1900-1901.] Dr Muir on a Peculiar Set of Linear Equations. 253 
which shows that it will hold also for the case of the 5 th order. 
The proposition is thus established. 
(6) The number of different kinds of terms in the final expansion 
of the determinant D of the n ih - order is evidently 
1 + 2 + C M 3 , + C n> 4 + • • • • 
which is equal to 
(l + l) ?l - C n>1 i.e. 2 n -n. 
(7) By dividing in every case the p th column by a p there results 
D ( a 1 ? a 2> a 3> a 4> a o ) 
a l a 2 a 3 a 4 a 5 
1 f 
1-1 
a 2 
1-1 
«2 
1 - 
1_ 
*3 
- 1 
1 -1 
1 -I 
a 5 
a 5 
= 4(1- 1 )(4- 1 )(4- 1 )(^- 1 ) 
r = 4 
D(a 1 ,a 2 ,a 2 ,a 4 ,a 5 ) = ^ “ a l)( X “ a 2)( X ~ a s)( X “ a 4)( X “ a s) 
+ (1 - ai )(l - a 2 )(l - a 3 )(l - a 4 ) . 
This when expanded contains a number of unnecessary terms, 
but it is useful as showing that when one of the variables is put 
