ON THE THEORY OP NUMBERS. 299 



(ii.) " If M admit of a primitive representation by the form (a, b, c) apper- 

 taining to the value £i of VD, mod M, the two forms (a, b, c) and 



are equivalent; and conversely, if these two forms are equivalent, M admits 

 of a primitive representation by (a,b,c) appertaining to the value £i of 



-/D, mod M." 



To establish the first part of this theorem, we observe that the assertion 

 that M admits of a primitive representation by the form (a, b, c) appertaining 

 to the value £i of \/D, mod M, implies the existence of four numbers 

 m, n, fi, V, satisfying the equations 



mv — w^ = l, 



am"+25mw+cw"=M, (k) 



anm-\-b\mv\-nfi'\-\-cnv=-Q,. _ 



If, therefore, we apply to (a, b, c) the transformation ' " , the resulting 



form will have M and fl for its first and second coefficients respectively ; 



^"— D 

 its third coefficient will therefore be — —. — , because its determinant must be 



M, a, — =-=— J are equivalent. And, 

 conversely, the equivalence of the two forms (a,b, c) and 



implies the existence of a transformation ' ^ of (a, b, c) into 



(M,a,^) = 



i.e. it implies the existence of four numbers m,n,fx, v, satisfying the equa- 

 tions {k) ; or, finally, of a primitive representation of M by {a, b, c) apper- 

 taining to the value 0, of VD, mod M. 



If (A, B, C) be a form equivalent to a form (a, b, c) by which 



M = arn^ + 2bmn + cri" 



is represented, and if *'^ be a transformation of (A B C) into (a, b, c), it 



is clear that 



(A, B, C)(am+(5n, ym + Sny=(a, b,c)(m,ny='M.. 



Two such representations of M by equivalent forms are called corresponding 

 representations ; and we may enunciate the theorem, " Corresponding repre- 

 sentations of the same number M by equivalent forms appertain to the same 

 value of the expression VD> mod M," the truth of which is evident from the 

 nature of the function Am;u + B[m»-f-n/:i] + Cw»', which is a covariant (in 

 respect of m, n and }i, v) to Ax^ -\-2^xy -\- Cy". 



To obtain, therefore, all the primitive representations of a given number 

 by a given form (a, b, c), we investigate all the values of ihe expression >JT>, 

 mod M. If £2i, iij,. . . . be those values, we next compare each of the forms 



( 





