1914-15.] The Determinant of an Orthogonal Substitution. 61 
the sum of the six determinants in question may be conveniently 
represented by 
I 1'2'34 1 + I 1'23'4 1 + I 1'234' | + 1 12'3'4 1 + 1 1 2'34' 1 + 1 123'4' ] . 
But by (!') of § 2 this sum 
roWj 
coh 
-t- 
row^ 
C0I4 
+ ....+ 
roWg 
colg 
row2 
C0I2 
rowg 
0 
0 
w 
row^ 
C0I4 
ttj a,2 0.3 
ft/^2^3/^4 
«2/^272^2 
71727374 
^1^2^3^4 
“3/^373^3 
^4/147434 
I ^ 1/^2 P i 1 ‘ I *^i72 I 1 “ 1/^4 1 * I ^1^2 I • • • • + i “3/^4 j • 1 yjS2 I 
+ I a^j2 \ • I I + 1 a^yg p + | a^y^ | • | | + ••••+! agy^ j • I y^Sg 
+ 
+ ! 7i^2 I • I ^ 3/^4 I + I 71^3 i • I «374 1 + I 7 i^4 I • I ^"3^4 I + + 
^ I *^ 1/^2 1 ^ ■*" "[1 ®l/^3 1‘1 ^^i 72 Ij" > 
{2 I “ 1/^2 I } - {i l‘l ""i73 1 } + ^2, {I H ^i 72 i} 5 
f J 11^172 ii<^i 72 n- 
73^^ 
12 
And as the first part of this is the square of Saxm2 , and the part to be 
subtracted is double the sum of the two-line coaxial minors of the second 
compound of | I ’ fbe result desired is reached with the help of (XIII). 
(14) The dwplicant of a four -line orthogonant of base a- is equal to 
(Saxnig - 2o-)2. (XYI) 
The duplicant of any four-line determinant | known to be 
2 I 1234 1 -1-22 I 1234'! 
and therefore, if the determinant be an orthogonant whose base is cr, the 
duplicant is by (VIII) and (XV) equal to 
2(7^ -f 2a-(Saxmj2 - 2Saxni2) 
+ (Saxni2^ — 2crSaxm4^ -1- 2cr^) , 
i.e. 
fSaxnig^ - 4orSaxni2 + 4or^. 
(15) The sum of the coaxial minors of every order in the case of 
2ttj 
^2 /^] 
«3 + 7i 
a 4 - f 3 i 
«2 + ft 
2/^2 
A's + 72 
/^4 + ^2 
«3 + 7i 
/^3 + 72 
273 
74 -<-^3 
A + ^2 
74 + ^3 
2S4 
