Mr Galbmith on Vanishing Fractions. 71 
get ^ ~ ^ 1 -f- a. Therefore, when mz=:% we see 
that as 71 approaches to unity, 
m /a \w 
a — x a— (g) 
approaches to (1 + 1) a = S as its limit. But when w = 1, 
a — a 0 
a — a 0 
- by subtraction, consequently in this 
case ^—9^ a. 
In like manner, if m=:S, then as n approaches to unity, 
/a \m 
^ 
^ — approaches to 3 a^. If m == % then it approaches 
n 
to 4 a^. And in general taking m itself, we may conclude, that 
a — a 
, when n:=z\ becomes 
^ m - 1 ^ 1 q- &c. to w terms) — a”* ~ i x x 1 = w 
Hence, the value of which in this and similar cases is mere- 
ly a symbol, shewing that we have arrived at a limit, is always 
thus obtained with as much certainty from the binomial func- 
tion under consideration, if we are not mistaken, though the pro- 
cess appears somewhat inductive, as any proposition in mathe- 
matics. If in the expression ^ we substitute .w for a, we 
have m.r’”” ^ the general fluxional coefficient of any variable a? 
raised to the power denoted by 7n. 
An unexceptionable method of demonstrating this proposition 
is the more to be desired, as it forms the basis of Landen’s Besi- 
dual Analysis, perhaps one of the best methods of establishing 
the principles of that calculus, which he considered equivalent 
to the Fluxionary or Differential Calculus. 
Edinburgh, April 1820* 
