ee 
ON THE THEORY OF NUMBERS. 509 
To Po +7,P, +7, p+ Ps = 0, and TQ +7, q +4, oF ™3%5 =1, which IREOVO 
that the two solutions form a fundamental set, 7. e. that the determinants 
PoP: PoPs| — (P,Q, R,8,T, Ul. 
Go i V2 
It only remains to show that the values of A, B, C, which are now sup- 
plied by the equations (Q’), are integral. Operating on the identity (I) with 
ik BE iio and also with a a # we find, by reasoning 
dx’ da dy’ dy” da dx'dy"’ dy” es 2 
similar to that which we have employed to establish the 5th conclusion, that 
2Ann', 2Bnn', 2Cnn', which are certainly integral numbers, are divisible by 
220’ if 34° ona 2S 
is uneven. In the former case A, B, C are evidently integral; in the latter, 
f 26 -- 2b'. ; : rae F 
either spel iar OS UB ONEN y te £- either m or m’ is even, and the quotients 
2B” 20 : 
——,.-——;, whence, again, 
mir mir 
are both eyen, and by é0' if either of these numbers 
of 2Ann', 2Bnn', 2Cnn' divided by 6d’ are ex 
mim 
A, B, C are integral*. 
109. Composition of several Forms.—It will now be convenient to extend 
the definition of composition to the case in which more than two forms are 
compounded. If a quadratic form, F, be changed by a substitution, linear 
in respect of n binary sets, into the product of x quadratic forms, 7, f, . - «fu 
wn 
so that F(X, Y)= Il (a, 47+2ba; y,+¢,y°), we shall say that F is 
‘= 
transformable into f,xf,x ..-f,; andif the determinants of the matrix of 
the transformation are relatively prime, we shall say that F is compounded 
of ff ---Jn . We shall retain, with an obvious extension, the notation of 
Art. 107. The invariant property of the determinant of F supplies the 2 
equations A? fat [af}; from which we infer (1) that D, d,, d,,... are 
to one another as square numbers, (2) that Dx’ is the greatest common 
divisor of the 2 numbers a IIm?. According as the equation A; f; 
= A IIf is satisfied by a positive or negative value of the radical, we 
shall say that f, is taken directly or inversely. Adopting this definition, we 
can enunciate the theorem— 
“If F be compounded of f,,f,,.-f,, and F’ be transformable into f, xf, 
x .. Xf, the forms being similarly taken in each case, F" contains F.” For 
we infer from (2) that D'k?=D, whence A’;=k'A,, or by the Lemmas 2 
and 1 of Art. 107, X'=aX+Y, Y’=yX+6Y, a, B, y, 6, denoting integral 
numbers which satisfy the equation ai—By=’. We thus obtain the equa- 
tion F’ (aX+,Y, yX+éY)=F(X, Y), whence, by Lemma 4, F’ is trans- 
formed into F by | fal | 
* Gauss shows that A, B, C are integral by substituting the values of p, »- +, Q+++5 
IM G192—%oo»_$ (YoPs+Po%s— Ti P2—P 42)» PiP2—PoPs» and observing that the results, 
after division by nn’, are integral. The values of p,... are always obtained free from 
any common divisor by the process in the text; but Gauss has to determine four new 
multipliers 9, 4, 9, 4,, to obtain from the formule (2) the exact values of qo, .. - 5 and not 
equimultiples of those values. M. Schlafli (Crelle, vol. lvii. p. 170) has shown that 
Gauss’s demonstration is connected with a remarkable symbolical formula. 
