294 



EEPORT — 1861. 



transformation, and the former one of the finite number of transformations 

 included in the formula 



t^V *1,2' "1,3- 

 0, 0, /i3 . . 



l.« 



"2,71 



3,n 



0, 0, /.„ 



(C.) 



in which /i^ X ju.^ X ...... X n,=ci, and 0^,j</J; (Phil. Trans, vol.cli. p. 312). 



To determine, therefore, whether the form /"^ can be transformed into /^ by 

 a transformation of modulus a, we apply to/^ all the transformations (C.) in 



succession, obtaining a series of transformed forms ^^, ^,, If none of 



the forms are equivalent to f^,f^ cannot contain^,; but if one or more of 

 the forms (p be equivalent tojf^,f^ will contain/'^, and all its transformations 

 into/j may be obtained as soon as the transformations of the forms <j) intoy^ 

 have been determined. This is tiie method proposed by Gauss for binary qua- 

 dratic forms (Disq. Arith. arts.213, 21 4) ; it is evidently of universal application ; 

 but the following modification of it possesses a certain advantage. Instead of 

 representing |T| by the formula |T| = |a|x|vl, we may employ the formula 

 I Tl = |z; 1 X |a], in which |t;| is a unit-transformation as before, and la|i9 

 one of the transformations included in the formula (C), where, however, the 

 inequality 0<^; ■</Xj is to be replaced by 0<A- •<"•; the transformations 

 thus defined we shall call the transformations (C). If we now apply toy^ 

 the inverse of each transformation included in (C), we shall obtain a 



series of forms (p^, (p^, (p^ of which the coefficients will not necessarily 



be integral numbers, because the coefficients of the inverse transformations 



are not necessarily integral. If all the forms (p^, cp^, be fractional, or if 



none of those which are integral be equivalent toy,,yj cannot contain y; 

 but if some of them be integral, and equivalent to/^, it is plain that/, con- 

 tains y^, and that all the transformations of/, into/ may be obtained by 

 means of the transformations of/, into those forms (p which are equivalent 

 to it. The advantage above referred to consists in the circumstance that 

 the rejection of the fractional forms <p diminishes the number of the problems 

 of equivalence which must be solved to obtain the complete solution of the 

 question proposed (compare Disq. Arith. art. 284, and note). 



81. Aiitomorphic Transformations — The unit-transformations by which a 



form passes into itself are the automorphics of the form ; thus 



2,3 

 1,2 



IS an 



automorphic of a;^ — Zy^. When every invariant of a form is zero, the form 

 may pass into itself by transformations of which the modulus is different 

 from unity; for example, x^—^xy+'^^y^, a binary quadratic form of which 



the discriminant is zero, passes into itself by the transformation 



3,2 

 1,2 



of 



which the modulus is 4. In like manner it is to be observed that when two 

 forms of the same sort have all their invariants equal to zero, it may happen 

 that each of them passes into the other by transformations of which the 

 modulus is not a unit. But in this Report we shall have no occasion to 

 consider these exceptional cases, whether of equivalence or of automorphism, 

 and we shall therefore employ these terms with reference to unit-transforma- 

 tions exclusively. If | T, | and | T^ | be automorphics of a form/ | T, | x | T, | 

 and j T, I X [ Tj I are also automorphics of/; so that, in particular, every power 



