1914-15.] Expansions of the Interpolation-Theory. 
187 
or 
^sin + 
7T w x-a-rw 
(4) 
represents a function which is cotabular with the given function f(x), hut 
which has no periodic constituents of period less than 2w. 
Now, in order to construct the expression (3) or (4), we do not need to 
know anything about f{x) except its YdluQS, f{a), f{a-\-w), f{a — w), etc., at 
the tabulated values of the argument. These values, however, are not 
peculiar to f{x), but are common to the whole set of cotabular functions. 
It follows that we arrive at the same expression (3) whatever function 
f(x) of the cotabular set we start from. The expression (3) is therefore an 
invariantive function of the cotabular set : and it may be regarded as the 
simplest function belonging to the set. We shall call it the cardinal 
FUNCTION of the set. 
§ 5. Examples of the determination of a cardinal function. 
We shall now work out two examples in order to show how in any 
given case the cardinal function may be obtained from the formula (4). 
Example 1. — Suppose that the given tabular values of the function 
f{x) are as follows : — 
/(o)=o, /(i)=-i, /(2)=i, /(3)=-i /{„)=oyi, 
Th 
/(-i)=i, /(-2)=-i, /(-3)=i — /(-„)=uyA\ — 
To 
The corresponding cardinal function is, by formula (4), 
1 . 
- sin 77 X 
7T 
1111 -1 
^r-l'^2(x-2)'^3(a:-3) ■^4(^-4)'^ ' ' ' ' 
1111 
x+l 2 (x + 2 ) 3 {x + S) 4 .{x + i)^ ' ■ ■ 
or, summing the series, 
or 
sin ttxV r'( 1 -x) V'{x +1)1 
~ 7TX Lr(i- 4 ” r(ic + i)J 
or 
sin ttx d , sin ttx 
— log 
TTX dX TTX 
or 
cos TTX sin TTX 
TTX^ 
X 
