OF EQUAL FOOTS OF A BINAEY QUAKTIC OE QUINTIC. 
579 
or, as this may be written, 
(1234, 1235, 1245, 1345, 2345X#,y) 4 ==-6I.HU + 9J.U, 
where HU is the Hessian of U, 
ac 
2 a.d 
ae 
2 be 
ce 
- b 2 
— 2 be 
+ 2 bd 
-3 c 2 
-2 cd 
-d 2 
6. That is, we have 
1234=( ac — b 2 , 
4. 1235=(2ad—2bc 
6. 1245=( ae+2bd-3c\ 
4. 1345=(2£c —2cd 
2345=( ce- d 2 
aJ-6 1, 9J), 
45X-6I, 9J), 
6<?X-6I, 9J), 
4d£ — 61, 9J), 
eX-6I, 9J). 
7. The determinants thus vanish if (I, J)=0, that is, for the root system 31 ; they 
will also vanish without this being so, if only 
/3J \ ac—b 2 ad— be ae + 2bd— 3c 2 be—cd ce—d 2 _ 
\2l / a 2b 6c 2d ~ ’ 
and we may omit the first member since if the remaining terms are equal to 
oj 
each other they will also be — — . The equations may then be written 
ac-tf, ad— be, ae-\-2bd—3c 2 , be—cd , ce—d 2 
a , 2b , 6c , 2d , e 
and the ten equations of this system reduce themselves (as it is very easy to show) to 
the seven equations 
(A, B, C, D, E, F, G) = 0, 
which, as above mentioned, are the conditions for the root system 22. 
8. It may be added that we have 
A 
B 
c 
D 
E 
F 
G 
1234 = 
c 
-4 b 
+ 3 a 
i. 1235 = 
c 
-3b 
+ a 
0 = 
d 
—3c 
+ a 
1245 = 
—e 
+ 4rf 
-3c 
0 = 
— e 
+ 6e 
—a 
0 = 
— d 
3c 
- b 
£.1345 = 
— e 
-f- 3d 
— c 
0 = 
— e 
+ 3 c 
-b 
£.2345 = 
— 3e 
+ 4 d 
—c 
where it is to be noticed that the four equations having the left-hand side =0, give 
