON CERTAIN FORMULAS IN THE EXACTLY THE SAME WAY AS WS WOULD OPERATE ON THEM TO DERIVE KRONECKER* S E QUL — ITIES. |N GENERAL IN THE DERIVATION OF AN 'DENTITY OF KRONECKER WE WE SEEK TO OBTAIN 8Y TWO METHODS THE CONSTANT TERM IN THE PRODUCT OF AN ELLIPTIC FUNCTION OF THE 2ND ORDER BY A »JACOBI' FUNCTION; ONE METHOD CONSISTS IN MULTIPLYING TOGETHER ON ELLIPTIC FUNCTION OF THE SECOND order and a Jacobi function; the other, in multiplying together AN ELLIPTIC FUNCTION OF THE FIRST ORDER AND AN HERMITIAN SERIES. BUT WE PERFORM THE SAME MULTIPLICATIONS IN THE DERIVATION OF CERTAIN OF LlOUVILLE’S FORMULAS WHEN WE OBTAIN EQUALITIES BETWEEN TRIGONOMETRIC SUMS; THESE ARE THE EQUALITIES BETWEEN THE COEFFICIENTS OF THE SAME P > WEftS OF q IN THE TWO SERIES REPRESENTING THE SAME FUNCTION. LET U^, FOR THE SAKE OF BREVITY, CALL SUCH AN EQUALITY,£J0UVille'S tTig-
mometric equality. Obsiqe x will be the function occurring in this equality. Thus, in liouville*s trigonometric equality we have equal ;TY OF THE COEFFICIENTS OF THE SAME POWERS.OF q; (N «ronecker's forMULA we HAVE EQUALITY OF THE COEFFICIENTS OF THE SAME POWERS OF ; IN THE CONSTANT TERMS OF THE SAME TWO PRODUCTS, FROM WHICH WE DERIVED Liouville’s trigonometric equality; THEREFORE kronecker's FORMULA MAY BE OBTAINED FROM LlOUVILLES TRIGONOMETRIC EQUALITY ii WE SELECT IN IT THOSE TERMS IN WHilCH THE ARGUMENT OF COsinCX IS
fqual to zero. This last may be done by a somewhat more complicat ED method: the theorem concerning an even function f( x) may first BE DEDUCED, AND THEN IN THIS THEOREM PUT f(0)=l AND f(x)=0 FOR X NOT EQUAL TO ZERO.
IT IS EVIDENT THAT ALL OF KRONECKER’S FORMULAS AND ALSO FORMULAS SIMILAR TO THEM CAN BE DEDUCED FROM LlOUVILLE’S FORMULAS.
Sxampls 3. Let us derive a theorem containing formulas (49), (50), AND (52) OF THE THIRD CHAPTER. PAGES 51 - F F SUGGEST THE SERIES TO BE TAK EN.
FIRST WE MULTIPLY TOGETHER THE FOLLOWING THREE SERIES:
/2:v	, .	.	2:'X	f	r-	. it6 1
y-jj-SgvX)* sin am-^- =2 ^ |q 4 ¿_ sin“5“xri
R=0	4o+3= i6
.	+ oc
for	--- 2
\ n
n = Lq ‘
n- -90
OZir	orv
A *»	•	2x	^
-—sin am-------'cos x = 2
n	rt
Ef’“' E t.».«
x + sin(i - 1)x
n*°	2n+l=i5
4K2k2 . 2Sxft , ,
-—r—sm^am—•§is(x)cos x
„2	n 2
7

/d + 5	\	/d+6 '
coSj—^— it- 1}x - co
s i-tt* + di + 1) x
/ i + 5	\	/'j+6	\ 1~)
+ cos/- dx + ljx - cos K- + ix - lj x
(07)
Let us multiply together the series
4 i v	»C
o
re '
sin2aia-
V( x) CO ;
n =	) i( 1 “ cos2ix)]>,
a31'	0=7(25-1)	J
*<•*»-»> *r®.
coax = ^LC52nx + cox2(n - l)xJL
n* l124
..,2 2
4‘i if
si n 2 8t.i
2'{x
&„(*/ 00 S X
=	l) + i(*n ‘l> 2^icos2nx + cos2(n-l)xJS
“ 4Z{qi(25'"1)+i(2n'*1)2I~d&cs2(n + i)x
+ coa2(n~i)x + eos2(n+i-l)x + ec s2( n - i - 1) xj^ .	(66 V-
Comparing the equalities (67) and (68) and following an argument
SIMILAR TO THAF USED IN THE PREVIOUS EXAMPLES, WE EASILY DEDUCE THE
following equality
ZJ'tr-H - '(“•*■
+ 1	+ f
/d+6
-*" j ' fftr
/ \ ^ / \
=	42i8{f(2n') ♦ f(2ii’ - 2)}: - 22da{f(2nf + 2da) + f(2n'--2daV
+ f (2 n' ;4- 2 i 8 - 2) + f(2n’ -2da-2)},	f(-x) * f(x).	(69)
LET N BE A POSITIVE INTEGER. THE FIRST SUMMATION IS EXTENDED TO ALL THE SOLUTIONS OF THE EQUATION
4a8 + IS + 2iiS! = S
WHERE n I S AN INTEGER, 1, 5»,-jl1# Sj, POSITIVE ODD NUMBERS SUCH d+5 IS DIVISIBLE BY 4; THEREFORE N IS OF THE FORM 4h + 1. TWO SUMMATIONS ARE EXTENDED TO SOLUTIONS OF THE EQUATION
( 69 * )
THAT
The other
4d‘a6a + (2 n ' - 1) 2 = H * 4b + 1,	(69")
WHERE 6a IS AN ODD POSITIVE NUMBER, da AND 0' POSITIVE INTEGERS.
Corollary. Let f(0)=l, and £(x)-=0 for x not equal to zero. In
THE FIRST SUMS OF EQUALITY (69) WS MUST CONSIDER TWO CASES*.
V/HERE a IS A POSITIVE INTEGER, AND 2a - 1 < 2p . LET 6j. *2b + 1; THEN ‘"ROM EQUALITY (69’) WE OBTAIN;
h -n2= p2+p+.b(2p + !)~a2+a, in case it = 2p+l, and li - n 2 * j? + p - 1 ♦ b(2p~ 1) - a2 + a, incase dt = 2p - 1.125
We RECOGNIZE' THESE EXPRESSIONS AS THE EXPONENTS IN EQUALITY (?)
ON PAGE 12', THEREFORE THE L.EFT HAND MEMBER OF (69) BECOMES IN 0 IT CASE
4[F(h) + 2F(b - l2) ♦ 2F(ii“2e) + 2F(h-3s) + .... 3 * R,
WHERE F(b) IS, AS USUALLY, THE NUMBER OF UNEVEN CLASSES OF THE DETERMINANT -h AND R IS EQUAL TO “29 ( IS ) I F h = IS WHERE ID IS ODD, AND 0 I F ID 5 0 ( MO 0 4.) .
IN THE RIGHT HAND MEMBER OF (69) WE MUST CONSIDER THE FOLLOW-ING THREE CASES:
n ’ = 1, n1 = d s, n’ - d 2 + 1.
1) o* * 1; then h = 126a, where 6a is necessarily ooo. We shall EMPLOY THE NOTATIONS OF CHAPTER III (PAGES H h 56). IF fc IS ODD THEN (ID DENOTING AN ODD NUMBER)
42i a = 4§(h) = 4${io).
If h = 4N + 2 - 2’e then
42d 2 = 3$(m).
a
If h =	2 ic WHERE 0£ > 1, THEN
T he	RESULT IS,	THUS,	THE	SAME AS THAT	AT	THE BOTTOM OF	P AGE	0 D •
	2) n* - ifi;	THEN	h * d	£ ( ^ 5 + “* l)]	THEREFORE			
				- 25d*f(2nf -	3ds)			
\ s	EQUAL TO Mt	N U S TWICE		THE SUM OP THOSE		DIVISORS OF h	W W 1 C H	Ah E
NOT	GREATER THAN /ft		AND	WHICH ARE OF	THE	SAME PARITY AS	THE	COM —
PL I MEN T AR Y DI V I SO R.
3) n* - d'a + 1; then h=ds(d2+&2 •*•!). Therefore
- 22d$f(2n' - 2i8 - 2)
IS EQUAL TO MINUS TWICE THE SUM Of THOSE DIVISORS OF h, SMALLER THAN l/h AND OF THE SAME PARITY AS THE CO MR L I M E N T AR Y DIVISOR.
It is Easy now to see that the two sums con si dered . i n 2) and 5) WILL GIVE
- 2$(ffi) + 2¥(m) and - 4$(n) + 4?(n),
FOR THE CASES ft = BS AND ft=40, AND ZERO FOR THE CASE ft - 2ffl . I N THAT WAY WE AGAIN DEDUCE EQUALITIES (11,	(12), AND (15) OF § 29.
