t 
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nis (eye 
1888, ] 99 [Hagen. 
{ (an tt rl 
Os == (— 1) + 1 - tO 2 eee 
vader zd)" Cr +x m(A + 1) 
This value substituted in (12) gives for 2 > 1 
fr -- 5] ( ie Lys +1 ir fod q 
B= lesa LS ota SY , = 2 > ee 
A= ( ) SQA +r) x(A+4) x(A) Gey 
r+o A . r=0 
(—1)a +1 
‘ine i) 
m(A) 
and for 2 = 0, since O, = 0, 
r= 00 | | 1y a of 1 
Bye Py jt Ney ~=1l—e; 
a Yr L (tr) re 0 mx) 
where e = 2.'718281828 + is the base of, the Mepierian system of loga- 
rithms, 
Substituting these values into the first of the formulas (11) we obtain 
A= 0 A= 7 4)SA+1 
y= SD Ba xi =l—e-+e 2 At 
A=0 Vel ™(X) 
Jonsequently the given series 
as ee uh, ai chess de ccm lewis io ake 
xai¢ + F+54+74 rae. 
is reversed in the following way : 
en ene ee Se pill, Gres x) 
oar o(1 hs a a HD 
Y= 0 * 
a Se Sige eens r ‘ 
=1—e oo ( L) x(t) 
if te) 
This last formula may be tested in the following way. The given series 
requires that we have at the same time }¢ = 1 and y = 0, consequently 
the reversed series requires the identity 
a 1 1 e—l1 
Le aie ARR TO hea? 
which may be verified without difficulty, 
Part III. 
The equations (10) and (10’) imply the approwimate solution of algebrate 
equations. Putting Aj==0 and assuming for % any constant quantity, 
Say a, we may write these equations in the following way : 
r=m é=- 
a= J Ay’, solution, y = 2 Bs a? (18) 
r=] é6=0 
The coefficients B are determined by the equations (8) and (9), We do 
not say (8’) and (9/), because the condition Bj =0 is not by necessity 
fulfilled in this case, although we have A,== 0. While in §1 we have 
Stated that the reversion of the series admits of but one root of the equa- 
tion (1), since there is but one way of developing y into a series of 
ascending powers of ¥*, we now have to say, that-all the m roots of 
