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LVIII. Inverse Interpolation by Means of a Reversed Series, 
By C. E. Van Orstrand *. 
THE formulas developed by Newton, Bessel, and Stirling 
for the direct interpolation of a value of a function from 
values tabulated at equal intervals of the argument, are con- 
sidered to be among the most important of their contributions 
to the science of applied mathematics. The converse problem, 
that of finding the argument when a value of the function is 
given, although of about equal importance, seems not to have 
received the thorough treatment winch mathematicians have 
given to the subject of direct interpolation. Apart from 
methods applicable only to special cases, there are two general 
methods f now in use which are to a certain extent satis- 
factory. Each of the methods is in reality based upon the 
same principle — that of diminishing the interval of interpo- 
lation. In the one case this is accomplished by computing 
values of the function at equal intervals of the argument for 
values preceding and following the required value. If the 
interval of the new series of tabular values is sufficiently 
small, the correct argument can be found by taking second 
differences into account. In the second case, the function 
and its first and second derivatives are computed for a value 
of the argument (n) true to the nearest tenth of a unit. It 
is then generally sufficient to write the interpolation formula 
as a Taylor's series involving only the first and second 
derivatives. The correction to the approximate value of the 
argument is then found by reverting this series of two terms. 
Each of the methods has the serious disadvantage of being a 
tentative process, and neither of them provides a satisfactory 
check on the computation, without the aid of additional 
quantities. 
As a means of avoiding the difficulties noted above, it is 
desirable to call attention to the use which may be made of a 
reverted series as a formula for inverse interpolation. To 
derive this formula, let the tabular values of the function 
(F_ 3 , F_ 2 , • • •) and its successive differences (the a% 6's . . .) 
be represented by the following schedule : — 
* Communicated by the Author. 
t Rice, ' Theory and Practice of Interpolation/ pp. 192-5. 
