304 
MATHEMATICS: E. HILLE 
Proc. N. A. S. 
ZZ Zl 
(6) 1R [wm] ~ J*M 2 + JW <*Ti = 0, 
21 21 Zl 
Z2 Zl 
(7) irm 
! + /w 2 ^k 2 + JWdr 2 = 0, 
where 
(8) 
dK = dKi + ^K 2 ; dT = dTi + id r 2 . 
We call relation (5) Green's transform of the differential equation^(l). 
This relation can be used in many ways for obtaining information con- 
cerning the distribution of the zeros of a function w(z), satisfying (1). 
Formula (5) enables us to assign regions, below called zero-free domains 
duo 
where there can be no zeros of w{z) or — . Some of these ways are in- 
dz 
dicated below. 
The four differential equations 
(9) dKi = 0, dK 2 = 0; dVi = 0, dV 2 = 0, 
define four families of curves; the Ki -family and the K 2 -family forming 
the K-net and the Ti and IYfamilies forming the T-net. The two 
families belonging to the same base net are orthogonal trajectories of 
each other. 
Take a solution w a {z) of (1) such that 
dw a 
W a (z) = K{z) w a {z) 
dz 
vanishes at a regular point a in the complex plane. Construct the Rie- 
mann surface on which w a (z) is single-valued and mark the two base nets 
on the surface. Further, draw all curves on the surface, starting from 
z — a which do not pass through any of the singular points of the differen- 
tial equation and which are composed of arcs of curves of the two base 
nets such that along the whole path one and the same of the following 
four inequalities is fulfilled, namely 
( dK 2 < 0, 
dT 2 < 0. 
We agree to smooth the corners of the path-curves, when necessary, in 
order to preserve the continuity of the tangent along the curve. Such 
curves we call standard paths and designate the four different kinds (dis- 
tinguished by their characteristic inequalities) respectively 
(11) 5K, + , SKf , 5K 2 + and SK7. 
The points on the surface which belong to at least one standard path, 
emanating from z = a, together form the standard domain D(a) of a. A 
r dKi > 0, 
dKi <0, 
dK 2 >0, 
(10) 1°< 
dVi < 0; 
2°- 
dV t >_ 0; 
3°< 
dT 2 > 0; 
4°- 
