ON THE THEORY OP NUMBERS. 



303 



by combining the equations (A) and (k), and will serve to express the relation 



Solving the 



X„Y,-X,Y„=nF(^,y)=^/(X„,Y„), 



aXA+i(XoY, + X,Y,)+cY„Y,=lF(x,y)=^/(X„Y,) 



for Xj and Yj, we find 



mX^=(T-bV)X,-c\JY„ 

 mY,=aUX„+(T+6U)Y„,- 



or, finally, equating the coefficients of x and y, 



Ta„-U(ia„+cy„), T/3,-U(6/3„+cg„) 



T-6U, -cU 

 aU, T+b\J 



yo' ^0 



(C) 



If we suppose the complete solution of the indeterminate equation 

 T^— DU-=»M" to be known, the formula (C) supplies us with a complete 

 solution of the problem, " Given one transformation of /into F, to deduce 

 all the similar transformations of/ into F." For if we suppose in that 

 formula that T and U denote indefinitely any two numbers satisfying the 

 indeterminate equation, it will appear (1) that every transformation of/into 

 F is contained in (C) ; (2) that every transformation contained in (C) is a 

 transformation of/ into F; (3) that no two transformations contained in (C), 

 and corresponding to different values of T and U, are identical. Only it is to be 

 observed that the transformations (C) are not, in general, all integral. They 

 are so, however, when e, the modulus of transformation, is a unit, a supposi- 

 tion which we have not yet introduced ; i. e. when the forms / and F are 



either properly or improperly equivalent : because — , — , and — are then 



m m m 



T'-LATT T* ATT 



evidently integral ; whence it may be inferred that —^ — and ~ are 



m m 



so too. 



90. Expression for the Automorphics of a Quadratic Form. — To find the 



automorphics of any quadratic form it is sufficient to consider the case of a 



primitive form. Putting then/=F, and taking for 



formation 



1,0 

 0,1 



yo» 



the identical trans- 



, we obtain from the formula (C) the following general exr 

 pression for the automorphics of/ 



y> ^ 



=^x 

 m 



T-b\], -cU 

 aU, T + 6U 



(D) 



where m=l, or 2, according as /is properly or improperly primitive. The 

 nature of this expression for the automorphics depends on the value of D. 

 If D be positive and not square, let us represent the least positive numbers 

 satisfying the equation T— DU^=m" by T^ and V^; we then have, by a 