§ 52, LIP TO THE PRESENT WE HAVE BEEN CONSIDERING llOUVILLE’S THEOREMS ON ODD AND EVEN FUNCTIONS OF ONE VARI ABLE. LlOUVILLE ALSO GAVE A SERIES OF THEOREMS OF THE SAME CHARACTER AS THOSE WE HAVE123
CONSIDERED ON FUNCTIONS OF SEVERAL VARIABLES. THE THEOREMS ON FUNCTIONS OF ONE V A FI ABLE ARE PARTICULAR CASES OF SIMILAR THEOREMS ON FUNCTIONS OF SEVERAL VARIABLES. IN THE DERIVATION OF SUCH THEOREMS ON FUNCTIONS OF SEVERAL VARIABLES WE MAY CHOOSE ONE OF FOUR WAY'S,
OF WHICH THREE ARE RELATED TO THE THEORY OF ELLIPTIC FUNCTIONS, AND THE FOURTH OF WHICH DEPENDS ON ELEMENTARY ALGEBRAIC CONSIDERATIONS WHICH ARE AT TIMES VERY COMPLICATED. WE SHALL CONSIDER ONLY THE FIRST THREE METHODS.
First of all the thoery of elliptic and Jacobi functions contains a large number of equalities between functions of several variables;
THESE ARE FORMULAS OF ADDITION OF THE ARGUMENTS, SO CALLED FORMULAS ON FIVE AND TWO LETTERS, AND THE LIKE. |T IS CLEAR THAT SUCH EQUALITIES may give theorems similar to those considered in the beginning OF THIS CH AP TER.
For such an equality may always be replaced by an equality between two algebraic sums of products which are expressible through TRiG-oN.iMETRic series.: (It is understood that sometimes the algebraic
SUM MAY CONSIST OF ONE TERM ONLY AND IT ALSO MAY BE THAT ONE OF THE MEMBERS OF THE EQUALITY WILL BE EQUAL TO 2 E RO. ) COLLECTING THE COEFFICIENTS OF THE SAME POWERS OF THE MODULUS q, WE SHALL OBTAIN AN EQUALITY BETWEEN TRIGONOMETRIC FUNCTIONS OF SEVERAL VARIABLES. SOMETIMES THIS EQUALITY, IF IT CONSISTS OF N 0 N-HO MO G E N EO U S TERMS, DIVIDES INTO SEVERAL EQUAUTI ES WHICH CAN BE OBTAINED BY REPLACING SOME OF THE VARIABLES U, V, t,... BY -U, -V, -o, ... FOR EXAMPLE, IF THERE OCCUR IN SERIES BOTH PRODUCTS OF TWO COSINES AND PRODUCTS OF TWO SINES,
THEN THE EQUALITY WILL DIVIDE INTO TWO EQUALITIES:	IN ONE THERE WILL
OCCUR ONLY PRODUCTS OF. CO SINES, IN THE OTHER ONLY PRODUCTS OF SINES.
Every analytic function of several variables may be, within certain LIMITS FOR THE VARIABLES, EXPRESSED AS TRIGONOMETRIC SERIES IF IT IS WELL DEFINED WITHIN THESE LIMITS. IN ORDER TO OBTAIN SUCH A SERIES.IT SUFFICES TO VARY ONLY ONE VARIABLE, LEAVING THE OTHERS CONSTANT; THEN THE FUNCTION WILL BE EXPRESSED AS A TRIGONOMETRIC -■SERIES, IN WHICH THE COEFFICIENTS WILL BE CERTAIN FUNCTIONS OF THOSE
variables which were fixed. Varying one of the variables in these
COEFFICIENTS LEAVING THE R EM AI N t N G ONES CONSTANT, WE OBTAIN AN EXPANSION OF THE COEFFICIENTS IN THE TRIGONOMETRIC SERIES, ThE COEFI-COENTS OF WHICH CONTAIN TWO VARIABLES LESS THAN DOES THE GIVEN FUNCTION. CONTINUING IN THE SAME WAY WE CAN EXPRESS THE FUNCTION OF SEVERAL VARIABLES AS A SUM, IN GENERAL INFINITE, WHOSE TERMS ARE PRODUCTS OF SINES AND 00 SINES OF THE FORM
nnx	giny	n2nz
h8 ' * * * '
WHERE X, y, Z, . . . . ARE ARGUMENTS OF THE GIVEN FUNCTION, 0, il1# tla,127
... ARE INTEGERS OF SUMMATION AND fc, ht» tig, ... TOGETHER WITH - b. -ht, -tla>... OENOTE THE LIMITS WITHIN WHICH THE GIVEN FUNCTION i REPRESENTABLE BY THE TRIGONOMETRIC SERIES.
Above we obtained an equality for certain trigonometric functions OF THE VARIABLES U, V, fc,...*; THESE EQUALITIES ARE COMPOSED OF HOMOGENEOUS TRIGONOMETRIC PRODUCTS, THAT IS, FOR EXAMPLE, THE VARIABLE U OCCURS IN EVERY TERM AS THE ARGUMENT OF COSINES. THE VARIABLE V OCCURS IN EVERY TERM AS THE ARGUMENT OF SINES ETC.
. . IN
WE SUS STITUTE 7, --,
h h
2-,	... IN PLACE OF U, 2,	7-,	f-,
l* 1
PLACE OF V,
h'	'	' lii in' hi
ETC.; WE OBTAIN THUS A SEQUENCE OF NEW EQUALITIES. L E
US DENOTE THE TOTALITY OF THESE EQUALITIES BY (A). IN THE SERIES REPRESENTING THE FUNCTION WE REPLACE X, y, Z, . . . BY.INTEGERS SO THAT THE ARGUMENTS OF THE TRIGONOMETRIC FUNCTIONS IN THIS SERIES WÍLL BE IDENTICAL WITH ARGUMENTS OCCURRING IN (A).	(, | T ,s ASSUMED
OF COURSE, THAT TRIGONOMETRIC FUNCTIONS IN THE SERIES ARE THE SAME AS THOSE IN THE DERIVED ARITHMETICAL THEOREM.) SUCH SUBSTITUTION WILL HAVE TO BE MaDE SEVERAL TIMES. WE SHALL OBTAIN SEVERAL EQUALITIES; LET US DENOTE THEIR TOTALITY BY (6). WE MULTIPLY BOTH MEMBERS OF THE EQUALITIES (A) BY THE COEFFICIENTS WHICH ARE CO MMO ¡' TO ALL OF THE SERIES (6), AND ADD THE RESULTS TOGETHER; WE OBTAIN AN EQUALITY (C) . WE SHALL WRITE THIS EQUALITY (C), F.ORMED BY TRIGONOMETRIC SERIES, SO THAT THE RIGHT HAND MEMBER WILL BE ZERO. BOTH MEMBERS OF THE EQUALITIES (B)	WE MULTIPLY	BY THE	COEFFICIENTS COMMON TO ALL THE EQUALITIES (A)	AND ADD THE	RESULTS TOGETHER.	WE OB-
TAIN THE EQUALITY (D), THE RIGHT HAND MEMBER OF WHICH BECOMES ZERO BY THE EQUALITY (C). THE LEFT	HAND MEMBER	OF (0)	CONTAINS A	GENERAL FUNCTION, AND THEREFORE THIS LEFT HAND	MEMBER	EQUATED TO	ZERO
WILL GIVE A LtOUVILLE THEOREM. ARGUMENTS OF THE FUNCTION IN THIS THEOREM ARE INTEGERS; THEREFORE THE THEOREM EXTENDS ALSO TO NUMERICAL FUNCTIONS AND ALSO TO SUCH ANALYTIC FUNCTIONS WHICH BECOME INFINITE, DISCONTINUOUS, OR ARE NOT DEFINED FOR NON-INTEGRAL VALUES OF THE ARGUMENT OR FOR INTEGRAL VALUES WHICH DO NOT OCCUR IN THE EQUALITY CONSIDERED.
The OTHER METHOD FOR DERIVING LlOUVILLE’S THEOREMS MAKES USE .
OF THOSE FUNCTIONS, STUDIED IN THE THEORY OF ELLIPTIC FUNCTIONS,
WHICH ALSO SERVE TO DERIVE SUCH THEOREMS IN ONE VARIABLE AS ARE
PARTICULAR CASES OF CERTAIN LtOUVILLE** ThEORSMS IN SEVERAL V A R-
A
IABLES. |N THE DERIVATION 0 F A L I 0 U V I L L E THEOREM ON FUNCTIONS OF A SINGLE VARIABLE, AN EQUALITY BETWEEN SUMS OF COSINES OR SINES IS OBTAINED; THESE COSINES OR SINES MUST MUTUALLY CANCEL OUT FOR THEY CONTAIN AN IN0ETERMINAHTE X. THE ARGUMENTS OF CANCELLING TERMS
* Sometimes we shall have several equalities of the same character; soe example II.
