638 
MAJOR P A. M ACM A HON ON THE 
1 
1 
1 —x .1 — ax.l —ax 1 .1 — ax i 8 .1 • 
■ctx i 
1 
+ 
ax 
+ 
ax~ 
+ 
ax 2 (1 — aa?) 
(1 — ax *') 4 1 _ 1 — x 1 —ax 1 —ax 1—ax.l— ax 2 
ax'' (1 — ax*) 2 
ax? (1—ax?) ox? ( 1 — ax 4 ) 
~T ,-“A-ITi + ZTN 13; + 
1 —ax. 1 — ax 2 
1—ax.l —ax? 
1 —ax . 1 —ax ’ 2 .1 — ay? 
The fractions, in the brackets { }, may be united in batches, but I prefer to leave 
them as written, as the law of development is shown the better. Also the fraction 
-—-—- may be cancelled if desired. I have not done so, in order to keep in touch 
with the arithmetic. 
Inspection of these expansions establishes the isolation therein independently. 
Art. 26. In the function 
_1_ 
1 — x . 1 — ax . ] — ax' 2 .... 1 — ax‘ 3 ’ 
the coefficient of a? 1 is, as is well known, 
1 — X e J . 1 — Xfi + l . ... 1 — X ( ‘ +p ~ 2 , 
- 
1 — X . 1 — X 2 .... 1 — X p ~ l 
Hence the coefficient of a? 1 x p ? b in the former is 
equal to the coefficient of 
xP?i b in the latter ; i.e., to the coefficient of x b* b(? b hi 
1 — x «. 1 — x'i +l .... 1 — xi + p~ 2 
(1 -~xf I - x 2 . ... 1 - xp~ 1 ’ 
which we know to be 
P + Q ~ 
V ~ 1 
, verifying our result. 
For a given value of s the partitions are enumerated by the coefficient of a’ I b s x p ' 1 in 
the product 
-—(i + ~^ x 
1 - x \ “ 1 - 
ax 
1 + 
abx 2 
1 — ax- 
abxP 
— axP 
or, the same thing, by the coefficient of (ax p Y 1 in the function 
a s ~ l . x ki + h + ■ ■ ■ + *•-1 
- --- 
1 — x . 1 — axP r, (1 — ax k ?) (1 — ax k *) ... (1 — ax h ~ l ) ’ 
wherein k lt Jc 2 ,. . . k s _ x denote any selection of s— I different integers drawn from the 
natural series 1, 2, 3, . . . p — 1. 
This coefficient, from previous work, has the value 
