630 Mr. C. E. Van Orstrand : Inverse 
Equation (2) may be put into a more convenient form 
for computation by rearranging the terms and putting 
ni = A¥-7-a' and r = ^ 1 -f-a / . Then 
n = n x + [ - rV + 2(rb>) 2 - $(rV) 3 + 14 (r&') 4 ] n x 
+ [«r ( - 1 + 5 r&' - 2 1 (rZ/) 2 ] V 
+ Lftf(-l + 6rftO + 3(«O r jn 1 » 
+ [_ r6 r]n 1 * + .... (3) 
Now substitute 
/ 1= -r&' + 2 K) 2 -5 (^)8 + 14 (r&') 4 
/ 2 = rc'[-l + 5W/-21 (r&') 3 ] 
/ 3 = rd! \- 1 + 6t#] + 3 (re') 2 
/ 4 =-re' 
in (3), and the equation for inverse interpolation, inclusive 
of fifth differences, is 
w=%+M+M 2 + />i 3 +/^i 4 . • • • ( 4 ) 
For backward interpolation % is negative. A slightly 
different form of (4) convenient for logarithmic computa- 
tion is 
n = n x +/>! + reffin* + rd'ftnf + 3 (re') V +M* ... (5) 
in which 
/y=-i-i-5w/-2i (r6') 2 
The expressions for the/'s, as obtained by developing the 
reverted series to the same order of powers as the given series, 
are represented by the following groups, according as each 
series terminates with the second, third, or fourth power : 
f=—rl)=-rb' + 2 (rb , ) 2 =-rb' + 2 (rb') 2 -5 (rb'f 
/ 2 == =—rc f = r</(— l + 5r¥) 
Analytically, there are an infinite number of the /'s and each 
contains an infinite number of terms. The number of terms 
required in any particular reversion is easily ascertained 
from the data. As a complete formula for inverse interpola- 
tion, applicable in all cases, the reversed series is unfortunately 
not always sufficiently convergent when a limited number of 
terms is used. However, the most important applications of 
reversion obtain when the numerical magnitudes of the 
successive orders of differences gradually diminish. It will 
be found for such functions that W/<0*1, and since n<2> the 
