1910-1 1.] The Less Common Special Forms of Determinants. 331 
we write any elements whatever, namely, 12, 13, . . . This change 
being made, the fourth and fifth are 
= ^12, 21. 34 43 ~2 12 23 24 41 
12 
13 
14 
21 
23 
24 
31 
32 
33 
41 
42 
43 
12 
13 
14 
15 
21 
23 
24 
25 
31 
32 
34 
35 
41 
42 
43 
45 
51 
52 
53 
54 
cover 
3, 6, 
24, 
20 
y 12 23 34 45 51 
212 23 31 . 45 54, 
added, nor any indication of a law including both results. The three of the 
other set are less general, namely, 
. 12 12 1 
21 . 
31 32 
1 1 
23 
^ 12 21 - 2 12 23 , 
where the E’s cover 3 and 6 terms respectively ; 
12 
13 
14 
1 
21 
23 
24 
1 
31 
32 
34 
1 
41 
42 
43 
1 
= 2 12 23 34 - 2 12 34 43 - 2 12 23 31 > 
1111 . 
where the X’s cover 24, 12, 8, terms respectively; 
12 
13 
14 
15 
1 
21 
23 
24 
25 
1 
31 
32 
34 
35 
1 
41 
42 
43 
45 
1 
51 
52 
53 
54 
1 
1 
1 
1 
1 
1 
-2, 12 23 34 45 ~2 12 21 - 34 43 
+ 2 12 23. 45 54 +212 23 34 41 
+ 2 12 23 31 . 45, 
where the Z’s cover 120, 15, 60, 30, 40 terms respectively. Here again no 
generalisation is attempted. 
(0 CENSUS OF TERMS IN SPECIAL DETERMINANTS. 
The first instance of the finding of the number of terms in the final 
development of a determinant of special form has already been drawn 
attention to, the investigator being Scherk, and the date 1825. 