** In this section, as in many others, by Liouvillef 3 theorems are understood not only theorems deduced by Liouvillo, but in general all theorems of a certain kind. ;MUST EITHER BE EQUAL OR HAVE THE SAME ABSOLUTE VALUE AND DIFFER IN SIGN. BUT THE ARGUMENTS CONSIST OF X MULTIPLIED SY LINEAR FUNCTIONS OF SOLUTIONS OF CERTAIN I N D E T E R.V! I N AN T E EQUATIONS OF THE SECOND DEGREE, OR SOLUTIONS OF TWO DIFFERENT EQUATIONS OF THE SECOND DEGREE. BUT IT MAY TURN OUT THAT SOLUTIONS OF THESE INDETERMINANTE EQUATIONS OF THE SECOND DEGREE WHEN GROUPED IN CERTAIN WAYS MAY BE CONNECTED BY SOME LINEAR EQUALITIES OTHER THAN THOSE WHICH FOLLOW FROM THE TRIGONOMETRIC FUNCTIONS SY WHICH TERMS THAT CANCEL OUT MAY BE MULTIPLIED. Thus, where before we had sums of cosines and sines in one
VARIABLE X, NOW, ON MULTIPLYING CANCELING TERMS, WE OBTAIN SUMS OF PRODUCTS OF COSINES AND SINES, IN WhtCH TO X ARE ADDED NEW VARIABLES y, Z, t, . . . ; IN BOTH CASES THE SUMS ARE EXTENDED TO ALL THE SOLUTIONS OF A CERTAIN KIND, OF THE SAMS I N D ET E RM I N AN T E EQUATIONS OF THE SECOND DEGREE.
Further procedure is identical with that of the first method.
It is evident that this second method can be modified in various ways. Thus, equalities in 2, 3, etc. variables may be given and from these equalities with a greater number of variables may be deduced.
IT IS CLEAR THAT THE METHOD OF DERIVATION REMAINS THE SAME.
. In GENERAL THE EQUALITIES BY LlOUVILLE, RELATED TO GENERAL PROPERTIES OF FUNCTIONS, ARE FORMED OF SUMS OF FUNCTIONS OF ONE OR SEVERAL variables; these sums are extended to different SOLUTIONS of in determinants equations of the second degree. The contents of these
THEOREMS FOLLOW FROM THE FACT THAT THE E EXIST LINEAR RELATIONS EITHER BETWEEN DIFFERENT SOLUTIONS OF THE SAME EQUAFION OF THE SECOND DEGREE OR BETWEEN SOLUTIONS OF TWO INDETERMINANTE EQUATIONS. IN THE MAJORITY OF THESE THEOREMS THE NUMBER OF LINEAR RELATIONS SUFFICES TO DETERMINE COMPLETELY CERTAIN SYSTEM OF SOLUTIONS ON AN INDETER— MINANTE EQUATION OF THE SECOND DEGREE EITHER THROUGH ANOTHER $Y gTEM OF SOLUTIONS OF THE SAME EQUATION, OR T Hr 10 UGH CERTAIN SYSTEM OF SOLUTIONS OF ANOTHER EQUATION OF THE SECOND DEGREE. IT IS CLEAR NOW THAT in THE DERIVATION OF SUCH THEOREMS IT IS POSSIBLE TO AVOID THE USE OF THE THEORY OF ELLIPTIC FUNCTIONS ALTOGETHER BUT THIS THIS THEORY EITHER SIMPLIFIES THE PROOFS OR HELPS TO DISCOVER NEW THEOREMS WHOSE EXISTENCE WOULD 6 E» V E R Y DIFFICULT TO GUESS OTHERWISE.
Finally, the third method for the derivation of Liouville’s
THEOREMS IS RELATED TO THE THEORY OF ELLIPTIC FUNCTIONS ONLY IN THAT THE SERIES OCCURRING IN THE DERIVATION CONTAINS AS PARTICULAR
CASES SERIES OF THE THEORY OF ELLIPTIC FUNCTIONS. THESE NEW SERIES. ALSO TRIGONOMETRIC, CONTAIN ALSO INDETER MINANTE q; BUT POWERS OF q
ARE MULTIPLIED BY TRIGONOMETRIC FUNCTIONS OF SEVERAL I N D E T E R 'V I
OR THEIR MULTIPLES (FOR EXAMPLE, THE TERMS HAVE THE FORM
qmC0 3iDiy CO s mflx, WHERE y AND x are I n d ET erm I n an te s) . TH I S METHOD129
HAS A STILL CLOSER RELATION TO THE THEORY OF,ELLIPTIC FUNCTIONS ON ACCOUNT OF THE FOLLOWING FACT. IN THE TRANSFORMATION OF SERIES STUDIED IN THE THEORY OF ELLIPTIC FUNCTIONS, INTO OTHER SERIES CERTAIN ALGEBRAIC OPERAFtONS ARE PERFORMED; THESE SAME OPERATIONS WITH FEW MODIFICATIONS MAY 8E REPEATED ALSO IN THE OASE WHEN 'WE INTRODUCE INTO THE SERIES (ACCORDING TO A DEFINITE L AW) CERTAIN TRIGONOMETRIC FUNCTIONS OF NEW VARIABLES; SUCH COMPLICATION OF TH' SERIES MAY AT TIMES RESULT IN SIMPLIFICATION OF THE COMPUTATION.
TO MAKE THE CONTENT OF THIS SECTION MORE CLEAR WE SHALL GIVE SEVEN EXAMPLES. |N TWO OF THESE EXAMPLES THE FIRST METHOD IS USED IN TWO OTHERS THE SECOND METHOD IS USED, AND IN THE REMAINING
three the third method; the third method is also used in the beginning OF ONE OF THE EXAMPLES WHOSE SOLUTION IS COMPLETED BY THE SECOND METHOD.
§ 53. From the addition formulas of the arguments of elliptic
FUNCTIONS ARE DEDUCED THE FOLLOWING TWO FORMULAS, WHICH MAY BE FOUND ON PAGE 113 OF THE SECOND EDITION OF THE TEXT BOOK OF DURE'g
2X(x + y) .	2:\ ( X - y)
sin am-----t-*-"- + sin am —
n
n
, . 2Xx 2Ky . 2Ky
2 sin am---cos am---- A am*--
n	n	n
2:i ( x + y )	.	2X(x~y)
si n am -—  -- - si n am ......
n	n
si n 2am	—■—si n JT
2Xy	2Xx
	co s	am ——
n	n
A am*
n
2Xx
g . 2	2:v x . g 2X y
- K^sm^am—>- sin^aD—£ n	n
From these equations we derive
iôkV	i 23', isa~'
~~~	
16X4k3	( otr
+ sn
2:L	>	2K
“r( x - y)> sn—-
ft	j	n
iy 2Xx 2Kx c s- — A *......
ît	n

The right hand member is obtained from the left hand MEM8ER by INTERCHANGING X AND y .
Let us now make use of equality (l) and another equality which is obtained FROM (1) BY DIFFERENTIATION:
4X2k — cos
am
n
A
•X7,V
am------
n
«5
n* 1
J- d cos ix^. 2n-l=i6
(71)
In place of (70) we obtain130
OC
^|qn"*2Sd"[ sind( x + y) ♦ sin d( x- y)] s in d'y co s d"xj
na 1
■ !!•
n-1

’d"[sin i(x+y)- sin i(x- y)]sind':xcc
cc s d "	.
n- 1
The inner sums are formed as follows: a certain positive odd number is decomposed in all possible ways .into the sum of three odd numbers
AND EACH OF THESE ODD NUMBERS IF DECOMPOSED INTO TWO POSITIVE FACTORS, i AND 6, J' AND	d" AND 6". THE SUMS ARE EXTENDED TO ALL
THE NUMBERS d, d * , d*' THUS OBTAINED.
Comparison of the coefficients of equal powers of q in the last
EQUAL 1 TY GIVES
2d" si n ix co s dy sin c ’ y The equality (72) may
%
co s d" x = 2d "sin dy co s ix sin d'x cos d"y
be written as
(72)
2d" { si n( d + d")x + sin(d - d")x} {sin(d + d’,)y - sin(d - i')y}
- 2d" { si n( d + d’ ) x - si n( i - d* ) x} {.sin( d+d") y + si n(d - d") y} . (72';)
Let us now take a function F(x,y) of two variables, odd with respect to X AND y, THAT IS SUCH THAT
?"(0,y)=0, F(x,0) = 0,	F(-x,y) = -F(x,y),	F(x,-y) = -F(x, y).
