ON THE THEORY OF NUMBERS. 293 



of \a\, which will be a transformation of the same type as \a\, will have all 

 its coefficients integral numbers ; so that in this casey^, which contains^,, is 

 also contained in it. When each of two forms is thus contained in the other, 

 they are said to be equivalent. Ify^ contain y,, andy^ cowia'm J\, f^ will con- 

 tainy; for if^ be changed into/^ by the transformation \a\, and f., into y 

 by the transformation \b\, it is clear that y will be changed into^ by a 

 transformation |T|, of which the constituents are defined by the equation 



'^i,j=%Aj+%J'^.j+ +^»V 



The transformation |T| is said to be conipomided o^ the transformations \a\ 

 and \b\, and this composition is expressed by the symbolic equation 



|Ti=!a|x|^'|, 



in which it is to be observed that the order of the symbols | a \ and 1 6 1 is not, 

 in general, convertible. When, in particular,yj is equivalent tof„, and f^ to 

 f^, y is equivalent toy^ ; i. e. forms which are equivalent to the same form 

 are equivalent to one another. All the forms, therefore, which are equivalent 

 to one and the same form, may be considered as forming a doss. All the 

 invariants of any two equivalent forms have the same values ; but it is 

 not true, conversely, that two forms which have the same invariants are 

 necessarily equivalent. Nevertheless it may be conjectured that all forms 

 of the same sort (i. e. of the same degree, and the same number of indeter- 

 minates), the invariants of which have the same values, distribute themselves 

 into a finite number of classes ; and this conjectural proposition is certainly 

 true for binary forms of all orders, and for quadratic forms of any number of 

 indeterminates. It is readily seen that if a number be capable of representa- 

 tion by one of two equivalent forms, it is also capable of representation by 

 the other; and that the number of representations is either finite for both, or 

 infinite for both, and, if finite, is the same for eacli. The general problem, 

 therefore, of the representation of numbers (which we have already enunci- 

 ated) suggests naturally tiie following, which we may terni that of the 

 equivalence of forms: "Given two forms (of the same sort), of which the 

 invariants have equal values, to find whether they are, or are not, equivalent, 

 and if they are, to assign all the transformations of either of them, into the 

 other." The number of transformations may be either finite or infinite ; if 

 finite, the transformations themselves, if infinite, general formulae containing 

 them, are required for the complete solution of the problem. 



When y is not equivalent to, but contains y, the invariants ofy are 

 derived from those ofy by multiplication with certain powers of the 

 modulus (i, e. of the determinant) of the transformation by which y is 

 changed intoy ; viz. if I be an invariant ofy, and if e and ?« be the orders 



mi 



of I, and ofy ory, the corresponding invariant ofy is a " I, a denoting 



mi 

 the modulus of transformation, and the number — being always integral. 



This observation enables us to enunciate with precision a problem in which 

 the preceding is included : "Given two forms, of which the invariants have 

 values consistent with the supposition that one of them contains the other, 

 to find whether this supposition is true or not, and, if it is, to find all the 

 transformations of the one form into the other." But, in every case, the 

 solution of the problem in this more general form may be made to depend 

 on the solution of the problem of equivalence. For every transformation of 

 order n, and modulus «, arises, in one way and in one only, from the com- 

 position of two transformations |fl| and \v\, of which the latter is a unit- 



