176 Transactions of the Boyal Society of South Africa. 
Integrating, we have — 
C + = = a function of (a? + 
Thus 0 is the velocity potential and S the stream function of a liquid 
moving irrotationally in two dimensions. Moreover, since along a 
stream line udy — vdx = 0, and along a cross line udx-\-vdy = 0, we 
see that S = constant, and 0 = constant are the respective equations 
of the stream lines, as defined in § 1, and the corresponding cross 
lines. 
The following case includes several of those which follow : — 
If— 
, > ■ j ^" n-1 
cos (n - 1)0 - i ?>m {n - 1)6 
(n - 1)7-"-' ' 
where r, 6 are the polar co-ordinates of z. Hence the stream lines are 
given by— 
^l-i ^^^ = constant. 
6. Construction of Models. — I have constructed models,* in which the 
modular surface is suggested by a wire framework, through which the 
stream lines drawn on the horizontal base are visible. The wires for the 
most part follow vertical sections and contour lines, which were previously 
traced on paper from the functional equation. 
In all cases except log z the functions are symmetrical about the real 
axis, and only the portion on one side of this is represented. This allows 
the framework to be attached to one side of a board, one face of which 
represents the vertical plane through the real axis. 
It is possible that plaster of Paris or plasticine might be found a better 
material than wire. 
7. Illustrated Examples. — The general characters of the models are 
shown in the following diagrams, which are accompanied by brief notes. 
In each case but the last the thick line in the upper half represents 
the section of the modular surface by a vertical plane through the real 
axis, while the lower half represents (horizontal) stream lines. 
A numerical factor k is introduced in most cases to reduce the eleva- 
tion suitably, while the stream lines are unaffected. 
* These were exhibited at the meeting. 
