126 Proceedings of Royal Society of Edinburgh. [sess. 
variable and beofthe form (x-a 0 )(x -a 1 ) . . . (x - a n ) ; then the 
coefficient of 1 . . . t" 1 in the expansion of 
/_lUn( n +l) n (yti, ♦ • •» tn) • t n ) 
V } m) • • • f (tn) 
is 
V < ^ ) ( a 0? a i> ■ • •> a n) 
^n(a 0 , a i, . . ., a n ) 
effect being given to the sign of summation by permuting in every 
possible ivay the elements a 0 , a x , . . a n . 
As we have seen above that 
^ 4 > ( a 0) a i > • » •» a n) 
^n(a 0 ,a lf . . a*) 
is the quotient of any rational integral alternating function by the 
difference -product of its elements, and that this quotient is often 
in request, it is important for- practical purposes to note that what 
this last theorem of Jacobi’s gives is the generating function of the 
said quotient. 
After giving a line or two to the case where m = n- 1, Jacobi 
returns to the general theorem and specializes in another direction, 
viz., by putting 
<)KVi *»)-#?•,• C 
Division of both sides by <f> is in this case possible, and the result- 
ing theorem is one of considerable importance : — 
The expression 
o 1 
n -m - 1 y 
a n-m-i a n 
X 1 - 
7m 
m+1 
( a i “ a o)( a 2 “ a o) • • • (an-an-i) 
which is the quotient of an alternating function by the difference- 
product of its elements is equal to the coefficient of 
(y+ l) (Vi+ i) ,-(y m -H) 
* • * ‘'rv. 
in the expansion of 
( tp t 1 )(t 0 — t 2 ) . . . (t n - 1 t n ) 
f(t 0 )f( tl ) . . . f(t m ) 
