Interpolation by Means of a Reversed Series. 633 
Substituting in equation (4) we compute n ; then substituting 
n in equation (1) we compute AFasa check. Thus, 
n, = + 0-4326284 a'n = + 61444*7 
f l7ll = -0-0096667 Vn 2 =+ 1405*4 
tW= + 0*0003463 c'n 3 =- 52*6 
t\n?= -0-0000029 d'n'= + 0*6 
^=-0-0000001 eV= 0-0 
n 
= S= +0*423305 AF = 2= + 62798*1. 
By means o£ the second method of inverse interpolation 
previously explained, Rice* finds 71 = 0*423303. This dis- 
crepancy is due to neglecting units in the seventh decimal 
place. The value of AF computed from this value of n and 
■Stirling's derivatives is AF = 62797*7. 
There are a great number of tabulated functions in which 
second differences are approximately constant and in which 
the term 2n 1 (rb') 2 is negligible. In such cases the following 
rule applies : Divide the increment of the function (AF) by 
the mean of the first differences (a 2 +^&o)? au( i likewise divide 
this quotient (n{) by the same quantity; then the product 
of these two quotients (n^) and one-half the second differ- 
ence (i& ) is the correction to be applied algebraically to the 
first quotient (n{) in order to obtain the required interpolation 
interval (n). The algebraic signs of the quantities need not 
be taken into account ; for it is plain that if the numerical 
values of the first differences are increasing, the quotient (n : ) 
is too large and the correction is therefore negative, while 
if the numerical values of the first differences are decreasing, 
the value of n ± is too small and the correction is positive. 
This method is very convenient when the numbers are so 
large as to require the use of a logarithm table in making the 
interpolation. For example, let it be required to find 6 from 
Vega's ten-place table when log sin = 8*910 7867 247. 
The tabular values are as follows : — 
9. 
log sin Q. 
A'. 
A". 
4° 40" 0" 
8*910 4038 653 
+ 2578 594 
10 
•910 6617 247 
2577 057 
-1537 
20 
•910 9194 304 
+ 2575 519 
-1538 
30 
•911 1769 823 
The quantity to be computed 
is 
n = n^ — 
rb'n^ 
* Loc. cit. 
p.195. 
