23 
three Acts, an Interact and an Exodion 
so that the things to be compared are the coefficient of ai in ^ —af+ l ani * 
1 _ x i+i 1 _ af+2 1 _ x i+j . 
the product of the vanishing fractions _ ^ ^ _ ^ 2 -, • ••, 1 * s 
readily seen to be a ratio of equality, so that J x — Xy Q.E.D. 
(24) (B) On the General Term in the Generating Function to Partitions 
into parts limited in number and magnitude, by Dr F. Franklin. 
To prove that the coefficient of ai in the development of 
1 . (1 - xi +l ) (1 - xi+ 2 ) ... (1 - xi +i ) 
(T- a) (1 - ax)(1 - ax 2 ) ... (1 - ax 1 ) 1S (1 -tf)(l -x 2 ) ... (1 -cd) 
I showed that the number of partitions of w into i or fewer parts, subject 
to the condition that the first excess (the excess of the first part over the 
second) is not greater than j, is the coefficient of x w in 
1 - xi +1 
(1 — #)(1 — x 2 ) ... (1 — x l ) ’ 
and in general that the number of partitions in which the rth excess (the 
excess of the first part over the (r — l)th) is not greater than j, is the 
coefficient in 
(1 - xi+ l )(1 - xi+ 2 ) ... (1 - xi+ r ) 
(I — x) (1 — a?) ... (1 - x l ) 
If we look at the question reversely, namely, the coefficient of ai in 
1 
(1 — a) (1 — ax) (1 — ax 2 ) ... (1 — ax 1 ) 
being known to be 
(1 - xi +1 ) (1 - xi+ 2 )... (1 - x'j +i ) 
(1 -¿c)(l -Ct?) ... (1 -at) 
if we ask what is the significance of the fractions 
1 - xi +1 (1 -^ +1 ) (1 -xi +2 )... (1 - xJ +r ) 
(1 - x) (1 -X 2 ) ...{l-x 1 )""’ (1 - x) (1 - x 2 )... (1 - x { ) 
the answer is immediately given by the generating function itself. For 
1 - xi +x 
(1 — x) (1 — x 2 ) ... (1 — x l ) 
1 
1 - xi +1 
(1 — x 2 )(l — a?) ... (1 — x 1 )’ 1—x 
(1-^)(1- 
= co. of aJ in 
1 
(1 — a) (1 — ax) (1 — æ 2 )(1 — æ 3 ) ... (1 — «*)