174 Transactions of the Boyal Society of South Africa. 
further be seen that the argument of the differential coefficient is the angle 
at which the stream line is cut by the line of constant argument. 
3. Divergence aiid Curvature of Stream Lines. — Suppose the stream 
Hues cut by a faniily of orthogonal trajectories, which we may call " cross 
lines." Let ds and dc be the respective (scalar) elements of arc of the 
stream line and corresponding cross line at z. If arg w = \p, we have 
to = he'^'>', whence — 
, dh . d\P ... 
^h dz^''' Tz W 
(a) Let dz be taken along the stream line. Then — 
arg c?^ — arg ^f; and \dz\-=ds, 
(2) 
therefore — 
dz _w 
ds " Ji 
Now lu' is constant for all directions at z. Hence, combining (1) and (2)- 
, dJi d-l ..js 
w'' = — + — (3) 
ds da 
(b) Let dz be taken aloug the cross line. Then — 
arg c/^ = arg (m/.^), 
\dz\ = dc and | iio \ = h, 
therefore — 
dc~ h 
dz . 10 ... 
Combining (1) and (4)- 
From (3) and (5)— 
, . dh 7 dxl 
to' ^ - I -^ + h — (5) 
dc dc 
dh_^^j^ d-iP _ ^ ^^^^ 
ds ds [dc dc 
Equating real and imaginary parts — 
1 dh_drip 1 dh_chi^ 
h ds dc' h dc ds' 
We may call the curvature of the cross hue, the " divergence " of 
the stream line at the point. 
