310 • REPORT 1861. 



And, since the same complete quotient will occur in an even or in an un- 

 even place alike in each development, it will be a root of the same denomina- 

 tion in the quadratic equation determining it in each development. The 

 period of equations will therefore be precisely the same for each develop- 

 ment ; and the same equation may be taken as the first equation of each 

 period. Consequently the periods of (a„, b^, aj, (A;,, B„, AJ are identical. 

 Two forms therefore are or are not equivalent, according as their periods 

 are or are not identical. To obtain the transformations of (a^, b^, aj into 

 (Ao, B„, Aj), when these two forms are equivalent, let the complete quotients 

 in the development of the first root of [a„, b^, aJ be Wp w^ , and let 



the convergent immediately preceding w„+j be — . Similarly, let ii„^j and 



— be a complete quotient and a convergent in the development of the first 



root of [ A(,, Bq, aj. Then, if w^=tiji (where /i^M, mod 2), all the trans- 

 formations of («(,, ip, Oj) into (A„, Bq, AJ are comprised in the formula 





xlT|x 



Pm-i» "m 

 Qm-i» Qm 



1 



j T I denoting any automorphic of the form corresponding to the equation of 

 which w^^, or ^jj+j is a root. 



It should be observed that a reduced form is always a form of its own 

 period. To prove this, we remark that reduced forms are of two kinds; 

 they are either such as («„, (i^, aJ, where the first root of [«„, /S^, aJ is 

 positive, or such as (aj, — /3j, a^), where the first root of [a^, — /S,, a,] is 

 negative. Now a reduced form such as (a^ j3^, a^) is evidently a form of 

 its own period, for the equation [a„, /3„, aJ is itself an equation of the 

 period in the development of its first root. And a reduced form such as 

 (ttj, — /3j, a^) is also a form of its own period. For if we develope the second 

 root of [a^, jSy a^], we obtain a period of equations of which [a^, /3i, a^J 

 is itself one. Let [a,, (i.^, a^'] be the equation immediately Ibllowing 

 [aj, /3j, a^] in this period ; then [a^, /3i, aJ is an equation occupying an even 

 place in the period of equations arising from the development of the first 

 root of [a^, ft^, a,^'], and consequently {a^, — /3j, aJ is a form in the period of 

 (a^, jS^, a.^) ; i. e. it is a form in its own period, because it is equivalent to 

 (a„ fi„ a,). 



It follows from this that no reduced form can be equivalent to a given 

 form, unless it occur in the period of that form. 



The inequalities satisfied by the roots of any equation of a period give 

 rise to certain inequalities which are satisfied by its coefficients. These in- 

 equalities (which are not all independent) are, 



(i) [aJ<2VD; [/3J<VD; K]<2VD. 



(ii) VD-[/3J<[aJ<:VD-F[/3J; 



(iii) VD-[/3J<CaJ<VD+[/3J. 



The same inequalities are, of course, satisfied by the coefficients of a reduced 

 form ; its middle coefficient is, moreover, positive. And conversely, every 

 form whose middle coefficient is positive and whose coefficients satisfy these 

 inequalities is a reduced form. 



94'. Improper Equivalence — Ambiguous Forms and Classes. — If it be re- 

 quired to find whether two forms (a, b, c) and (a', 6', c') of the same positive 



