THE PARTITION OE NUMBERS. 
113 
(ii.) We change the symbol B into the symbol Q, and find 
D m2 JfF ft (m 1 ; l Al 2 k *...i h ) 
V*2 h *...i*.k x \k a \...k i \ 
= X Fj {rn l ; \ kl 2 kl ... i ki ). F fi (m 2 ; l Al 2 /t2 ... i ki ) 
Qi Al Q/u..Q; 
/Ci 
l Al 2 /c ‘ 2 ... i ki . /cj! Jc 2 \ .. 
• 4!’ 
(iii.) Lastly we change the symbol Q into the symbol U and find that 
D ms X F t K ; 1 *> 2* 2 ... i *0 . F ? (m 2 ; 1 *‘ 2* 2 ... i'«) 
U/''' 
1^2**... t**. /c,! Jc 2 \... ki\ 
= X~F b (m 1 ; l Al 2* 2 ....F s (m 2 ; l* 1 2* 2 ...f l ' i ).F !< (m3; l Al 2* 2 ... i^) 
TJ 1 Al U/ 2 ...U 
ki 
l Al 2* 2 ...i h . \ ! 4!... ^!’ 
and herein putting U - ! = IJ 2 = ... = TJ* = 1, we reach the conclusion that the multi¬ 
partite number m 1 m 2 ...m s has 
X 
m ,; l Al 2* 2 ... . F q (m 2 ; l Al 2* 2 ... i ki ) . F„ (m 3 ; 1*»2* 2 ... i*) 
l Al 2* 2 ... i ki . k x ! k 2 ! ... 
partitions of the nature we are considering. 
YOL. CCXYII.-A. 
R 
