120 Proceedings of Royal Society of Edinburgh. [sess. 
If f> be any rational integral function of m + 1 variables , II tlieir 
difference-product , and f be a function of the (n + l) th degree in one 
variable and be of the form (x - a 0 )(x - a x ) .... (x-a n ), then 
when m > n no single term of the expansion of 
n^,^, . . t m )^>(t 0 ,t 1 , . . ., t m> 
f(t 0 )f(t 1 ) f(t m ) ' 
according to descending powers of t 0 , t 15 . . ., t m , can contain nega- 
tive powers of all these variables. 
To prove it, lie of course uses the identity 
1 . 1 
f(x) t,e (x-a^x-aj . ... (x- am) 
1 + 1 1 
f(a 0 ) . (x - a 0 ) f\af . (x - a ,) + + f{a m ) . (x - a m ) ’ 
and thus changes the expression into the form 
v ( a o) • (*o a o ) f ( a i) • (*o a i) 
f 1 1 
1/ (« 0 ) • (^1 — a o) f ( a i) • (^1 “ rt l) 
1 ) 
f {fm) • (^o ~ a m)^ 
+ - -1 l 
f\<*>n) • ( ti ~ Ct n ) i 
< 1 1 1 } 
x \f\a 0 ) . (t m - a 0 ) + f(a 1 ) . (t m - af + ’ * ' ’ + f\a n ) . (t m - a n )\ . 
He then says that the result of performing the multiplication 
of these bracketed factors is to produce terms of the form 
Uf 
f\a)f\b) . . . f\p) . (« 0 - «)(<! -b) ... (t m -p) ’ 
where each of the m + 1 quantities a, b, . . ., p is necessarily one 
of the n + 1 quantities « 0 , a v . . ., an, and where, therefore, on 
account of m being greater than n, the quantities a, b, . . ., p can- 
not be all different. But terms of this form can be changed into 
f n \ 1 _ 1 ) 1 
f(a)f(b) . . f'(p) ’f-b-f + altQ-a t x - b j ‘ (t 2 - c)(t 3 - d)...(t m -p) 9 
which shows that in the case of two of the quantities a, b, . . ., p 
being alike, say a and b , the second factor would become 
n 
