1900-1901.] Dr Muir on a Peculiar Set of Linear Equations. 255 
aggregate of terms independent of a 4 are symmetric functions of 
the remaining variables. This implies that the order in which 
a 2 , a 3 , a 4 , a 5 are written in N(a 1 } a 2 ,a 3 ,a 4 ,a 5 ), is of no conse- 
quence. 
(10) Expressing the subsidiary determinant, /( a 2 ,a 3 ,a 4 ,a 5 ), of 
the preceding paragraph in terms of the elements of its first row 
and their complementary minors, we find that the latter have the 
same form a# 1ST, and that the determinant is equal to 
- a 2 N(a 2 ,a 3 ,a 4 ,a 5 ) - a 3 N(a 3 ,a 4 ,a 5 ,a 2 ) - a 4 N(a 4 ,a 5 ,a 2 ,a 3 ) 
- a 5 N(a 5 ,a 2 ,a 3 ,a 4 ). 
There thus results 
o 
N ( a i,a 2 ,a 3 , a 4,a 5 ) = ai D(a 2 ,a 3 ,a 4 ,a 5 ) - ^ | a 2 N(a 2 ,a 3 ,a 4 ,a 5 ) . (iq) 
(11) Again, expanding the said subsidiary determinant in terms 
of binary products of the first-row elements and the first-column 
elements, we find it 
= - aP(a 3 ,a 4 ,a 5 ) - a^D(a 2 ,a 4 ,a 5 ) - . . . . 
+ a 2 a 3 
a 3 
a 4 
a 5 
+ a 3 a 2 
a 2 
a 4 
a 5 
a 3 
1 
a 5 
a 2 
1 
a 5 
a 3 
a 4 
1 
a 2 
a 4 
1 
+ a 2 a 4 
a 4 
a 3 
a 5 
+ a 4 a 2 
a 2 
a 3 
a 5 
a 4 
1 
a 5 
a 2 
1 
a 5 
a 4 
a 3 
1 
a ? 
a 3 
1 
+ 
= - 2a!D(a 3 ,a 4 ,a 5 ) + 2« 2 a 3 (a 2 + a 3 )(l - a 4 )(l - a 5 ) . 
Now it is easily shown that 
- 2dJD(a 3 , a 4 , a 5 ) = - 2 a 2 + - 22aU 3 a 4 a 5 , 
and that * 
Sa 2 a 3 ( a 2 + a 3 )(l - a 4 )(l - a 5 ) = ™ 2 2 a 2 a 3 a 4 + 32 a 2 a 3 a 4 a 5» 
