Analytic Functions of a Complex Variable. 
175 
Thus, as we proceed along a stream line the gradient of the line on the 
modular surface, which lies vertically over it, is proportional to the 
divergence, while as we cross a stream line from concave to convex 
the gradient is proportional to the curvature of the stream line. 
Hence a straight stream line indicates that the modulus is stationary 
as we cross it, and parallel stream lines indicate that the modulus is 
constant as we proceed along them. 
4. Stream Lilies at Zero Points. — If f(z) has a zero at a, we may in 
general iput f{z) = {z — a)(l>{z), where ^{z) is finite. Then — 
arg/(^) — arg(^ — a) = arg 0(a) ultimately, 
and is approximately constant for a small circle round a. 
Hence the stream lines make a constant angle with the radii drawn 
from the zero. 
Infinite points will not be discussed generally, but special cases occur 
in and log z, treated below. 
5. Determination of Stream Lines. — One or more of the following 
methods may be used : — 
(1) At a point of a stream line whose angle of contingence is -^z — 
■ih = arg ID = arg f{x + iy) , 
and it may be possible to plot at a number of arbitrary points from the 
principle that — 
arg tv^ + arg = arg (iv^iv^ , 
and to sketch the stream lines in an approximative manner. 
(2) Solution of the differential equation — 
^ = tan arg f{x + iy) . 
(3) The method of conjugate functions,* which is the method most 
generally useful. 
Let— , 
w = u-\-iv, 
then — 
dz _dx + idy _ udx + vdy . udy - vdx 
w u + iv ~ + ^ + 
Both the real and imaginary parts of the last expression can be shovm 
dz 
to be complete differentials. Hence we may write — = dC + idS, where 
C and S are real functions of x and 
y- 
Vide acknowledgment at end of paper. 
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