of any Functions of Multinomials. 6 1 
I shall take no notice of any numbers, which divide the dif- 
ferent terms, till the end of the operation ; having shewn, in 
Article 5, that it will be sufficient then to place the product 
2.3.4....^ under every jttth power. 
There can be no difficulty in perceiving, that all the com- 
"...(m — 1 ) "...m 
binations in may be derived from those terms in T , 
/ 
that are multiplied by the powers of c , in the following man- 
/ 
ner. First, diminish the exponent of c by one. Then, dimi- 
nish the exponent of one of the other quantities by one, and 
multiply by the quantity that has the next greater number of 
l"...r\a l"...s\b 
strokes. For, if [ c J x P x { c I be one of the combinations 
"...(m — 1) " ...in 
in , there must necessarily be in the combination 
, l"...r\a 1) l"...s\b— i 
r x 1 r / x P x r x\cj ; and from this the former 
one is derived, in the manner above-mentioned, by taking 
, "...(S~l) 11 ...s 
away the c and changing c into c . 
We will next see if there be any quantities that it would be 
,b l"...[p — q) \ r 1 11 ...p \ s 
superfluous to make vary. Let txPxU / x l c ) be 
"...m 
a term of , we find from it, by the prescribed operation, 
"...(m — 1) 
the two terms of 
,b - 1 t"..(p-q)\r 1" ...p\s — — 1 S-i) ,b-i I".. [p~q)\r-i 
c xPxl c 1 x\ c I x c andr xPx\ c I 
"...{p — q+ 1) l"...p\s "....m 
xc x\r J but will also have the combina- 
,b t"...(p- q ) w-_l — g-j- 1 ) "-(/>- 0 i"...p\s- 1 
tion rxPxlr ] x c x c x \ c J , from 
which the combination («) may be got by diminishing by 
