388 Prof. H. J. S. Smith on the Orders and Genera of [Feb. 28, 



meration, a Table is given in the Paper for finding the supplementary 

 characters of any proposed form. 



2. A Table is also given for forming the complete generic character of 

 any proposed form. This Table is intended to serve the same purposes, 

 in the theory of ternary quadratic forms, for which the Table of Lejeune 

 Dirichlet is available in the binary theory (Crelle, vol. xix. p. 338). The 

 Table, like that of Lejeune Dirichlet, distinguishes between the possible 

 and impossible generic characters and the Paper contains a complete 

 demonstration of the criterion by which they are distinguished. 



3. Besides the particular characters relating to uneven primes dividing 

 the discriminant, it is convenient, in those cases in which there is no sup- 

 plementary character, to consider a certain particular generic character 

 which does not appear to have been regarded as such by Eisenstein. This 

 character is termed in the Paper the simultaneous character of the form 

 and its contravariant : its existence is demonstrated ; and its introduction 

 as an element of the complete generic character is justified by its use in 

 the distinction of possible and impossible genera. 



4. It has been proposed to define a genus of forms as consisting of all 

 those forms which can be transformed into one another by substitutions 

 of which the coefficients are rational and the determinant a unit. It is 

 desirable (in the case of quadratic forms) to add to this definition the 

 limitation that the denominators of the fractional coefficients are to be 

 uneven, and prime to the discriminant. And it is shown, in this Paper, 

 that two ternary quadratic forms are or are not transformable into one 

 another by such substitutions, according as their complete generic cha- 

 racters do or do not coincide. 



5. The preceding observations apply equally to the cases of definite and 

 indefinite forms. These two cases are included in the same analysis by 

 means of a convention as to the signs of the two numbers defined by 

 Eisenstein, and termed in this Paper the arithmetical invariants of the 

 ternary form. The first invariant of a form is the greatest common 

 divisor of the first minors of the matrix of the form ; the second invariant 

 is the quotient obtained by dividing the discriminant by the square of the 

 first invariant. According to the convention adopted in the Paper, the 

 second invariant has the same sign as the discriminant ; and the first 

 invariant has or has not the same sign as the second, according as the 

 form is definite or indefinite. 



6. The latter part of the Paper is occupied exclusively with the theory 

 of definite and positive forms. In the case of these forms, the weight (or, 

 as Eisenstein has termed it, the density) of a class is the reciprocal of the 

 number of automorphics (of determinant + 1 ) of any form of the class ; 

 the weight of a representation of a number by a form is the weight of the 

 form, i. e. the weight of the class containing the form ; the weight of a 

 genus or order is the sum of the weights of the classes comprised in the 

 genus or order. In his Memoir, Eisenstein has given (but without demon- 



