302 REPORT— 1861. 



obtain a single transformation of one form into the other, and (3) from a 

 single transformation to deduce all the transformations, the last only admits 

 of being treated by a method equally applicable to forms of a positive and 

 negative determinant. We shall therefore consider it first. The solution which 

 Gauss has given of it (Di<q. Arith. art. 162) depends on principles which 

 are concealed (as is frequently the case in the Disquisitiones Arithmeticas) 

 by the synthetical form in which he has expressed it. We shall not therefore 

 repeat the details of his solution, but shall endeavour to point out the basis 

 on which it rests. 



Let/=(a, 6, c) {x,yy be transformed into F=(A, B, C) (a;, 2/)° by two 



different, but similar transformations, 



ao>/3o 



To' ^0 



and 



71' ^1 



t. e. by two 



transformations of which the determinants are equal in sign as well as in 

 magnitude to the same positive or negative number e. Let also, for brevity, 



Xo=a<,a;+/3oy, Y„=7„a;+S„y, X^=a^x+^,y, Y^=y^x+Z^y, 



so that /(Xo, Y„) =/(Xj, YJ = F(r,?/) ; we have then the algebraical theorem — 

 "The homogeneous functions V{x,y) and X^Yj— XjY„ differ only by a 

 numerical factor, not containing x or y." 



The truth of this theorem is independent of the supposition that the 

 coefficients of the given forms and given transformations are integral num- 

 bers. Its demonstration is implicitly contained in the formulae given by 

 Gauss ; or it may be verified more indirectly by the consideration, that if w 

 be a root of the equation a-\-2bw-\-cw'^=0, we have, simultaneously, 



£l denoting in each case the same root of the equation A + 2Ba+Cn-=0, an 

 assertion which would not be true, if the equal determinants a,^^a~l^oya ^"^ 



ai^j— /Sjyj were of opposite signs. Hence the equation Yq + °o^^ = Vi + ^i^. 



coincides with the equation A + 2BQ + Ci2'=0; i.e. X^Y,— X^Yj, is identical 

 (if we neglect a factor not containing x or y) with F(a:, y). 

 Comparing this conclusion with the identity 



[F(^,y)]^=XXo,Y„)x/(X,YJ= I 



[aX„X, + i(X„Y, + X,Y„)+cY„YJ=-D(X„Y-X,YJ,r * ^^^ 



we obtain a second result of the same kind — 



« The function aX„Xj + i(X„Y, + X,YJ+cY„Y^ differs from V{x,y) only 

 by a numerical factor not containing x or y." 



Let m be the greatest common divisor of A, 2B, and C ; U and T the 

 greatest common divisors of the coefficients of or, xy, and y"" in X^Yj— X^Y^ 

 and aXoXj+6(XoYj + X,Yj4-cY„Yj respectively ; m being a positive integer, 

 but the signs of U and T being fixed by the equations 



F(a;,y)_ XJ-X,Y„ _ aX,X, + &(X,Y, + X,YJ+cY,Y, ,,. 



m U T ' • * ^^ 



which are implied by the two algebraical theorems that have preceded ; the 

 numbers T, U, and m will satisfy the equation T"—DU'=»r, which is obtained 



