1914-15.] The Determinant of an Orthogonal Substitution. 59 
the condensed total is 
+ &2 + Cg + + lb + Cg + D4) - I a^^gCgd^ I , 
as was to be proved. 
In the special case where the four-line determinant is an orthogonant 
whose base is cr, it is known that 
Aj + I >2 + Cg + + 62 + Cg - 1 - , 
and 
I I = Cr^ y 
Consequently the sum of the fifteen coaxial minors in question is then 
o-(Saxmi2 _ ^ (XIII) 
(12) The sum of the three-line eoaxial minors of the duplicant of a 
four-line orthogonant of base a- is equal to 
2Saxmj(Saxm2 - 2cr) . (XIV) 
If the orthogonant be | 1 ? fbe first of the coaxial minors in 
question is 
+ f^i 0-3 + 7i 
+ 2^2 A + 72 
I “3 + 7l /^3 + 72 ^73 j 
and the three others are got by performing simultaneously the cyclical 
substitutions 
1,2,3, 4 2,3, 4, 1 
o 
— an operation which we may denote by S. Now, by the theory of 
duplicants the said leading coaxial minor is equal to 
2 I «i^ 273 I + 5 ^1 j 7i ^ ! /^27 s 1 5 ■“ 1 /^lys I > I /^i72 1 ) 
+ ^(^2 j ^2 ’ 72 ^ “ 1 “-273 I ’ 1 ®'-i73 1 ’ “ I ®'i72 1 ) 
+ 2 (ttg } /I 3 5 73 ) ! «-2/^3 ! » “ I I > 1 ^1/^2 i ) i 
consequently the sum of this and the three others is 
I ®"i/^273 I + ^2 I 73^4 I + I /^2^4 ! + I ^73 I )} 
- |a2( 1 a284 I + I tt2yg | )| 
-22{a3(|a3bl + |a3/?2l)} 
“2{'"4(|a473l + l“4/^2l)}- 
