Certain Determinant Identities Arrived at hy H. v. Koch. lOS 
(6) Eeturning to (X), and recalling that 
A 
, i. e. for (x,j)g, 
we readily transform the equality into 
^(1) q(2) 
Ap Ap+i 
A second simplification is effected by using for Aq the notation |aig|, 
which needs no definition ; and a third would be got if the C's could be 
replaced by something more readily recalling the fact that C^^'"* is the minor 
of Ap+r got by deleting the last row and the^th column, or the minor got 
by differentiating Ap+j-with respect to ap+r,p- The symbols 
suggest themselves for this, and, adopting the first, we obtain 
K^p)7i = {Xp)p r — I {Xp + i)p+'i H r- n {Xp + 2)p+2 
I'^ip I l^-i, p + 1 1 
-.... + (- 1 
which is readily seen to contain nothing but determinants, ranging from the 
j9th order to the ^^.th, and all of them minors of the array 
Ctj^-j^ .... dlji 
djil .... CLnn '^^w 
Also, the same must be true of (Y) and (Z), since the only fresh symbol- 
appearing in these equalities is Bp, which we have seen defined as a minor 
of \ain\ ; in fact, the array with which (Y) and (Z) are concerned has one 
column fewer than the previous array, being the array of lai„.| 
(7) Having these facts in view, one naturally asks how (X), (Y), (Z) 
can be established by using merely the properties of determinants ; and 
clearly a helpful reply is possible when one recalls that particular instances 
of the case of (X) where is 1, namely, the case 
+ 
^12*1^11^2! 
1*^11^22 • • 
1 ~ «11 
^lli^ll"22l 
*^12^23 
1 ct-j^-j^cx.^Q'Z/'g 
"ll'^22^33 
have more than once turned up in past work, and been proved in the way 
desired. One such case, for example, has been used by Kronecker* 
and another by Grlaisher.f Further, in regard to the latter, there is on 
* 'Monatsb. Akacl. d. Wiss.' (Berlin), Jahrg. 1874, pp. 2]4-5. 
t ' Monthly Notices E. Astron. Soc.,' xxxiv (1874), pp. 311-334. 
8 
