1888 - 89 .] Dr T. Muir on the Theory of Determinants. 439 
a t + a x 4 + . 
. . + a k t k + a k+l t k+1 + . . 
. + a. t —u. 
n n ’ 
a' + a/ t x + . 
• • + % V + <v+ A+i + • • 
• + °n *»=“l> 
A + + . 
. . j*> t . 
• • + 7c + a yfc+l Tfc+l + • * 
. + =w 7 , 
ran k ’ 
o (B) i + + . 
, A n) / i /y( W ) / i 
• • + a k h + a k+ 1 **+i + ’ ' 
. + 
n n n i 
gives rise to the system 
A u + A Ul + . . 
■ + \ u k + \+ 1 u k + i + • • ■ 
■ + A w ^ n = R . t 
A' u + A^ u x + . . 
• + V u k + A ’t+i u k+ 1 + ■ • • 
+ A u — R . t 
n n 
A + A<\ + . . 
• + A i ’“i + A *+l “i+1 + • • • 
+ A 9^U = R . t. 
n n k 
A” M + AJ°«] + . . 
■ + A k\ + A *+i“*+: 1 + - • • 
+ A w w =Y..t 
n n n 
in which 
R = 2 ± aa' x , 
4’°> A n ) = ^± aa \ 
a (w_1) 
a (n-\y 
Then taking only the first k + 1 equations of the first system and 
eliminating t, t v . . . , 4-u there is obtained 
C A +C WA + 1 + • • • • +C Jn = I)u + D l“l + • • • + ( X ) 
where the multipliers D, D 1? . . . , D*, by which the elimination is 
effected, are 
(-1 
( - l) 7m 2 ±aar" a k-i , 
+ 2±«a / 1 a" 2 ...a ( ^J ) , 
and consequently by denoted 
, > » (*-i) (fe) 
2,±aa 1 a 2 - a \ > 
2 ± aa\a '\. . > 
2±aa> // 2 ...« ( ^J ) a ( * ) . 
