GROWE: CANONICAL FORMS OF THE BINARY QUINTIC AND SEXTIC. 203 
S=1, b394+-3(63.3 b1.6+b22 $5.7) —b35 bistobag 5.9: 
II Prosrem: To find the covariant C, upon which to take the 
ground points in order to reduce the sextic (a,d,c,d,e,f,¢)(xy)® to 
‘the form where 4 and fare zero. 
(a? d—jabc--26%) is the first coefficient of the Jacobian (DE 
(ac—*) is the first coefficient of the Hessian (H). 
(a) is the first coefficient of the sextic (a). 
(a®*/—sabe+ 2acd—6bc® +862 d¢) is an octavic covariant CL. 
2(a?d—3abc-+ 263) (ac—b*)—a® (a®f—sabe+2acd—6bc? |-8627)—=o 
when 6 and // are zero. 
.'. the first and last term of (2HJ-—a?L) disappears when ¢ and 
f are zero. 
This is the form of Brill and Stephanos. 
This covariant is of order 5 in the coefficients and 20 in the vari- 
ables. 
Theorem 2: A non-singular sextic may be brought by linear trans- 
Jormation to the canonical form (@,0,¢,d,e,0,¢)(xy)® in ten different 
Ways. 
This 20th degree covariant is further found to be the Jacobian of 
the sextic (a) and a covariant S, of the 16th degree, whose second 
term disappears when 6 and fare zero. 
(a) is the first coefficient of the sextic (a). 
(we—gbd-|-3c*) is the first coefficient of a quartic covariant Cran 
(ac—6*) is the first coefficient of the Hessian (H). 
Forming the combination (2a?K—-H?): 
2a* (ace—gbd+-3c°) —(a%c®—2ab2¢-+-b4)== 
2a%e Sarda-| sarc? \-2ab®c—64 
Using the operator D, 
a ed ad a a a 
on 2 52 5 ashe ee ley Ci Sada Me 
(s Me ae ag a i os) 
To obtain the second coefficient, we have: 
(4a3f—2ga*be- goa? cd--8ab?d—goa®cd 
{+-20abC® +2003 c-|-65( - —)==0 
when 4 and / are zero. 
(oq ee (2a PEIN 2) 
III Prositem: To find the covariant C, upon which to take the 
ground points in order to obtain the canonical form of the sextic in 
which ¢ and ¢ are zero. 
