0* 



order n is included in the formula 



=| A | X | e|, in which | e j is an unit 



ON THE THEORY OF NUMBERS. 331 



These converse propositions may he demonstrated by means of the differ- 

 ential equations satisfied by the elliptic functions : by a similar process we 

 obtain the following equally important theorem : — 



" If A is any quantity, real or imaginary, other than zero or positive unity, 

 there exist values of w, having the coefficient of i in their imaginary parts 

 different from zero and positive, which satisfy the equation <p 8 (o>)=A." 



When n is an uneven integer other than 1, the formula of transformation 

 is 



) v(|,fl)=Te"^ M m , B (^, w ), (36) 



in which to and n are determined as before, and T is a homogeneous function 



of order ~ of the squares of two of the functions 6^, v (x, w). "We need 



S3 



not occupy ourselves here with the determination of A and T, but shall 

 confine ourselves to the consideration of the modulus and multiplier alone. 

 Eepresenting by <5(n) the sum of the divisors of n, every binary matrix of 



a, b 



c, d 



y,0 



Tc, y' 



divisors of n, and h representing any term of a complete system of residues, 

 mod y. It is thus sufficient to consider a system of <b(n) transformations of 

 order n, since all others arise from compounding transformations of the first 

 order with the transformations of that system. If we take, in particular, the 



system of transformations, w= ■ - — , corresponding to the matrices 



y 



, fi T' , (since n, and therefore y is uneven, we may take a system of 



residues, mod y, of which every term is divisible by 16), we have for the 

 determination of the transformed modulus, the fundamental theorem *, 



" The quantities ( - Ws2)=( - \ <j> ( y , ) are the roots of an equation 



of order 4>(n), in which the first coefficient is unity, and the other coeffi- 

 cients are rational and integral functions of <j>(J) having integral coeffi- 

 cients." 



This equation is termed the modular equation of the transformations of the 



nth order ; designating f(w) by u, and (- \ $ |2^iZ_ — j by v, we shall re- 

 present it by f(n, u, v)=0, or more simply by f(u, v)=0. The function 

 f(u, v) is characterized by the following, among many other properties, 



matrix, and | A | one of the <b(n) matrices 



, y and y being conjugate 



/(«, ,)=(-l)*<«> n(|) x /((|),, «), 

 /(«, ,)=(- 1)*« ng) x (*)•« f(K 1) 



(37) 



* M. Hermite, Sur la th^orie des equations Modulaires, p. 36 ; M. Joubert, Comptes 

 Rendus, vol. 1. p. 774 ; or, in a separate reprint with the title " Sur la Theorie des Fonc- 

 tions Elliptiques, et son application a la Theorie des Nombres," p. 21. The demonstration 

 of this theorem for the case in which n is a prime, is contained in Sohnke's important 

 memoir "JEquationes modulares pro transformatione functionum ellipticarum," Crelle, 

 vol. svi. p. 97. From this particular case, the truth of the theorem for any value of n is 

 inferred without difficulty. 



