502 Proceedings of Royal Society of Edinburgh. [july 18, 
S 4 ( ± a 1 b 2 ) is an impossibility, as when there are four indices a 1 b 2 
does not satisfy the condition required of a typical term ; indeed, 
Cauchy notes that the number of indices in any term must either be 
the total number or 1 less. 
The number of permutations being even, it is clear that the 
number of + terms K a , K^, . ... is the same as the number of 
negative terms Ka, K^, (x. 2) 
a generalisation of a remark of Vandermonde’s. 
Further, since K a , Kp, . . . are all the terms that arise from an 
even number of transpositions, and Ka, K m , .... all those that 
arise from an odd number of transpositions, it is plain that any 
single transposition performed upon each of the terms of the 
function 
(K a + K/3 + K y + . . . . )-(Ka + K m + K v + . . . . ) 
must change it into 
(Ka + K^ + K„ + . . . . ) — (K a + K|S + Iv y + . . . . ) 
— this is, in fact, the proof that it is an alternating function — con- 
sequently each of the parts 
K a + K|3 + Ky + ...., 
Ka + K^ + K v + ...., 
belongs to the class of functions which have only two different 
values. 
Also it is evident that if throughout the function any particular 
index be changed into another and no further alteration made , the 
resulting expression must be equal to zero , (xii. 5) 
a theorem regarding alternating functions which is the generalisation 
of a theorem of Vandermonde’s. 
We have lastly to note, that the criterion which determines 
whether a particular K belongs to the class K a , K^, .... or to the 
class Ka, K^, .... is incidentally shown to be reducible to a more 
practical form. For example, if the term be K 0 , and it be derivable 
from K , say, by the change of the suffixes 1, 2, 3, 4, 5, 6, 7 into 
3, 2, 6, 5, 4, 1, 7, that is to say, in Cauchy’s language by means of 
the substitution 
1. 2. 3. 4. 5. 6. 7 
