202 KANSAS UNIVERSITY QUARTERLY. 
A binary mic may be interpreted geometrically as a system of 
points “on a line. When the nic is written in full, the ground 
points (see Clebsch, Aigebriische Formen, Chap. II,) are in no 
special way related to the points of the mic; but, in order to 
simplify the form of the ic, we may, without any loss of gener- 
ality, choose any two points on the line as ground points. 
Since changing the ground points is equivalent to a linear trans- 
formation, canonical forms of the mic may be obtained by choos- 
ing the ground points of the system of binary coordinates on certain 
covariants of the sic. 
I Prostem: To find the covariant C upon which to take the 
ground points in order to reduce the quintic (a,4,c,d,¢,f)(ay)5, to 
the form where 4 and ¢ are zero. 
«xy==0 is the equation of the ground points. 
If the ground points are to be on C, then « and » must divide 
out, or the coefficients of the first and last terms of Care zero. The 
problem then becomes: 
To find the covariant C, the coefficient of whose first (and neces- 
sarily last) term is zero, when 4 and ¢ are made zero. 
(a) is the first coefficient of the quintic (a); 
(ac—6*) is the first coefficient of the Hession (H); 
(ae—4bd--3c*) is the first coefficient of a quadratic covariant (I); 
a” (ae—gbd-+-3c”)—3(a* c®—2ab%¢ |-$4\=0 
when @ and ¢ are zero. 
-'. the first (and last) term of the r2ic (a#I1—3H®) disappears 
when # and ¢ are zero. 
11, C==(a#I— 3H), 
This covariant is of order 4 in the coefficients, and 12 in the 
variables. Since theory, shows that the points of this covariant 
are associated in pairs, we have six different pairs of points which 
may be chosen as ground points. 
Thus we have six different pairs of linear factors of the covariant 
C which may be used to transform the quintic to this canonical 
form. Hence we have 
Theorem 1: A non-singular binary guintic may be brought by linear 
transformation to the canonical form (ax®-+-rocxy® 4-rodx®y34.fy>) 
in six different ways. ‘ 
It may be interesting to know that C, in turn, is found to be the 
Jacobian of the quintic and another covariant §, whose second 
term disappears when 4 and ¢ are zero, and is of the ninth degree. 
Using Bruno’s tables: 
