60 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Here the first double cyclical sum is recognised to be 
2Saxm3 ; 
the second 
= 2 ^^ -la^iSaxing — j 1 ~ 1 “■i 73 i ~ I ^ 1^4 1 )j" 
= 2Saxmj . Saxni2 - ^2 + 73 + ^4) ~ + ®-37i + ‘^4^i)| 
= 2Saxmj . Saxiri2 - 2 |g-i^Saxm^ - a^(roWj.cob)| ; 
= - 2^ jas^Saxm^ - a3(roWj.col3)| ; 
the third 
the fourth 
and the fifth 
= - 2^|a^2gaxnij - a^irow^.cob)! . 
Summing these five expressions and noting that 
o 
- 2 ^ ^2 ^ _ ScrSaxnij , 
and that 
2^ |aj(roWj.cob) + a2(roWj.col2) + a2(roWj.col3) + a4(roWj.cob)| 
O 
= 22(a^roWj2 + a2row^.row2 + a3roWprow3 + a^row^.row^), 
o 
= 22airoWi2^ 
= 2Saxm^ . cr , 
we obtain the total 
2Saxni3+ 2Saxnij.Saxm2 - ScrSaxm^ + 2o-Saxm^ , 
which, since Saxm 3 = crSaxm^^ , is equal to 
2SaxnijSaxm2 - 4o-Saxnq , 
as desired. 
(13) The sum of the six determinants formahle from a four-line 
orthogonant of base a- by deleting a pair of columns and inserting in 
their places the corresponding rows is equal to 
Saxni2^ - 2crSaxm^2 + 2 <t^. (XV) 
If the rows of the orthogonant be denoted by 
1,2, 3, 4 
and the columns by 
r, 2', 3', 4', 
