252 Proceedings of Royal Society of Edinburgh. [sess. 
the elements of the last row and their complementary minors, it 
is readily seen to he 
1 
a 3 
a 4 
1 
-« 2 
1 
a 3 
a 4 
1 
~ a 3 
1 
a 2 
a 4 
1 
” a 4 
1 
a 2 
a 3 
1 
a 2 
1 
a 4 
1 
a l 
1 
a 4 
1 
a l 
1 
a 4 
1 
a l 
1 
a 3 
1 
a 2 
a 3 
1 
1 
a l 
a 3 
1 
1 
a l 
a 2 
1 
1 
a l 
a 2 
1 
1 
a 2 
a 3 
a 4 
1 
a l 
a 3 
a 4 
1 
a l 
a 2 
a 4 
1 
a l 
a 2 
a 3 
1 
where the cofactors of cq, a 2 , a 3 , a 4 are like functions of a 2 , a 3 , a 4 ; 
cq , a 3 , a 4 ; oq , a 2 , a 4 ; cq , a 2 , a 3 respectively. Taking any one of 
them, say the first, we see that it is transformable into 
1 - a 2 a 3 - 1 
1 -a 3 a 4 — 1 
1 -a 4 
a 2 a 3 a 4 1 
and therefore 
= (l-a 2 )(l-a 3 )(l-a 4 ). 
The subsidiary determinant above referred to is thus seen to be 
= - 04(1 -a 2 )(l -a 3 )(l -a 4 ) - a 2 (l - a 3 )(l - a 4 )(l - cq) 
- a 3 (l - a 4 )(l - a 1 )(l - a 2 ) - a 4 (l - cq)(l - a 2 )(l - a 3 ) , 
= - 2 4 «i + - 32 4 a ia2 a 3 + 4cqa 2 a 3 ct 4 \ (S 2 ) 
and consequently we have 
D(a l ,a 2 ,a 3 ,a 4 , a 6 ) = a 5 ( - 2 4 cq + 22 4 cqa 2 - 32 4 cqa 2 a 3 + 4a 1 a 2 a 3 a 4 ) 
+ D(a 1 ,a 2 ,a 3 ,a 4 ) . 
If, therefore, the proposition hold good in regard to the case of 
the 4th order, that is to say, if 
D(cq,a 2 ,a 3 ,a 4 ) = 1 - 2 4 cqa 2 + 22cqa 2 a 3 - 3a 1 a 2 a 3 a 4 , 
— and this is easily verified — we shall have 
D(a 15 a 2 ,a 3 ,a 4 ,a 5 ) = 1 - ( 2 ^^ + a- 2 ^) + 2 ( 2 4 a 1 a 2 a 3 + a 5 2 4 a 1 a 2 ) 
- 3(a 1 a 2 a 3 a 4 + a 5 2 4 cqa 2 tt 3 ) + 4cqa 2 a 3 a 4 a 5 , 
= 1 - 2 5 a 1 a 2 + 22 5 a 4 a 2 a 3 - . . . . (§ 3 ) 
