Inverse Interpolation by Means of a Reversed Series. 629 
T 
F(T) 
A' 
A" 
A'" 
Aiv 
AV 
t — ow 
F-3 
t - 2w 
F-2 
"-3 
»-* 
t -o, 
F-i 
«_2 
ft^l 
C -2 
*-l 
t 
F 
°-l 
*o 
<?1 
^0 
t +U) 
*, 
ftx 
<*. 
t 4-2w 
F 2 
a.. 
ft 2 
( 'o 
z 1 +3w 
F 3 
a s 
Put also 
1/ X 
, C 
a=a ~6 + 30 
c =9 (Cl + C_J 
7 / ^o d 
2 24 
« =2^i+ g -0 
/ c e 
C ~ 6 24 
AF=F,-F 
24 
120. 
Then, using Stirling's expression for the successive derivatives, 
because o£ their simplicity and rapid convergence, and writing 
the interpolation formula as a Taylor's series, 
£F = a'n + b'n 2 + c f ?i 3 + d'n* + e / n 5 + . (1) 
The solution of the problem consists in finding the value of n 
when AF, a, b', c', . . . are given. To accomplish this, we 
revert equation (1) by means of the usual method of equating 
coefficients, or by means of the expression obtained by 
Professor McMahon for the general term of a reverted series*, 
and obtain 
AF Z//AFy ro/^'Y c'-\(£F\* 
""" IT" SKIT ) + L 2 \?) -a>\{ir) 
\_ \a J a a! a \a J a \ a J J \ a' J 
* Bull. Am. Math. Soc. iii. p. 170 (1893-1894). 
Phil. Mag. 8. 6. Vol. 15. No. 89. May 1908. 2 U 
(2) 
