508 REPORT—1862. 
In order, therefore, to find a form, F, compounded directly or inversely of two 
given forms of which the determinants are to one another as two squares, we 
have to find eleven integral and two fractional numbers, satisfying the equa- 
tions (Q) and (Q’), in which a, 6, ¢, a’, 6', c', and the signs of n and x’, are 
alone given; the numbers p, P, P.Ps> %o 4%, V2 Yq» being further subject to the 
PoP. P2Ps| ave to admit of 
a f 0% Ws Ws ~ 
no common divisor. To determine n and n', we observe that the six deter- 
minants satisfy the identical relation PU—QT+RS=0; from which we 
infer, first, that P, Q, R—S, R+8, T, U must be relatively prime, 
if P, Q, R, 8, T, U are to be so; and secondly, substituting for the determi- 
nants their values given by the first six of the equations (Q), that dn? =d'n?. 
Denoting by a’ and 6 the greatest common divisors of P, R—S, U and of 
Q, R+58, T, so that 6 and 6’ are relatively prime, we have evidently 
condition that the determinants of the matrix 
Uy 
6 — aes , ; 
n= +—, n=+-—; the positive or negative signs being taken according as 
v1) Mm 
f and f enter the composition directly or inversely ; and the absolute values 
of 6 and a’ being determined by the equation 6? d'm?=6" dm". The fractions n 
and w’ being thus ascertained, the values of P, Q, R, S, T, U are known from 
the equations (Q): these values are all integral: for P,Q, R—S, R+5S, T, U, 
this isevident from the equations (Q), and may be proved for R and § by 
means of the identity PU-QT+RS=0. We have next to assign such 
values to the constituents of the matrix | oP: P2 Ps , that its determinants 
may acquire the known values of P, Q, R, ST, U. To do so, it is sufficient * 
to obtain a fundamental set of solutions of the indeterminate system, 
v,U —xv, T+a, S=0 
—«,U +a, R—v,Q=0 s 
, To Be” ae BAO 7 “Wer De ee) ee 
—w#, S+a,Q-—x, P =0, 
which is equivalent to only two independent equations. From the skew 
symmetrical form of the matrix of this system, it appears that if 6, 6, 6, 6, be 
any multipliers whatever, any four numbers (a, «, «, 7,) proportional to 
6,P+6,Q+46,R 
—4,P 46,849. T = 
—§,Q—6,8 +6,U . . . . . . ( ) 
—),8 7 To 
will satisfy the system (S), and in addition the equation 
6, +6, 4,46, 7,+4,7,=0. 
Assigning, then, to 6, 6, 4,4, any arbitrary values whatever, let ¢, 9, 9,9, be 
four numbers relatively prime, and proportional to the four numbers (2); let 
also 7, Qo +7, 9, +72 Go+7,9,=1; and employing x, 7, 7,7, in the place of 
§,9,4,4,, let us represent by p,p,p,p, the solution of (8) thus obtained. 
We have thus two solutions of (8), satisfying respectively the relations 
2 
* For a solution of the general problem, “ To find all the matrices of a given type, of 
which the determinants have given values,” see a paper by M. Bazin, in Liouville, vol. xvi. 
p. 145; or Phil. Trans. vol. cli. p. 302. For the definition of a fundamental set of solu- 
tions of an indeterminate system, see 2bid. p.297. It may be observed that the analysis 
‘of Gauss, which is exhibited in the text, is applicable to any matrix of the type 
2X (n+2). ; 
