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XII. On the Orders and Genera of Ternary Quadratic Forms. By Heney J. Stephen 
Smith, M.A., F.B.S., Savilian Professor of Geometry in the University of Oxford. 
Received February 21, — Read February 27, 1867. 
Eisenstein, in a Memoir entitled “ Neue Theoreme der hoheren Arithmetik” *, has 
defined the ordinal and generic characters of ternary quadratic forms of an uneven 
determinant ; and, in the case of definite forms, has assigned the weight of any given 
order or genus. But he has not considered forms of an even determinant, neither has 
he given any demonstrations of his results. To supply these omissions, and so far to 
complete the work of Eisenstein, is the object of the present memoir. 
Art. 2. We represent byf the ternary quadratic form 
ax 2 +a'y s +a"z 2 +2byz+2b'xz+2b"xy ; (1) 
we suppose that f is primitive ( i . e. that the six integral numbers a, a /, a ", b, b\ b" admit 
of no common divisor other than unity), and that its discriminant is different from zero ; 
this discriminant, or the determinant of the matrix 
a, b", V 
b\ a\ b , 
b 1 , b , a" 
( 2 ) 
we represent by D ; by Q we denote the greatest common divisor of the minor determi- 
nants of the matrix (2) ; by OF the contravariant of f or the form 
(dai'-Vy+iai'a- b'*)y*+{aa! -b" 2 )z 2 \ 
+ 2(b’b" - ab)yz + 2(b"b - a!b')zx+2 (W - a"b")xy ; J 
. (3) 
we shall term F the primitive contravariant off , and we shall write 
F=A# 2 -f- A'v/ 2 + A V + 2B yz+ 2B 'xz + 2B "xy (4) 
If D = Afl 2 , A is an integral number, and the discriminant, contravariant, and primi- 
tive contravariant of F are respectively OA 2 , Af and f. The numbers O and A are 
arithmetical invariants of f; i. e. they remain unaltered when f is transformed by any 
substitution of which the determinant is unity and the coefficients integral numbers. 
We shall accordingly describe the primitive form f, and the class of forms containing f 
as a form, and class, of the invariants [O, A]. Similarly, F is a form of the invariants 
[A, O], and the class containing F is a class of those invariants. The relation between 
the forms f and F is reciprocal ; and this reciprocity extends throughout the whole 
* Ckelle’s Journal, vol. xxxv. p. 117. 
2 N 
MDCCCLXVII. 
