of any Functions of Multinomials. 55 
This rule is, however, more simple in the enunciation than 
the practice ; on which account I proceed to a 
5. Second Method. We might obtain one from equations 
(2) and (5), but, as it would be somewhat worse than the 
last, I omit it; and substitute in equation (3), the values of 
B B 
n- 2 > n- v &c. given by (5). We find thus 
C‘ “1“ 
( 7 )> 
or, if we consider under the form H __, = /3 + /3 c -f- p c* -j- js c 3 4 -, 
b Pb ' /*C0« /*(*),„ pi) ml 
n-fj n-1 c ' J r\j fic+i.a.fj p r-f,&c (8)* 
Where any number of strokes under the fs denotes that the 
fluxional coefficients of f (c) therein contained, must be de- 
pressed so many orders. 
T v ( 2 ) 
m this expression, we must neglect in /3 all terras contain- 
. " . ( 3 ) „ m 
rag c, in /3 all those containing c or c, and so on. Let it be 
required, for an example, to find the coefficient of x 7 , from that 
of x 6 given in article 3. We shall have, after neglecting such 
terms as are above specified, 
^ /■' , \ . . a I" !" "i mm 
n (c) c-rf (c)cc; (i =J(c )- ; /3 =/ (c) 
and by performing the operations indicated by equation (8)3 
we find 
U , x / min in 
J (c) c c +/ (r) 
1 
-2. mu mi 
— c +/ (c) 
1 
c 3 "" iiiii 
— 6’ 4-J (V) 
/ 11 llll 
"1=- 6’ £ 6* 
in 
/ 
^ 2 , II III 
+ -CC 
1 /i2 
+ *- 
, 'b 
2 
— J— 6’ 
2-3 
,» ' s 
• c -4- / (c) — - — 
2.34 » u v y 2.3. 4.5 
* +/ (0 — “ 
1 J K ' Z.3.4....7 
C 3 C 2 - 
2.3 2 
* See Note III. at the end. 
