ON THE THEORY OF NUMBERS. 



307 



either [26] =a, or a=c; in which cases only one of the two forms satisfies 

 the special conditions. In every case, therefore, there exists one, and only 

 one, reduced form in each class. 



To obtain the reduced form equivalent to a given form, we form a series 

 of contiguous forms, beginning with the given form and ending with the 

 reduced form (Disq. Arith. art. 171). Two forms of the same determinant, 

 (a, b, c) and {a!, b', c'), are said to be contiguous when c=a', and b+b'^0. 



ent; for ifb+b'=na', the 

 0,-1 



1, 



mod a'. Two contiguous forms are always equivalt 



former passes into the latter by the transformation 



Let, then, («„, b^, a^) be the given form of det. — A, which is supposed not 

 to satisfy the general conditions for a reduced form. Let b^+b^^fji^a^, — b^ 

 denoting the minimum residue of b^, moda,, so that f^^il— Qi ; and let a^ 



7 2,. 



represent the integral number -^ . The form (oi, ^i, Oa) ^'^^ ^^ ^°^' 



tiguous, and therefore equivalent, to (a,,, b^, a,). Let a third form, (a^, b^, a,), 

 be similarly derived from (a^, b^, a^, and let the series of contiguous 

 forms (a^, b^, a^), (cij, b^, a.^), (a^, b^, a^), . . . he continued until we arrive 

 at a form (a„,6„, a„^i), in which c„^,>ia„. We shall certainly arrive at 

 such a form, or we should have a series of numbers a^, a^, a^, . . . . all re- 

 presented by the form («„, b^, a J, and yet continually decreasing for ever ; 

 whereas a form of negative determinant can acquire only a finite number 

 of values inferior to any given limit. The form (a„,6„, a„^j), in which 

 '^n—"n+v satisfies the general conditions for a reduced form. For by the law 

 of the series of forms [26, J ;^a„ ; and since a < a«+i, we have also 



Again, the process can always be terminated in such a manner as to give a 

 form satisfying the special conditions for a reduced form. If fl„=«n+i, and 

 6„ is negative, instead of stopping at the form (a„, J„, a^), we continue the 

 process one step further and obtain the reduced form (o„, — b^, a„). If 

 — 26,^=a,^, instead of the form («„, b,„ a„4.,),we take the form (a^i—b^, a^+j), 

 which is contiguous to («„_i, 6„_],a„),for the concluding form of the series. 

 The transformation | T,i | by which (a^, b^, Oj) passes into the equivalent 

 reduced form («, 



where 



X 



0,-1 



l» Hn 



or if we represent the successive cobvergents to the continued fraction 



-i 1 



/'a' 



Hi 



Wo '*! Wu 



Qj=/ii/j2~l> . . . , we may express 



|Tnl = 



T„ I by the formula 



"n-V "n 



x2 



