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ME. C. W. MEEEIFIELD ON A NEW METHOD OE APPEOXIMATION 
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a — ^ third approximation will evidently be b—'^=c, and so forth. If we apply 
this method to the pure equation the convergent terms which we obtain are as 
follows : — 
j {n — \)a”' +p 
6-— rtn ’ 
^ (w— 1) |(m— + 
The second method is that of the reversion of series ; it is sufficiently discussed by 
Arbogast * 
The third method was suggested to me by Mr. Cayley’s remark that Mr. Sylvester’s 
third approximation is a particular case, for n— 2 , of the common form (of the books on 
the binomial theorem) j ^ • o,, approximately, « being a first approxi- 
mation. In order to gain generality, and thereby symmetry, I shall pass from the par- 
ticular form \/N to the more general 9~*N by the following Lemma: — 
Let N=«o+«i^+«52^+<^3-^^+ , (1-) 
and let x^, X 3 , x^ be determined by the system of equations. 
N=«.(l+^;x.) 
( 2 .) 
and so forth; also let 'N—ao=fJtj, and 
1 = ((^— )(Ao-f Ai«+A2^-j-A3a:^+ 
1 
(3.) 
then 
Aq a 1 Aq 
— /y* - — i rp — _2 /y» 
*^1 > ^ »^2 5, 9 »^3 \ .... . 
Aj Ag Ao An 
For, if we substitute these values in the equations (2.) after placing them in the 
following form, 
ix=^ — a^—aiXi -\ 
— I C^‘) 
®1^3“b®2^3^2”l“®3^3®2®U 
and so forth, we obtain 
^}^() "h ^^Aq ^jAg “1“ ^gAj -f- CI^\q 
^ ^1 «2 '^3 
?=&c., (5.) 
which are the same equations as we should get by multiplying the two series in (3.) and 
equating to zero the coefficients of x and of its powers. The coefficient Ao=-, 
obviously. 
* Calcul des Derivations, pp. 288-296. 
