312 REPORT — 1861. 



which we shall represent by (/?„, qo,p^), (p^, 5'i, Jfj)' &c. Ifj9^, 9'vi'x+i) ^^ 

 any form of this period, the associate of (p/^, q^^, P)^^i), i- e. the form 

 Ot'a+i' 9k^ P\)' ^''' ^^"^ ^^ ^ form of the period, and the indices of these two 

 forms in the period will differ by an uneven number, because the signs of the 

 numbersjo^,/>;^^j,. ..are alternate. From this we can infer that there will 

 be two different forms in the period, each of which will be immediately pre- 

 ceded by its own associate; so that the type of the period will be 



(Po^ 9o^Pi)y (Pi^ QvPoJi • • • • (Pk-v %-vPk\ 



(Pk> %-vPk-l)' (Pk-V %-vPk-2\ • • • (Pl. ffo'i'o). 



where for simplicity we have supposed that {p^, q^^p^ is one of the two 

 forms which is preceded by its associate; the other is (jt>;j, qi--i'> 2^k-\)- 

 These two forms are ambiguous, for it follows from the contiguity of each 

 form to that which precedes it, that 2«/„ ss 0, mod p^ ; 2gj_ j=:e 0, mod jfj^.. We 

 arrive therefore at the conclusion that the period of every ambiguous class 

 contains two ambiguous forms ; either of which enables us, as in the case of 

 forms of a negative determinant, to obtain all the improper automorphics of 

 any form of the class. 



Gauss has shown (Disq. Arith. art. 164), by an analysis which it is not 

 necessary to explain here, that ify contain F both properly and improperly, 

 an ambiguous form contained iny, and containing F, can always be assigned. 

 This theorem comprehends the result which we have incidentally obtained in 

 this article, that every ambiguous class contains ambiguous forms. (See also 

 a note by Dirichlet, in Liouville, New Series, vol. ii. p. 273.) 



95. The important theorem, that for every positive or negative determi- 

 nant the number of classes is finite, is a consequence of the theory of reduc- 

 tion. To establish its truth, it is sufficient to employ the reduction of Lagrange 

 (art. 92), which is applicable to forms of a positive determinant having inte- 

 gral coefficients no less than to forms of a negative determinant, and which 

 shows that in every class of forms of determinant D there exists one form at 

 least the coefficients of which satisfy the inequalities [26]<[a], [2/>]<[c]. 



These inequalities give, if D be negative, ac^ D, [^]-^V — ; and if 



3 3 



D be positive, [ac]<D, [5]<V_. The number of forms whose coeffi- 



5 



cients satisfy these inequalities is evidently limited; therefore, dfortiori,ihQ 



number of non-equivalent classes is finite. 



To construct a system of representative forms of det. D, we have only to 

 write down all the forms of det. D whose coefficients satisfy the preceding 

 inequalities, to which we may add [«]^[c]. If the determinant be 

 negative, it only remains to reject the forms which do not satisfy the special 

 conditions; if it be positive. Me must examine whether any of the forms 

 which we have written down are equivalent; and, if so, retaining only one 

 form out of each group of equivalent forms, we shall have the representative 

 system required. 



A few particular cases of the theory merit attention from their simplicity. 



If D= — Ij there is but one class of forms, represented by x'^-\-if ; and by 

 the theorems of arts. 87 and 90, the number of representations of any uneven 

 (or unevenly even) number by the form x°-\-if is the quadruple of the ex- 

 cess of the number of its divisors of the form 4«. + 1, above the number of its 

 divisors of the form 4« + 3. (See .Jacobi in Crelle's Journal, vol. xii. p. 169 ; 

 Dirichlet, ibid. vol. xxi. p. 3. In counting the solutions of the equation 



