ON TH^ THEORY OF NUMBERS. 311 



or negative determinant are or are not improperly equivalent, it will suffice 

 to change one of them, as (a, b, c), into its opposite (a, —b, c), and then to 

 solve the problem of proper equivalence for (a, — b, c) and (a', b', c"). If 

 it be found that these two forms are properly equivalent, let | T | represent 

 any transformation of the first into the second; then the improper trans- 

 formations of {a, b, c) into (a', b', c') will be represented by the formula 



0, -1 ^i-^i" 



It may happen that two forms are both properly and improperly equivalent 

 to one another ; when this is the case, each of the two forms, and every form 

 of the class to which they belong, is improperly equivalent to itself, i.e. 

 admits of improper automorphics. A class consisting of such forms is said 

 to be ambiguous (^classis anceps — classe ambigiie). An ambiguous form is 

 a form (a, b, c) in which 2b is divisible by a ; if 2b=fj.a, the ambiguous 



1, h 



form is transformed into itself by the improper automorphic q' — 1 » ^"'^ 



if I T I be the general expression of its proper automorphics, all its improper 



automorphics are included by the formula „' _^ X |T|. Every ambiguous 



form belongs to an ambiguous class, and, as we shall presently see, every 

 ambiguous class contains ambiguous forms. 



To complete the theory of equivalence, we shall here briefly indicate the 

 solution of the problem, " To decide whether a given form is improperly 

 equivalent to itself or not, and if it is, to find its improper automorphics." 



When the determinant is negative, it follows from the principle that two re- 

 duced forms cannot be equivalent, that no reduced form, the opposite of which 

 is different from it and is also a reduced form, can be improperly equivalent 

 to itself. Hence the only reduced forms which have improper automorphics 

 are those in which b=0, or 2b = a, or a=c. In the two former cases the 

 reduced form is ambiguous, in the latter it has the improper automorphic 



, * , , and is moreover contiguous and therefore equivalent to the am- 

 biguous form (2a— 2b, a— b, a). These considerations supply a sufficient 

 criterion for deciding whether a form of negative determinant is equivalent 

 to itself or not. If it is, its improper automorphics are given by the formula 



|T|x| r |x|T|~' ; ITj denoting the reducing transformation of the given 

 form, and | r | any improper automorphic of the reduced form. For forms of a 



positive determinant, we observe that if (a„, /Sj, a^), (a^, — jS^, a^), 



('^2it-i' ~P2k-v "'■o) be the period of (a, b, e), the period of (a, —b, c) is 



(^0' -(^2ic-v "■2k-i)> (aai-i. l^2k-2' «-ik-^^ («i'/5o» «o)- For (a, -b, c) 



is equivalent to (a^, —(S^k-v '^2k-i)f because (a, b, c) is equivalent to 

 (°''2k-v ~(^2k-\' "0)5 ^^^ by * known theorem, the period of equations in 

 the development of the second root of (a, b, c) is [a„, — fi2k-v °^2k-il> 



C^iifc-P -p2k-2' a2i-2]»"--[*i'-/5o. ^o]. tbe equation [a„,—^2fc-pa2it-i] 

 occupying an even place in the development ; this period is therefore the 

 period of equations in the development of the first root of [a^,— /32i_i» °^2k-il > 

 i,e.theperiod(ao,— ^2i_i, a2j,_i), (a2j,_i, (3.2k_2, a2ft-2)» • • (*i' /^o. *o) 's the 

 period of (a^, —fiik-v '^2k-i)' O'"' which is the same thing, of («, —b, c). If we 

 now suppose that (a, b, c) is improperly equivalent to itself, it will be properly 

 equivalent to (a, —b, c) ; and these two forms will have the same period. 



