1818,] Prof. Anglin on Theorems mainly Alternants. 393 
cd. . . or' 
7t{c. d e ... X),. 
7T consisting of the n- \ factors {cde . . . l)^, (JAe . . . /),,, . . . , 
{bed . . . k),.j by giving to r the values 1, 2, 3, . . ., n - 2 ; and 
for the former series of functions, the values of A will be 
represented by 
7t(c d e . . . 1 ),. 
where ?’ = 1, 2, 3, . . ., ?i- 3. 
We further observe that if r==0 the value of A for the first of 
the above series becomes ^.n integral form; and if r = 0 and 
n-2 the corresponding values of A for the second series are 
respectively (b + c + d+ . . . +iy~^ and b c d . . . 1, the two 
remaining integral forms. Also the greatest admissible value of r 
is obviously n - 2 since any factor in tt, as (c c? e . . . T), consists of 
only n-2 letters. The two results, therefore, are 
and 
la. .. a”~^, 
lb. .. b^-\ 
la... a”“^, 
17;... b^-\ 
a’’"^(5 cd . . 
■ lt~" 
7t(^c d e . . 
h^~'^{a cd . . 
•0.. 
• iV 
77{g d e . . 
■l)r 
{bed . . 
. . 0"-‘ 
/r+1 
7t(c d e . . 
{acd. . 
■hr 
■ C" 
7r{c de . . 
■l)r 
= be. . .1) 
— ± ^{a be. . . 1), 
where r = 0, 1, 2, . . . , ?i-2; and these include all the integral 
and fractional functions considered above. 
9. We will now establish these two identities, by a general 
method which applies alike to alternants of any order. 
It is obvious that 
(ab c . . . l)^ = a(b cd . . . -\-(b c d . . . Z),. , 
