7° 
Mr. Knight on the Expansion 
<*■> @> 7 > &C. 
2.3. 4. ...m: and the fluxional coefficient* tp , that multi- 
plies any term, will have for the left hand figure over it that 
number which is the sum of the exponents of the c’s ; for the 
next figure on the same side that number which is the sum of 
the exponents of the e’s ; for the third that which is the sum 
of the exponents of the d’s ; and so on. 
The only difficulty then is to find these combinations (with- 
out the possibility of missing any, or the trouble of finding 
the same more than once ) by some regular process of deri- 
vation. 
A rule was given in Art. 8, when we were considering the 
similar problem in the case of one multinomial, for deriving 
all the combinations in B, in which the sum of the strokes is 
r, from c as origin of derivation. 
The same rule will apply here, but instead of the one origin 
c r , we have, in the case of two multinomials, the origins 
y s — I 
r—% 
1 
c t c e, c e , 
2 r — 2 r— 1 r 
c e , c e , e . 
Let us consider any particular origin as c n e m . I denote the 
term derived immediately from c n (by the rule in Art. 8,) by 
A c n ; and the terms derived from this last, from the same 
rule by A 1 2 c n ; those got from A 2 c n by A 3 c n ; and so on. 
It is evident that all the possible combinations (of the kind 
ct, (3, y, &C. r 1 
* represents the fluxional coefficient of <pi F (c),f (e), V ( d ), Sec. r 
of the order a-f /3+y+, Sec. where the fluxion has been taken a times with respect to 
t , @ times with respect to e, and y times with respect to d ; and so on. 
