512 REPORT—1862. 
by the equations n= / 2 [eo 2 and, representing by p the greatest 
common divisor of an’, a'n, bn'+b'n, we obtain A, B, C from the system 
Ante 
an __ ab’ 
bei vila 
Ghiptegs mod A. 
iP p 
bn'+b'n pee 66'+Dnn' 
B B j 
2 —— I) 
ot A 
These formule, which are applicable to every case of composition, and are 
therefore more general than the analogous formule given by Gauss (Disq. 
Arith., art. 243), are due to M. Arndt*, who has also given an independent 
investigation of them, though our limits have compelled us here to deduce 
them from Gauss’s general solution of the problem of composition. That 
(A, B, C) is transformed into (a, b,c) x (a b' c!) by the substitution 
3 ej beg A ee vy! + b6—Bn se Wada ee +6 hy. 
pe a a ad 
LYS an' vy’ +a'n a'y+(b'n+bn')yy', 
may be inferred from the vdfues of p,,...9,, .++3 or may be verified 
directly by observing that 
pl AX + (B+ VD)Y)=[ae+(b+4+n¥o D)y] x [a'a'+(0'+2' / D)y']. 
111. Composition of Forms—Method of Dirichlet——Lejeune Dirichlet, in 
an academic dissertation (“ De formarum binariarum secundi gradus com- 
positione,” Crelle, vol. xlyii. p. 155), has deduced the theory of the composi- 
tion of forms from that of the representation of numbers. The principles of 
this method are applicable to any case of composition; but Dirichlet has 
restricted his investigation to properly primitive forms of the same deter- 
minant D. Let (a, 6, c), (a’, b,c’) be two such forms; let M and M’ be two 
numbers prime to 2D, and capable of the primitive representations M=am? 
+2bmn-+en*, M'=a'm? + 2b'm'n'+ cn", by the forms (a, b,c) and (a',b’, c’) 
respectively ; also let these representations appertain to the values w and w’ 
of 7D, so that w°=D, mod M, w?=D, mod M’, and so that the forms 
2 
(a, b,c), (a', b',c') are respectively equivalent to the forms (at o,— V ”) ; 
* Crelle’s Journal, vol. lvi. p. 64. In the new edition of the Disq. Arith. (Géttingen, 
1863), a MS. note of Gauss is printed at p. 263, containing the congruences by which B 
is determined in the case of the direct composition of two forms of the same determinant. 
The account of the theory of composition in the preceding articles (106-109) differs 
from that in the Disq, Arith. (arts. 284-248) chiefly in the use which is here made of the 
invariant property of the determinant. <A different mode of treatment of Gauss’s analysis 
is adopted by M, Bazin, in Liouville, vol. xvi. p. 161. 
In Arts. 108 and 110 we have endeavoured to supply the analysis of a problem which 
Gauss, as is not unusual with him, has treated in a purely synthetical manner (Disq. 
Arith., arts. 236 and 242, 243) ; and it is for this reason that we have introduced the con- 
sideration of fundamental sets of solutions of indeterminate systems, which are not ex- 
plicitly mentioned in the Disq. Arith. It is perhaps singular that Gauss does not employ 
the identity PU—QT+RS=0; it was first given by M. Poullet Delisle, in a note on Art. 
235 in his translation of the Disq. Arith. 
