AND GENERA OF TERNARY QUADRATIC FORMS. 
267 
generic character which does not satisfy it belonging to no forms whatever. The demon- 
stration of this important theorem will occupy the next articles ; it is, however, requisite 
to show in the first place that the enumeration of the supplementary characters in Table 
II. is in accordance with the Table I. of Art. 4. For the Tables B and C this is evident ; 
in Table A it is necessary to attend to the signification of the symbol T, which serves 
to represent the simultaneous character of f and F (as has been already explained 
fzi 
in Arts. 6 and 7) in those cases (marked S in the Table) in which neither (—1) 2 nor 
F— 1 
(— 1) 2 is a character, but which also appears in every compartment of the Table 
without exception. 
■ izl f-i 
(1) When ( — 1) 2 and ( — 1) 2 are both characters (cases P in the Table), T is not 
fzl 
an independent character, because its value is determined by the values of ( — 1) 2 and 
F— 1 
( — 1) 2 . It is retained in the Table only because it serves to express the criterion of 
possibility. 
/-I F — 1 
(2) When ( — 1) 2 and T, but not ( — 1) 2 , are incribed as characters, T represents 
F— I /- 1 A,-t 
the character ( — 1) 2 , if ( — 1) 2 =( — 1) 2 , and is simply +1 ( i . e. not a character 
/- 1 A— i 
at all), if ( — 1) 2 = — ( — 1) 2 . This is in accordance with Table I., according to 
F— 1 
which, in the cases under consideration, ( — 1) 2 is or is not a character, according as 
( — 1)~ =( — l) -2 ”, or = — ( — 1)~. Similarly, if ( — 1)~ and T, but not (— 1)~, 
/-i 
are inscribed as characters, T represents the character ( — 1) 2 , or is not a character at 
all, according as ( — 1) _ 2 _ =( — 1) 2 ? or = — ( — 1) 2 ; which again agrees with Table I. 
In these cases, marked Q in the Table, the symbol T supersedes the asterisks and 
obelisks of Table I., and also serves to express the criterion of possibility. 
/-I F 3 — 1 F-I 
(3) When ( — 1) 2 and (—1) 8 xT, but not ( — 1) 2 , are characters in the Table, 
F-l , F 2 — 1 F 2 — 1 
(-1)" 
, or 
■l) -2 * and(- 
according as 
1) 8 X T, but not 
(— 1) 8 x T represents the character ( — 1) 
/-I A,-l A, — 1 
(— 1) 2 =( — 1) 2 , or — (— 1) 2 . And again, when ( 
( — 1) 2 , are characters in the Table, (— l) - ® - XT’ represents the character ( — 1)~ 2 ~ +_8_ , 
_ F— 1 0,-1 Q|- 1 
or (—1) 8 , according as ( — 1) 2 = (— 1)~ 2 , or —( — 1) 2 . 
The result in these cases (marked R in the Table) is again in accordance with Table I. ; 
and the use of the symbol T is the same as in the cases Q. 
P - 1 F 2 — 1 
Thus the units T, ( — 1) 8 xT, (— 1) 8 xT, which properly represent simultaneous 
characters of the forms f and F, are employed, in the cases Q and R of the Table, to 
represent supplementary characters. This use of these symbols is admissible, because, 
when I2A is even (as it is in the cases Q and R), every representation of an uneven 
number by f or F is simultaneous with the representation of uneven numbers by F or f. 
