-On New Canonical Forms of the Binary 
Quintic and Sextic. 
md 
BY. BESSIN OH. GROWE, 
INTRODUCTORY NOTE. 
The reduction of the non-singular binary cubic and quartic by 
linear transformation to their canonical forms (a@,o0,0,@)(«y)* and 
(@,0,@,0,e)(xy)*, was achieved by Cayley early in the development 
of the Invariant theory. In the year 1882, Brill (Math. Annalen, 
XX., p. 330,) and Stephanos, in the ‘‘Mémoire sur les faisceaux de 
formes binaires ayant une méme Jacobienne,” (Tome XXVII, of 
the Savants Ltranger s of the Académie des Sciences, 1883) showed 
that the sextic was reducible by linear transformation to the form 
(@,0,¢,¢@,¢.0,¢)(ay)®. In Elliott’s Algebra of Quantics, 1895, Chap. 
XIII, is to be found a discussion of Hammond’s canonical form of 
the quintic (a@,4,0,0,¢,/)(xy)®. (See last paragraph of the preface). 
[In a paper on Hessians and Steinerians of Higher Orders in the 
Geometry of One Dimension, published in the Annals of Mathe- 
matics, Vol. XI, page 121, I gave the following theorem: Zvery 
non-singular quantic of odd degree may be linearly transformed so that 
i Y i i A Neat Vs 
tts two middle terms shall vanish, this may be done tn distinct 
: 2 
ways. I was in possession of this result at least a year before the 
appearance of Elliott’s book. 
Miss Growe undertook, at my suggestion, to find the two new 
canonical forms. of the quintic and sextic, which theory showed 
must necessarily exist. 
There is still lacking a general theory of these canonical forms 
for higher quantics. Such a theory must involve the notion of 
Jacobians and Cremonians of higher orders in some way analogous 
to my theory of Hessians and Steinerians of higher orders. 
H. B. Newson. 
(201) KAN. UNIV, QUAR., VOL, VI, No. 4, OOT.. 1897, SERINS A. 
