1901 - 2 .] Dr Muir on the Theory of Jacob ians. 
187 
and that therefore, if the factor B on the right-hand side do not 
vanish, the other factor 
^ f 0/",, 
m+ 1 ^n+m 
must vanish, he takes some pains to obtain a new identity and 
then applies this very reasoning to it. Denoting the cofactors of 
• . , K > he 
6, b 
i> 
6„in 2±W r ... C 1 ’ b y*. X 
points out that as none of the (T s involves f n the same is true 
of the A’s. On the other hand, the b' s do involve f n , but only 
one of them, viz., b k , involves the differential-quotients of f n with 
respect to x n+7c , this differential-quotient being in fact the last 
element of all and having B for its cofact n\ In this way it appears 
that on the left-hand side of the identity 
the cofactor of — 
dx 
is Ak-B . 
If in the same manner we take ^ + 
n+k 
0/ 
dx„, dx, 
Vi 
denote * the cofactor of 
dx*. 
m+1 
dfn_ 
dx n . 
and 
in it by p k , it must follow that on 
but while c x occurs in each element of the first row on the left, and d 2 
similarly in the second row, e 3 does not so occur in the third, and consequently 
the cofactor of c x d 2 e 3 on the left takes a different form from that given by 
Jacobi. 
The first derived theorem in its general form may be enunciated as follows : — 
If there be two determinants D and A of the n th order such that the last n - m 
columns of D are the same as the first n - m columns of A, and if there be 
formed a square array of new determinants by supplanting each of the first m 
columns of D by each of the last m columns of A, the determinant of this 
square array of the n\ th order is equal to 
(_l)m(w+l) Dm— iA . 
To illustrate the second derived theorem we may equate cofactors of f x g 2 
where we formerly equated cofactors of e-J^g^ the result clearly being 
| a x b 2 c 3 d 4 e 5 | 
| a 2 b 3 c 4 d 5 | 
1 a x b 3 c 4 d 5 | 
| a x b. 2 c 3 d A e 6 ( 
I a 2 b 3 c 4 d e | 
I h c 4 d 6 1 
| a x b 2 c 3 cl 4 e 7 | 
| a 2 b 3 c 4 ^7 I 
| a x b 3 c 4 d 7 | 
- I (hb 2 c 3 d 4 | 2 . | a 3 b 4 cyd 3 e 7 1„ 
The next of the series would be got by equating cofactors of g x . 
* This is not the same as putting, with Jacobi, 
V_L§^_ 3/l dU _ §/» dfy dfn 
dx m ■ dxm+ x ■ ‘ ' dx n+m ^dx n + ^ 1 dx n+1 + * * ' + dx n+m 
for the determinant on the left being of the (%+l) th order there should be 
n + 1 terms on the right instead of m + 1. 
