252 Proceedings of the Royal Society of Edinburgh. [Sess. 
then — ~ is the leading element in the determinant reciprocal * to the 
square of the determinant 
b + x 
c i 
0 
0 
... 0 
0 
d 1 
b ± + x 
C 2 
0 
... 0 
0 
0 
d 2 
b 2 + x 
C B 
... 0 
0 
0 
0 
d% 
b 3 + x 
... 0 
0 
0 
0 
0 
0 
• • • d n b n 
+ x 
where c p c 2 , . 
. . d v 
d 2 > • • 
. are 
any 
quantities 
such 
^2^2 — ^2’ " ’ * Cnd n — a n . 
We shall now proceed to deduce an explicit expression for 
we consider the equations 
Y 0 = (& + x)z 0 + ' 
Y 1 = d i z 0 + (b 1 + x)z 1 + c 2 z 2 
Ym d 2 z 1 + (b 2 + x)z 2 + c 3 z 3 
Y n d n z n _ x -f- (b n "l - xi)z n 
0 = (b + x)y 0 + C\Vi — % 
0 = d^y 0 + (6 X + x)y 1 + c 2 y 2 - z 1 
(15) 
c 1 d 1 = a 1? 
dS t ij 
~dx ' 
(16) 
0 = d n y n _ i + (b n + x)y n - z n 
it is evident that (Y 0 , Y 1? Y 2 , . . . Y n ) are derived from (y 0 , y lf y 2 , . . . 
y n ) by the substitution which corresponds to the square of the matrix 
b + x c 1 0 0...0 0' 
d x b Y + x c 2 0 ... 0 0 
0 d 2 b 2 + x c 3 . . . 0 0 
0 0 0 0 . . . d n b n + x 
(17) 
and therefore the leading element in the reciprocal of the square of this 
matrix is the value of yf Y 0 derived from equations (16) when Y v Y 2 , 
* If D = | a n , a 22 , a 33 , . . . a nn | is any determinant, and if A rs denotes the co-factor of 
aZ in D, then the determinant | A u , A 22 , A 33 , . . . A nn | is called the adjugate of D. This 
nomenclature follows that of Cauchy’s original memoir, and is always used by Sir Thomas 
Muir in his extensive writings on determinants : some writers have improperly used the 
word reciprocal in the sense of adjugate. We use the word reciprocal to signify the de- 
terminant I ^2, ^3, . . . I , which is the determinant of the matrix reciprocal to the 
matrix (ot’u , (^22 ? • • • ® nn )• 
