The So-called Vahlen Relations between the Minors of a Matrix. 701 
In this connection it is important to note that there is considerable 
advantage in writing the vanishing trinomials of the Monge type as 
Pfaffians : thus 
I 2 
8 
14 
0 = 111 2 
They can then be grouped in interesting sets of five, namely, the five 
primary minors of a quasi-Pfafl&an. Written in this way the complete set 
of thirty is : 
1 1 2 3 4 I 
11 12 13 
14 15 j 
16 ! 
I 3 6 8 10 
12 14 16 
17 19 
20 
and from the thirty the chosen set of ten is got at once by taking the nine 
which involve 1 and any one of the nine which involve 20. 
= 112 5 
11 14 15 
17 18 
20 
115 6 7 
11 12 13 
17 18 
19 
I 4 7 9 10 
13 15 16 
18 19 
20 
48 
