230 report — 1859. 



of any order with integral coefficients. In this general point of view, these 

 two theories are hardly more distinct from one another than are in algebra 

 the two theories to which they respectively correspond, — the Theory of 

 Equations, and that of Homogeneous Functions ; and it might, at first sight, 

 appear as if there was not sufficient foundation for the distinction. But, in 

 the present state of our knowledge, the methods applicable to, and the re- 

 searches suggested by these two problems, are sufficiently distinct to justify 

 their separation from one another. We shall therefore classify the researches 

 we have to consider here under these two heads; those miscellaneous investi- 

 gations, which do not properly come under either of them, we shall place in a 

 third division by themselves. 



(A) Theory of Congruences. 



4. Definition of a, Congruence. — If the difference between A. and B be 

 divisible by a number P, A is said to be congruous to Bfor the modulus P; 

 so that, in particular, if A be divisible by P, A is congruous to zero for the 

 modulus P. The symbolic expressions of these congruences are respectively 



A=eB, mod P, 

 A=0, mod P. 



Thus 7 = 2, mod 5; 13e=— 3, mod 8. 



It will be seen that the definition of a congruence involves only one of 

 the most elementary arithmetical conceptions,— that of the divisibility of 

 one number by another. But it expresses that conception in a form so 

 suggestive of analogies with other parts of analysis, so easily available in 

 calculation, and so fertile in new results, that its introduction into arithmetic 

 (by Gauss) has proved a most important contribution to the progress of the 

 science. It will be at once evident, from the definition, that congruences 

 possess many of the properties of equations. Thus, congruences in which 

 the modulus is the same may be added to one another; a congruence may 

 be multiplied by any number ; each side of it may be raised to any power 

 whatever, and even may be divided by any number prime to the modulus. 



5. Solution of a Congruence. — If (j> (x) denote a rational and integral 

 function of x with integral coefficients (we shall, throughout this report, 

 attach this meaning to the functional symbols F,f (p, &c, except when the 

 contrary is expressly stated) ; the congruence <j> (x)=0, mod P, is said to be 

 solved, when all the integral values of x are assigned which make the left 

 hand number of the congruence divisible by P; i. e. which satisfy the inde- 

 terminate equation (J)(x)=Pg. It is evident that if x=a be a solution of 

 the congruence f(x)=0, every number included in the formula #=a-f-juP 

 is also a solution of the congruence. But the solutions included in that 

 formula are all congruous to one another and to a. It is proper, therefore, 

 to consider all these congruous solutions as identical, and in speaking of the 

 number of solutions of a congruence to understand the number of sets of 

 incongruous solutions of which it is susceptible. To assign, by a direct 

 method, all the solutions of which a proposed congruence is capable, is the 

 general problem which, in the Theory of Numbers, corresponds to the 

 problem of the solution of numerical equations in ordinary algebra. But 

 the solution of the arithmetical problem is attended with even greater 

 difficulties than that of the algebraical one ; and the attention of geometers 

 has been turned with more success to the improvement of the indirect or 

 tentative methods of solution, and to the discovery of criteria of possibility 

 *or impossibility for congruential formulae, than to their direct solution. It is 

 to be observed that, by virtue of a remark already made, the tentative 



