5 $ 
Mr. Knight on the Expansion 
/ 
i> ii n uni in 'a in T*( 1 2 * * ) m i, hi in, 
P C’ — J (c) C c + f (c) - C ; 2 J p r == f (c) c c 
( 6 ) "... 7 
2.3.4 5-6 / ft c • =/(c) c ; these being added give the 
t-X mil ° 7 
coefficient of .r 7 , which was required. 
To find n from equation (7) requires the use of both 
fluxions and fluents; in (8) we are without the fluxional 
process ; but, in its place, have the trouble of observing the 
numeral coefficients of each term : there is, however, a way 
of avoiding the mention of fluents, and the necessity of pa}- ing 
attention to these coefficients. If we consider equation (3) 
and the mode of expansion derived from it, it will be evident. 
that whenever we have any power (as the with) of c or c or c, 
or &c. it must be divided by the product 2.3.4 —-m. This fol- 
lows from the manner of finding the fluent of such a quantity 
,r , ,m 
as c c , and the consideration that we cannot arrive at c with- 
out having passed through all the lower powers, and repeated 
the fluential process at each. Hence results the following 
(2) (3) 
rule ; ( where by / 3 , / 3 , &c. I mean these quantities after we 
have neglected in each of them the terms that have been al- 
ready specified). 
(0 
(2) 
Omitting all the denominators , multiply n _ J by c; /3 by c; (1 
w ( 3 ) 1111 
bye; /3 by c; and so on: add these products together, and zvherever 
1 v 
there is any pozver of c , c, &c. as the mth,put the product 2.3.4....^ 
for a denominator . 
6 « Third Method « If we substitute in equation (3) the 
