1888.] Prof. Anglin on Theorems mainly Alternants. 391 
where h refers to c, cZ, . . . , 1; while the expression 
I | or ^\abc . . . 1) . 
Pm 
1 Z Z2 . . . Z'^-2, L 
where A denotes any one of the coefficients of ^i(b, d, . . . 1) 
in the above n- \ expressions, with similar corresponding expres- 
sions for B, Cy . . . , L. 
7. The foregoing expressions obtained for A are in reality only 
certain of the values which A is capable of having, without inter- 
fering with the validity of the identities. The possible general- 
isation we leave for a future communication j and it will suffice at 
present to note that the identities may be established in a manner 
different from the^ foregoing, and better suited for making 
extensions. 
We observe that there are only two integral functions denoted 
by A in each case. Taking the first of these, we have for the nih. 
order 
Hence we have 
1 a A 
1 b B 
1 c C 
= he ... 1) . . . (IV)i, 
1 a a^, .. . a” - 
1 a . . . , a”"-, bed . . . I 
1 & . . . , acd . . . Z 
a 
X abc . . . I 
111%..., abe ... k 
a a? a? . . . 1 
b b^ b^ . . . 1 
= l^\abc . . . 1). 
I P P .. . p-\ 1 
Taking the second, we have 
