278 
PROFESSOR H. J. S. SMITH OIST THE ORDERS 
form we shall henceforward understand a positive form of negative determinant, by 
a ternary form a positive and definite form; and we shall occupy ourselves in the 
remainder of this memoir with the determination of the weight of a given genus or 
order of such ternary forms. 
A ternary form has always 1, 2, 4, 6, 8, 12, or 24 positive automorphics, i. e. automor- 
phics of which the determinant is a positive unit. The weight of a form is the reci- 
procal of the number of its positive automorphics ; so that a form and its contravariant 
have the same weight ; the weight of a class is the weight of any form contained in the 
class ; the weight of a genus or of an order is the sum of the weights of the non-equi- 
valent classes contained in the genus or order. When a number is represented by a 
ternary form, the weight of the representation is the weight of the ternary form. The 
weight of a binary form, or class, is also the reciprocal of the number of its positive 
automorphics ; thus the weight of a binary form is always except when the form 
either is, or is derived from, a form of determinant —1, or an improperly primitive form 
of determinant — 3 ; in these excepted cases the weight of the binary form is J and ^ 
respectively. When a binary form is represented by a ternary form, the weight of the 
representation is the product of the weights of the two forms. 
To determine the weight of a given genus of ternary forms, we avail ourselves of the 
principles introduced into arithmetic by Gauss and Dirichlet, and employed by them 
to determine the number of binary forms of any given determinant. Let (/", F) repre- 
sent a given genus of ternary forms of the invariants [O, A], and either of the properly 
primitive order, or of that improperly primitive order in which f is improperly and F 
properly primitive. Let /j, f 2 , ... or (f) denote a system of forms representing the 
classes of the given genus ; F 1? F 2 , ... or (F), the primitive contra variants of those 
forms. Let M represent any positive number, prime to 20A and satisfying the generic 
characters of F ; when (f \ F) is a properly primitive genus, O being uneven, and A 
uneven or unevenly even, we shall also suppose that M satisfies the congruence OM=l, 
mod 4 : the numbers designated by M will be subject to the restrictions here stated 
throughout the whole investigation. Lastly, let L be a positive quantity which we shall 
afterwards suppose to increase without limit ; and let T be the sum of the weights of the 
representations by the forms (F) of all the numbers M which do not surpass L. The 
quotient T 4- L^ approximates to a finite limit, when L is increased without limit. Of 
this limit, we shall obtain two distinct expressions, the one containing as a factor the 
weight W of the genus (/' F), the other not containing that factor, and depending on 
the arithmetical relation which subsists between the sum of the weights of the represen- 
tations of a given number M by the forms (F), and the sum of the weights of the properly 
or improperly primitive binary classes of determinant — £2M. A comparison of the two 
expressions will then give the required weight of the genus (f, F). 
Art. 14. The first determination of the limit of the quotient T 4 -L 4 depends on the 
following auxiliary propositions, in which F represents any form of the system (F). 
(1) If h is an uneven prime dividing A, F acquires a value prime to h for c) 2 (£— 1) 
systems of values of x, y , z, mod 
