of determining Stream Lines and Equi potentials. 263 
IV. 
We have hitherto passed over without mention the pecu- 
liarities relating to points of equilibrium — these are points 
at wldcli the first space-rate of the potential vanishes in 
all directions. In the neighbourhood of these the chequers 
become unusually large, and if any chequer goes right up to 
an equilibrium point it will not have the shape characteristic 
of its neighbours, but will take a peculiar form of its own. 
There are several diagrams of this in Maxwell's ' Electricity 
and Magnetism.' See, for example, vol. ii. fig. xvii. 
Now if V be expressed in terms of rectangular coordinates 
n and v lying in the plane of the graph with their origin at 
the equilibrium point, then linear terms in V must vanish, 
and we have 
Y = Au 2 + Bco + Co 2 + Eu* + Firv + Guv 2 + terms of higher 
degree. 
Now let us make V 2 V vanish. 
For guiding lines parallel straight and normal to the plane 
of the graph 
V 2 V= |^ + ^ = 2(A+C)+<6E+2G)+<2F-f6H). 
When the graph is on a plane passing through an axis about 
which there is symmetry of revolution and u is normal to 
this axis, we must add to the above value of V 2 V the term 
r ov r r r 
where r is distance from the axis. 
Now when the point considered is not on or close to the 
axis, it will be possible to put in so many chequers that the 
first two chequers in any direction from the equilibrium 
point require for their measurement so small a range of u 
u v 
and v that the fractions -, -, &c, will be small, and therefore 
r r 
the additional terms which come in for symmetry about an 
axis may be neglected, and we have the same form for V 2 V 
in both cases. 
Further, since Mercator's projection does not alter the 
shape of any small pieces, V 2 V will have the same form in 
the neighbourhood of an equilibrium point on the Mercator's 
plan of the distribution on a sphere. 
