so that κ'+x" = I, we shall have

@»»<»>= @»<»>= »<. @»<»<=»>=; .d

0—(-my) ~/ic snu, Bmw) Per u,6m< v) VK, nu, and, supposing for simplicity that iw is a real negative quantity,

1. 0002 =2K, w119092 =2iK', to ='iK'/K,

the notation being that which is now usual for the elliptic functions. It is found that

25 sn 2Ku=2§ J¥ -Q sin (2.9-I) 1ru,

1r 1 I “Q

K so, 1

-5; cn 2Ku =2Z%é-éqcos (2s - I)1ru,

I

K °° ~

~; -dn 2Ku = ~§ +227-L- cos 2s1ru.

From the last formula, by putting u =0, we obtain ®

I +4§ j% =3§ =000”(I +2q+2q'+2qP+- - -)“, and hence, by expanding both sides in ascending powers of q, and e uating the coefficients of q", we arrive at a formula for the number of ways of expressing n as the sum of two squares. If 5 is any odd divisor of n, including I and n itself if n is odd, we find as the coefficient of g" in the expansion of the left-hand side 42(-I)“(6'1)§ on the right-hand side the coefficient enumerates all the solutions n = (=|=x)2+(=*=y)', taking account of the different signs (except for 02) and of the order in which the terms are written (except when x2 =y”). Thus if n is an odd prime of the form 4k-i-I, Z(-1)5(8 )=2, and the coefficient of q" is 8, which is right, because the one possible composition n = zz'-l-b” may be written n = (ia)2+ (=|=b)2 = (ib)2-1-(¢a)2, giving eight representations.

By methods of a similar character formulae can be found for the number of representations of a number as the sum of, 6, 8 squares respectively. The four-square theorem has been stated in § 41; the eight-square theorem is that the number of representations of a number as the sum of eight squares is sixteen times the sum of the cubes of its factors, if the given number is odd, while for an even number it is sixteen times the excess of the cubes of the even factors above the cubes of the odd factors. The five-square and seven square theorems have not been derived from q-series, but from the general theory of quadratic forms.

68. Still more remarkable results are deducible from the theory of the transformation of the theta functions. The elementary formulae are

0n(u, w-l-1) =e"i/“0n(u, w), 010(1l, w'l'Il=@"i/4910(14, <v)» 9o1(1»¢» w'i'I) =900(14. QF), 000(u1 w'l'1) =9ox(14, w), 5-iriuz/wall (2» "!') = "1;/ iW01l(u1 01), w w

Cdiuz/waio (2, -l> = V "if-v910(14, 02), £0 (D

e-1|-iu2/w9w(E' .l) =, /"' ¢¢, ,'90, (u, w), 0) C0

e'"i“2/"'000 (3, -3) =w/ -'Lw900(u, on), where 4 -iw is to be taken in such a way that its real part is positive. Taking the definition of /c given in § 67, and considering x as a function of w, we find

x(w +I) =10102/0012 =iK(<.o) /l<'(r.o), = 2 z= I

rc((D) 001 /000 K

For convenience let »<2(¢.i)=a: then the substitutions (w, w~|-I) and (w, -w“) convert zr into <r/(a-1) and (I -iv) respectively. Now if a, B, y, éare any real integers such that 115-H-y=1, the substitution o, (aa>+5)/(-yt.>+6)1 can be compounded of (w, w+I) and (w, -nfl); the effect on a will be the same as if we appl a corresponding substitution compounded of [<r, 11/(0-I)»} and £3 I -n]. But these are periodic and of order 3, 2 respectively]; therefore we cannot get more than six values of a, namely

0, I, ,' l J. fl, I,

is-1 I -a 17 a

and any symmetrical function of these will have the same value at any two equivalent places in the modular dissection (§ 33). Their sum is constant, but the sum of their squares may be put into the form

2(<f2-<1-4-1)

a'2(a - I)2 3

hence (af -a+ I)3 +<r'(¢r - I)2 has the same value at equivalent places. F. Klein writes

]=4C'12 ¢7 i'I)3 .

27<r2(¢r-1)2

this is a transcendental function of w. which is a special case of a Fuchsian or automorphic function. It is an analytical function of q2, and may be expanded in the form

J=;%n~2+v44+¢1q2+f2q4+ . .z

where cl, c2, &c., are rational integers.

