58 Proceedings of the Koyal Society of Edinburgh. [Sess. 
= 2 |ai/?2| + 2|ai73|+ .... +217384! - | 4 or-(ai + ^2 + 73 + ^ 4 P| 
= 2 Saxni 2 - 4 (t + Saxm^^ , 
as desired. 
( 11 ) For the investigation of the duplicant of a four-line orthogonant 
two properties of the general determinant of the fourth order are useful. 
The first is that if \ a4b2C3d4 [ he any four-line determinant, then 
o^iAi -h 62B2 + ^303 + 
= l«lV 3 ^ 4 l + I I I l^ 3^4 ! I + |l«F 3 llM 4 l| + | I « A I I Vs I ] * 
In proof of this nothing seems available save an inelegant verification. 
The second is that the sum of the tiuo-line coaxial minors of the 
second compound of any four-line determinant | a^bgCgd^ j is equal to 
Saxm^.Saxm3 — Saxm^. (XII) 
The second compound in question being the six-line determinant 
«1&3 1 
. . . 
1 1 
«l'^2 1 
1 ^F3 1 
■ i <*3<’4 1 
1 
1 ^1^3 1 
• . . 
• 1 «3'^4 1 
the coaxial minors to be summed are fifteen in number, namely, 
1 1 aA 1 1 “ 1^3 1 
l«i^2l l«AI 1 
1 1 ^2^4 1 1 ^3^4 1 
1 1 1 1 ^1% 1 
J 
IM 2 I l«l<^4l 1- • 
. . . , 1 1 c^d^ 1 1 eyd^ j 
These are equal to 
«i I I , af af^d^ | , . . . . , df\ b^c^d^ | , 
every one of the fifteen, except three, being expressible as the product of a 
diagonal element of the original determinant and a coaxial three-line 
minor of the same ; and the three which are not so expressible being 
l«A! 1 ^ 3 1 
I ^ 1^4 I I ^ 2^3 
I I cijh^ I I \ I I I I 
I ! ^1^2 1 I ^3^4 I 5 I I ^1^3 1 1 '^''2^4 
Without the latter three the sum is 
^l( • + ^2 + C3 + II4) 
+ &2(A + • + C3 + D4) 
+ 1 ^ 3 (A + ^ 2 + • +^4) 
+ + ^2 + C3 + . ) ; 
and, since for the sum of the said three we can by (XI) substitute 
cijAi + ^ 2^^2 4* ^ 3^3 + ^4114 ~ 1 (ifcpgdj^ I j 
