1914-15.] Expansions of the Interpolation-Theory. 
181 
X YIII. — On the Functions which are represented by the Expansions 
of the Interpolation-Theory. By E. T. Whittaker. 
(MS. received May 14, 1915. Bead June 7, 1915.) 
§ 1. Introduction. 
Let f{x) be a given function of a variable x. We shall suppose that 
f{x) is a one-valued analytic function, so that its Taylor’s expansion in any 
part of the plane of the complex variable x can be derived from its Taylor’s 
expansion in any other part of the plane by the process of analytic 
continuation. 
Let the values of f{x) which correspond to a set of equidistant values of 
the argument, say a, a-\-w, a — w, a-\- a — 2w, a -f Zw, .... be denoted 
by /oj /p /-u A /- 2 ’ /s’ shall suppose that these are all finite, even 
at infinity. Then denoting (/i-/,) by (5/,, (/ 0 -/- 1 ) by (Sf.-Sf-.) by 
etc., we can write out a ‘‘table of differences” for the function; the 
notation which will be used will be evident from the following scheme : — 
Argument. Entry. 
a - 2io 
/-2 • • 
8/-i 
a -10 
/-I 
¥-i 
S!/-i 
¥f-i- ■ ■ ■ 
a 
fo 
¥i 
SVo 
. ¥t'o ■ ■ 
SS/J .... 
a + 10 
h 
• 
a -1- 2io 
/2 • • • 
7 
Now it is obvious that /(a?) is not the only analytic function which can 
give rise to the difference-table (1) : for we can form a new function by 
adding to f(x) any analytic function which vanishes for the values a,a-{-w, 
a — w, a-\-2w, .... of the argument, and this new function will have 
precisely the same difference -table as f(x). All the analytic functions 
which give rise in this way to the same difference-table will be said to be 
cotahular. Any two cotabular functions are equal to each other when the 
argument has any one of the values a, a-[-w, a — w, a-^2w, . . . ., but they 
are not equal to each other in general when the argument has a value not 
included in this set. 
