Interpolation by Means of a Reversed Series. 631 
doubly infinite series will always converge*. The cases in 
which the series fails to converge with sufficient rapidity are 
therefore exceptional and are easily discovered. In fact it is 
only necessary to see if the terms at the end of each hori- 
zontal series, such as 14 (rb'yn^ 21 re' (rb') 2 n 2 , . . ., are small 
in comparison with n im 
The quantities (a f b f c f d'e') are easily computed when the 
first derivatives are tabulated. Representing the functional 
values, the first derivatives, and the successive differences of 
the first derivatives by the scheme : 
T 
F(T) 
wF' 
A' 
a" 
A /ii 
^iv 
t — 3w 
F -3 
a_ 3 
i 
t -2w 
F 
a 2 
P-3 
— 2 
7-2 
t — w 
F -l 
«-i 
P-2 
y- 1 
7o 
?-2 
t 
Fo 
a o 
0-1 
2-1 
e_i 
ft 
e o 
t-j-OJ 
*i 
°i 
1 
ft 
ft 
7i 
*, 
e i 
t+ 2oj 
F 2 
F 3 
«2 
a 
y 2 
^2 
these auxiliary quantities are represented by the simple 
forms: — 
2 12 
*" 6 72 
<?= 
24 
120 
To 
when Stirling's formulas for the successive derivatives are 
used. Evidently, 
*- ^ 
™ 3 F ''' ™ 4 F iv 
6 ' ~ 24 ' " 120 
The computation of a', b', c', d', e' . . . can be checked in various 
ways. In general it is quite sufficient to duplicate the work. 
An independent check is of course obtained by computing 
* Harkness & Morley, ' Treatise on Theory of Functions,' p. 116. 
2U2 
