390 
Proceedings of Boyal Society of Edinburgh, [june 18, 
the coefficients of if (bed) in the above expressions, with similar 
corresponding expressions for B, C, D. 
Further, we shall find the values of the expressions denoted by 
(2345), taken in the order of the first, fourth, third, and second, to 
respectively become 
bedefihede), liffibcde), 
h h 
4 
1 7^l 
n 
1 b 
1 
fifede ) , 
Tiy hc^ 
4 
1 b Z?2 
and Kfifcde) 
1 \ ^2 
0 1 
-f n 
1 h. lies 
0 1 
, where h refers to &, c, cZ, e ; 
while the expression 
Hence we have 
— becomes ] | or f(abcde). 
1 a cd A 
\b b^ B 
Ic c2 a 
1 d dd dd D 
= if(ahcde)^ 
le E 
where A denotes any one of the coefficients of f{bcde) in the 
above expressions, similar corresponding expressions holding for 
B, a, A E. 
Generally, in the case of n letters a,b,c,.,.,l, n — \ 
expressions denoted by (234 . . . n), beginning with the first and 
going in order from the last to the second, respectively become 
(bed . . . l)f{bcd . . . Z), hf~^fihcd . . . Z) , 
h\ 7^2 
n-\ 
1 b 
h\ 7^-2 7^3 
s 
1 b &2 
f{bcd...l) 
1 
-^n 
1 \ 
, ^i{hcd ...1) 
1 ^2 
0 1 \ 
-^n 
1 h^ ^2 
0 1 7ii 
, and fibed . . .1) 
h-^ 7^2 . , . 7i,j_2 
n-l 
1 b W . . . b^-^ 
1 h-^ 7^2 • • 7i,j_3 
1 h^ . . • hn-^ 
. . . 
-^n 
... 
0 0 0... 1,7^1 
0 0 0 . . . 1, 
3 
3 
