Colloquium publications.

110

THE BOSTON COLLOQUIUM.

be any convergent series defining a function 0,(x) which is regular at a.. Then 02(x) = 1(x) + +(x) is a function which has at
ai the same singularity as cb(x). The combination of the series
for 02(x) and for *(x) by Hadamard's process gives the function
f2(x) = (ao + c0)bo + (a,+ c)bx- + (a-+c2)b2x2 + * =f(x) + f(x),
in which
f,(x) = cb cb + c c1b x + c a2 +.
Now since 01(x) is regular at a%, when compounded with r(x)
it must give a function fj(x) which is regular at yj. It follows
that f2(x) and f(x) have the same singularity at 7i,. Thus the
nature of this singular point is not altered by any change in +(x)
or Jr(x) which does not affect the character of the points ai and 3..
It depends therefore solely upon the character of the singularities
compounded.
Complications arise only when there is a second pair of singularities a~,,,3 such that.ij = a.i.S = caz.
Clearly the resultant singularity is then dependent upon both
pairs. Their effects may be so superimposed as to create an ugly
singularity, or they may, on the other hand, so neutralize each
other that..j is a regular point. Very simple examples of the
latter occurrence can be easily given. It seems probable that
when 7ij is but once a product of an a by a /3, it must always be
a singular point, but this has not yet been proved. Its demonstration will greatly enhance the value and applicability of Hadamard's theorem, for then it can be stated in numerous cases, not
what the singular points off(x) may be, but what they actually are.
A detailed study of the nature of the dependence of the singularity y, upon a, and i3 would probably be both interesting and
profitable. Borel examines the case in which ac and ij. are poles of
any orders, p and q, and shows that 7. is then a pole of order
p + q - 1. It can, furthermore, be easily shown that whenever
a. is a pole of the first order, 7yij is the same kind of singular point