322 
ME. CAYLEY’S NOTE ON THE EOEEOOING PAPEK. 
Now V”0'", or ^ M X ^ ^ V’O'', 
or 
( log(f+A ))'Q''’ W X coef. in (^—rj 
e'-l 
Hence, putting R=— the last-mentioned formula will be true if, as regards the 
term which contains t^, we have 
1"^ _j_ ^"| r 3. 1 1 
1 [ 0 ] W [ 1 ] [« - 1] (^ + 1 ) [2] [^ - 2] (n 2) 
R-"= 
R^— &c 
•}’ 
the series on the right-hand side being continued up to the term in R*. This formula 
e‘ — l 
is, in fact, true if R, instead of being restricted to denote — denotes any function 
whatever of the form l + cf +&c., and it is true not only for the term in but for 
all the powers of t not higher than And, moreover, R"” may denote any positive or 
negative integral or fractional power of R. In fact, the formula (assuming for a moment 
the truth of it) shows that the expansion of any power whatever of a series of the form 
in question, can be obtained by means of the expansions of the successive positive 
integer powers of the same series : the existence of such a formula (at least for negative 
powers) was indicated by Eisenstein, Crelle, t. xxxix, p. 181 (1850), and the formula 
itself, in a shghtly different form, was obtained in a very simple manner by Professor 
Sylvester in his paper, “ Development of an idea of Eisenstein,” Quart. Math. Jomm. 
t. i. p. 199 (1855); the demonstration was in fact as follows, viz. writing 
R-=(l+E=l)-=H-y(R-l)+^;^(E-l)’+&c.: 
if we attend only to the terms involving powers of t not higher than the series on the 
right-hand side need only be continued up to the term involving (R — 1)®, and the right 
side being thus converted into a rational and integral function of R, it may be developed 
in a series of powers of R (the highest power being of course R'^), and the coefficients of 
the several powers are finite series which admit of summation ; this gives the required 
formula. But there is an easier method ; the process shows that the series on the right- 
hand side, continued as above up to the term involving is, as regards w, a rational and 
integral function of the degree x ; and by Lagrange’s interpolation formula, any rational 
and integral function of n of the degree x^ can be at once expressed in terms of the 
values corresponding to {x-\-l) particular values of n. The investigation will be as fol- 
lows : — Let R denote a series of the form 1 and let R“ denote the develop- 
ment of the nth. power of R, continued up to the term containing if, the terms involving 
higher powers of t being rejected. R“, R\ R^, «&c., and generally R*, will in like man- 
ner denote the developments of these powers up to the term involving if, or what is the 
same thing, they will be the values of R“, corresponding to w=0, 1, 2,..s. By what 
precedes R" is a rational and integral function of n of the degree x, and it can therefore 
be expressed in tenns of the values R“, R‘, R^, ...R'^, which correspond to w=0, 1, 2, . x. 
