88 
Mr. Knight on the Expansion 
Rule. 
To find B from B in a function of any functions of the multinomials c -f- c x -f- 
n— i n 
II I II I II 
cx l -\-,e-\-ex-{-ex' 1 -\-,cl + dx-\-dx' , --\-, Sec. ist. Consider only those terms, 
/ II III 
in B, which contain some of the quantities c, c, c, Sec. ; reject all the terms in which 
n 
the last of these letters are raised to a higher than the first power : reject also ( ij 
there be more than one multinomial ) such terms as contain none of the above men- 
/ 
tioned quantities but the first power of c. Change, generally, in each remaining 
"'...m "...(m — i) 
term, the last of the c’s as c into c • and take the fluent with respect to this 
quantity. 
I II III 
adly. Neglecting those terms, in B, into which c, c, c, See. enter, consider those, 
n 
I II III 
of the remainder, which contain e, e, e. Sc c. rejecting all those terms in which the last 
of the e's are raised to a higher than the first power. Those terms must also be rejected 
C if there be more than two multinomials ) which contain none of the e's but the first 
, "...m 
power of e. Change generally, in the remaining terms, the last of the e’s as e into 
"...(m- i) 
e • and take the fluent, with respect to this last. 
I II III / II III 
3<Jly. Neglecting the terms into which c, c, c, Sec. e, e, e, Sec. enter, consider those, 
/ II III 
of the remainder, which contain d, d, d, Sec. and proceed as before. — And so on. 
This rule has no difficulty, whatever may be the number of multinomials. 
The words in italics, were inserted to make the rule include the finding of B from 
i 
B; they are of no use when n is greater than one. 
Similar rules for multinomials of higher orders are formed with equal ease ; being 
the reverse of those that -have been given for direct derivation. 
