Lamb and Batchelor define the stream function  as follows.   
Using the expression derived above for the total volumetric flux, , this can be written as
In words, the stream function  is the volumetric flux through the test surface per unit thickness, where thickness is measured perpendicular to the plane of flow.
The point  is a reference point that defines where the stream function is identically zero. Its position is chosen more or less arbitrarily and, once chosen, typically remains fixed.
An infinitesimal shift  in the position of point  results in the following change of the stream function:
From the exact differential 
so the flow velocity components in relation to the stream function  must be
Notice that the stream function is linear in the velocity. Consequently if two incompressible flow fields are superimposed, then the stream function of the resultant flow field is the algebraic sum of the stream functions of the two original fields.