of determining Stream Lines and Equipotentials. 259 
we may draw on this plane the sections of tubes formed by 
the surfaces a and ft in such a way that —5 is proportional to 
v 
4ttV 
The sections of these tubes will in general not be rectangles; 
in fact, the angles and ratio of sides of the chequers formed 
by the traces of a and ft on the plane (f> = constant will both 
now depend on the orientation of the chequer as well as on 
its distance from the axis. It will therefore be necessary 
to make a chart of standard chequers in various orientations 
at a number of distances from the axis. Plate XII. is such 
a chart. The rectangles in the right-hand margin represent 
normal cross sections of the tubes formed by the surfaces 
a and ft. In a line with each of these are five sections of a 
tube of the same size and shape by the plane of the paper, 
when the angle between one face of the tube and the normal 
to the axis of the screw is successively 0°, 22^°, 45°, 67^°, 
90°. In order to be clearly visible these parallelograms are 
drawn quite large. What each really represents is the 
shape of an infinitesimal chequer situated at the central 
point of the large one. Practically the difference will not be 
important. 
Now this standard diagram can be covered by a sheet of 
tracing-paper, and two intersecting families of lines drawn on 
the tracing-paper in such a way that the parallelograms formed 
by them are everywhere similar to the chequers underneath, 
which have the same distance from the axis and the same 
orientation on the paper. Then if this tracing-paper plane 
rotate round the axis and slide along it so as to follow the 
guiding lines, the equipotential lines on the paper will 
sweep out the contours at equal intervals of V in space in 
such a way that V 2 V = and the other family of lines will 
sweep out stream-surfaces. 
A quantity which it is frequently necessary to determine- 
is the magnitude of the flux 
-\/f£MSMSK- 
Since H a / 4^ 
