of Edinburgh , Session 1885-86. 
827 
c M , d n , e w , . . . l n , that is, m - 2 times, and similarly for the other 
factors in the difference-product of a, b, c, . . . Z, it follows that 
P m = | a°b 1 c 2 d s . . . ir-*?*-- 1 | M 
m — 2 
or Z, c, ... Z) . 
Secondly , when the indices of the minors are not consecutive. 
Before proceeding to the wth order, we shall, for greater clearness, 
first consider a particular case — say, the fifth order. 
In the determinant 
| Wtf 1 1 , 
since from a known result in the theory of alternants the minor 
|&W| = £» (bcde)h' r _ 3 , 
we have by § 1 
Iwwi -.{»(&)*) | (r ^ 4) (r “ 3) |, 
with like results for the other four minors of the determinant. 
Hence the value of P m in 
is equal to P w in 
| a?Wc 2 d r e n | 
| a°b 1 c 2 d i e n | 
multiplied by the product of 
1 a 
(r - 4) (r - 3 ) 
and four like expressions with 5, c, d, e respectively in the first row 
instead of a ; a result which may be represented thus : 
Again, in 
P TO = @(abcde ) . H 
1 
(r - 4) 
a 
(r-3) ' 
| | , 
since it has been previously established that the minor, 
(r - 3)' (r - 2)' 
f | 5°cW | = £(&afe) 
(2-3)' (2-2)' 
which by § 1 
1 a a 2 
(r-4) (r-3) (r- 2) , 
(2-4) (2-3) ( 2 - 2 ) 
= £*( bcde ) 
