254 Mr. L. F. Richardson on a Freehand Graphic way 
functions of position on its surface, and all conditions being 
constant throughout the thickness of the shell at any point of 
its surface. Take a solid bounded by a surface of the shape 
of the shell and draw small rectangles at numerous points of 
the surface, so that their chequer ratio = -, %-, — a — 
1 breadth across now 
is directly proportional to the product of the thickness and 
-conductivity at each point of the shell. For then the flow 
through each chequer will be the same. Suppose that these 
standard chequers are in some distinctive colour, say red. 
Now lay off in black the boundary conditions of the special 
problem and draw a black chequerwork to have the same 
chequer ratio as the red at each point, much as was done for 
symmetry about an axis. The standard red chequers need 
not be connected so as to form two systems of orthogonal 
lines but may be scattered anyhow over the surface, all that 
is necessary is that they should be sufficiently small and 
numerous. 
Or it may be convenient to use a projection of the surface 
as was done in the case of the spherical shell above. 
Case (c). — When there exists no family of surfaces normal 
to the guiding lines. Without pausing for generalities we 
will proceed at once to : — 
Screw symmetry about an axis. — Let us discuss this with 
the aid of cylindrical coordinates r, <£>, z. At a point P on 
the axis OZ let a perpendicular be drawn extending to 
infinity. This perpendicular, which is to project only on one 
side of the axis, is imagined to revolve round the axis and 
slide along the same with proportional velocities. In one 
rotation round the axis let it move I along the axis. Then 
the line sweeps out a surface, at all points of which the 
expression z — 5- </> is constant. Let us put z — — <£ = a>. 
Then as co varies we pass from one of these screw surfaces 
to another formed by shifting the first parallel to z. The 
range of the coordinate w is from to I. The intersections 
of o> = const, with the cylinders r — const, are a family of 
screw-threads. 
Let drj be an element of distance measured along any 
.... ,. xl , drj length of turn of screw 
screw guiding line, so that -j- = — -, = 
a function of r only. And let us make V a linear function of 
distance along each screw-thread so that ~ — = a function 
of r only. OV 