Let F(x, y) be continuous ane. finite within the limits -ti, +h for x and -g, +g for y. Then an argument similar to that used on page 126 will convince us that F( x,y) may be expressed as a double.infinite
F(x, y)
Z Z‘
n=1 p= 1
. IVÎÎX . DTXy
si n —— s i n-R— , p n g
(73)
There is no neeo. to determine the coefficients A
n, p
In equality (72’) we put x = nn/h, y = pn/v where n and p range
OVER ALL INTEGRAL VALUES FROM 1 TO	THEN WE MULTIPLY BOTH MEMBERS
OF THE EQUATION BY A„ „ AND SUM OVER ALL INTEGRAL VALUES OF n AND p
n / p	1
between l and «. In equality (73) we substitute the following e i ght
S \9TEM$ 1)	0 F X	VALUES of X = d + d", y	and y: = d ♦ d';	2)	X =	d- i",	y =	i + d ’ ;	
3)	X	= d + d", y	- d - d* ;	4)	X =	d - d",	y =	d - d';	
5)	X	= ¿ + d», y	= d 4 i”;	3)	X «	•d + d ’,	y =	d - d";	
7)	X	= d-d’, y	= d + i";	3)	X s	d - d ’	y =	d - d ",	
in which	a,	d ’ , A N D d "	take successively			ALL THOSE	: VALUES WHI CH		THEY
have IN	equality (72).		It is evi	DENT	THAT	IN THIS	WAY	WE OBTAIN	THE
sum
formula131
^d"{Kcl+cl^d+d,) * F( d-d”, d+i1 ) - F( ~i+ d",d-d ' ) - F (d-d% d-d' )} -
2n-l= dô+d'S* ?+i'0"
^"{FU+dVi+i") + F(d+d', i-i”) -f(d-i',d+d") - F( d-d Vd-d"M . (74 2n-l= dô+d'ô' + d!' 8'
Equality (7 4) is given by liouvulle on page 33 s of the 3k d voli.
OF THE SECOND SERIES OF HIS JOURNAL (JOURNAL DE MATHEMATIQUES P U H ET APPLIQUEES),! N THE THIRD VOLUME LI O U VILLE PUBLISHED SIX PAPER? RELATING TO THE SUBJECT MATTER OF THIS SECTION; FORMULA (74) IS G EN IN THE SIXTH P APEfi.
Example 10. we shall make use of the well known formual co3 am (u + v) = cos amu cos aicv - sin «nsa sin amv à am (u + v),
WHERE
orr v
/0* i. A
_ 2:£y
ix	n
IN THE DERIVATION OF TWO ARITHMETICAL THEOREMS. FOR MULTIPLY BOTH MEMBERS OF THE EQUALITY BY SK8k*/n3. LAS (1),	(2),	(3) OF THIS CHAPTER AND FORMULA (4)
TER AND NOTING THAT
THAT PURPOSE W„ USING THE FORM-. OF THE FIRST C H
<? oe 43 V.. ,V-(2r		GC ^— r	I 2r-i , h l_ *
_ E ** / K — j	1 y* to 1 ►-* »		
r * 1		1	2 r-1*16
( ms LAST FORMULA IS OBTAINED FROM FORMULA (4) OF THIS CHAPTER IF WE PUT X=IX/2 IN THE LATTER AND REPLACE q BY /Iq) , WE EASILY DERIVE • THE FOLLOWING FORMULA:
6-1
(-1) 2 d'
2n= dô+d* 6».
co s i( x + y)
r—	6-1
2n=iô+ d' 6*
b2zl
2 CO s
ix cc s J y
+
r~ {hi 6*-l ,6"-l 4 2_ (-1) 2 *	2	~^”co3 Jxcos i'y
2n=d5+d,ô'+2d"ô"
y~ sin ix sin J y 2n=dô + d'ôf
1 N
i"
■' f
L
6 "-l
(-1)	2 sin dx sin d'y ccs2i"( x + y).
2n=i& + l*ô'+2d,,ô”
THIS FORMULA d, 6, i f 5', AND 6" ARE POSITIVE ODO NUMBERS, ARE POSITIVE INTEGERS; 0 IS CONSTANT.
a n d n ;
Changing the sign of x, we obtain another equality similar to ti first one. Adding and subtracting both equalities, we obtain, after
A FEW SIMPLE OPERATION, PASSING FROM TRIGONOMETRIC FUNCTIONS TO Afiu132
TRAfiY FUNCTIONS AS WAS SHOWN HOW TO DO IN EXAMPLE 9,
__	6-1 r	6*-l	T	6-1 CT’-l 5!'-l
2n= dS+d'5'	J 2n=d&+d'6»;+2d!'&‘'
— £hl
2 {f(i- 2d", d' - 2d") - f(i“ 21", d' + 2d") - f (d + 2d", d' - 2d”) + f(i+ 2dyi' + 2d")}, (75) 2n=16+1* 6'+2d"6"
6-1 1 1 (-1) 2 i' fi(^d) -	{ fx( i - 2d'% i' ~ 23isO
n= 16+ d' 6'	2n=d5 + d':6'+2d"6"
+ f1(d-2d%i,+2i") + f i(i + 2i", d'-2iM) + f i( d+2i", d' •* 21")} .	(76)
The functhons f(x,y) and fx(x,y) in equalities (7$) and (76) have
THE FOLLOWING PROPERTIES:
*(-x,y) = f(x,y),	f(x,~y) = f(x,y).
it
	fi(-x,y) = -fi(x,y), i i( x, - y) =	-fi(	•x, y).		
§ 54.	Example 11. On page 119 we derived formula (60):				
	1^6 22 ai n ( d ' + 2 0 ' ) x = 2 si n~--—	X,			(77)
WHERE THE FIRST SU MM AF ION IS EXTENDED TO ALL		THE	PO SI T 1 7 £	CDD	V AL-
ÜES OF *1	1 6' AND ALL THE INTEGRAL VALUES OF n'	SATISFVlN fi		THE	EQUATION
	2n + 1 = 2n'8 + d'ô'				(77')
vtf h E R E n	IS CONSTANT, AND THE SECOND SUM MAT ION	1 S	EXTENDED	TO	ALL
T H £ POSI	TIVE 0 D D V AL UES OF d AND 6 AND EVERY	0 DD	V AL U E 0 F	III,	S AT-
1 S F Y 1 NO	THE EQU AT 1 ON				
	4n + 2 = ta 8 + dô.				(77")
The second sum' of equality (77) differs from the first only in
THAT IN THE SECOND SUM THE COEFFICIENT OF X IS NECESSARILY POSITIVE AND IN THE FIRST IT MAY BE EITHER POSITIVE OR NEGATIVE; THEREFORE THE TERMS OF THE SECOND SUM CANNOT CANCEL ONE ANOTHER, AND EVERY $!NE OF THE SECOND SUM WILL CANCEL ONE SINE OF THE FIRST SUM. THUS FOR EVERY
WE HAVE A CORRESPONDING 1^20*, SUCH THAT ALSO
ÎJ
i + 6
= d ' + 2 n '
(78)
LET US ALSO TAKE EQUALITY (57):
d'-1	d- 1
22(-l)~co s( i ' - 2n' (x =• 2(-1)~^co s m x ;	(79)
THE SUMMATIONS BEING TAKEN ACCORDING TO THE SAME LAW AS THAT FOR THE EQUALITY (77). EQUALITY (79) SHOWS THAT FOR CERTAIN VALUES OF ■- * i' > AND tl' THERE SHOULD EXIST THE EQUALITY ¡8 = i'-2fl-' OR 5-2o', AND FOR CERTAIN OTHERS ID “ — i ' + 2 0 ' OR -S'+2o' .. ,Let
133
m * ±(ô' - 2tv ' ;,	= i* ♦ 2n * ;
WE HAVE THEN
4n + 2
= ir, + it * (£' - 2n* r + ( i' + 2n ')
i> *	«
-b'
= 6' 2 + i' R-i«n»ô’:+ 4n» i* + 3o* 5 -
i - 6\2
IT IS EASY TO SEE THAT IN ORDER TO OBTAIN FORMULA (77*) FROM THE ABOVE EQUALITY WE MUST PUT
Z.._-5 V
/
- (2n ' + i* — Ô'-)“.
Preceding reasoning will give us those linear equalities by the HELP OF WHICH WE CAN PASS FROM EQUATION (77") TO EQUATION (11').
