or 
or 
1 
1 
Proceeding now to determine the partial fractions of Ae 
x 
we know by Cavcar’s formula that 
lie 1 1 
EEOC EN EEE 
DE) (vO) 
where the double parentheses denote that the residues must be taken 
for all the roots of ®(z)=0, viz. for all the roots of the equations 
124 = 0, ls =O dee 0. 
By developing the factor 
1 2 gt 
we get immediately for the required coefficient of 2” 
a] 8 
pra. oe an od BLL = 5 W, 
on REKEN en 
where 
1 
Toes 
Hi (lazy (Lae)? (Let em) en 
or, restoring the original form of d(z) 
1 lee 
UL 
t 
ert (l—zn) (Lea) (lee) ( (1--2)) 
where the residue is to be taken for all the roots of the equation 
OME eee (6) 
Representing one of these roots by o and putting 
Olea 
the preceding value of W,, takes this form 
7 gn pnt t 
pe Sb aes a 
at 
(1—o% e—% 1) (l—o% lem Aad) (ij AG 7 ) ((6)) 
pent (20) 
wherein the summation must be extended to all the roots @ of the 
equation (6). 
The term W, may be further developed, for the corresponding 
equation (6) 
1—-z=0 or 1—z=0 
shows that the only root is g==1. Therefore W, reduces to 
N ent t 
a E 
(l—e at) (1—e CA) eb (l—e ")(() 
Proceedings Royal Acad. Amsterdam. Vol. XXII. 
