326 ON SYSTEMS OF LINEAK INDETEEMINATE EQUATIONS AND CONOEUENCES. 
precisely the same as in the case of common whole numbers ; while the expression for 
the number of the solutions (when the condition of resolubility is satisfied) is simply 
tlie norm of m. 
But without entering into the deA^elopments which this extension of the subject of 
this paper would require, we shall confine ourselves to an application of the result of 
the last article to a demonstration of the fundamental principle in the arithmetical 
theory of complex numbers, that the number of incongruous residues for any complex 
modulus is represented by the norm of the modulus. 
Let cc be one of the roots a„ osa, ... of the equation F„(^) = 0, which is supposed to 
be of n dimensions, to be irreducible, and to have all its coetficients integral, and that 
of its first term unity. Let also be the complex modulus under consideration; 
its norm, which we shall symbolize by N, is defined by the equation 
N=N. = 
Consider the Na,,., residues (incongruous mod. N) which are included in the formula 
R2n-2(a)j (9^0 
Avhere E 2„_2 denotes an integer function of 2n—2 dimensions; it is evident that every 
complex number is congruous, for the modulus <J5„_i(a), to one at least of these 
residues. If E and E' be any two (the same or different) of the same residues, it is 
also plain that the congruence 
E=E', mod. <?„_,(«) 
will, or will not, be satisfied, according as it is, or is not, possible to assign two functions 
of ,T, F„_,(.r) and <P„_ 2 (a’) having integer coefficients, and satisfying the equation 
F„(a’)<P„_2(^)+F„_i(^)<P«_i(^)=E(^)— E'(. r) (100.) 
This equation is equivalent to a system of 2n — 1 linear equations, in which the 
unkno'v\Ti quantities are the 2n—l coefficients of <p„_ 2 (^) and F„_i(^), and of which the 
determinant is the dialytic resultant of F„(^) and (p^_ i. e. the norm of <p„_,(a) orN. 
If then we suppose E(«) to represent any given residue included in the formula (99.), it 
will appear from the theorem of the last article that the equation (100.) is resoluble for 
^27!-2 (Afferent values of E'(^), ^. e. that every complex number is congruous, for the 
modulus <p„_i(a), to precisely of the residues contained in the formula (99.), 
or that the number of residues, incongruous mod. <p„_i(a), is precisely N. 
It is, however, proper to observe, that a complete demonstration of this important 
theorem has already been given by Professor Sylvester (see a paper signed “Lanavi- 
censis,” in the ‘ Quarterly J ournal of Pure and Applied Mathematics,’ vol. iv. p. 94 
and 124). 
