250 Proceedings of Royal Society of Edinburgh. [sess. 
(2) The general set of n equations having this peculiarity is 
(01+*l) + 
92 x 2 *b 9z x z 4- 
. . . + 
9 n x n = 0 \ 
01*1 + (02 + * 2 ) + 03*3 + 
• • . + 
O 
II 
i 
01*1 + 
02*2 + (03 + *3) + 
9n x n = 0 
01*1 + 
p 2 ^2 "b 9b x s “b 
• • • + (ffn + x n) = 0 > 
and the number of derived sets of n equations in which the same 
function persistently appears is 
1 + C Wil + C n>2 + • • • + Q n<n 
i.e. 2 n . 
These n X 2 n equations may also be viewed as consisting of 2 n ~ x 
expressions for each of the 2 n magnitudes x x , x 2 , . . . , x n , g x , g 2 , 
• • • j 9 n • 
(3) When n = 3 the persistent function, <£(a, /3 , y), is 
a 
P 
7 
1 1 
13 
7 
P 
1 
7 
-4- a 
1 
7 
y 
P 
1 
\ a 
P 
1 
and, generally, 
^( a i» a 2 » a 3 > . . .,a n ) = 
a i 
a 2 
a 3 • * 
. a n 
1 
a 2 
a 3 • 
• • a n 
a 2 
1 
°3 * * 
• a n 
a l 
1 
a 3 . 
• • 
a 3 
a 2 
1 . . 
. a n 
-r 
a l 
a 2 
1 . 
• • a n 
a 2 
a 3 . . 
. 1 
a l 
a 2 
a 3 ‘ 
. . 1 
A study of these two determinants, which are both functions of 
a x , a 2 , . . . , a ?l , and which may therefore be conveniently denoted 
by 
br(a l ,a 21 ...,a n ) and I)(a 1 ,a. 2 ,...,aJ 
is thus desirable. 
(4) Taking D first we see that it may be defined as a determinant 
in which all the elements of the principal diagonal are unity and 
iy, lohich no two non-diagonal elements situated in the same column 
