ON THE THEORY OF NUMBERS. 309 



(Ag, Bj, A,), the corresponding roots of the quadratics 



i.e. those which are connected bv the relation w=i- — —, are of the same, 



a + pit 



or of opposite, denominations, according as the equivalence is proper or im- 

 proper. Let the first root of the equation [cr^,, 6^,, Cj] be developed in a 

 continued fraction, of which all the integral quotients are positive except 

 the first, which has the same sign as the root. In this process we obtain a 

 perfectly determinate series of transformed equations, each having a com- 

 plete quotient of the development for its first or second root, according as it 

 occupies an uneven or an even place in the series, counting from the pro- 

 posed equation inclusive. The complete quotients eventually form a period 

 of an even number of terms ; there exists therefore a corresponding period 

 of transformed quadratic equations, which will be of the type 



Cao» /3o> ajj [.a-v ^v ^2]' C*2' Z^^, a^'], [a2fc-i' ?ik-v «o]' 



Every equation of the period has one of its roots positive and greater than 

 unity, the other negative and less in absolute magnitude than unity ; and if 

 we suppose (as we shall do) that we begin the period with an equation 

 occupying an uneven place in the series of transformed equations, the positive 

 root of any equation of the period will be its first or second root, according 

 as it occupies an uneven or an even place in the period. 



To apply what has preceded to our present problem, we require the fol- 

 lowing lemma (see sect. 2 of Dirichlet's memoir, or M. Serret in Liouville, 

 vol. XV. p. 153). 



" If w and ii be two irrational quantities connected by the relation 



w=^iX-— -, where a, /3, y, ^ are integral and aS— /3y= + l, the develop- 



ments of w and il in a continued fraction will ultimately coincide, and the 

 same quotient will occupy an even or an uneven place in both developments 

 alike, if a2— /3y= + l, but an even place in the one, and an uneven place in 

 the other, if aS — (^y'=- — !•" 



A quadratic form (»„, ji^, a J of positive determinant, is said to be reduced* 

 when the roots of [ao'/^o' ''^J ^""^ ^^ opposite signs; the absolute value of 

 the first root being greater, that of the second less than unity. A series of 

 reduced forms equivalent to any proposed form (a^,bg,aj can always be 

 found. For, if the first root of la^, b^, aj be developed in a continued frac- 

 tion, and if its period of equations (beginning with an equation occupying 

 an uneven place in the series of transformed equations) be represented as 



before by [a„, /3„, aj, [a^, fi,, aj [aai-P f^2k-i> «»]> the forms 



(a„ /3„, aj, (a^, -/3„ a,), (a^i-,, -l^^jc-v «o) ^iH be all reduced and 



all equivalent to (a^, b^, aj. These forms, so deduced from the develop- 

 ment of the first root of the equation [a„, b^, cj , we shall term the period of 

 forms equivalent to (a^, b^, a J, or, more briefly, the period of («„, b^, a J. It 

 will be seen that each form of the period is contiguous to that which precedes 

 it, and that the first is contiguous to the last. 



We can now obtain a complete solution of our problem. If («;,, &„, Oj) 

 and (Ag, B„, A J are equivalent, the first roots of [a^, bg,a^'] and [A„, B„, Aj 

 will be corresponding roots, and the developments of these two roots will 

 ultimately coincide, giving one and the same period of complete quotients. 



* These reduced forms are not to be confounded with the reduced forms of the last article. 



