TRANSACTIONS OF SECTION A. 441 
part of any root lies between B, and Bn, and the absolute value of the imaginary 
part hes between y, and yn. 
The first part of this theorem is due to Hirsch,' but the second seems new. 
I have attempted also to obtain some connection between the invariant factors 
of the determinant |~—A! and those of? |b—Ac|, but hitherto without success. 
I have constructed, however, certain examples which show that the connection (if 
there is one) cannot be obtained by any very simple method, 
5. On the Zeroes of Two Classes of Taylor Series. By G. H. Harpy. 
The problem of obtaining reasonably precise information as to the nature of 
the zeroes of an integral function defined by a Taylor series is one which may 
fairly be said to be generally impracticable, at any rate with the analytical 
machinery at present at our disposal. The utmost which has been accomplished 
in this direction is the determination of certain limits for the increase (crozssance) 
of the moduli of the zeroes when corresponding limits for the increase of the 
coefficients are given, these limits being found by the determination, as a pre- 
liminary step, of limits for the increase of the function itself. For this reason it 
seems to me that the results proved in this paper may be of interest to those who 
are engaged in the study of the general theory of integral functions, They are 
essentially results concerning particular cases, but they are very much more precise 
than any results so far furnished by any of the general theorems. 
Of the two classes of functions dealt with, the second is perhaps the more 
interesting in itself. On the other hand, the determination of the zeroes of the 
first class seems of more theoretical interest, as being effected directly from the 
Taylor series, whereas in dealing with the second class a preliminary transformation 
of the series into the more easily manipulated form of a definite integral seems to 
be essential. 
A. Functions formed by selecting terms from the exponential serties.—The 
general form of such functions is 
X APY 
AO) pp pies 
ae ACO 
where (7) is a positive and continually increasing function which is integral for 
all integral values of x. It is easy to see that (in the ordinary notation) all such 
functions satisfy the inequalities 
Ke" |/r SM (r) Se". 
Moreover, if the increase of (x) is regular and sufficiently rapid, the nature of 
the zeroes associated with the essential singularity at infinity may be determined 
with much precision. The essence of the proof is to determine a series of circles 
on each of which the behaviour of f(A) is completely dominated by that of one 
term. 
Suppose, e.7.,° p(z)=n*. Then it can be proved that if 
As | 
(N°)! 
ia 
where |e. as Hence it follows that, round the circle 7 = N°, 
1 Acta Mathematica, |.c., p. 367. 
? Since the invariant factors of |/—A| and |¢—A| are known to be linear, it seems 
quite hopeless to make any connection between these and 'a—A\, 
* The increase of ~(”) =n? is not quite rapid enough, | ’ 
r=N3, N7N, fO)= Dt ea) 
