Vol. 7, 1921 
MATHEMATICS: A. B. COBLE 
337 
the fact that their coefficients are, respectively, the one- and three- row 
minors of A . The four comitants of the second degree above are evidently 
self-dual in meaning and therefore should involve the two-row minors of 
A. Indeed we find that the 36 coefficients of the four comitants (6r) 4 
(00 4 > (t0 4 > (ctY> 5 are linearly independent in the 36 two-row minors of 
A . The only remaining comitants of the second degree, not linear in the 
minors of A, are the two sextics Si(r), 52 (t). 
The 36 two-row minors of a four-row determinant A } though linearly 
independent, must satisfy a system of quadratic relations. It is not hard 
to see that there are precisely 41 such quadratic relations. If we take 
the above four comitants and form all of their comitants of total second 
degree in the coefficients of the four we have a set of comitants with 666 
coefficients which is the number of quadratic combinations of the 36 
minors. Hence the coefficients of this new set of comitants must be 
connected by a system of 41 linear relations due to the existence of the 
41 quadratic relations among the minors. These relations will be expressed 
by the existence of a system of syzygies of the second degree in the co- 
efficients of the four comitants. We find that this system of syzygies is 
(bry {b'rY W f y = 6[(ct)*]«; (bb'Y (0'*) 4 = 6[( 7 0 4 ] 2 ; (bb'Y (M* 
(&'t) 2 WY = 6(cc') 2 (cr) 2 (c'r) 2 ; (bb'Y WY W (p't) 2 = 6( 77 ') 2 
(yty ( 7 '0 2 ; (brY (pyY = 6d.(crY; (be)* (pty = 6 5.( 7 0 4 ; (bb'Y WY = 
6( 77 ') 4 = 6(cc'Y = 36 5 2 . 
The 41 coefficients of these syzygies furnish the quadratic relations. The 
syzygies themselves determine for given (br) A (fit) A the three remaining 
forms (cr) 4 , ( 7 /) 4 > 5 to within a change of sign of any two. 
12. The fundamental combinants of F, F. We have had occasion at times 
to consider such pencils of cubics in / as are determined by the members 
(ari) 3 , (aty, (ar 2 ) 3 (atf) 3 (n^^). If h, h belong to the same member of 
this pencil then the fundamental combinant of Gordan for the pencil is 
= (an) 3 (ah) 3 (ar 2 ) 3 (ahY 
(a'nY (a%Y (aV 2 ) 3 (a't 2 y 
Evidently it expresses also that n, r 2 belong to the same member of the 
pencil determined by the members (ar) 3 (ati) 3 (ar) 3 (a:^) 3 (Zi+fe). We call 
r the fundamental combinant of F, and r the corresponding fundamental 
combinant of F. Bach is expressible in terms of the two-row minors of A 
and therefore in terms of the four comitants of 11. We find that T, 
f = (r,r 2 ).(hh) { (bn) 2 (br 2 Y (phY (^ 2 ) 2 =fc (hhY-^Y (cr 2 Y ± (riT 2 ) 2 .- 
(yhY (yhY + (W 2 ) 2 .(t 1 t 2 ) 2 .5} 
where the upper sign holds for T and the lower sign (together with the 
factor A/9) for r. Let us call the four terms within this brace, taken 
with plus signs, K, L, M, N, respectively. Then the involution of binary 
cubics determined in 7 by the form (irx) (dr) (U) z for variable r has for 
fundamental combinant (when x is replaced by ri, r 2 ) K + L—M—N 
