1899-1900.] Dr Muir on the Theory of Alternants. 
125 
n-m and m + 1 elements respectively, and all permutations of the 
elements of the second class are to be taken. In this expression, 
however, another substitution can be made by reason of tho 
identity 
0 1 n-m - 1 
U(a 0 ,a v . . ., an-m-l) 0 1 * * * ^n — m — 1 
i' ^ 2 j p 
where under the sign 2 all possible permutations of the indices 
0, 1, . . ., n-m -l are to be taken. When this substitution has 
been made, we shall consequently have to take every possible per- 
mutation of both classes of elements. But to take every possible 
separation into two classes and permute the elements of each of 
the classes in every possible way is the same as to take every 
possible permutation of all the elements. Our result will there- 
fore be 
0 1 n-m - 1 ,/ N 
a 0 QL . . . a 1 • <p{an-m,a n -m+V • • •> a n) 
H =* (_l)«m+l)2— p ’ 
if it be understood that under the sign of summation all possible 
permutations of a 0 , a v . . . a n are to be taken : and this is what 
we set out to prove. 
The case where m = n is then considered, because of its special 
interest. The first expression obtained above for H becomes in 
this case 
-7 P . <£(«,„«,, . ■ an ) 
“/'KVK) • • •/(«*>)’ 
where under % all permutations of a 0 , oq, . . ., a n are to be taken. 
Making in this the substitution which is possible by reason of the 
identity 
• • • /(« - (-i)^+ i >P 2 , 
we have 
H = (_l — •> a n ) ; 
The formal enunciation of the result thus obtained is : — 
If <J> be any rational integral function of n+ 1 variables , II their 
difference-product , and f be a function of the (n + l) th degree in one 