Let Os consider these linear relations more carefully. We first
TAKE THE SYSTEM	.
i* : + 2n ' = *	, is = ô * 2n1	2u 1 + i ’ - 5' =
I f? T	___ _______r. r,.,	r.r.c* r r. r, !	A
i ’ ID Ô ' , A N D n ' :
i =4n * +2 i* - 6f , t - b',
b « b'
2n ' = 6 — in,
= »a +
2 *
i - 6	
- 5» * -, O	(,8C)
	
i IB AND THEN	FO R
2 ti ' ;	(51)
5 •	(.82)
Equalities (81) show that for every i', S'-, and o' there will exist corresponding i, b, and e if and only if
	4n ' + 21 * - 5 ' >	r\ J •				(.83)-
Equalities (82) show that we can pass		FROM	EQUATION	(IV)	TO	EQU A-
TION (77’) to equation	(77") if only ic	+ (i	-b)/3 IS	P 0 S 1 T 1	V E,	FO k
(1-5) ■/ 2 'IS NEXESSAHILY ODD.						
But i fib + ( 1 - 6) /2	is negative, we	TAKE				
1 + 6 i' + 2(1* = 2	m s - 5 * + 2 n ',	i - 2	- = -2n* -	¿'•+6*	• f	(£4)
1 = 5', as	= 2-n* - 5', 6	* 4n	' + 2 i * -O'	f		(85)
6' = d,	2n' = id + d, d	t ~ ,	d-5 ¿U			(33)
Equalities (S3) and (8	6) SHOW us THAT	THE	ONLY CASE	IN WHICH		WE DO
not have a linear relation BETWEEN THE solutions of equations (77!) AND (77") IS THAT IN WmICH
4n' + 2d* - 6» <0.	(87)
The limiting condition (87) exists only; for transition from (77") TO (11'), AND NOT FOR THE TRANSITION FROM (11') TO (77");
THAT IS, AS( WE PARTLY ANr i CIPATEO, FOR EVERY IS, 4, AND 6 TH-tRE EXIST CORRESPONDING fi',4f aN0 6', BUT NOT FOR EVERY	AND 6* DO
THERE EXIST CORRESPONDING ID, i, AND 3.134
LINDER CONOIITIONS (87) THE TERMS OF THE FIRST SUMS OF THE EQUALITIES (77) AND (79) [IT IS SETTER TO REPLACE d'■ BY 5' IN (79)1 MUST CANCEL WITH 'OTHER TERMS OF THE SAME SUMS. LET i',	AND i i',6i*> AND
1i T S E TWO SOLUTIONS OF EQUATION (77f)- LET US TAKE SYMMETRIC EQUALITIES
d' + 2n' = -dj» - Siu', 6' - 2n1 = 6t' - 2tV,
2n': + d’- 5'	= 20*» + dt':- 6i'-.	(38)
WE NOTE THAT THE LAST EQUALITY OS OBTAINED FROM THE EQUATION
2x2 ♦ (2x + &i' ~2iu’:)(“2x - dt‘ - 2 n t' ) = 2o i' 2 + dx ' bx' *
OF WE SOLVE FOR X AND PUT fiV EQUAL TO ONE OF THE RIITS THUS OBTAINED.”
From (88) we have
dx' = d', 2n i’ = -2n ’ - 2d*,	6X* - -4n,-2d': ♦ 6* .	(39)
From equalities (89) we see that equalities (88) are always possible
I F ONL Y
4o1 + 2d* -S' <0,	41U' + 2di,;- &x' < 0;
HENCE WE CONCLUDE THAT (88) ARE THE EQUALITIES WHICH WE SOUGHT AND WHIVH DETERMINE UNIQUELY THOSE PAIRS OF TERMS OF THE FIRST SUMS OF THE EQUALITIES (77) AND (79) WHICH CANCEL ONE ANOTHER, FOR IT IS EASILY VERIFIED THAT THE TERMS WILL ft E EQUAL IN ABSOLUTE VALUE AND OPPOSITE :IN SIGN,
FROM EQUALITIES (80),	(84), AND (88) WE SEE THAT EVERY TERM OF THE
FIRST SUM OF (77) MAY BE MULTIPLIED BY TRIGONOMETRIC FUNCTIONS OF THE
variables	2n')y and (2n’+ i' - 6' )z, and every term of the second
SUM BY THE SAME FUNCTIONS OF THE ARGUMENTS IHy AND (d-&)z/2. SUCH MUL-LIPLIERS, HOWEVER, MUST NOT CHANGE SIGN IF THE THE SIGNS OF BOTH y AND Z ARÉ CHANGED AT THE SAME TIME.SUCH MULTIPLIERS ARE
d - 5
!)• eos(5' - 2ii')y cos (2n’ +d'-6')z,	cos my cos—z,
¿ - 5
2) si n' 6' - 2n ' ) y sin (2n ' + d' - 5*:) z, sin my sin—z.
Equality (77) may, thus, be replaced by the following two equalities: 22sin(df;+ 2n1) x cc s (6' - 2n’ ) y cc s ( 2n' + j' - 6’)-z „ . d + 6	d - 6
= ¿sin- ,-,-x cos my cos -	- z,	(90)
*2i	Á
22sin(i,;+ 2n')x sin (5»'-2nV)y sin (2n’,+ d' — 5f) z
_ . i + 5	d - 6
= Zsin- -x sin my sin ——z.
(91)
*V E CONSIDER IT UNNECESSARY TO REPEAT AGAIN THE METHOD FOR DEDUCING THE FORMULA
22f(J'*2n', 5t-2n\ 2»'.d'-6') = jf fcji, ¿gij, (92)
FROM FORMULAS (09) AND (01). IN THIS FORMULA f(x,y,z) IS.AN ANALYTIC135
Oft NUMERICAL FUNCTION WELL DEFINED AND FINITE FOR ALL INTEGRAL V AL UES OF X, y, AND 2 WHICH OCCUR IN FORMULA (92). THIS FUNCTION ALSO SATISFIES THE CONDITIONS:
f(~x, y;z) = -f( x, y, z),	f(x,-y,-z) = f(x, y, 2) .	(92’ 'i
Equality (92) is given by Liouville on page 42 of the seventh
VOLUME (SECOND SERIES) OF "JOURNAL DE MATHEMATIQUES PURES ET APPLIQUES" IN HIS ANSWER TO HER Ml TE.
§ 55.Example 12. There is known in the theory of elliptic functions A VERY ELEMENTARY METHOD DUE TO JACOB! BY MEANS OF WHICH E-QUALITY (4) CAN BE OBTAINED FROM EQUALITY (1). THIS SIMPLE METHOD MAY ALSO BE APPLIED TO OTHER MO RE COMPLICATED SERIES OF THE TYPE:
A
B
r~j- eyi /q 1
¿4i _
* .........

r*.l
1	. 2r-
1 - q Ô
(93)
Hr
r= i
yi./ ër- 1 e V q
3-yi^T
Ê r-1 gyi .
q e	1
Sr
q e
_£_____\
-1 -2ylj
cos(2r - l)x.
P	p
Let us determine A' +3 . We shall obtain terms of three kinds,
GROUPING INTO THREE SUMS. I F 'IE DENOTE BY d AND d1 POSITI VE ODD NUMBERS WE OBTAIN
1)
2 y ii/ d+ d'
e y q
q e )(1
d Eyi »
q e )
( sin dx sin d' x
+ cos dxcos d’x )
2)
I&r
d ¿r u
2 y i,/" î V q
d + d 1
d 2yi w d 2yi\ q e ) ( 1 - q e )
co s( d — d’ ) x,
II-
A A *V

* d
,_f. d - S3 y i V ,,	« -
7^1 ” q e ) ( 1 — q e
_——cc s( d - d ’ ) x, d'-2yix	' *
3)
1/d + d'
v m.
d 2yi q e
) ( 1 “ q d
vi~00 s( d + d’
e ‘y )
)x
Here d + d* and d-d' are even; let us collect all the terms with cos2nx. The first sum gives
P q e
n Syi
•	a
¿(l--qVyi)(l-qs""e2Jrl)
. 2q V ”
1 * q'
HI
d« 1
d gyi
1 - q e
1
- q
in * d
q___
in*’
e136
n £ y i r= n
2 q e *s~
~ 7	~ ¿_
2r- 1
„	-- 4L— - 2r*i 2yi *
1 - q irtl - q e
\S THE COEFFICIENT OF C0s2uX.' THE SECOND SUM GIVES
n -2 yi r= n	. 2 r- l
(94)
2 q e	q
„	2n~ / ■	2 r-l -pyi *
1 “ q r= i 1 " 4 e
(95)
The third sum gives a finite sum
r = n
_ V.
