184 Proceedings of Royal Society of Edinburgh. [sess. 
Next, - /3j£f being used to stand for 
y\ ± m <fi ¥i-i # 3/„-i 
^ “ 0a? 0a; 1 0^-i bx n+k dx i+1 dx n _f 
it is sought to find an equivalent for % ± /3/3 S .... Clearly 
- ft® is the determinant formed from B by using x n+li as an inde- 
pendent variable instead of % ; but for our purpose it is of more 
importance to note that it is the cofactor of in b k \ for then 
dx { 
the determinant under consideration, viz., 
2±/< 
P. 
i(m) 
being thus the cofactor of 
<fn ' 04+1 
dx dx 1 
n+m 
dx r , 
in the t left-hand member of the identity just found, it only 
remains to seek the cofactor of the same expression in the right- 
hand member. Now, in the first factor, B m , of this right-hand 
member the expression does not occur at all, and in the other 
factor, which is transformable into 
( _ 2yi(m+l) 
df 0/j 
dx, 
m+l bx m+ 2 
If n—\ bj n . . . ‘bJ n+m 
bx m+n dx dx m 
I % e 5 f 6 I | a 2 e 5 f 6 1 
I ^1 e 5fe I I <%/6 1 
I C 1 e 5fe I I C 2 e h/6 I 
I ^1 I I ^2 e oR I 
I «3 I I a 4 ! 
I 4 05 R I I 4 I 
I c 3 e 5 / 6 | 1 a 4 e 5 f 6 I 
I 4 e s/6 I I 4 e 5/6 I 
I ®i02 e 3^4 g 5/6 I * I g 5/6 I 3 . 
where it is viewed as an extensional of the manifest identity 
a x a 2 a 3 a 4 
h h h h 
C 1 C 2 C S C 4 
d x d 2 d 3 d 4 
I a x b 2 c 3 d 4 | . 
The theorem in its general form maybe enunciated as follows : — If from 
the determinant 1 ai, n +m+i r 1 there be formed all minors of the (n + l) th order 
which have | a ]n I for the cofactor of their final element , and these be orderly 
arranged in square array , the determinant of this square array of the (m + l) th 
order is equal to 
I U\n I m • | a\ji+m+\ I • 
