Hagen.] 94 
Parr I, 
nj m. 5 
7 
§1. Given the series * == » A. y', to find the reversed series y= Y Bs x8, 
f= 0 é= 0 
where », and B are unknown. Replacing y by the latter series, we obtain 
t = Mm 8 en r 
xX = JA, | 2 Bs x8 or, applying Waring’s formula, 
3 tl 
r=0o Oo 
ge ay B,“B,“.. . By 2 
Pe ae Bice: Ot) oe ge: Ped SE, OO y ate 10 cole Ry te. vin apo te Oy 
Lee 0 a,x 7 Ay) Ty). + .7( Ay) 
In the formula, z(r) = 1:2°3... r, according to the notation of Gauss, 
and the series ay a,.+. a, represents all such combinations of the num- 
DOre Or Ly wey Oy sy ee satisfy the condition 
Go taka t... +a, =r 
(ing 
The last formula is an identical equation and, according to the theorem 
of Indeterminate Coefficients, may be resolved into the following con- 
ditions : 
1. Oase. Ou + lay -+ 2a ae as ++ ba, = 0, 
This equation admits of the following combinations ; 
do Oy Oy ds r 
Oe MO Oia Meg 
Le rhe Oi One oo 
2 0 0 0 2 
hence we have the condition 
r=m yt r=m 
2 By <= 0 or gy ere Ob (1) 
r=0o mr) r=o0 
Though this equation is of the mth degree with regard to B, yet for 
the reversion of series, but one of its roots is fit, because there is but one 
way of developing y into a series of ascending powers of %, and indeed we 
find that B, is to satisfy still another condition, 
Putting y = 0 we have ¥ = Aj, hence 
Same 
2 Bs Ao se '0. 
é=0 
In the special case A, == 0 we have By ee 0, 
2. Case. Oa + lay + 2a, +... + pay = 1. 
This case admits of the following combinations : 
ao ay Ay ay 4 T 
0 1 0 0 0 1 
al a 0 0 0 2 
2 1 0 0 
{April 6, 
