56 
Proceedings of the Poyal Society of Edinburgh. [Sess. 
gives 
“1“ ^3 1 + I ^1^3 1 + . . . . + 1 1^ , 
as desired. 
(6) In an n-line orthogonant of base a- the sum of the coaxial minors 
of the order hears to the sum of those of the (n — order the ratio 
1 ; Or, in symbols, 
Saxm^_,. = or=”~’'SaxTn^ . (VI) 
This follows from the relation which exists between complementary 
minors of an orthogonant. When n is 6 the examples are 
Saxnig = cr® 
Saxnig = o-^ . Saxnij 
Saxm^ = or . Saxm2 . 
(7) If the rows and columns of an orthogonant of base a- be denoted by 
^ 1 ’ ^2 ’ • • • • 5 n 5 
Cl , C2 , . , C,„ , 
and the rows and columns of its adjugate be denoted by 
then 
j ’ • • • • 5 , 
2(K.iCi = O'” ^(Saxnii^ - 2Saxni2) . 
For, since any element of the adjugate is equal to 
a,,, 
it follows that 
and therefore 
■h-rsh-sr ^ • ^rs^sr 5 
2EA = 
whence with the help of (V) the desired result is obtained. 
(VII) 
(8) The sum of the n determinants formable from an n-line ortho- 
gonant of base cr by deleting a column and inserting the corresponding row 
is equal to 
cr=”~^(Saxm2 “ 2Saxnii) . (^Iff ) 
By (I) the sum in question is 
^^n-i(roWj.coli + row2.col2 +.... + row„.col„) , 
to which, again, it is only further necessary to apply (V). 
