102 
Proceedings of Royal Society of Edinlurgh. [jan. 6, 
we have 
15 16 17 18 
25 26 27 28 
35 36 37 38 
- 
14 16 17 18 
24 26 27 28 
34 36 37 38 
-h 
14 15 17 18 
24 25 27 28 
34 35 37 38 
45 46 47 48 
45 56 57 58 
46 56 67 68 
14 15 16 18 
14 15 16 17 
- 
24 25 26 28 
-f 
24 25 26 27 
34 35 36 38 
34 35 36 37 
47 57 67 78 
48 58 68 78 
14 15 16 17 18 
_ 
1 . . 14 15 16 17 18 
24 25 26 27 28 
' . 24 25 26 27 28 
34 35 36 37 38 
34 35 36 37 38 
45 46 47 48 
45 46 47 48 
56 57 58 
56 57 58 
67 68 
67 68 
78 
78 
the zero-elements of the Pfaffian occupying as before the places of 
those elements of the axisymmetric determinant, which do not 
occur in the identity, and which, in this case, are three in number 
(v. § 4). 
7. The next theorem in regard to vanishing aggregates of deter- 
minants lends itself readily to formal enunciation. It is as 
follows : — 
If any tivo determinants A and B of the n*^ order he taken, and 
from them two sets of determinants he formed, viz., first, a set of n 
determinants, each of which is in one row identical with A, and in 
the remaining rows loith B, and secondly, a set of n determinants, 
each of inhich is in one column identical with A, and in the remain- 
ing columns with B, then the sum of the first set of determinants is 
equal to the sum of the second set. 
Let the two determinants A and B be 
^2 ^3 
^6 
^3 
h h \ 
Cl Cg Cg 
} 
C4 Cg Cg 
then the first set of determinants derived from them is 