2q
2q
/	, . pr-l 2yi\/,	2 n-2 r + l -2yiv
.TjCl - q e )(1 - q	e )
n r= n r~	2r-l 2yi	2r-l -2yi
1 - q
r.I
r* 1
Sr-l 2yi
1 +•-—L-z---t—r—t +
- 2r-l 2yi	2r-l~2yi
1 - q e	1 - q e
] _ 3ogn J ' “ l-q8n
0	«	2yi	VC_n	2r_1
2q a	y	q
.	Pn	/	gr-
1	-	q	Jr i - q
_ n - 2y i r=n
2 q e	—
2r- i
Vyl
1 -q
2n
E, (99)
OOMPAFHNG EQUALITIES (94),	(9 5), AND (96) WE SEE THAT THE COEFFICIENT
üf cos2nx, if n is not zero, is
2 a q n
1 -q
2 n
¡F, HOWEVER, n IS ZERO, THERE WILL BE NO THIRD SUM. WE MAY 'WRITE THUS
=®	n
A 2	o2
A + 3
2f(y) - 2 Z-^
co s2nx,
' r - 1 P y i
«»> ■ ZtTI
P r-l -2yi
q e
,ti(i -q
^(1-4'
:r-i -2yiN2*
(97)
(97')
A, 3, ANO A2 + 32 MAY BE GIVEN ANOTHER FORM;
cc
A = 2 y~^i/q" r 1 ¿_cas sin dxl r=1	3r-l=dô
oc
3 =. 2i <[l/q"^	* ^)~sin ôy cos dx^J-
r*l	Ov._-|-r)A

(93)
2 r-l=dô
A® +
32 = 4 ^qn ^¡~ [co s ôy cos ô ' y si n dx si n d'x n=1	2n=d5+ i16 '
- sin ôy sin ô'ycos dxcos d'xjj^.
(99)137
INTERCHANGE of X AND y WILL RESULT IN A CHANGE OF SIGN IN (99,. AND THEREFORE
00	B
f(y) *	--~gnC0S2ny>
nTl1	q
WE HAVE THUS AN IDENTITY:
°° ■» Y (qy £cos(d — d' )x cos (6 + 6f )y - cos(d + d* )x cos (6 — 6' )yjV
n=1 2n=dÔ+d,ô'
ÙC
- 7 —'''"""'zn )c0s3ny - cosSnxV.
I ~ Q V	J
nar H
(100'
Ot
Let n =2 m,where m is odd and a = 0, 1, 2, 3,then the co efficient OF qn IN THE RIGHT HAND MEMBER OF EQUALITY (100) MAY EV1DEN LY BE WRITTEN AS
a v-
«♦l
a+ a
2	2^ d[cos2 dy - cos2 dxj .
ftl=d6	;
WE THEREFORE HAVE THE FOLLOWING EQUALITY:
7jcos(df ~d")x cos (6' ♦6")y. - cos(d* + d")x cos (6* — 6")y] 2n=d‘ .6' +d"6"
a +1	ol+ l
a r*- , . a + i ,	a+i .
,= 2 J d(cos2 dy - cos2 dx)
ni=.d6
( 101)
Theorem. If ts, d, b, d'., 6', d", 6" are odd positive numbers and a is a non-negative inteier; if f(x,y) is a function finite and defined for all thé integral values of x and y considered; if this function satlafiss the conditions
f(-x,y) = f(x,-y) = f(-x,-y) = f(x,y), then the following equality is true:
2[f(d‘ -d", 5«;+ 6”)	f(d'+d\ ô*-ô")]
a+1
l
* 2 2d[f(0>2 d) - f(2	d,0)j,
( 102)
in which the first sum is extended to all solutions d', 6', d", Ô" of the equation
CC+ 1
2	» >	d'.&>	dog,)
in vhtoh a is lises, ana tits second sun. to all solutions i ana &
m » d6.	( 102 ")
Equality (10?2) contains in itself two equalities given by Liou-VILLE ON PAGES 151 AND 199 OF THE THIRD VOLUME OF "JOURNAL DE (V AT H-138
EMAT I QUES -P URES ET APPLIQUES" (2ND SERIES); THE EQUALITY IS THE SUBJECT MATTE* OF HIS 1st AND 2nd PAPERS.
Example 13. I now want to indicate another very simple example of
THE GENERALIZATION of THE METHODS of DEDUCTION TKAt ARE USEO IN THE THEORY OF ELLIPTIC FUNCTIONS. THIS TIME THE GENERALIZATION WILL BE RELATED TO THE DERIVATION OF HERMITIAN SERIES, fhIS GENERALIZATION WILL GIVE US SEVERAL VERY INTERESTING THEOREMS# THE PECULIARITY IN THE CHARACTER OF HERMITIAN SERIES,IS DUE TO T H E FACT THAT IN THE RESULT OF MULTIPLYING TOGETHER TWO SERIES THERE OCCURS A FACTOR WHICH IS A PARTICULAR VALUE OF A JACOBI FUNCTION; IT APPEARS THAT A SIMILAR FACTOft APPEARS ALSO IF THE SERIES IN QUESTION ARE REPLACED BY MORE GENERAL SERIES; BUT THIS COMMON FACTOR IS NO LONGER A PARTICULAR VALUE of a 'Jacobi function but rather the same function with an indeter-
MIN A N T € ARGUMENT.
Denoting by ¡3 an odd number, let us multiply the series A of formulas (93) or (98) by
3	9
1# * , 4
^ q4 ccsmxeosmy = ^ q3"' c:s!Dx(emyi + e~myi), (103)-
m= l	1
T H £ SERIES B OF THE SAME FORMULAS 8 Y
D -
^ Am
sin id x sin ray = j
m~ i	m-î
= 2i£ ,1* si
qï- ¡in«i(8,,i-e','‘).	(104)
.WE SHALL PERFORM THE MULTIPLICATION BY TWO METHODS AND SHALL DETERMINE THE VALUE OF AC - 3D.
;'The first method gives
°c
AC - 3D = 3 £ ,**<£.iaW m)xcos(6-m)y ,	(10 5)
<5=0
WHERE THE SECOND SUM IS EXTENDED TO ALL SUCH SOLUTIONS OF THE EQUATION
m* + 2d6 = 4g + 3,	CIOS')
FOR WHICH d AND 6 ARE POSITIVE AND ODD, AND ffl IS ODD AND POSITIVE OR NEG AT I VE.
|N THE DETERMINATION OF THE VALUE OP AC - 3D'BY THE OTHER METHOD YE SHALL FOLLOW STEP BY STEP THE METHOD OF PAGE 52
AC - 3D = ^¡~ Jq4* sin (2f ? 1 -a)x
^^qi*'.sin(2rM*»)'x{!^-
■f/qsrr'ie<*M,srl	./ Sr-l -( m + i)yi V q e
*1 1 - f*'1	+ ' gr-l-gyi 1 - q Ô
(l/q2l-V("‘I,jrl	,/ gr-l ( m-l) yi , ^ e 		__
Xl ! -	„ gr-l -gyi 1 “ q • e139
Let es determine the coefficient of sin2nx in the right hah:
MEMBER OF THE PRECEDING EQUALITY.IT IS EASY TO GIVE TO THIS COE' F I C I E N T THE FOLLOWING FOR M !
yv<
All»".
gr-.l)
r = i
( ?,*-1 ) yi / -n(sr-i) -gnyi	n(2r-l) 2nyi,
(, q	e	- q________ô_____^
,/ - 2 r ♦ 1 yi /Vr-1 -yi
v q	e - yq e
-(2r-l)yi, -n(2r-i> gnyi	n ( 2 r- 1 )	- 2 nyl
é	K q	a - q	a '
./-2rVl yi ./ ¿"r-'i -yi
V q	a v q e
J
Performing the division ib the preceding two fractions we may wri"
THE COEFFICIENT, AFTER A SIMPLE GROUPING OF THE TERMS, IN THE FOR:.
2	cos(2r-2)y ♦ qr cos2ry^
r* 1
* ,-*l	[V’*”
+ q“(r	“£?
	-x n * 2q
HAVE	THUS
cos(2r--4)y + q^r + 1^ cc s(2r + 2);yJ + r~n* ccs2(r-n)y + qi r + n-1^ cos2(r * n - 1)yj|
-|(2n-l)*J
4	• r\ 4
~	» sj
AC - BD
= 2Sfi(y) ^q11 sln2nx|j} 4 + q”4 +	, . ... «■ q ^ n ^ ^
n* 1
1 /2Sft , w , x .	2*i x
= « \ T $<,(y'$,,( x) Ksin auj-----------
¿0 V rc	^	ft
The preceding equality may be given another form if we multiply TOG-ETHER §^(y) AND AN HERMITIAN SERIES:
AC - BD
= y <fqS + 4Y~ sini“~^)x ccs2nyV;	(107)
a™o	1	<*
thh last sum is extended to all solutions of the equation
4ne + d6 = 4« + 3,	(107,f,).
FOR WHICH j and 6 are POSITIVE AND ODD, AND G IS AN INTEGER.
