ON THE. THEORY OF NUMBERS. 507 
Multiplying together the two resultants A and A’, we obtain an identity, 
which we shall write at full: 
(Po G1 —Pr Wo) @ + (Po Ya Pa Yo+P 2 1. Ps Va) PY + (Po Is —P 2) "| 
x (Po q2—P» Yo)? +( py Is —P 3 Lo +P, V2— Po %) vy ip, Y—Ps q)y") 
(U1 %— 0%) (Po te +p,xy'+p,v'y+p, yy'y 
+ (9. Pst PoUs— U1 Pa— Yn Pr) (Po ee’ +p, vy +p, vyt+p,yy') --~ + () 
X (Gore +9, cy’ +9, cy +9, yy’) 
+(P,P.—Po Ps) (% awa! +4, xy’ +h w'y + qs yy’). 
Comparing this identity with the equation AA’=nn' ff’ =m F, we find by 
Lemma 4 
% ae Gs —VoPs +P fees YoPi1_P\ P» oot nn' Gh (Q'.) 
The 5th and 6th conclusions relate to the order of the form compounded 
of two given forms. The equation 
AX?4 2BXY + CY? =(aa* + 2bay + cy’) x (av? + 2b'x'y' +c'y’?) 
shows that M divides mm’. But also mm’ divides Mk’. For operating on 
& ae P 
SM Capa ap Caeely, we find 
ody 
the equation just written with —, 
2 2 
aX on! 7 1G oe LT ey 
da? ‘dx dz 
dX dX dX dY ,dXd dY d¥ ) é 
2[a dx dy ls (as at dy 7) +e dx a= ORs: ax 
dX? apex X16 dY* 
dy? dy dy dy* 
Whence AA’, 2BA’, CA’, and consequently Mo, are congruous to zero, 
mod mm’. Similarly Md*=0, mod mm’; i.e. mm’ divides Mk*?, If then 
k=1, i.e. of F be compounded of f and f', M=mm'. (Gauss’s 5th con- 
clusion.) 
Again, if in the congruences (j) we take m'm as modulus instead of mm’, 
we may omit the factor 2 in the second congruence, and may infer that AA?, 
BA’, CA* are all divisible by m'm, i.e. that mm’ divides #17", or fH, when F 
is compounded of f andj’. It is also readily seen that {#1 divides mm’ and 
mm’; whence observing that m=m or 3m, m'=m' or im’, f41=M or 1M, 
according as f, 7’, and F are derived from properly or improperly primitive 
forms, we conclude that if f and f' be both derived from properly primitive 
forms, the form compounded of them is also derived from a properly primitive 
form; but if either f or f be derived from an improperly primitive form, the 
form compounded of them is derived from a similar form. (Gauss’s 6th con- 
clusion.) 
In the transformation of F into fx/’, the form f is said to be taken 
directly or inversely, according as the fraction n is positive or negative. And 
similarly for f’ and n’. 
108. Solution of the Problem of Ne Spans —It appears from the identity 
(I) that if A, B,C, p, p, P, Ps %o %, Ys Ys» _be integral numbers satisfying 
the nine equations (Q), the form (A 7 C) (X, Y)’ will be transformed into 
the product of the two forms (a, b, c) (w, y)° and (a', 5’, c’) (2’, yy by the sub- 
stitution X=p,we' +p, ry +p, ye' +P. Yy's Y=qve' +g, vy +9, ye +4,yy/. 
A = 0, mod mm’. 
