204 Proceedings of Royal Society of Edinburgh. [sess. 
11 
12 
13 
-12 
22 
23 
- 13- 
23 
11 
12 
13 
14 
-12 
22 
23 
24 
-13 
-23 
33 
34 
- 14 
-24- 
34 
• 
11 
12 
13 
14 
15 
-12 
22 
23 
24 
25 
- 13- 
-23 
33 
34 
35 
- 14- 
-24- 
-34 
44 
45 
-15 - 
-25- 
-35- 
-45 
• 
1113 11-23! 
— 11-23 22-131 
11-14 
-11-24 
11-24 
22-14 
-11-34 -[1234] 
11-34 
[i 234] 
33-14 
-r 11-14 , 
11-15 11-25 11-35 
- 11-25 22-15 [1235] 
-11-35 -[1235] 33-15 
- 11-45 -[1245] -[1345] 
11-45 
[1245] 
[1345] 
4415 
-11-(15) 2 , 
11 
12 . 
. . 15 
16 
-12 
22 . 
. . 25 
26 
- 15 - 25 . 
. . 55 
56 
-16-26 . 
. .-56 
11-16 11-26 11-36 
- 11-26 22-16 [1236] 
-11-36 -[1236] 3316 
-1146 -[1246] -[1346] 
-11-56 -[1256] -[1356] 
11-46 11-56 
[1246] [1256] 
[1346] [1356] 
44-16 [1456] 
[1456] 55-16 
11-(16) 3 . 
Of course, this mode of writing does not at once suggest any 
better mode of proof, but it makes clear the general theorem, 
which consequently may he enunciated as follows : — 
“ A skew determinant of the n th order icliich has a zero for the 
last element of its main diagonal may , if multiplied by 11 •(n) ,l_3 be 
transformed into a skew determinant of the (n - l) th order , which 
has for its first row the last column of the original determinant 
multiplied by 11, for its main diagonal the main diagonal of the 
original determinant multiplied by In, and for the element in every 
other place rs situated between these two lines the Pfaffian [lrsn]. 
The rest of the paper deals with inverse matrices , and with the 
application of them to the problem afterwards known as the 
automorphic transformation of a quadric. 
