of determining Stream Lines and Equipotentials. 257 
be true throughout the whole region, with the stated pro- 
• • "dV 
visions as to the value of ^ — . The geometrical meaning of 
this result is that if we draw any infinitesimal rectangle 
normal to one of the screw-threads and draw screw-threads 
through each of its four corners, then the infinitesimal tube 
thus formed will be everywhere rectangular in normal cross 
section, and more than this, the rectangle will have the same 
ratio of length to breadth and will be of the same size at all 
points along the tube. For if we consider one pair of opposite 
faces of the tube as equipotentials and the other pair as 
lines of flow, then these properties are seen to follow 
from the fact that V 2 V is constant along a screw-thread 
when V is constant along the same. And indeed these pro- 
perties are immediately obvious from the appearance of the 
system. 
Consequently, if we take any family of surfaces a passing 
through the guiding screw-threads, there will always be, an 
orthogonal family of surfaces /3, also passing through the 
screw-threads. If the surfaces a are the contours drawn at 
equal intervals of the potential V the surfaces /3 are Stream- 
surfaces. And ~^ may be named the " Chequer Katio * r 
consistently with what has gone before. If V 2 a is to vanish 
we must have 
1/ li \_ 
P^ a \ JJ x (l en o tn of portions of successive screw-threads intercepted I — u - 
^ between two stream-lines lying on the same stream-surface) ' 
Since the screw system is uniform the length of the 
portions of successive screw-threads intercepted between two 
stream-lines lying on the same surface /3 can be proportional 
to nothing else than the length of one turn of the screw- 
thread at the radius considered. For the two stream-lines 
in question must by symmetry make equal angles with planes 
normal to the axis of the screw, at each pair of points lying 
on the same screw-thread. So that the projection of the 
distance between the said pair of points onto the axis of 
the screw will be always the same fraction of I as the points 
move from one screw-thread to another. 
Now the length of an arc ds of a screw-thread beino- 
s/dz 2 + iW = dOy/~ + r' 