69. Suppose, now, that a, b, c, d are rational integers, such that dv(a, , b, c, d)=I and ad-bc=n, a positive integer. Let (aw+b)/ (cw-f-d) =w'; then the equation ](w') = ](w) is satisfied if and only if w'—w, that is, if there are integers a, [3, 'y,5suchthata5-]3'y===1, and (uw-i-71) (7w+5) " (€w+<i)(¢1w-HS) =0-If we write, b(n) =nII(I +p'1), where the product extends to all prime factors (p) of n, it is found that the values of w fall into ¢(n) equivalent sets, so that when w is given there are not more than //(n) different values of ](w'). Putting ](w')=]', ](;v)=], we have a modular equation

f1(]'» J) =0

symmetrical in J, ]', with integral coefficients and of degree , b(n). Similarly when dv(a, b, c, d) =r we have an equation f-, (], ]) =o of order 1p(n/-rf); hence the complete modular equation for transformations of the nth order is

F(J′,J) =${\displaystyle \Pi }$f_r(J′,J) =0,

the degree of which is ${\displaystyle \Phi }$(n), the sum of the divisors of n. . Now if in F(J′,J) we put J′,J, the result is a polynomial in J alone, which we may call G(]). To every linear factor of G corresponds a class of quadratic forms of determinant (K2-4n) where »<'<4n and K is an integer or zero: conversely from every such form we can derive a linear factor (J - a) of G. Moreover, if with each form we associate its weight (§ 41) we find that with the notation of § 39 the degree of G is precisely EH(4n-xz)-en, Where 6f, =I when n is a square, and is zero in other cases. But this degree may be found in another way as follows. A complete representative set of transformations of order n is given by w'=(aw+b)/d, with ad=n, 0Eb<d; hence

Gt1>=11 no-

and by substituting for]'(w) and I their values in terms of q, we find that the lowest term in the factor expressed above is either q 2/1728 or q"l° /'1/1728, or a constant, according as a<d, a>d or a=d. I-fence if v is the order of G(]), so that its expansion in q begins with a term in q'2" we must have V =z<1 -d) +2 fi) =zd+>:a

d > V n d> V n a>V n

d> V n

= 22d

extending to all divisors of n which exceed √ n. Comparing this with the other value, we have

${\displaystyle \Sigma }$H(41Z-K2)=22d+€n =${\displaystyle \Phi }$(n) +${\displaystyle \Psi }$(n),

as stated in § 39.

70. Each of the singular moduli which are the roots of G(J) =0 corresponds to exactly one primitive class of definite quadratic forms, and conversely.

Corresponding to every given negative determinant -A there is an irreducible equation ${\displaystyle \psi }$/(j) =0, where j = 1728 ], the coefficients of which are rational integers, and the degree of which is h(-A). The coefficient of the highest power of j is unity, so that j is an arithmetical integer, and its conjugate values belong one to each primitive class of determinant -A. By adjoining the square roots of the prime factors of A the function , b(j) may be resolved into the product of as many factors as there are genera of primitive classes, and the degree of each factor is equal to the number of classes in each genus. In particular, if {I, I, § (A+x)} is the only reduced form for the determinant -A, the value of j is a real negative rational cube. At the same time its approximate value is exp - zri-ii;-Q +744 = 744-effV A, so that, approximately, e"V 4=m3-I-744 where m is a rational integer. For instance effv 43=8847367 3-9997775 . .= 9503+744 very nearly, and for the class (I, I, 11) fihe exact value of j is-9603. Four and only four other similar determinants are known to exist, namely -11, -19, ~67, ~-163, although thousands have been classified. According to Hermite the decimal part of e1fV 163 begins with twelve nines; in this case Weber has shown that the exact value ofj is -218- 3-53-233-293.

71. The function(n3 is the most fundamental of 2 set of quantities called class-invariants. Let (a, b, 6) be the representative of any class of definite quadratic forms, and let w be the root of ax2-{-bx-I-c=0 which has a positive imaginary part; then F (w) is said to be a class invariant for (a, b, c) if FC'-;%' -§)=F(w) for all real integers a B, »y, 5 such that aB-/3'y=I. This is true for j(w) whatever w may be, and it is for this reason that j is so fundamental. But, as will be seen from the above examples, the value of j soon becomes so large that its calculation is impracticable. Moreover, there is the difficulty

of constructing the modular equation f1(], j') =0 (§ 69), which