506 REPORT—1862. 
In their enunciations we represent by X, Y, «, y, four functions, homo- 
geneous and linear in respect of each of the v binary sets, é, ,, &,n,,..-&, in 
by | ‘ | and | 4 | the matrices composed of the coefficients of X, Y and a, y, 
respectively; by (P, Q, R), (P’, Q’, R’) quadratic forms of which the coefii- 
cients are any quantities whatever ; and by & an integral number. 
(ii.) “If X, Y, x, y, satisfy the n equations included in the formula 
dX dY dXdY_, (dx dy dxdy 
dé, In, dy; dE, (aan dy; AS" 
the matrices | a | and | satisfy the equation 
2? 
ee 
(iii.) “ The greatest numerical common divisor of the 7 resultants 
aX d¥_ aX ay 
dé, dy, dn; 4; 
is equal to the greatest common divisor of the determinants of ae 
(iv.) “If the n resultants of X and Y be not all identically equal to zero, 
the equation PX*+2QXY+4RY*=P’X*+2Q’XY+RY? implies the equa- 
tions P=P’, Q=Q’, roe ee ¥ 
107. Gauss’s Six Conclusions.—Let F, f, f' represent the forms (A, B, C) 
(X, Y)*, (4,5,¢) (a, y)*, (a, 5', ¢) (a, yf, of which the determinants are 
D,d,d'; let also M, m, m' be the greatest common divisors of A, 2B, C, of 
a, 2b, c, and of a’, 2b',c’; $4, m, m’, the greatest common divisors of A, B, C, 
of a, 6, c, and of a’, b,c’, respectively. Supposing that F is transformed in f x f’ 
by the substitutionX=p, vw'+p,cy'+p,y+p,yy',Y=q,cx'+q,ry'+9q,vy 
+q,yy', let us represent the two resultants 
dX dY_dXdY = dX dY dX dY 
dxdy dy dx dx' dy dy’ dx' 
by Aand A’; the six determinants of the matrix ie aj = 3) (taken in 
0 41 72 13 
their natural order) by P,Q,R,S,T,U; the greatest common divisor of 
these six numbers by /, and the greatest numerical common divisors of 
A and A’ by 6 and @’, so that (Lemma 3) & is the greatest common divisor of 
é and é’. 
From the invariant property of the determinants of F, f and f’ we infer 
amet Fi ara f°, Di?=d'm’*, DP? =dm*. 
Hence the quotients ss and ¢ are squares. (Gauss’s Ist conclusion.) Also 
D divides d'm? and dm. (Gauss’s 2nd conclusion.) But & is the greatest 
common divisor of é and 6’; therefore Dk’ is the greatest common divisor of 
d'm? and dm?. (Gauss’s 4th conclusion.) Let oon ee and let the 
signs of x and n' be so taken that A’=n'f, A=nf"; these two equations are 
equivalent to the six following :— 
ae ars a a owes ee res em) 
(Gauss’s 3rd conclusion.) 
