Dr T. Muir on the Theory of Determinants. 
541 
^ I ^lV.S ^4 l .l I 
I «lV 3 ^ 4^5 I I «lV 3^4 I I «lV 3 ^ 4^5 I 
_j. I 0^1 Vs i I Q^ 1 ^ 2 ^ 3/4 I 
I I I ^ 1^3 ^4 I 
^ I (^A l . l Q^ 1%/3 I 
I 1 I I 
_j_ .1 ^1/2 i 
64 1 0^462 i 
/i 
The general result, as stated by Schweins, is that 
(-) 
^w+w+l\ 
”+’ II B Aj . . . A„_iA„+i. ■ ■ 
ai . 
A, 
^w+m+1 
^n+m+ly 
_ "y _j_ -^0*+^) 
. vm+1 j 00 ,^(n+m+l) 
-^m+r ^ ’ 
where L 
and = 
ai . 
(w) = ll ^1 
A„_iA^ 
0 
A, 
ai . 
Ai 
^W+JW-l\ ’ 
A ) 
— 1 / 
^n+m - 
^-f-m \ 
■h-n+m-l ^ / 
ai . 
Ai 
^n+m \ 
•h-n-\-mj 
■ (W-) 
Since the expression thus expanded is itself one of the L’6-, viz., 
Lm+ 2 — readily seen by transferring the B from the beginning 
to the end, and denoting it by — and since Lo”^=l, the 
identity may equally appropriately be written with 
end of the right-hand member, and looked upon as the recurring 
law of formation of the L’s in terms of the Y’^. This Schweins 
(71) (71) 
does, giving indeed the result of solving for • 
The Second Section, consisting of five chapters, and extending 
to 30 pp., is devoted to a special form of determinants, viz., those 
already partly investigated by Cauchy, and afterwards known as 
alternants. 
