- 
NOTES ON LAORANOE’s THEOREM. 
3 
or the coefficient of h m k n l p ... is 
1 
[m] m [n] n [p] p ... 
the last of the series m, n, p being always zero; e.g. in the coefficient of h m k n , 
/ d\p 
account must be had of the factor ( (frc) n or (\Jrc) n . The above form is readily 
proved independently by Taylor’s theorem, without the assistance of Lagrange’s. If in it 
we write h = k, &c., a = b = &c., and (f> = -yjr = &c. =/, we have F (a + h/(a + hf(a +...) = Fx, 
where x = a + hfx. Hence, comparing the coefficient of 1l s with that given by Lagrange’s 
theorem, 
1 - — f»\ = s L 1 
w W-n = s Ipgs № [pF ; ** • 
(10) 
d 
where m + n + &c. = s, and as before Fa, fa, h ave been replaced by F, f 8. By 
comparing the coefficients of B m F, 
1 
W 
s — t 
s 
iff = 2 { 
1 
(fi*/* - *) W') 
(ID 
where n+p+...=t, the last of the series n, p ... always vanishing. The formula (10) 
deduced, as above mentioned, from Taylor’s theorem, and the subsequent formula (11) 
with an independent demonstration of it, not I believe materially different from that 
which will presently be given, are to be found in a memoir by M. Collins (volume n. 
(1833) of the Memoirs of the Academy of St Petersburg), who appears to have made 
very extensive researches in the theory of developments as connected with the combina 
torial analysis. 
III. 
To demonstrate the formula (6), consider, in the first place, the expression 
where p + q — k. Since 
q </>P 
[p] p [q] q 
{(8 p f p+s ) (8 q f~ p ~ s ~ d )}, 
1 _1 / 1 1_ 
b? [q] 9 ~ k \[p -1? _1 [q] q + b? [q -1] 3-1