THE PARTITION OF NUMBERS. 
109 
which, written in Cayley’s notation, is 
We have 
and if, in F A (J), we write 
we must reach the expression 
1 (1) J l (2) J -1 (i) J ' 
Tf /J\ S? _ J... JV‘ _ 
{sj + s)p T 
isr f ■ 
(k+l)(k + 2) ... (k+j) j written (-? + 1 )(.;+ 2 ) ••• (.?+&) 
( 1 )( 2 ) ... (j) other ™ wnttei1 (1) ( 2 ) ... (k) 
For the rest the enumeration, connected with this x’estriction, proceeds pari passu 
with the special cases involving the values of j, oo } i } 2, which have been already 
examined. 
Art. 19. Coming now to the last stage of the investigation we have for considera¬ 
tion the partitions of multipartite numbers in which the constituents of the multi¬ 
partite parts must be drawn from the particular set of numbers 
u lt u 2 , ... u s . 
The partitions involved in the first function lb of the infinite series 
Uj, U 2 , U 3 , ... U*, ... 
must involve the numbers u 1} u 2 , ... u s , and no others. 
Thus 
Ui = 1 + (uj) + (mj 2 ) + ( u 2 ) + (u*) + (u^) + (u 3 ) + ... ad inf, 
U 2 = 1 + (2+ {2u 1 2u l ) + (2 u 2 ) + (2u 1 2u 1 2u 1 ) + (2u 1 2u 2 ) + (2 u 3 ) + ... ad inf., 
Tb = 1 + (iu i ) + (iujiuj) + ( iu 2 ) + (iu 1 iu 1 iu i ) + ( iu x iu 2 ) + (iu 3 ) +... ad inf. 
Clearly 
D !il U 1 = D K2 U 1 =...=D Ks U i = U 1 ; 
and any other Hammond operator causes lb to vanish. 
Also 
D i Jb = D i Jb = ... = D f ,,Ib = U t ; 
and any other Hammond operator causes lb to vanish. 
