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GKAPHICAL EBPEESENTATION OF SOME OF THE SIMPLEE 
ANALYTIC FUNCTIONS OF A COMPLEX VAEIABLE. 
By E. T. Littlewood, M.A., B.Sc. 
(Eead October 19, 1910.) 
1. A uniform function iu^f{z), where s = x + iy, may be completely- 
represented in a model in the following manner : — 
Modulus. — At each point of the (horizontal) z plane erect a vertical 
line of height h equal to | lu \. The upper extremities of these lines will 
form a surface which will be called the "modular surface." 
Argument. — Construct a family of curves in the z plane such that the 
tangent to one of them at any point is parallel to the vector w correspond- 
ing to that point. It will be shown in § 5 that these curves are the 
" stream lines " of a liquid moving irrotationally in a certain manner in two 
dimensions. Hence they will be called the "stream lines" of the 
function to. 
The idea of the "modular surface," and the complete representation 
of a function by means of a model I believe to be original. 
The modular surface and stream lines of an analytic function have 
certain simple and interesting relations, as will now be shown. 
2. Contour Lines and Lines of Greatest Slope. — Let a system of con- 
tour lines and their orthogonal trajectories, the lines of greatest slope, be 
traced on the modular surface. The orthogonal projections of the contour 
lines on the z plane are evidently lines along which h, i.e., \tv\, is con- 
stant, or "lines of constant modulus," while the projections of the lines 
of greatest slope are their orthogonal trajectories. 
It can easily be shown from elementary principles that the latter curves 
are also lines of " constant argument," and, since arg tu is constant, it is 
evident that — 
dh \dw\ , I 
Thus the gradient of the line of greatest slope vertically above any point 
represents the modulus of the differential coefficient at that point. It will 
