Dr Muir on General Determinants. 
699 
1907-8.] 
C after omitting a row from the first array and a row from the second, and 
that therefore his second theorem is unnecessary. Further, his mode 
of procedure leads him to an expression for a multiple of the minor, 
namely, for 
(M-m + 1)^ , 
oc rs 
and making an oversight similar to Hesse’s of 1853, he does not divide both 
sides by n — m + 1. 
His third theorem, 
C -- 0C 8C + VC_ 0C SC 0C 
dc rs da rl db st da r2 db s2 3 a rm db sm ’ 
is more worthy of note. The proof of it depends essentially on substituting 
for C in the first factor of each term of the right-hand member its equivalent, 
0C , dC 
-lr— + C r2 ~ + • 
Mrl 
dc r 
. . + c r 
ac 
% bc Tm 
in which, it is important to note, the differential-quotients are necessarily 
all independent of a rl , a r2 , . . . . The said right-hand member can then be 
transformed into 
0C ( h 0C v 0C 
06 sl \ 11 0C rl 21 0C r2 
,7 0C\ 
• • + bmix 
0C rm / 
0C /, 0C 7 0C^ 
00 s2 \ 12 0C rl 22 0C r2 
7 0C\ 
• • + ) 
oc rm / 
SC A. SC , , 0C 
XT-fS!,,— + ■ • 
ob sn \ oc rl oc r 2 
, 7 0C' 
• • + V mn —— 
c6 rm . 
which, if addition be performed columnwise, becomes 
0 + 0 4- ... + C - — +0-i- ... + 0 , 
dc rs 
because of the fact that the theorem 
0C 0C 
C rl^T~ + C r2X 
A 0c, 
+ c. 
00 
0C«« 
when 
| r = s 
( r + s 
holds in reference to the a ’ s and &’s as well as to the c’s — a fact which 
should be noted for other purposes, and which is readily seen to be justifi- 
able if we view C in its composite form AB and bear in mind that the 
operation 
0 
0 . 0 
