Analytic Ftmctions of a Complex Variable, 
181 
Stream lines, derived from- 
dz 
C+*S 
_ r dz 
~~ J COS z 
cos z 
= log (sec 5; + tan z). 
given by- 
sinh y 
cos X 
constant, 
a family of curves intersecting at 
zeros. 
8. 
Here- 
?^ = log z. 
h^ = {\og r)2 + 0^ 
277 
0 
Fig. 10. 
Modular surface, a continuous spiral surface, winding upwards round 
a vertical axis through the origin. 
When r = l, = a helix of unit pitch. It can be shown to be a line 
of greatest slope, its projection on the z plane being a line of constant 
argument ^tt. 
From this helix the surface 
curls up inwards round the axis 
and becomes asymptotic to it. 
Vertical sections through real 
axis (Fig. 10), obtained when — 
Stream lines (Fig. 11) plotted 
from — 
tan 
6 
log r* 
Fig. 11. 
Discontinuity along real axis 
apparent only, as stream lines 
belong to different parts of sur- 
face. 
[In the model the upper part is made wholly of wire tied by helical 
pieces and suitable vertical supports.] 
