WITH A 16-NODAL QUARTIC SURFACE. 
159 
[662 
662] 
>arently arbitrary 
her on. I recall 
its u, u', are* 
3 which are odd 
Pi, • • • , Q\ etc., 
stants, depending 
ich are however 
parameters: the 
ear functions of 
ns between the 
ling 12 squares, 
•e linearly inde 
ed he seems to 
m five squares. 
deux variables 
364—468, that 
e exist sixes of 
each of the 
e three squares, 
equations, and 
by the following 
Strahlensysteme 
14 
15 
16 
14 
15 
16 
6 
7 
8 
2 
3 
4 
11 
10 
9 
12 
9 
10 
9 
12 
11. 
In fact, to show that any four of the squares, for instance 1, 9, 13, 8, that is, 
P 2 , P 2 2 , R 2 , R s 2 , are linearly connected, it is only necessary to show that the determ 
inant of coefficients 
a, 
~/3, 
~7, 
8 
a, 
-¡3, 
7, 
- 8 
7, 
- 8, 
— a, 
/3 
7, 
8, 
a, 
/3 
is = 0, or what is the same thing, that there exists a linear function of the new 
variables (X, Y, Z, W), which will become = 0 on putting for these variables the values 
in any line of this determinant: we have such a function, viz. this is 
/3X + aY-8Z-yW, 
or say 
[1] (0, -S, -y)(X, r, Z, W). 
This function also vanishes if for (X, Y, Z, W) we substitute the values 
8, ~ V, ¡3, - a, 
8, 7, /3, a, 
which belong to 7, 6, that is, S 2 2 and S 2 respectively. It thus appears that 1, 9, 
13, 8, 7, 6, that is, P 2 , P 2 2 , R 2 , R- 2 , S 2 , S 3 2 , are a set of six squares having the 
property in question. I remark that the process of forming the linear functions is 
a very simple one; ive write down six lines, and thence directly obtain the result, thus 
1 
a, 
-/3, 
~7, 
8 
9 
a, 
-/3, 
7, 
-8 
13 
7» 
-8, 
~ a, 
/3 
8 
7, 
8, 
a, 
/3 
7 
8, 
~7, 
A 
— a 
6 
8, 
7, 
A 
a 
/3, 
a, 
-8, 
-7: 
viz. /3, a, 8, y are the letters not previously occurring in the four columns respect 
ively: the first letter /3 is taken to have the sign +, and then the remaining signs 
are determined by the condition that, combining the last line with any line above it 
(e.g. with the line next above it /38 + ay — 8/3 — ya), the sum must be zero. 
We find in this way, as the conditions for the existence of the 16 sixes respectively, 
[1] 
(/3, 
a, 
-5, 
-7)№ 
Y, Z, 
W) = 0, 
[2] 
(«> 
-/3, 
- 7> 
S)(X, 
Y, Z, 
W) = 0, 
[3] 
(«> 
/3, 
~ 7> 
- S)(X, 
Y, Z, 
W) — o, 
[4] 
(/3, 
- a, 
- 8, 
y)(X, 
Y, Z, 
W) = 0,