266 Mr. L. F. Richardson on a Freehand Graphic way 
arithmetic. And although it would doubtless be possible to 
carry out the necessary operations, yet it would almost cer- 
tainly be quicker and more accurate to use arithmetical finite 
differences altogether, writing in the numerical values of V 
at a set of points on the paper and adjusting these numbers 
until the finite difference equation is satisfied, — in a manner 
which may be described in a future paper. In view of this 
I will not attempt to elaborate freehand methods for V*V = 
a given function of V and of position. 
There are, however, eertain common space distributions 
which may be treated graphically with simplicity although 
they do not satisfy V 2 V = 0. 
Firstly, when the conductivity is a continuous function of 
position, and the direction of the flux is normal to the contours 
of a potential, and the magnitude of the flux is the maximum 
space-rate of the potential multiplied by the conductivity, 
and the flux has no divergence. For example : the flow of 
heat and electricity in isotropic but non-homogeneous bodies, 
or the soakage of water in a saturated subsoil the upper 
layers of which are more porous than those below. Let K 
be the conductivity and suppose that it is constant along each 
guiding line but varies from one such line to another. 
Then, when the lines have a family of surfaces normal to 
them we must have 
9 /H^.H r .K\_ 
~da\ H a 
).. 
in order that the flux shall not diverge. This is very easily 
assured by preparing the paper with standard chequers 
TT 
having their chequer ratio ~ proportional to H r . K. In 
fact, we have an example of this in Section III a above ; 
for circular symmetry about an axis may be regarded for 
this purpose as flow between parallel planes in a medium 
having conductivity directly proportional to the distance 
from the axis. And reciprocally. 
Similarly in the case of screw'syminetry, standard chequers 
are to be prepared having 
4ttV 
^ proportional to Ka /l + 
Two other cases can probably be treated freehand, namely, 
the flow of heat in bodies where the conductivity varies with the 
temperature, and, of great practical importance, tJie distribution 
