oj determining Stream Lines and Equipotentials. 241 
Now the curl of the vector velocity is defined as the line 
integral round a small circuit divided by the area of that 
circuit — that is in this case by the area ABCD which will in 
the limit be equal to the mean of the areas of the two 
adjacent chequers. So that we have : — 
difference of successive chequer ratios in a direction 
t ~ , , . ., , perpendicular to velocity 
curl of the velocity =k *-— -, 
J mean chequer area 
If the velocity has no curl the chequer ratio must not vary 
along any line normal to the flow. It may vary from one 
normal to another, but if on the other hand we prefer to make 
it constant all over the field, then at any point the distance 
between successive normals will be inversely as the flow, so 
that these normals will be contours drawn at equal intervals 
of a velocity potential. 
To return : since the fluid is incompressible the condition 
for the existence of a stream function is satisfied, and since 
the stream-lines are drawn so that the flow between each 
successive pair is the same, it follows that these stream-lines 
are the contours drawn at equal intervals of a stream 
function -v/r. Now it is proved in works on Hydrodynamics 
that -^-V + ^-~ is equal to the curl of the vector velocity. 
Therefore : — 
difference of successive chequer ratios in a direction 
~b 2 ^r ~& 2 ty 7 perpendicular to the contours of ^ 
~b'''~ ~d l f 2 " mean chequer area 
And since i|r may be any one-valued function of position on 
the plane, it is seen that all hydrokinetical considerations have 
been eliminated from the above equation, which is purely a 
proposition in differential geometry. The only implication 
being that contours are drawn at equal intervals of y]r what- 
ever be its physical meaning. 
To draw chequers freehand so as to satisfy a difference 
relation of this sort between the chequer ratios is likely to be 
toilsome, and we will here consider only the case when 
v 2 -^=o. 
Supposing then that a chequerwork has been obtained in 
which the chequer ratio is everywhere the same and in 
which the given boundary conditions are satisfied, then by the 
uniqueness of the solution of ^rj + ^-y = this chequer- 
work gives us what we want. 