Equating the coefficients of the same powers of q in equalities (10 5) AND (107), WE OBTAIN
, ■ d 1 +6i
. 11 + o x	r—
/ 3in~ ~xcos2oy -2) sin( i + id) x cc s (6 - m)y,	(30
3)140
Thsorsm. If i, 5, ¿1, and are odd positive numbers, in an odd number, n an integer, and if a function f(x,y> has the properties
f(-x,y) = -f(x,y),	f(0>y) = 0,	f(x,-y) = f(x,y),
£ hen the followiny equality holds:
2^~f(i+in, 6-a) =	¿i + ^i
2n
(109)
in which .the sums are extended to ail the solutions of the equations
m S + 2d6 = 4* +3,	4‘02+di6t = 4g ♦ 3;	(109' )
where 4s + 3 is a positive number.
Example 14.From equalities (105) and (106) we can deduce in an
EXTRAORDINARILY SIMPLE WAY ANOTHER ThEOREM; TO DO THIS LET US MUL TIPLY BOTH MEMBERS OF THE TWO EQUALITIES BY $ (0):
2 y~ q gy~ ai n ( i + in ) x cc s ( & -- in ) y 4s + 3=m2+4ii2«f-2i0
,g=o *-
Ki< . 2Kx„ ,	*
— sin ain — Ss(y) 9g(x)
= 2S3(y)Sg(x) ^

2 n -
T31" jxJ ■ 2H
2n-l=d6
g = o
s* §
* £X
y~~ sin(d+n)x cos 2ny 4g+3=m +4n +2iô
Equality (1) of this chapter is used in the derivation.
Equating coefficients of the same powers of q we obtain
2sin ( i + id) x{eos( a - in) y - cos2ny} = 0.-	(110)
Theorem. If i and 6 are positive odd numbers, is an odd number, o an integer, if
fVx,y) * -f(x,y)>	f(0,y) = 0,	f(x,~y) = f(x,y),
then the follovjini equality holds:
2[f(d+m, 6-in) - f(i+is, 2n) 1 = 0,	(111)
in which the summation is extended to all solutions of the equation
is2 + 4nS + 2d6 - 4g + 3; where 4g + 3 is a positive number.
§ 56. Example 15, let us take two series
+ oc
+ JC
= yy qn co s 2nx cc s 2ny ,	• F = i q& sin 2nx sin 2ny. (112)
n* - 00
n=s - oc
By the method used in Example 13 we easily obtain141
AS - BF
n* 1
2 ^ cos(5' - 2 n ’) y sin ( d' + 2n> )x 2n-l=2n' 2 + d' 5’
n* 1
r—	. d+6
) cos ray sm-r-x ,
^- j	j
4n-2=ffl2 + d&
(113
WHERE d’ ,	6’, d, AND 6 ARE OOO AND POSITIVE, ¡11 IS ODO, POSITIV,-.
or negative, n' is an integer, 2n - 1 is a positive number constant FOR THE INNER SUMS.
From equality (113) follows
sinCd* +2n* )jc cos (&» -■2'n,)y « sin— x cos ray .; (IK. 2n-l=2n'2+d ’6*	4n-2~m2+d6
Equality (114) [and therefore (115)] may be deduced from equalit
(yO) BY PUTTING Z = 0•
LET US MULTIPLY THE SECOND AND THIRD PARTS OF EQUALITY (113) by Ss(0):
2 L
£-0
t“1
^ sin(d + 2n)xcos(6 — 4g+3=m2 +4n2 +2d6
2n)y
Kk . 2Kx N . — si n am - -S (y) & (x)
K	it 2	ö
= 2ftß(y),V3C) 1_
n= i
/‘gn^T V- .	,
v q	) si n dx
2n-l=d6
a 2
JZ_
£ = 0
L.
in(d + 2n)x cos ra^
4g + 3=m2 + 4ns +2dS
From the last equality follows
__
^ sin(d -2n)x[cos(& + 2n)y - cosray] = 0*	(115)
4g+3=ra 2+4n 2+2d5
In equality (H5)d and 5 are positive and odd, m an odd number, n an integer, 4g + 3 a positive number. From equality (11s) we shall deduce one general theorem of Liouville containing a function OF FOUR VARIABLES. |N EQUALITY (115) TERMS MUST CANCEL IN PAIRS, FOR THEY CONTAIN IN DETERMINANTE X AND y. THERE ARE 4 WAYS
in which sin(d - 2n)x cos(6 +2n)y may cancel with some term equal
To IT IN ABSOLUTE VALUE BUT OPPOSITE IN SIGN; LIKEWISE THERE ARE
4 ways to cancel sin( d - 2n) x cos ray. But it may happen that the whole expression
sin(d-3n)x[cos(6 +2n)y - cosray]	(113)
CANCELS WITH THE EXPRESSION
sin(d'-2n')x[cos(5’+2n')y - cos ra'y]
(117)IN WHICH d',	n-', AND rn' DENOTE ANOTHER SYSTEM OF SOLUTIONS OF
THE EQUATION
2 2
m + 4n + 3d& = 4g + 3;
IN THIS OASE THERE CAM BE 16 CASES, OF WHICH, HOWEVER, NOT ALL ARE POSSIBLE. WITHOUT GOING INTO DETAILS WE SHALL PROVE THAT FOR EVERY
system d', 6', m', 2n' it ispossible to find a system d, 6, m, 2n SUCH THAT THE EXPRESSIONS (116) and (117) WILL BE EQUAL IN ABSOLUTE VALUE BUT OPPOSITE IN SIGN. SUCH CORRESPONDING SYSTEMS WILL BE DETERMINED BY ONE OF THE SYSTEMS OF EQUALITIES:
d -	2n =	-	d'	+ 2n'
d +	2n =		6*	+ 2n'
	m	s	m'	
	LET	u s	S € T Z	
FUNCTION			0 F	d, 0,n
OF	d',	6'		2n', m
BO	FOR	d,	6	, 2n,
II
d - 2 n a d' — 2n' 5 + 2 n = m' a = 6' + 2n'
HI
d- 2n = d' - 2n'
5 + 2n = -m* m * -(6’ + 2n<).
ALSO SOME OTHER LINEAR
d, 6, 2n, and in in terms
THE EXPRESSION THUS OBTAIN-
2 2 2 2
m + 4n +2d6 = m'	+ 4n'	+2d'6'
W:E SHALL F I NO IN EACH OF OUR THREE CASES A QUADRATIC EQUATION; OF THE TWO ROOTS OF THIS EQUATION WE SHALL TAKE ONLY ONE. WE SHALL OBTAIN THUS, AN ADDITIONAL EQUALITY FOR EACH OF THE THREE PRECEDING
systems:
I
I
d + m = F ROM THE
d' + m’
d + m
THREE S Y ST EM S I, II ,
I
= d' + m '
HI WE OBTAIN H
IH
d + m = - d' - m' .
HI
ffi = m' d = d'
2n = 2d* - 2n*
6 =	6« + 4n* -2d’
na a 6 * + 2n * d = d* + m' - 5' - 2 n1 2 n = m * - 6 ’
6 = 6*
, = -5' - 2n* d = 6* + 2 n' - d* - m* 2n = 5* + 4n' - 2d' -m* 6 = 2d' - 6* - 4fl».
Careful consideration of the preceding three systems shows us that the first system will always determine d, 6, 2n, and m if the
CONDITION
&' + 4n' > 2d';
holds. The only condition for the second system is
d' +m* > 6' + 2n' ;
AND THE ONE FOR THE THIRD IS
6' + 4n' < 2d', d' +m' < 6' + 3n'Thus for every system d', 6', 2n' and m' there exists a corresponding system d, 6, 2n, m. The question arises as to whether tc one system d, 6, 2m, m there will correspond two systems d', 6*, 2n,,m'. The question will be answered if we determine dV, 6', 2r and m' through d, S, 2n, and m by the help of I, II, HI. then i ‘
WILL APPEAR THAT FROM SYSTEM J ONLY SUCH d, 5, 2n, ffl CAN B E DETERMINED WHICH SATISFY CONDITION
ô + 4n > 2d;
FOR SYSTEM II WE SHALL HAVE
d + m > Ô + 2n;
FOR SYSTEM HI WE SHALL HAVE
6 + 4n < 2d, d + m < 6 + 2n.
We have thus proved that the expressions (116) and (117) cancel out entirely in pairs.
LET US NOW NOTE THAT SYSTEMS I, I , HI CAN BE GIVEN THE FOLLO
1NG form:
I
d—2n=.-(d' — 2n') 6,+2n-m = 6' +2n'-m' 6+2n+m ». 6' + 2n' +m' d + m » d ' + m'
II
d - 2n * d' — 2n' ô+2n-m = -(ô,+2n,-m') ô+2n+m = 6* + 2n' + m‘ d + m = d' + ra'
III
d - 2n =* d' — 2n ' ô+2n-m » S'+2o'“m’' ô+2n+m =s-(0'+2n,+m') d + ra = - ( d ' + m ' ) .
