98 {April 6, 
Hagen.) 
Here we have exactly the case of $3 and the formulas (9/) will at once 
give the coefficients of the series 
HT é = 
| y = J Bs (¥ — Ap)’. (10”) 
} $6=0 
$5. When it is required to have y developed into a series of ascending 
powers of ¥ itself, we may proceed in the following way. Let the given 
ail series be written in two different ways, 
| r=m r=m 
s= YA y’ and ¥—A,= 2 Ay’, i 
r=o r=] | 
and consequently also the reversed series | 
$= 0 b= w | 
y= Bs xo andy = 2 Os; (% — Ay). (11) | 
8=0 8=0 | 
The values of the 0 are given by the formulas (9/), provided that we | 
write (instead of B, as has been explained in $4, while the coefficients B | 
are still unknown. Developing (3 — A,)é by the Newtonian formula, | 
we get 
ne ) 
3! 
(3 — Ag) = (— 198 2 1p (7) Ava xe; 
A=0 
HH and equating the two series (11), we obtain 
i I 8 =e 0 A = 00 6 re 
| BBs x8 =F (— 1) 4 YC 18 Cs (2) Ads 
I] §=0 A=0 é&=0 A 
1 | and finally by the theorem of Indeterminate Ooefficients, 
HI b= BS i | 
il Ba ee (yy (—1)8( ) Os AeA (12) 
} | é=A A 
| This formula may be transformed, by changing the index 9 = 4 +-r1, 
| thus : 
r=@ th 
Bie ie (aw BAS ee C cL "Yon day Ae (12/) 
r=0O A 
i ‘ Aq (Att 
{| where instead of ( M4 ) we may write ( ss ys The convergence of 
HH} | 
{| i the series (12) will depend upon the coefficients @ and must be examined q 
| in each special case; in general we can state that it always converges, a 
ath when we have 
Tim 108 on 1 
| ame se Ay 
| Heample.—Let it be required to reverse the series 
| Keli tytey tay 
Here is As = ; - (J) > 0) and A, = 1, hence we obtain from (9/), writing 
O instead of B, 
C= — 
1 1 1 
| Dry 7, =10,= —, - O, = 
{| Oyen, Oj 5 Os (2) OC, = + x(3)' x(4y’ etc., 
and in general (except Cy) 
