1888.] Prof. Anglin on Theorems mainly Alternants. 387 
K(/)(&, c, d, e), c, d, e), K<^(a, h, c?, e), 
~K(f)(a, b, c, e), K<^(«, b, c, d), 
where K denotes j a^p-^c^d^e^ and where 
then 
fK(n h r d\ = I I 
^ ' I «0^2^3^4 I I ! I S^1^2^4 I I «0^1^2^3 ! * 
(2345) - (1345) + (1245) - (1235) + (1234) 
I «i^2^3^4% I ^ 
1 «0^2^3^4% I 1 I I «0^1^2^4% I I «0^1^2^3^5 I I I 
(HI)'. 
3. It is thus evident that the theorem is a general one, and 
(employing elements formed from the n letters a^b^c, . . . 1) 
may he enunciated as follows : — 
If (123 . . . n-1) denote any one of the following n expres- 
sions 
K^(6, c, c?, . . . Z), 'Kcfi(a, c,d,. . . Z), K(f>(a, b,d,. . . Z), 
...... Kcfi(a, b,c, . . . k\ 
where K denotes | . . . Z„_i and where 
<jf)(a, b,c,...k) 
I ^1^2^3 • • • ^’w-l I ^ 
n I a^pif^ . . . k^_^ I 
n I ajb^e^ . . . kn_-^ | consisting of n~l factors formed by taking 
- 2 together the suffixes 1, 2, 3, . . then 
(234 . . . 72) -(134 . . . 72) -i- (124 
- +{-iy-^ (123 
72 ) 
. . 72 
1 ) 
■ ln\ 
n I 
ln\ 
(IV)', 
n I . . . ln \ consisting of n factors formed by taking 72 - 1 
together, the suffixes 1, 2, 3, . . . , 72. 
4. If in the theorems (I)', (II)', (HI)', • • • , (IV)' we put the 
last of the series of elements &o, Cq, . . . in each case equal to 1 
and the others each equal to zero, all the expressions involved in 
each theorem reduce to those in the first set of theorems (I), (II), 
(III), . . . , (lY) respectively of the former paper ; while the last 
expression denoted by (123 . . . ) in every case becomes infinite, 
and is therefore inadmissible. By this substitution, therefore, the 
