ON THE THEORY OP NUMBERS. 305 



which the determinant is not a positive square, by the following process 



(Crelle, vol. xxiv. p. 324'). If "'' L be any rational autoniorphic of/, we 



have evidently 



a(ai,f + 2bxi/+cf)= lax+(b+ VD)y] lax+(b~~ VD)y], 



= l(ax+ lb+ VD]y>+(ai3+ [i+ VD]%] X 



[(«a+ lb- VD]y> + (ai3+ [6- VD]S)^], 



an equation which, for brevity, we may write 



and which implies one or other of the two following systems : — 



{9"\ n n =P P • Pi=Ma- P^— ^2 



If (1) be the system which is satisfied by 



a., ft 



y, d 



,Iet|i=l[T + UVD], 



V 1 



-^=— [T— UVD]. T and U denoting rational numbers, and m still repre- 



2 



senting the greatest common divisor of a, 2b, c. These assumptions are 

 legitimate, because^ and ^ contain no irrationality but VD, and are con- 

 jugate with regard to VD- Substituting in the equations 



Pi 9i »» 



for p^,p^', q^, q^\ Pj, P^; Q^, Q^; the expressions which these letters represent, 

 and equating the rational and irrational parts, we find 



a,/3 

 y, ^ 



1 

 :— X 



m 



T-iU, -cUi 

 aU, T+iU I 



In this expression T and U satisfy the equation T'— DU'^=m^, because 



/JjP2=PjP2. From this we infer thata^ — /37=1 ; further, if we now introduce 



the condition that a, /3, y, § are to be integral and not merely rational 



a lb c 

 numbers, it will follow, because y, I— a., — /3 are integral, that — U, — U, — U 



m m m 



are also integral; i.e. that U itself, and consequently T, is integral; so that 



the formula at which we have arrived coincides exactly with the formula 



(D). The system (2), treated in a similar manner, leads to the conclusioa 



a^ — y/3=— 1 ; whence it follows that that system can be satisfied by no 



proper automorphic of/. 



This method, as Diriciilet observes, has the advantage of putting in a 



clear light the difference between proper and improper automorphism. A 



proper automorphic changes each of the two factors, into which the form 



may be decomposed, into a multiple of itself by a complex unit of the form 



— [T+U VD] ; whereas improper automorphics, which only exist for parti- 

 cular kinds of forms, change each factor into a multiple of the other. A 

 1861. X 



