196 
Proceedings of Boyal Society of Edinburgh. [sess. 
[rs] would have stood for the cofactor of ( rs ) in A . From these by 
addition and subtraction and by utilizing the fact that u r + v r = 0 f 
two others are obtained, viz., 
2x 1 A = 0 +([12] -[21 ])ra 2 + ... + ([Ira] - [ral])ra n ' 
2x % A - ([21] - [12]K + 0 + . . . + ([2ra] - [>2]K _ 
I 
2x n A = ([ral] - [lra])?q + ([ra2] - [2ra])ra 2 + ... + 0 J 
and 
0 = 2[ll]ra 1 + ([12] + [21])ra 2 + ... + ([lw] + [ral])ra„ ] 
0 = ([21] + [12])«! + 2[22]k. 2 + . . . + ([2 n ] + [k2])m„ j_ 
0 = ([ral] + [1 w])mi + ([ra2] + [2ra])ra 2 + . . . + 2\nn\u n ). 
Then follows the very curious sentence — curious, that is to say, 
logically — 
“ The comparison of these three systems gives either 
A = 0 
* 
[12] = [21] . 
[Ik] = [»1] ] 
to 
1 — « 
T 
| bO 
* 
T 
it 
i bO | 
[ral] = [Ira] 
[»2]-[2»] • 
* 
[11 = 0] 
[12] + [21] = 0 . . 
. . [lra] + [ral] = 0 ] 
[21] + [12] = 0 
[22] = 0 
. . [2ra] + [ra2] = 0 
[ral] + [Ira] = 0 
[ra2] + [2ra] = 0 . . 
[rara] = 0 J 
and consequently either a symmetrical skew determinant of 
an even order or a ” [symmetrical skew] “ determinant of an 
odd order vanishes.” 
Temporarily setting aside the latter portion of this sentence we 
see that what is considered to be proved is the proposition that If 
A be a zero-axial shew determinant , then either 
(I) A = 0 and [?*s] = [sr], 
or ( 2) [rr] = 0 and [rs] = - [sr]. 
t Along with this fact Spottiswoode associates the statements that 
Wi + w 2 + • • .+u n = 0, v l + v 2 + . . . +v n = Q, which are manifestly in- 
correct. 
