i 
ON THE THEORY OF NUMBERS. 519 
be negative, the three forms will belong to different classes; and there will 
be three times as many properly as improperly primitive classes. From this 
statement, however, we must except the determinant —3, which has one 
properly and one improperly primitive class. 
It will be found that the properly primitive class or classes, into the double 
of which a given improperly primitive class can be transformed, and which in 
turn can be transformed into the double of the given class, are also the class 
or classes which compounded with the class { 2, 0, a produce the given 
class. Thus every improperly primitive class is connected either with one or 
three properly primitive classes (see Art. 98, note, and Art. 118). 
114. Composition of Genera.—Let f and f' be two properly primitive 
classes of det. D, m and m’ two numbers prime to one another and to 2D, 
and represented by f and f’ respectively; then mm’ is represented by fx’. 
Hence the generic character of fx /' is obtained by multiplying together the 
values of the particular characters of f and f’. For those generic characters 
which are expressed by quadratic symbols this is evident, since 
“) (=) (=) 
— j=—[ — ) Xi — };5 
P P 
and it is equally true for the supplementary characters, since it will be found 
that 
mm!—1 m'—1 m2m!2—1 m*2—1 m/2—] 
m—1 
eee ta (rt)? (1) 8 ti) ® x(—1) 
The genus I’, in which fx/’ is contained, is said to be compounded of the 
genera y and y’, in which f and f’ are contained; and this composition is 
expressed by the symbolic equation r=y xy’. It will be seen that the 
composition of any genus with itself gives the principal genus. 
The same considerations may be extended to improperly primitive classes. 
Thus, if f and f' be respectively properly and improperly primitive, m and m’ 
uneven numbers prime to one another and to D, represented by f and $7", 
the genus of the improperly primitive class, fx f', may be inferred from the 
number mm’, i.c. it is obtained bythe composition of the generic characters 
of fand f’. Or, again, if f and f’ be both improperly primitive, so that the 
class compounded of them is the double of an improperly primitive class, the 
generic character of this improperly primitive class is obtained by compound- 
ing those of the two given classes. 
It follows, from these principles, that the number of classes in any two 
genera [of the same order] is the same. For if ,, ,,..., be all the 
classes of any genus of properly or improperly primitive forms, F, a class 
belonging to any other genus of the same order, and @ a properly primitive 
class satisfying the equation ®,x@=F,, the classes’ ®,x¢,..-.- nx 
are all different, and all belong to the genus (F); consequently (F) has at 
least as many classes as (@), and vice versd (®) has at least as many as (F), 
i. e. they both contain the same number of classes. 
115. Determination of the Number of Ambiguous Classes, and Demon- 
stration of the Law of Quadratic Reciprocity—The number of actually 
existing genera of properly primitive forms cannot exceed the number of 
properly primitive ambiguous classes. For let x be the number of classes 
in each genus, & the number of actually existing genera, so that kn is the 
number of properly primitive classes; let also 1, A,, A,, . . . A,-1 be the pro- 
perly primitive ambiguous classes. Every class produces, by its duplication, 
a class of the principal genus; and if K be a class of the principal genus 
