250 Mr. L. F. Richardson on a Freehand Graphic way 
'Example of Symmetry about an Am.— Byerly in his 
' Fourier's Series and Spherical Harmonics,' p. 230, sets 
the following problem : — " A cylinder of radius one metre 
and altitude one metre has its upper surface kept at tem- 
perature 100°, and its base and convex surface at the 
temperature 15°, until the stationary temperature is set up. 
Find the temperature at points on the axis 25 cm., 50 cm., 
and 75 cm. from the base, and also at a point 25 cm. from 
the base and 50 cm. from the axis." To solve this the first 
thing necessary is to prepare a chart bearing chequers of the 
appropriate shape for each distance from the axis. The 
graph of any solution of y 2 V = symmetrical about an axis 
would serve this purpose. For example several of the figures 
out of Maxwell's * Electricity and Magnetism ' would do. But 
I preferred to prepare a standard chart by ruling equidistant 
parallel equipotentials normal to the axis of revolution, and 
then stream-lines parallel to the axis at distances from it 
proportional to the square roots of the natural numbers 
0, 1, 2, 3, 4, 5, &c. The cross section of the cylindrical 
shell enclosed between successive stream-lines is then the 
same for every pair, and the chequer ratio proportional to 
the distance from the axis. This having been done in red 
ink, a sheet of tracing-paper was pinned over it, the section 
of our given cylinder was drawn in black and equipotentials 
and lines of flow were drawn in pencil. These were then 
rubbed out and amended with the aim of making the pencil 
chequers everywhere very similar to the red rectangles 
underneath. When improvement became slow, the blurredlines 
were made firm and definite with ink and the chequers con- 
sidered individually and marked as to whether they were too 
square or too thin. The lines were then drawn on a clean sheet 
of tracing-paper, the chequers again examined individually, 
and finally the lines fixed in ink (see fig. 3). Coordinate lines 
were then ruled and the values of V at their intersections were 
read from the graph. This process, from the ruling in of 
the given contour to the determination of V in numbers, 
took me four hours. The analytical method would perhaps 
have been more rapid in this case ; but for an irregular 
shaped contour with an irregular boundary distribution 
the freehand solution would still take about the same 
time, while analytical methods may be almost indefinitely 
tedious. 
