472 
ON SOME NUMERICAL EXPANSIONS. 
[289 
Suppose in general that 
<f>x = x + B x x 2 + C x af + ..., 
and let it be required to find the r th function cf) r x. It is easy to see that the 
successive coefficients are rational and integral functions of r of the degrees 1, 2, 3, &c. 
respectively ; we have in fact 
<f> r x = <f)°X 
+ j {(fix — <f>°x) 
+ ~ 9 (4> 2æ — 2^> 1 « + (f>°x) 
+ ^ (<£ 3 & — 3 <\rx + o(f) l x — (f>°x), 
&c., 
and by successive substitutions, 
<p°x = X, 
fix = x + B x x 2 + G x x 3 + D x o^ + ..., 
cf) 2 x = x + 2B x x 2 + (2B x 2 + 2(7j) x 3 + ( B x 3 + oB x C x + 2D x ) oc* + ..., 
(f) 3 x = x + 3 B x ot? + (6 B 2 + 3(70 a? + (9 B x 3 + 15B X C X + 3A) «r 4 + .... 
Whence 
<j?x = x 
+ rB x x 2 
+ {(r 2 — r) B x 2 + r A} ^ 
+ {A 3 — f r 2 + §r) B x 3 + (fr 2 — fr) B X C X + rD x \ x? 
+ &c. 
It would, I think, be worth while to continue the expansion some steps further. 
2, Stone Buildings, W.G., Oct. 2nd, 1859.