1914-15.] The Determinant of an Orthogonal Substitution. 57 
(9) The duplicant of a three-line orthogonant whose base is cr is equal to 
2o-V-Saxmi)2. (IX) 
The duplicant 
2ai + ag + 
‘^(^2 /^3 72 
a.3+7l ^3 + 72 % 
of any three-line determinant | I known to be 
2{|«,^27 sI + 2»-A}' 
consequently, if the determinant be an orthogonant whose base is a, the 
duplicant is 
which by (V) is equal to 
and by (VI) is equal to 
2|o-t -1- o-hSaxm^^ - 2Saxni2)| , 
!|ot2 + o-TSaxnij^ _ 2cr=Saxmj)| , 
^.e. 
^.e. 
2|o-t - o-^Saxm^j- 
2(rTo-* - Saxm^)2 . 
(10) The sum of the two-line coaxial minors of the duplicant of an 
n-line orthogonant whose base is <j is equal to 
Saxnq2 ^ 2Saxni2 - na . (X) 
Taking, for shortness’ sake, the case of the orthogonant \ ^ 
see that the sum in question 
2aj 
tt2 + /?i 
2^1 as + 71 ^ 
.. + 1 
74 + S3 
^2 /^1 
2/^2 ' 
«3 + Ti 273 ! 
1 74 + S3 
284 
- la-1^2 + ^“i 73 + • • • • + ^73^4 
— 2tt2/?;^ - - .... - 27^83 - 
4 - af - 1 - af - 1 - af 
+ / 3 ^ 2 + . + 
+ yf+yf+ • + 74 " 
+ V + V + V + • 
= 2 a^l 3 o + 2 a 273 273S4 
-f2 |a^^2l + 2 la^ygl-f- .... + 2 | 7384 | - |4(r - (a^^ + ^^2 .^^2 + g^2)| 
