]93 
1914-15.] Expansions of the Interpolation-Theory. 
The first question proposed in § 1 is thus answered ; and the answer 
to the second question follows from it, since we have seen in §§ 4-5 how 
the cardinal function may be constructed analytically. 
§ 8. Conclusion. 
The cardinal function may be regarded from many different points of 
view. We defined it originally as that (unique) function of the cotabular 
set which has no singularities in the finite part of the plane and no 
constituents whose period is less than twice the tabular interval w. But 
the result of § 7 shows that it might be defined as the sum of the central- 
difference expansion formed with the given set of tabular values : or 
(what amounts ultimately to the same thing) it might be defined as the 
limit, when r->oo, of that polynomial in x of degree 2r which has the 
values /q, /^, f^, /_ 2 , when the argument has the values 
a, a + ia, a — w, . . . . , a-{-rw, a — riu respectively. When we regard it 
from this latter point of view, we see the underlying reason for the 
absence of singularities in the finite part of the plane and of short-period 
oscillations. 
The introduction of the cardinal function seems to necessitate some 
reconstruction of ideas in the general theory of the representation of an 
arbitrary function by a series of given polynomials, say 
f(x) ^ cIqPq(x) -1- aii?i(ir) + ^ ad inf. 
Our ideas on the subject of these expansions have hitherto been based 
chiefly on the study of the two best-known cases, namely, Taylor’s 
expansion 
f{x) = aQ-\- afx - a) + afx - a)‘^ + af^x- aY + . ... , 
and the expansion in terms of Legendre functions 
f{x) = af?fx) + a^fx) + afdfx)+ .... 
Now it so happens that in both these special cases the roots of the given 
polynomials are either all concentrated in a single point (as in Taylor’s 
expansion) or else everywhere-dense on a finite segment of the real axis 
(as in the Legendre case, the roots of P,i(^r) when n~^cc being everywhere- 
below a. In the case of an interpolation-formula such as Newton’s, namely, 
f{a + nw)=fQ + nfi + 
n(n-l) 
2 ! 
n(n-l)in-^) 
use is made only of the tabulated values of f(x) for the values a, a + w, a + 2w, .... of the 
argument, and no use is made of the tabulated values of f(x) for the values a — w, a — 2w, 
a — 3w, .... of the argument ; in such cases a wholly different theorem holds, which I 
hope to give in a later paper. 
VOL. XXXV. 
13 
