192 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
and when r increases indefinitely this tends to the value 
sin TTfi 
. 7r(z — a) 
sin — ^ - 
lu 
so that the integral to be studied is essentially 
r C(z)dz 
sin 7T?z 7 r(z - a) 
Iz - a — 7UV) sin - 
Jy w 
. ( 8 ) 
The question as to whether this integral tends to zero or not depends 
fundamentally on whether C{z) becomes infinite to a lower or higher order 
than sin when the imaginary part of 2 ; tends to infinity. Now a 
simple periodic function like sin \z becomes infinite to the same order as 
e^y, where y denotes the modulus of the imaginary part of 2 :: and we have 
seen that the distinguishing property of the cardinal function C(z) is that 
the periodic constituents into which it can be analysed all have periods 
greater than 2w : so, combining these statements, we see that C(^) becomes 
infinite to an order less than , whereas sin — — becomes infinite to 
IV 
Ly. 
the order 6^ 
Thus the factor 
sin 
•(^ - 0^) 
w 
of the integrand tends to zero when the imaginary part of 2 ; tends to 
dz 
either positive or negative infinity : and as the other factor may 
be written idO, where 0 denotes the vectional angle of the point 2 ; measured 
from the origin a + tiw, we see by a proof of the kind usual in analysis that 
the integral (8) vanishes.^ The equation (7) now becomes 
C(a + nw) =/o + 7^S/x + 
n{n-\) {n+l)n(n-\) 
2 ! 
3 ! 
+ + ad. inf., 
which shows that the function rejpresented by the expansion is the cardinal 
function of the cotabular set which is associated with the given 
function f(x).f 
* The manner in which the characteristic properties of the cardinal-function are 
required in order to ensure the vanishing of this remainder-term is v^ery remarkable. 
t It should be noted that the interpolation-expansion considered is a “central- 
difference ” formula, i.e. it makes use of all the tabulated values of f{x) both above and 
