302 
ON THE CONDITIONS FOR THE EXISTENCE 
[400 
4. The values of the determinants are 
1234 = 3 x 
1235 = 3 x 
1245 = 
1345 = 3 x 
2345 = 3 x 
a?ce + 1 
a 2 de — 1 
a 2 ë z - 1 
abe 2 — 1 
ace 2 + 1 
o?d 2 - 3 
abce + 4 
abde + 2 
aede + 4 
ad?e — 1 
ab 2 e — 1 
abd 2 + 1 
ac 2 e + 9 
ad 3 — 3 
b 2 e 2 - 3 
abed +14 
ac 2 d — 3 
acd 2 - 9 
b 2 de + 1 
bede + 14 
ac 3 — 9 
b 3 e - 3 
b 2 ce — 9 
6c 2 e — 3 
bd 3 - 8 
b 3 d - 8 
b 2 c 2 + 6 
b 2 cd + 2 
b 2 d 2 + 8 
bed 2 + 2 
c 3 e — 9 
c 2 d 2 + 6 
5. The syzygetic relation with (/, J) is given by means of the identical equation 
= -6I.HU+9J. U, 
y*> 
— 4 xy 3 , 
Qx 2 y 2 , 
- 
¿c 4 
a , 
36 , 
3c , 
d 
b , 
3c , 
3 d , 
e 
a, 
36 , 
3c , 
d , 
b, 
3c , 
3 d , 
e , 
or, as this may be written, 
(1234, 1235, 1245, 1345, 2345$a, y) 4 = - 6/. HU + 9/. TJ, 
where HU is the Hessian of U, 
ac + 1 
ad + 2 
ae + 1 
be + 2 
ce + 1 
b 2 -1 
6c -2 
bd + 2 
cd — 2 
d 2 - 1 
c 2 - 3 
6. That is, we have 
1234 = ( ac- b 2 
4 . 1235 = (2ad- 2bc 
6 . 1245 = ( ae + 2bd 
4 . 1345 = (2be - 2cd 
2345 = ( ce — d 2 
, o$—6/, 9J), 
, 46][- 6/, 9/), 
3c 2 , 6c $-6/, 9/), 
, 4d$- 6/, 9 J), 
, c$-6/, 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 
SJ \ac — b 2 ad — be ae + 2bd — 3c 2 be — cd ce — d 2 
26 
6c 
2 d 
/SJ \ 
and we may omit the first member Uj =), since if the remaining terms are equal