of any Functions of Multinomials . 77 
Let it be required from B to find successively B , B , See. 
°>3 *>3 2 >3 
We saw, when treating of a simple multinomial, that 
' C " C C ) 3 
B — cp [c) 0 , + <p (r) f 1 x 0 2 + <p [c] \2ii! whence, by the rule, 
0,3 ’ 5 2.3 
B — $ ( C ) l , 3 + $ ( 6 ) [ 1,0 X 0,3 + 1,1 X 0,2 + 0,1 X 1,2 } + <P ( C ) 
I >3 
{ « X o!, x x } + ? (c) ,‘ c X (F ; B = i (c) * 
2-3 2,3 
+ ^ W { 2,0 x o! 3 + no x n 3 “1” Z !i x o!a 1 L i!i x 1,2 + o!i x z! 2 } 
+ ^ M { ^ x o! 3 + no x ni x 0 % + i!o x 0,1 x 1,2 + 
2 >0 Oj I 
0,2 
+ 
Eil + x 2 ‘, } + 9 ( 0 { y x l 
C C 1 c 
i x 0,2 r 1,0 
( c y 1 c \ 3 ( c y t c y 
— x I I + 20 x ~ ( + £ (0 X Ifil/ . It is not 
2 ’ 5 2.3 ^ ' 2 2.3 
ne- 
cessary to calculate all the coefficients we may want by direct 
derivation ; when we have got a few, in this manner, we may 
find the rest by the inverse method which is much easier. M. 
Arbogast has put the twenty-eight first terms in a table ;* of 
these there was need to calculate only four directly , as I shall 
show hereafter. But, to give an example of this inverse pro- 
ceeding, let it be required to find B from B just now given. 
2,2 
2,3 
/ B 
Equation (a) becomes in the present case B = / — 1 1 - 
m — u,n-~u \ c J 
\ t*”» / • 
= Q M 2,2 + ? ( c ) { no x i,z 
whence B = B 
2,2 2—0,3 — ! 
* Calc, des Deriv. p. 127 . 
