(101) 
NOTE ON CERTAm DETEEMINANT IDENTITIES AEEIYED 
AT BY H. V. KOCH. 
By Sir Thomas Muik, LL.D. 
(1) In Koch's first paper on the solution of an infinite set of linear 
equations,* he devotes a section (§2, pp. 114-117) to the establishment of 
certain identities which he required for use in connection with the main 
subject of his investigation. These identities, for some reason — probably 
because of the medium of publication, or because the title of the paper gave 
no indication of them — have not received from students of determinants the 
attention which they deserved. 
(2) The first of them appears as a deduction from the solution of two 
sets of linear equations, namely, the set 
^ r — n 
arlXi + ar29:2 +....+ arn'Xn = Ur \ 
J r = 1 
and the set 
ttrlXi + ar2X2 +....+ ar„Xn + + + i = Ur + l [ 
jr=l 
which are seen to be so related that the first set may be viewed as f ormable 
from the second by striking out from the latter the last equation and the 
last term of all the other equations. In the case of two sets of equations 
so related, it is clearly necessary to be able to distinguish between the value 
of an X in the one set and the value of the same x in the other set. Let us 
therefore denote by 
the value of Xp in the n-line set of equations ; we can then assert that 
Koch's first subject for inquiry is 
and that his initial lemma is 
(Xp)n + \ - (Xp)n = (-1)'^-^ + !. _P . (Xn + l)n + l 
An 
where 
* "Om uppldsningen af ett system lineara likheter mellan ett oandhgt antal 
obekanta," 'Ofversigt af K. Vetenskaps-Akad. Forhandl.' (Stockholm), xlvii, pp. 109- 
129. 
