OF THE COMPOSITIONS OP NUMBERS. 
845 
and thence 
F iVilh) 
oih + Pi-'^ Oh 4- ! oj^i + i)2-2_ (Ih ~t 1) • 
- l)!(i?3 - !)■ 
I + ilh+V^-^V- __ 
^ •2\{p,-2)\{v,-2)\ ••• 
until one of the denominator factorials becomes zero.^ 
13. There is another method by which the value of F {pil^^Pz - • • p) can be easily 
obtained from the numbers /{pippp^ ■>•,'>') which compose the value of F . ..). 
Writing /ij, + A -3 + /ig + . . . = H, we may write the generating function 
= 2 F {p^p.2p ^. . .). {2hPzP^ . . .) 
where ippppp^ .. •) denotes the symmetric function 
and 
Let 
W = tf{pip.p^ . . . {PMh • • •)• 
dr — da, + + 1 + 2 + • • • 
Dr = ^ (9^?i + cti da^ + dffj + • • -Y) 
Dr being an operator of the order obtained by symbolical multinomial expansion, 
as in Taylor’s theorem of the Differential Calculus, and not denoting r successive 
performances of linear operations. The effect of the operation of D^ upon a symmetric 
function of the quantities a^, . . . expressed in the notation of partitions, is well 
known ; it obliterates one part r from every symmetric function partition which 
possesses such a part, and causes every other symmetric function to vanish. 
Also Dr (r) = 1. 
Hence 
D^H'- = tfi'PdhPi {p-dhPz • • •)• 
To evaluate D^H*" we require the well-known formulm 
leading to 
= -f H). 
When operating upon a function of H, is equivalent to when p, is uneven, 
and equivalent to — di when p is even. Hence, with such an operand, we have the 
equivalences 
Tables for the verification of formulae will be found post pp. 898-900. 
