514 REPORT—1862. 
a' v' +(B+w)y'=0, mod p; whence, eliminating # and w' from the congruence: 
Y=0, and observing that 2w is prime to M and therefore to p, we find 
yy'=0, mod p. If y is divisible by p, we infer, from the congruence X=0, 
that 2’ is also divisible by p; but the congruences satisfied by w and a’ 
give in this case the contradictory results o=+B, o=—B; i.e. y is not 
divisible by p, and similarly it may be shown that y' is not divisible by 
The congruence yy'==0, mod p, is therefore impossible ; or the represen- 
tation of MM' by (aa’, B, C) is primitive. (2.) Let Q' be the value of /D, 
to which this representation appertains ; and let p be any divisor of M; then 
Q/ satisfies the congruences aa’ X+(B+Q/)Y=0, (B—Q’')X+CY=0, 
mod p; and it will be found, on substituting the values of X and Y, that 
these congruences are also satisfied by w; whence it follows, since either 
X or Y is prime to p, that Q'==w,mod p. Similarly, if p be a prime divisor 
of M', Q'=.w', mod p; or ©’ is a solution of the congruence Q?=D, mod 
MM’, comprehending the solutions w and w’. Hence Q'==Q, mod MM’, and 
2 
the form (ane, Q, a is equivalent to (aa’, B, C), because either of 
them is equivalent to (nr, QO; — . The equivalence of all the forms 
a? —D 
is therefore demonstrated. 
included in the expression ( MM’, Q, wr 
It will be seen that Dirichlet’s method may be applied to the composition 
of any number of forms, and that the theorems of Art. 109 present them- 
selves as immediate consequences of his definition of composition. 
112. Composition of Classes of the same Determinant.—We shall now con- 
sider more particularly the composition of classes of the same determinant D. 
We represent these classes by the letters f, ¢, . . - , and we use the signs of 
equality and of multiplication to denote equivalence and composition respec- 
tively *, The following theorems are then immediately deducible from the 
six conclusions of Art. 107, and from the formule of Art. 110. 
(i.) “If f be a properly primitive class, fx ® is of the same order as ®.” 
(ii.) “A class is unchanged by composition with the principal class.” 
In consequence of this property, it is sometimes convenient to represent the 
principal class by 1. 
(iii.) “The composition of two opposite+ properly primitive classes pro- 
duces the principal class.” 
If, then, f denote any properly primitive class, we may denote its opposite 
by f-!, and we may write fx f-!=1. 
(iv.) “If f be a given properly primitive class, and ® any given class, the 
equation F x f=® is always satisfied by one class, F, and by one only ; viz, 
by the class F=® x f-!.” 
(v.) “If®,, ,,..be all different classes, and f be a properly primitive 
class, fx ©,, fx ®,, . . are all different classes,” 
(vi.) «A properly primitive ambiguous class produces by its duplication the 
principal class ;” for an ambiguous class is its own opposite, Conversely, if 
¢°=1, i.e. if be a class which, by its duplication, produces the principal 
class, ¢ is a properly primitive ambiguous class; for we find ¢*x @-1=¢71, 
whence ¢=9~!, or @ and its opposite are properly equivalent. 
_ * Gauss uses the sign of addition instead of that of multiplication; thus /X¢ is /+¢ 
in the Disq. Arith., and f” is nf. The change appears to have been introduced by his 
French translator, and to have been acquiesced in by subsequent writers. 
‘+ Two classes which are improperly equivalent are called opposite, because they con- 
tain opposite forms (see Art, 92). ; ; 
