Analytic Functions of a Complex Variable. 177 
1. to = k{z - a). 
No diagram shown, but modular surface a right circular cone, with 
axis vertical and vertex at a. 
Stream lines, found from C + iS= f^*^"^' ^ family of straight lines 
radial at a, 
2. (i.) tv = k{z^-a^). (Fig. 1.) 
Modular surface rises from two 
conical zeros, and tends to a para- 
boloid of revolution at infinity. 
Stream lines, found from — 
a system of coaxial circles intersect- 
ing at zeros. 
^ Fig. 1. 
(ii-) w==k{z^-\-a^) (Fig. 2.) 
Modular surface, the above turned through a right angle. 
Stream lines, a system of coaxial 
circles, of which the zeros are limit- 
ing points. 
By shifting the origin along the 
real axis the general quadratic func- 
tion — 
reduces to form (i.) or (ii.) according 
as — 4^ is positive or negative. 
[In the models constructed, 
^• = •1 and a — 5, unit of length 
being a centimetre.] 
Fig. 2. (iii.) w=^kz^. 
Cases (i.) and (ii.) reduce to l^his when a = 0, and origin becomes a 
double zero. Modular surface a pafaboloid of revolution. Stream lines a 
family of circles touching real axis at origin. 
