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PROFESSOR H. J. S. SMITH ON THE ORDERS 
Art. 8. The complete generic character of a form or class is the complex of all the 
particular characters attributable to the form or class, and to its primitive contravariant, 
including their simultaneous character, if any. And two forms, or classes, which have 
the same complete generic character are said to belong to the same genus. But not 
every complete generic character that can be assigned a priori , is the character of any 
really existing genus of forms. The annexed Table will serve, in the case of any given* 
order, to distinguish those complete generic characters, which are possible, i. e. to which 
actually existing genera correspond, from those which are impossible. 
In this Table 0 2 and A* are the greatest squares dividing H and A ; the quotients 
fl fl 2 , A-4- A 2 are respectively represented, if uneven, by O, and A 1? if even by 20, 
and 2A„ sothatO ,and A, are always uneven and not divisible by any square; and 
are any primes dividing O, and A,, cu. 2 and c) 2 are any uneven primes dividing O s and A 2 , 
n'F+l A'F+1 
but u 2 must not divide 0„ nor must \ divide A, ; lastly, T - is still the unit ( — 1) 2 ’ 2 , 
n,F + l • A,F+1 
or, which is the same thing, the unit ( — 1) 2 ‘ 2 jjfand F in the exponents of these 
units denoting uneven numbers simultaneously represented by the forms f and F. 
The Table A of properly primitive generic characters contains twenty-five compart- 
ments corresponding to the twenty-five cases indicated in its margin ; the Tables B and 
C of improperly primitive genera contain three such compartments each. In each com- 
partment are inscribed all the particular characters which make up the complete generic 
character of a form coming under the case to which the compartment corresponds ; the 
symbols {j)’ \J ) f h as a particular character with respect 
to every prime u l or &>. 2 , F a particular character with respect to every prime cq or 
Each compartment is divided into two parts by a vertical line, and the particular 
characters (one of which in Table A either is or contains T) placed to the left of this 
line are subject to the condition that their product is equal in Table A to the unit 
Q, + l. A,+ l # Oj-1 Oj + I A, + l 
( — 1) 2 ‘ a ? i n Table B to the unit ( — 1_)~ s x (— 1) 2 ’ 2 , in Table C to the unit 
A 2 -l fl| + 1 A t + 1 
( — l) - x( — 1) 2 2 . If a=+ 1? or — 1, according as O is of the form Q^ 2 , or 
2*0, 0 2 , and if, similarly /3= + l, or —1, according as A is of the form A, A®, or 2A,A 2 , 
we may express this condition in Table A by the equation 
/ 2 -i liri / f \ /F\ n > +1 A i +1 
B x(4)x(^)=(-l) ..... (11) 
and in Tables B and C respectively by the equations 
n ,-! n,+i a, 
(-«^x(^)x(0=(-i)^ 
(-^x(^)x(|)=(-i )V +e V' 
( 12 ) 
(13) 
The condition distinguishes the possible and impossible genera, every generic cha- 
racter which satisfies it being the character of an actually existing genus, and every 
