of determining Stream Lines and Equipotentials. 265 
an axis the term 
4tt-V ck 2 47rr- v 
And therefore writing 
^s*-* 
we have 
V=a(RV- r) + 6 . «u+^(R% 3 -3^) + A(v>-3ItVv). 
A simpler way of looking at the matter is to consider a tiny 
plane element normal to the guiding screw which forms the 
line of equilibrium. The normals to the surfaces &>=const. 
lie in this plane. If dS M distance along such a normal, then 
j "-'<V(M-K^) w V 
I 2 
+ 4ttV' 
Substituting this in the expression of V 2 Y in terms of r and 
co we have 
just as if S w was z in circular symmetry about an axis. 
From this we see that the appearance of the equilibrium 
point on a small plane element normal to the guiding screw 
will be exactly similar to the forms already dealt with. Its 
appearance on a plane which passes through the axis of the 
screw may be sketched without much difficulty by comparing 
the chequers in the right-hand margin of the standard chart 
with their projections as drawn in the middle of the chart. 
Y. Equations other than Laplace's. 
It has been shown above that in order to solve the equation 
^-^- + ^—2- = any given function of Y, x, y, 
a relation between differences of chequer ratios has to be 
satisfied. And the same will be found to be true for the 
other forms of the equation V 2 Y = a function (of Y and of 
position) which can be treated by two coordinates. A 
difference relation of the sort referred to would involve the 
comparison of each chequer with a standard set having 
graded chequer ratios, followed by the calculation of V*Y by 
