86 
Mr. Knight on the Expansion 
NOTES. 
Note I. The rule in article 8 may be differently enunciated thus. 
- 1 ) "...m , 
To find ^ take the jiuxional coefficient of 4- with respect to c ; and of this 
jluxional coefficient take the fluxion with respect to the last quantities ; change gene - 
"...m "...{m- fi) 
rally c • into c • and take the fluent with respect to this last. 
2 dly. If the last quantity hut one be that which precedes the last in the number of 
strokes make it vary in the same manner and take the fluent. 
This is simpler than the rule in article 8, and more conformable to the mode of 
expression made use of in other parts of the paper. 
Note II. In looking back on what I have written, I am apprehensive it may be 
thought that I have affected too great brevity in the last pa agraph of ar.icle zi. 
That the reader may have no difficulty, the following problem is added, to illustrate 
what was said in the passage alluded to. 
Problem. 
To find at once B in the expansion of a function of two functions of double multino- 
m,n 
mials. 
It is plain that B must contain all the possible combinations of c’s and e’s (see the 
m,n 
notation of article 21) that can be formed with this condition; that the number of 
left hand strokes be m ; the number of right hand strokes n. Every rth power must 
a,@ 
be divided by the product 2.3.4. .r. And the fluxional coefficient , that mult ; lies 
each term, will have, for the left hand figure and over it, the sum of the exponents of 
the c’s in that term ; for the right hand figure /3 the sum of the exponents of the c’s. 
Now to get all the combinations of the kind mentioned above, with their proper 
divisors, we must plainly take, for origins of derivation, all the terms of the following 
product, when actually multiplied. 