Let us note also that the four equalities
d-2n = -(d* - 2n' ), S + 2n - m = -(S' + 2n' - ra' ), 6 +2n + m = -(S' + 2n' + m'), d + ra = -(d' + m'),
OF WHICH THE LAST ONE IS DERIVED FROM THE EQUALITY
m2 + 4ns + 2d6 » m* 2 + 4n'8 + 2d'S'
BY TAKING INTO CONSIDERATION THE FIRST THREE, ARE I N CO N C I CT E N T, FOR THEY LEAD TO THE CONDITION d=—d'. TAKING ALL THIS INTO CONSIDERATION WE CAN DEDUCE THE FOLLOWING THEOREM.
Thborsm. If d and 6 are odd positive numbers, m an odd number n an Integer, and if f(x,y,z, t) has the properties
f(-x,y, z, t) = f(x,-y,z, t) = f(x,y,-z,-t) = -f(x,y, z, t), f(x,0,z,t) = 0,	f(x,y,Q,-t) » -f(x, y,0, t),
f(x,y,-z,0) = -f (x,y, z,0),	f(x,y,0,0) = 0,
then the following equality holds;
2f(d-2n, 6+2n-m, 6+2n+ra, d+m) = 0,	(118)14.4
In this last the sum Is extends! - to all the solutions of the equation
m2 + 4ns + 2d6 = 4g ♦ 3*,-where 4g + 3 is a positive number.,
Indeed, with the help of one of the systems I, II, lit we can find
FOR EVERY
f(d-3n, &+3n-m, 6+2n+®, d+m)
A CORRESPONDING VALUE
f(d'-2n', S'+Sn'-m', S'+2n,+in', d'+tn’)#
EQUAL TO THE FIRST ONE IN ABSOLUTE VALUE BUT OPPOSITE IN SIGN,
Equality (118) is investigated by Liouville in his thirteenth and
FOURTEENTH PAPERS ON GENERAL NUMERICAL IDENTITIES INVOLVING ARBITRARY FUNCTIONS; THESE TWO PAPERS ARE PRINTED in the 9TH- VOLUME OF
Liouvill’s journal (2ieme Seri<=})• Liouville derives there first, two
PARTICULAR FORMULAS WHICH CAN SF OBTAINED FROM (118) BY OMITTING IN TURN THE ARGUMENTS 6 +2n +ffi AND d+m, AND THEN, AT THE END OF THE FOURTEENTH PAPER, HE CONSIDERS THE GENERAL FORMULA (118).
§	57, IN ALL THE PRECEDING EXAMPLES THE FUNCTIONS IN THE THEOREMS
OF LIOUVILLE.WERE EITHER EVEN OR ODD; IT MAY EVIDENTLY HAPPEN THAT THE SAME THEOREM HOLDS TRUE FOR BOTH EVEN AND ODD FUNCTIONS, IT IS EVIDENT THAT IN THIS PARTICULAR CASE THE PRECEDING METHODS REMAIN USEFUL,* HOWEVER, IN PROVING SUCH THEOREMS ANOTHER SIMPLER METHOD, ALSO BASED ON THE THEORY OF ELLIPTIC FUNCTIONS, MAY SOMETIMES BE EMPLOYED. IN THE SECOND CHAPTER WE CONSIDERED EQUALITIES BETWEEN TWO OR MORE DOUBLE SUMS. COMPARISON OF THE COEFFICIENTS OF THE LIKE POWERS IN THESE WILL EVIDENTLY LEAD TO EQUALITIES WHICH IN THE MAJORITY OF CASES CONSIST OF SUMS OF DIFFERENT PuW6ftS OF -1. Vv £ MAY PUT THESE SIMPLE NUMERICAL FUNCTIONS IN A FORM INVOLVING AB SO L U T EL Y ARBITRARY FUNCTIONS; THE REASONING INVOLVED IN THE DERIVATION OF SUCH FUNCTIONS HAS MUCH IN COMMON WITH THE REASONING IN THE EXAMPLE 15.
A GENERAL EXPLANATION CAN GIVE ONLY A REMOTE IDEA OF THIS METHOD OF DERIVATION OF THEOREMS; ! SHALL THEREFORE REPLACE THIS EXPLANATION BY A PROOF OF A THEOREM OF LIOUVILLE GIVEN 8Y HIM IN HIS JOURNAL IN
April, 1870. The theorem was proved by Zolotarev, a member of the
ACADEMY, IN THE BULLETIN DE L’ACADEMIE DE ST. PETERSB0URG, TOME 16, ’¡0.2, PAGE 86. WE SHALL MODIFY SOMEWHAT THE BEGINNING OF THE PROOF
of Zolotarev, for it is desirable to indicate the connection between
HIS METHOD AND THE THEOREMS OF CHAPTER It.
Liouville' s Theorem., Let m be a number of the form 41 + 1; let us denote by i and ij certain odd positlue numbers, by to, s, ut and sx
arbitrary integers.. Let a function F( x)' have deuinlte finite values for all the integral values of the argument considered. Then145
v(-l) 3+5(i *"'lVF(u) =	3lF(J iV,	(li ’
uh ere the sums ar e extended to all so Lut io fis i} s, ii> iù x for a giuen in of the equations
i2+w2+16s2 = in, ii8 + w12+.8s12 = m..	(119 :
respectively.
Proof. From equality (15) of the second chapter (page 45) we
MAY WRITE
f_ f_ (-1V’**1'(1Ï)
•i=l 8««*	8l = “,0è
PUTTI N G
ivi	2 -i -s ^ 2	"NI f «. * 2 , ^ 2
i>] - j. . + 1^3 >	N T - JL i + G ^ i • .
WE MAY WRITE EQUALITY (120) IN THE ABBREVIATED FORM
(120'ì
¿MV

N
IT IS EVIDENT THAT
A = !
N
L.
N !
(121)

, a' , = s(-DsS
N
(122)
WHERE THE SUMS ARE EXTENDED TO ALL SOLUTIONS OF THE CHARACTER SPECIFIED IN THE THEOREM, OF THE EQUATIONS
i 2 + 16 s2 = 11, it2 + 8sx2 = N'.-	(122')
R E SP ECT1VEL Y.
Let us consider now two sums:
2(-l)s+l(l *1)F*(W) = 2A f(w), S(-l)3lP(Wi) = 2 A' ,F(u)l),(123)
N K	M'	N
WHICH ARE EXTENDED TO ALL THE SOLUTIONS OF EQUaT!ONS (ll9f)
WE SHALL CONSIDER SEPARATELY THREE CASES.
1) Let u)x = u); then
P(«0 • F<»>, C-1^('S'° -	N'
In THIS CASE THE EQUALITY (121) GIVES
A' , = A ,
N	N
FOR THE COEFFICIENTS OF THE SAME POWERS OF q IN THAT EQUALITY MUST BE EQUAL TO ONE ANOTHER. THUS IN THIS CASE
A n,F(wi) = AnF(u>)..146
2)	LET IT BE SO that FOR A CERTAIN U)x THERE IS NO W EQUAL TO IT.
!n THIS CASE m-Mi = Nf can be represented by THE FORM li8+Ss1*,
BUT NOT BY THE FORM i*+16s8, THEREFORE q TO THE POWER N’-l WILL NOT OCCUR IN THE LEFT HAND MEMBER OF (121) AND HENCE IN THIS CASE A*' ,=0. WE SEE THUS THAT THE TERM -A * H i F(d t) VANISHES BY ITSELF.
3) Those terms in the first sum of (123) for which there is no
THEM AL SO V AN I SH.
CONSEQUENTLY
2A‘h.F(iO = 2AKF(u>V,	(124)
QUOD ERAT DEMONSTRANDUM.
IT is CLEAR FROM THE TEXT OF the PRECEDING proof that every theorem of Jacobi which belongs to the class of theorems considered in
the SECOND CHAPTER, WILL GIVE A THEOREM SIMILAR TO THE FOREGOING ONE. IT IS EVIDENT THAT THE METHOD OF DEDICATION ITSELF MAY BE GENERALIZED CONSIDERABLY. SUCH INVESTIGATIONS ARE, HOWEVER, NOT THE OBJECT OF THIS BOOK:	IT WAS INTEN ED TO PRESENT HERE IN AS COMPLETE A FORM AS
POSSIBLE THOSE MOST IMPORTANT APPLICATIONS OF THE THEORY OF ELLIPTIC FUNCTIONS WHICH HAVE BEEN MADE BY DIFFERENT INVESTIGATORS UP TO DATE.
Note., The Lectures of Professor Bliss on Tha Problem of La^ran^e In t’ne Calcul us of Variations have been similarly duplicated by me and can be had by writing. They contain a table of contents and four chapters forming 76 pages in all bound also in press-board. price, SU 75 dus oosta^e.