52 
Mr. Knight on the Expansion 
, p i u 
of c. In like manner, / B c ' will be the sum of all those 
’ J n— 2 
II 
terms, in the same coefficient, that contain c and its powers ; 
f* i "- m 
and, in general, / // ’ the sum of those that have for factors 
...m 
the powers of c . 
Hence is derived an easy method of finding any coefficient, 
when we know those that precede it : for if these partial 
values be united, there arises 
ii i ii i hi 
® + f,L / + JM +* &c (3). provided that 
we neglect 
in n _ 2 all those terms which contain c, 
in all those which contain 
in „ B „ all those which have 
n — 4 
/ // 
c or c y 
1 n in 
c or c or c , 
and so on ; whence it happens, that many of the B’s will be 
neglected entirely, and the chief part of the operation will 
always be in the first term / . From equation (3), we 
1 n 
find the first part of the expansion of/ (c 4- c x -f- c x* -f ) 
to be 
fO)+f{c)cx+f(c)c 
I 
II r 7 
+/W4 
I III 
x '+f( c ) c 
II I II 
+f(c)cc 
+/(*) 
c 6 
2 -3 
/ Hll 
* 
II 
II f 1 III r z 
+J(c){cc + -} 
/ 
III E 2 * 11 
+/(r)-fr 
/ 
1111 ^4 
+/( c )r^ 
/ lllll 
x*+f(c)c 
If f I nil If III -J 
J rJ (c)[cc -{-cc | 
1 11 
III r r z III I r z 
+/(0{T e + ‘f } 
C * 
+/(r) -r 
mil 
(0 
c* 
2 . 3 . 4.5 
