284 
A THIRD MEMOIR ON QUARTIC SURFACES. 
[454 
158. The before mentioned irrational equation may be written 
VT.5 + 7276 +VO =0, 
and by symmetry we see that also 
7179 + 72.10 + 73.11 = 0, 
7l .13 + 72.14 + 7.3715 = 0; 
viz., these are three equations each containing the planes 1, 2, 3, which are three of 
the planes belonging to the node 1 ; the other three planes in any such equation 
(for instance, the planes 5, 6, 7, in the first equation) being three planes belonging to 
another node. Instead of the planes 1, 2, 3, we may have any other three planes 
belonging to the node 1; and instead of the node 1, any other node; but each 
equation belongs to two nodes: the number of equations is thus 
rir! x 16 x 3 - 2 > = 480 - 
159. To obtain the planes belonging to any such equation, combine any two of 
the outside right-hand lines of the diagram, these contain in common two numbers 
the places of which are interchanged; striking these out, we have four columns, and 
taking out of these any three columns, we have the corresponding sets of planes. For 
instance, lines 1 and 2 contain 78 and 87 respectively; striking these out, the lines are 
1, 9, 13, 6; 
2, 10, 14, 5; 
whence we have the sets (1, 9, 13) and (2, 10, 14); viz., there is an irrational equation 
of the form 
7l 72 + 79.10 + 7l3.14 = 0, 
but it is probably necessary to introduce constant factors along with the products 
1.2, 9.10, and 13.14 respectively. There are ¿16.15, =120 pairs of lines, and each 
line gives 4 equations; in all 120 x 4, = 480 equations, as above. 
160. I stop to remark that Kummer gives for his 13-nodal surface an equation 
containing three arbitrary constants, say A, /¿, v, such that, putting one of these = 0, 
we have the 14-nodal surface; putting two of them each =0, the 15-nodal surface; 
putting all three of them = 0, the 16-nodal surface. The equations for the 16-nodal 
surface and that for the 14-nodal surface, made use of by Kummer, are, in fact, those 
deduced as above from the equation of the 13 (a)-nodal surface; and the like form 
might have been used for the 15-nodal surface. But the form actually used by 
Kummer, as presently appearing, is an equivalent form not thus deducible from the 
equation of the 13 (a)-nodal surface.