424 
Proceedings of Royal Society of Edinburgh. [sess. 
/= S, 
II 
o 
II 
O 
II 
JO 
O-J 
il 
JO 
/= 3, 9 = 1, 
II 
o' 
II 
o' 
Jl 
cT 
II 
t—H 
II 
II 
JO 
II 
O 
o-j 
II 
JO 
II 
qo 
II 
o 
II 
i— < 
?S*> 
II 
JO 
II 
JO 
II 
o 
II 
h= 1, 
II 
JO 
IT 
JO 
/- 1, 
II 
JO 
II 
JO 
7c = 1, 
OnJ 
II 
JO 
o' 
II 
o' 
II 
II 
JO 
o 
If 
i-i; 
et par suite, une fonction alternee du cinquieme ordre renfer- 
mera sept especes de termes.” 
The next question considered is as to the number of terms of a 
given cyclical species which exist in any determinant of the n iYl order. 
The species being characterised by / groups of one index each, g 
groups of two indices each, h groups of three indices each, &c., the 
required number of terms is denoted by 
N /, • 
Now all the terms of the species will certainly be got if we 
write in succession the various permutations of the n indices 
0, 1, 2, 3, . . . . , n - 1, and then in the usual way mark off each 
permutation into the specified groups, viz., first / groups of one 
index each, then g groups of two indices each, and so on. As a 
rule, however, each term of the species will, in this way, be obtained 
more than once. For, if we examine in its grouped form the 
particular permutation, which was the first to give rise to a certain 
term, we shall find that changes are possible upon it without entail- 
ing any change in the term. For example, the set of groups 
(0) , (1), (2,3), (4,5, 6), 
instanced above as corresponding to the term 
^ 0 , 0 ^ 1 , 1 ^ 2 , 3 %, 2 ^ 4 , 5 ^ 5 , 6 ^ 6 , 4 ’ 
might be changed into 
(1) , (0), (2, 3), (4, 5, 6) 
or (1), (0), (3, 2), (6, 4, 5) 
or 
which, while still corresponding to the term 
<* 0.0 <* 1,1 <* 2,3 <* 3-2 <* 4,5 <* 5,6 <* 6,4 
