282 A THIRD MEMOIR ON QUARTIC SURFACES. [454 
down the equations of the 16 singular planes, and thence to deduce the coordinates 
of the 16 nodes; viz., 
singular planes 
are 
and the 
nodes 
are 
(1) 
x = 0, 
(1) 
( o , 
-ß, 
7 > 
«'«"/37 ), 
(2) 
y =0, 
(2) 
( « > 
0 , 
“7> 
ß'ß"7« ), 
(3) 
* = 0, 
(3) 
( -«» 
ß , 
0 , 
7 7" a ß )> 
(4) 
w — 0, 
(4) 
(«V', 
ß'ß", 
77"» 
0 
X 
(5) 
X-w = 
0, 
( 5 ) 
( 1 > 
0 , 
0 , 
0 
), 
(6) 
Y — w = 
0, 
(6) 
( o , 
1 , 
0 , 
0 
), 
(7) 
Z —iv = 
0, 
(7) 
( o , 
0 , 
1 , 
0 
X 
(8) 
P = o, 
(8) 
( o , 
0 , 
0 , 
1 
X 
(9) 
X'-w = 
= 0, 
(9) 
( o , 
-ß', 
/ 
7 * 
«"«/3V), 
(10) 
T-w = 
0, 
(10) 
( «' , 
0 , 
-7. 
ß"ßi* 
X 
(11) 
Z' -w = 
= 0, 
(11) 
(-«', 
ß', 
0 , 
y'ya'ß' ), 
(12) 
P' =0, 
(12) 
(«"a, 
ß"ß ; 
u 
7 7 > 
0 
X 
(13) 
X"-w = 
= 0, 
(13) 
( o , 
-ß", 
// 
7 . 
aa'ß’Y 
), 
(14) 
Y" - to -- 
= 0, 
(14) 
( «" , 
0 , 
— y", 
ßßY*' 
'X 
(15) 
Z" -W-. 
= 0, 
(15) 
(-«", 
ß", 
0 , 
yy'oTß" 
X 
(16) 
P" = 0, 
(16) 
( ««' > 
ßß', 
77 > 
0 
X 
where the nodes and planes are numbered as by Kummer; and by means of his 
(differently arranged) diagram of the relation between the several nodes and planes, 
I was enabled to form the following square diagram, which exhibits this relation in, 
I think, the most convenient form. To explain this, observe that in the upper and 
left-hand margins, the numbers refer to the nodes; in the body of the table, and in 
the right-hand margin to the planes, the table shows that for the node 1, the 
circumscribed cone is made up of the planes 1, 6, 7, 8, 9, 18 ; and that the remaining 
15 nodes are situate on the nodal lines of this cone, the node 2 on the intersection 
of the planes 7, 8; the node 3 on the intersection of the planes 6, 8, and so on; 
and the like as regards the other lines of the table.