Analytic Functions of a Complex Variable. 
179 
Stream lines plotted as in (i.). Also tend to Bernoulli's lemniscates at 
infinity, as in Fig. 6. 
The above illustrate a cubic 
equation, (i.) with three real roots, 
(ii.) with two imaginary roots. 
(iii.) iv = kz^. (Fig. 6.) 
Cases (i.) and (ii.) reduce to 
this when — 
a = b = 0. 
Origin a triple zero. 
Modular surface a paraboloid 
of revolution of third degree. 
Stream lines, given by — 
sin 2d' 
constant, 
Fig. 5. 
a family of Bernoulli's lemniscates. 
5. w^ = k2. 
(Pig. 7.) 
Modular surface obtained by rotating half a parabola about tangent 
at vertex. 
Stream lines, given by r (1 — cos 0) = constant, a system of parabolas 
with origin as focus. 
A case of a multiform function, the two sheets of the modular 
surface corresponding to the two branches being coincident. Note 
that one circuit round origin 
reverses direction of stream line, 
i.e., increases argument by tt. 
Fig. 6. 
Fig. 7. 
