THEORY OF THE PARTITION" OF NUMBERS. 
G39 
lienee the portion of this function which consists only of powers of ax p is 
(5 _ 1) \{ axP ) s ~ l + (1) ( axP ) s + ( 6 t X ) + • • • J 
or 
tp — 1\ (ax?)** 1 
\s — 1/(1 — axPy 
Hence also from the function 
a s ~ 1 x h+h+ •»• + f *-i 
1 — as** (1 — ax 1 ' 1 ) (1 — ax 1 '- 2 ) ... (1 — ax 1 ' 1 - 1 ) 
we can isolate the portion 
Ip— 1\ i ax? V- 1 
\s- l) \1 - ax p ) 
Ex. gr. for p = 3, s — 2 we can verify that 
ax* 
1 — ax s 
can be isolated from 
ax 
+ 
ax~ 
1 — x. 1 — ax 1 — x . 1 — a 
Y'7’2 
Art. 27. Before generalising the foregoing it will be proper to give another corres¬ 
pondence between the compositions and partitions of unipartite numbers which leads 
readily to theorems concerning the generating functions of partitions when the parts 
are unrepeated. 
Writing down any composition of the unipartite p, viz. : 
(piPiP 3 • • • • > Vs), 
we can at once construct a regularised partition, viz. :— 
(Pv Pi + Pv Pi + P2 + Pv - >P) 
t 
of the number sp x + (s — 1 )p 2 -f- (s — 2 )p 3 + ....+ p s . 
The correspondence is between the compositions of p into s parts and the partitions 
of all unipartite numbers into s unequal parts limited in magnitude to p and possess¬ 
ing a part p , 
The numbers whose partitions appear are the natural series extending from 
