162 REPORT — 1860. 



prime modulus. Since in any modular function we may omit those terms 

 the coefficients of which are multiples of p, we shall always suppose that 

 the coefficient of the highest power of a; in the function is prime top. 



If F(x)=f(x)xf(x), mod p, f(x) and/ 2 (,r) are each of them said to 

 be divisors of F(x) for the modulus p, or, more briefly, modular divisors of 

 F(#), or even simply divisors of F(x) when no ambiguity can arise from this 

 elliptical mode of expression. If a be a function of order zero, i. e. an integral 

 number prime to p, a is a divisor, for the modulus/?, of every other modular 

 function ; so that we may consider the p— 1 terms a v a. 2 , a 3 , . . . Op_i, of a 

 system of residues prime to p, as the units of this theory, and, in any set of 

 p— 1 associated functions 



a l F (x), a 2 F(x) a p - x F(x), 



we may distinguish that one as primary in which the highest coefficient is 

 congruous to unity (mod p). 



If F(x) be a function which is divisible (mod p) by no other function 

 (except the units and its own associates), F(x) is said to be a prime or irre- 

 ducible function for the modulus p. And it is a fundamental proposition in 

 this theory, that every modular function can be expressed in one way, and 

 one way only, as the product of a unit by the powers of primary irreducible 

 modular functions. The demonstration of this theorem depends (precisely 

 as in the case of ordinary integral numbers) on Euclid's process for finding 

 the greatest common divisor, which, it is easy to show, is applicable to the 

 modular functions we are considering here. For, if fa (x) and fa (x) be two 

 such functions [the degree of fa(x) being not higher than that of fa(x)], 

 we can always form the series of congruences 



fa(x)=q 1 (x) i^^ + r^^x), modp, 

 fa(x)=q 2 (x)fa(x) + r 2 fa(x), modp, 



in which r v r 2 , . . . denote integral numbers, q x (x), q 2 (x), . . . modular func- 

 tions, and fa(x), fa^(x), .... primary modular functions, the orders of which 

 are successively lower and lower, until we arrive at a congruence 



fak(x)^q k(x) fak+i (x) + r k <t> k+2 (x), modp, 



in which r£=0, mod p. The function <j>k+\ (x) is then the greatest common 

 divisor (mod p) of the given functions fa(x) and fa(x); and, in particular, 

 if faic+i(x) be of order zero, those two functions are relatively prime. We 

 may add that, if R be the Residtant of fa(x) and ^ 2 (^), the necessary and 

 sufficient condition that these functions should have a common modular 

 divisor of an order higher than zero is contained in the congruence R=0, 

 mod p* — a theorem exactly corresponding to an important algebraical pro- 

 position. From the nature of the process by which the greatest common 

 divisor is determined, we may infer the fundamental proposition enunciated 

 above, by precisely the same reasoning which establishes the corresponding 

 theorem in common arithmetic. Similarly, we may obtain the solution of 

 the following useful problem : — " Given two relatively prime modular func- 

 tions A m and A„, of the orders m and n, to find two other functions, of the 

 orders m— 1 and n — 1 respectively, which satisfy the congruence 

 A m X n _! — A„X m _i=l, mod p." 



* See Cauchy, Exercices de Mathematiques, vol. i. p. 160, or M. Libri, Memoires de Mathe- 

 matique et de Physique, pp. 73, 74. But a proof of this proposition is really contained in 

 Lagrange's Additions to Euler's Algebra (sect. 4). 



