1914-15.] Expansions of the Interpolation-Theory. 
191 
Consider the algebraical identity 
1 
+ 
nw 
+ 
n{n — \)w‘^ 
z- a-nw z - a {z- a){z - a-w) {z-a + w){z - a)(z -a-w) 
(n + l)n{n — \ )w^ 
+ 
+ 
{z - a + tc){z - a){z - a - w){z - a - 2io) 
n{ii^ - - 2^) ... . - (r - l)^}(w - 
{z — a){{z-aY-iv‘^)[(z-aY‘-2‘^w^) .... [{z - - rhv‘^} 
7^(7^2- 12)(^2_22) .... (n2 - r2)w;2-+i 
{z - a){{z - ay - w‘^) .... [{z - ay - rhv'^]{z - a - nio) 
Let f{x) be the given function, and let C(x) be the corresponding cardinal 
function. Multiply the identity (6) throughout by (z), and integrate 
Ztt'I 
with respect to 2 ; round any simple contour y which encloses all the points 
a, a-\-w, a — w, a-\- 2w, . . . . , a -f- rw, a — rw, a + nw. 
Now we have 
1 i C{z)dz 
' y ^ 
±.f 
27rijy 
z- a-nw 
Q{z)dz 
■ibT- 
^Trijy 
C(z)dz 
z - a 
2lw^ 
{z — a){z — a- w) 
C{z)dz 
27^^ jy {z-a-\- iv){z - a){z -a-w) 
Thus the equation (6) becomes 
C(« + nw) =/„ + «8/i + 
= C(a + nw) 
= C(a)=/„ 
= C(a + w) - C(a) =/, -/, = 8/j 
= 8%, etc. 
3 ! 
?^(?^2 - 12)(?^2 - 22) .... {n^^ - {r - \)‘^]{n - 
+ 2H ^ 
+ 
-f — 
^ 7 ^^ Jy (z — 
7^(7^2 - 12)(^2 _ 22) 
. (?^2 - 
27rijy (z - a) {(z - a)^ - .... {(z - a)^ - rVj(z - a ~ nw) 
We have now to investigate the value of the last term in the right-hand 
side as r increases indefinitely. Since C(z) has no singularity in the 
finite part of the plane, we are free to extend the contour as much as we 
like. We can suppose it to be a circle of very large radius, whose centre 
is at a + nw. 
Now the integrand, apart from the factor 
C(0) 
z— a— nw 
may be written 
n\ 1 
r 
22 
1 - 
i 1 _ i i I I 
V IV ) \ \‘^ W ^ J I 22^(;2 j • ' ' ■ [ 
{z - a) 2 
